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概率论与数理统计中的英文单词和短语概率论与数理统计Probability Theory and Mathematical Statistics第一章概率论的基本观点Chapter 1Introduction of Probability Theory不确立性indeterminacy必定现象certain phenomenon随机现象random phenomenon试验experiment结果outcome频次数frequency number样本空间sample space出现次数frequency of occurrencen 维样本空间n-dimensional sample space样本空间的点point in sample space随机事件random event / random occurrence基本领件elementary event必定事件certain event不行能事件impossible event等可能事件equally likely event事件运算律operational rules of events事件的包括implication of events并事件union events交事件intersection events互不相容事件、mutually exclusive exvents/互斥事件/incompatible events互逆的mutually inverse加法定理addition theorem古典概率classical probability古典概率模型classical probabilistic model几何概率geometric probability 乘法定理product theorem概率乘法multiplication of probabilities条件概率conditional probability全概率公式、全formula of total probability概率定理贝叶斯公式、逆Bayes formula概率公式后验概率posterior probability先验概率prior probability独立事件independent event独立随机事件independent random event独立实验independent experiment两两独立pairwise independent两两独立事件pairwise independent events第二章随机变量及其散布Chapter2Random Variables and Distributions随机变量random variables失散随机变量discrete random variables概率散布律law of probability distribution一维概率散布one-dimension probability distribution 概率散布probability distribution两点散布two-point distribution伯努利散布Bernoulli distribution二项散布 / 伯努Binomial distribution利散布超几何散布hypergeometric distribution三项散布trinomial distribution多项散布polynomial distribution泊松散布Poisson distribution泊松参数Poisson theorem散布函数distribution function概率散布函数probability density function连续随机变量continuous random variable概率密度probability density概率密度函数probability density function概率曲线probability curve平均散布uniform distribution指数散布exponential distribution指数散布密度函exponential distribution density 数function正态散布、高斯normal distribution散布标准正态散布standard normal distribution正态概率密度函normal probability density function数正态概率曲线normal probability curve标准正态曲线standard normal curve柯西散布Cauchy distribution散布密度density of distribution第三章多维随机变量及其散布Chapter 3 Multivariate Random Variables and Distributions二维随机变量two-dimensional random variable结合散布函数joint distribution function二维失散型随机two-dimensional discrete random 变量variable二维连续型随机two-dimensional continuous random 变量variable结合概率密度joint probability variablen 维随机变量n-dimensional random variablen 维散布函数n-dimensional distribution functionn 维概率散布n-dimensional probability distribution边沿散布marginal distribution边沿散布函数marginal distribution function边沿散布律law of marginal distribution边沿概率密度marginal probability density二维正态散布two-dimensional normal distribution二维正态概率密two-dimensional normal probability 度density 二维正态概率曲two-dimensional normal probability 线curve条件散布conditional distribution条件散布律law of conditional distribution条件概率散布conditional probability distribution条件概率密度conditional probability density边沿密度marginal density独立随机变量independent random variables第四章随机变量的数字特点Chapter 4 Numerical Characteristics fo Random Variables数学希望、均值mathematical expectation希望值expectation value方差variance标准差standard deviation随机变量的方差variance of random variables均方差mean square deviation有关关系dependence relation有关系数correlation coefficient协方差covariance协方差矩阵covariance matrix切比雪夫不等式Chebyshev inequality第五章大数定律及中心极限制理Chapter 5 Law of Large Numbers and Central Limit Theorem大数定律law of great numbers切比雪夫定理的special form of Chebyshev theorem特别形式依概率收敛convergence in probability伯努利大数定律Bernoulli law of large numbers同散布same distribution列维 - 林德伯格independent Levy-Lindberg theorem定理、独立同分布中心极限制理辛钦大数定律Khinchine law of large numbers利亚普诺夫定理Liapunov theorem棣莫弗 - 拉普拉De Moivre-Laplace theorem斯定理第六章样本及抽样分布Chapter 6 Samples and Sampling Distributions统计量statistic整体population个体individual样本sample容量capacity统计剖析statistical analysis统计散布statistical distribution统计整体statistical ensemble随机抽样stochastic sampling / random sampling 随机样本random sample简单随机抽样simple random sampling简单随机样本simple random sample经验散布函数empirical distribution function样本均值sample average / sample mean样本方差sample variance样本标准差sample standard deviation标准偏差standard error样本 k 阶矩sample moment of order k样本中心矩sample central moment样本值sample value样本大小、样本sample size容量样本统计量sampling statistics 随机抽样散布random sampling distribution抽样散布、样本sampling distribution散布自由度degree of freedomZ 散布Z-distributionU 散布U-distribution第七章参数预计Chapter 7Parameter Estimations统计推测statistical inference参数预计parameter estimation散布参数parameter of distribution参数统计推测parametric statistical inference点预计point estimate / point estimation整体中心距population central moment整体有关系数population correlation coefficient整体散布population covariance整体协方差population covariance点预计量point estimator预计量estimator无偏预计unbiased estimate/ unbiasedestimation预计量的有效性efficiency of estimator矩法预计moment estimation整体均值population mean整体矩population moment整体 k 阶矩population moment of order k整体参数population parameter极大似然预计maximum likelihood estimation极大似然预计量maximum likelihood estimator极大似然法maximum likelihood method /maximum-likelihood method似然方程likelihood equation似然函数likelihood function区间预计interval estimation置信区间confidence interval置信水平confidence level置信系数confidence coefficient单侧置信区间one-sided confidence interval置信上限置信下限U 预计正态整体整体方差的预计confidence upper limit confidence lower limitU-estimatornormal populationestimation of population variance置信度方差比degree of confidence variance ratio第八章假定查验Chapter 8Hypothesis Testings参数假定假定查验两类错误统计假定统计假定查验查验统计量明显性查验统计明显性parametric hypothesis hypothesis testingtwo types of errors statistical hypothesis statistical hypothesis testing test statisticstest of significance statistical significance单边查验、单侧one-sided test查验单侧假定、单边one-sided hypothesis 假定两侧假定两侧查验明显水平拒绝域 / 否认区two-sided hypothesis two-sided testing significant level rejection region域接受地区acceptance regionU 查验F 查验方差齐性的查验拟合优度查验U-testF-testhomogeneity test for variances test of goodness of fit。
sen 1997 多维贫困指数构建方法
多维贫困指数是用来衡量一个地区或国家的综合贫困程度的指标，它综合考虑了多个维度的贫困表现，例如收入、教育、健康等。
以下是一种常见的多维贫困指数构建方法，称为“联合稳定性方法”：
1. 确定指标维度：确定要考虑的贫困指标维度，例如收入、教育、健康等。
2. 确定指标变量：为每个指标维度确定衡量指标的变量，例如收入指标可以使用人均收入水平、教育指标可以使用受教育程度指数等。
3. 标准化指标变量：对每个指标变量进行标准化，使得它们都具有相同的量纲。
标准化方法可以使用z分数标准化或者最小-最大标准化等。
4. 确定权重：为每个指标变量确定权重，反映了其在多维贫困指数中的相对重要性。
权重可以通过专家咨询、主成分分析等方法确定。
5. 计算贫困指数：将标准化后的指标变量与对应的权重相乘，然后对所有维度的结果进行加权求和，得到一个综合的多维贫困指数。
6. 敏感性分析：进行敏感性分析，检验不同权重组合对多维贫困指数的影响，以检验构建方法的稳定性和可靠性。
需要注意的是，多维贫困指数的构建方法可能因地区和数据可用性的不同而有所差异。
因此，在构建指数时，需要根据具体情况灵活选择指标变量和权重，并进行必要的实证研究和检验，以确保指数的准确性和可靠性。
公平评估函数"公平评估函数"这个术语没有一个统一的定义，因为它可能因上下文和特定应用的不同而有所变化。
然而，在一般意义上，我们可以将其理解为用于量化或评估某个系统、算法或过程在多大程度上实现了公平性的函数或指标。
在许多领域中，如机器学习、资源分配、法律和政策制定等，公平性是一个重要的考量因素。
公平评估函数可能包括多个方面，如：1. 均等性（Equality）：衡量不同组别之间结果的相似性。
例如，在机器学习的上下文中，均等性可能意味着不同人口统计群体之间的错误率应该相似。
2. 无偏性（Unbiasedness）：衡量系统是否对不同的输入或用户群体表现出偏见。
无偏性可以通过比较不同组别的平均结果来评估。
3. 机会均等（Equality of Opportunity）：在某些特定条件下，衡量不同组别获得有利结果的概率是否相等。
例如，在招聘算法中，机会均等可能意味着具有相同资格的不同候选人被选中的概率应该相同。
4. 公平性差距（Fairness Gap）：衡量不同组别之间结果差异的量度。
公平性差距越小，系统的公平性就越高。
设计一个公平评估函数时，需要明确以下几点：1.定义公平性：首先，你需要明确在你的应用中公平性的具体含义是什么。
这通常涉及到识别相关的用户群体和潜在的偏见来源。
2.选择指标：根据你的公平性定义，选择一个或多个能够量化公平性的指标。
3.权衡：公平性通常与其他目标（如准确性、效率等）存在权衡关系。
设计一个公平评估函数时，需要明确这些权衡，并决定如何在它们之间进行折中。
4.验证和调整：在应用公平评估函数时，需要验证其有效性，并根据反馈进行调整。
这可能涉及到收集和分析数据，以及与受影响的利益相关者进行沟通。
八年级生物与生态环境英语阅读理解25题1<背景文章>The tropical rainforest is one of the most diverse and complex ecosystems on our planet. It is a world filled with an astonishing variety of plants and animals.Plants in the tropical rainforest are highly adapted to the warm and humid environment. For example, the large and broad - leaved trees, such as the kapok tree. It can grow extremely tall, reaching towards the sunlight above the thick canopy of the forest. Its trunk is thick and strong, providing support for its large branches. The leaves are large to capture as much sunlight as possible for photosynthesis. There are also countless vines and epiphytes. Vines climb up the tall trees, using them as support to reach sunlight. Epiphytes, like orchids, grow on the branches of other plants instead of in the soil. They get water and nutrients from the air and rain.The animal life in the tropical rainforest is equally diverse. There are colorful birds, such as the toucan. The toucan has a large, brightly colored beak which is not only used for eating fruits but also for attracting mates. Monkeys are also common inhabitants. They are highly agile, swinging from tree to tree. They mainly feed on fruits, nuts and small insects. Insects are in abundance here too. Butterflies with their beautiful wings flit amongthe flowers. Some insects, like ants, live in large colonies and have complex social structures.The relationships between these organisms are intricate. For instance, many plants rely on animals for pollination and seed dispersal. Bees and butterflies pollinate flowers as they move from one to another in search of nectar. Fruit - eating animals like monkeys help to spread the seeds of plants far and wide.The tropical rainforest ecosystem is of vital importance. It is often called the "lungs of the earth" because it absorbs a large amount of carbon dioxide and releases oxygen through photosynthesis. It also helps to regulate the earth's climate. In addition, it is a huge reservoir of biodiversity, providing a home for countless species. Losing the tropical rainforest would mean the loss of many unique plants and animals, and would havea far - reaching impact on the global environment.1. What is a characteristic of the kapok tree in the tropical rainforest?A. It has small leaves.B. It is short.C. It has a thick and strong trunk.D. It grows in cold areas.答案：C。
SPSS 专业技术词汇、短语的中英文对照索引% of cases 各类别所占百分比1-tailed 单尾的2 Independent Samples 两个独立样本的检验2 Related Samples 两个相关样本检验2-tailed 双尾的3-D (=dimensional) 三维-->三维散点图AAbove 高于Absolute 绝对的-->绝对值Add 加，添加Add Cases 合并个案Add cases from... 从……加个案Add Variables 合并变量Add variables from... 从……加变量Adj.(=adjusted) standardized 调整后的标准化残差Aggregate 汇总-->分类汇总Aggregate Data 对数据进行分类汇总Aggregate Function 汇总函数Aggregate Variable 需要分类汇总的变量Agreement 协议Align 对齐-->对齐方式Alignment 对齐-->对齐方式All 全部，所有的All cases 所有个案All categories equal 所有类别相等All other values 所有其他值All requested variables entered 所要求变量全部引入Alphabetic 按字母顺序的-->按字母顺序列表Alternative 另外的，备选的Analysis by groups is off 分组分析未开启Analyze 分析-->统计分析Analyze all cases, do not create groups 分析全部个案，不建立分组Annotation 注释ANOV A Table ANOV A表ANOV A table and eta (对分组变量)进行单因素方差分析并计算其η值Apply 应用Apply Data Dictionary 应用数据字典Apply Dictionary 应用数据字典Approximately 大约Approximately X% of all cases 从所有个案中随机选择约X%的个案Approximation 近似估计Area 面积Ascend 上升Ascending counts 按频数的升序排列Ascending means 按均值升值排序Ascending values 按变量值的升序排列Assign 指定，分配Assign Rank 1 to 把秩值1 分配给Assume 假定Asymp. Sig.(=Asymptotic Significance) (2-sided) 双尾渐近显著性检验Automatic 自动的Automatic Recode 自动重新编码Axis 轴Axis Title 横轴名称BBack 返回-->上一步Backward 向后-->向后剔除法Bar chart 条形图Bar Spacing 条形间距Bars Represent 条的长度所代表的统计量Based on negative ranks 基于负秩Based on time or case range 在给定范围内随机选择个案Below 低于Between Groups (Combined) 组间值Binomial 二项分布Bivariate 双变量的-->双变量相关分析Bivariate Correlations 双变量相关分析Both files provide cases 是由外部文件和当前文件二者都提供个案Bottom 底部-->下边框Break Variable 分类变量Brown-Forsythe Brown-Forsythe检验Browse 浏览CCache 贮存Cache Data 数据暂存Calculated from data 根据数据计算Cancel 取消Case 个案Case Label 个案标签Case Processing Summary 个案处理概要Casewise 以个案为单位Casewise diagnostics 个案诊断表Categories 分组数，分类变量，分类模式Categorize Variables 变量分类Category 类别-->单元格数据类型Category Axis 分类轴Cell 单元格Cell Display 单元格显示Cell Percentages 在每个单元格中输出百分比Cell Properties 单元格属性Cell Statistics 单元格中的统计量Center 中心-->居中，中间对齐Central Tendency 集中趋势Change 变化，更改Change Summary 改变汇总函数Chart 图表，图形Chart Type 统计图类型Chart values 作图数据选项Charts 确定生成的图形选项Chi-square test 卡方检验Chi-square（χ2）卡方Choose 选择Choose Destination Location 选择安装路径Classify 分类Clear 清除Close 关闭Cluster 群，组，分组Clustered 分组的-->分组条形图Clustered bar charts 分组条形图Cochran's and Mantel-Haenszel statistics ：Cochran 统计量与Mantel-Haenszel 统计量Cochran's Q Cochran Q 检验Code 编码Coefficient 系数Coefficient statistics 系数统计Collinearity 共线性Collinearity diagnostics 共线性诊断Column 列-->单元格中个案数占列总数的百分比Columns 列数-->显示宽度Comma 逗号-->带逗号的数值型变量Common 共同的Compact 最低要求安装Compare 比较，对比Compare groups 分组对比Compare Means 比较平均数Compare variables 比较变量Complete 完成Compute 计算Concordance 和谐Condition 条件Confidence Interval 置信区间Contingency coefficient 列联相关的C 系数Continue 继续Contrast 对比Control 控制Convert 转换Convert numeric strings to numbers 将数值型字符转换为数字Copy 拷贝Copy Objects 拷贝对象Copy old value 拷贝旧变量Correlate 与……相关-->作相关分析Correlation 相关，关联Correlation Coefficient 相关系数Correlations 皮尔逊（Pearson）相关系数Count 个案数，次数，频数Counted value 用于计数的值Covariance 协方差Covariance Matrix 协方差矩阵Covariance ratio 协方差比率Creat new data file 创建新的数据文件Create Time Series 生成时间序列Criteria 标准（复数）Criterion 标准（单数）Cross-product 叉积Cross-product deviation 叉积离差Cross-product deviations and covariances 叉积离差与协方差Crosstab (=Cross-tabulation) 交叉列表Crosstabs 交叉列表（列联表）分析Cum.(=Cumulative) % of cases 累积百分比Cum.(=Cumulative) N of cases 累积频数Cumulative Percent 累计百分比Cumulative Sum 累积和Currency 货币-->货币型变量Current 目前的Current Selections 当前选择Current Settings 目前设置Current Status 目前状态Curve 曲线Curve Estimation 曲线估算Custom 自定义安装Custom Currency 自定义货币记法Cut 剪切Cut point 切分点Cut points for N equal groups 将数据分为N个个案数相同的组的切分点DData 数据-->数据文件的建立与编辑Data in Chart Are 图中数据为（用于统计量模式的选项）Data View 数据(编辑)窗口Database 数据库，数据文件Date 日期-->日期型变量Decimal 小数-->按小数点对齐Decimals 小数位数Define 定义Define Clusters by 以……定义分组（确定分组变量）Define Dates 定义日期Define Dichotomy 确定二分值Define Groups 确定分组Define Ranges 定义范围Define Sets 定义多选变量集Define Simple Bar 定义简单条形图Define Slices by 以……定义扇形块（确定分块变量）Define Variable Ranges 定义变量范围Degree 度，程度Degrees of freedom 自由度Delete 删除Deleted 删除掉-->删除未选个案，将残差删除Dependent 非独立的，依赖的-->因变量Dependent List 因变量/分析变量列表Derive 推导Derived 推导出的Derived axis 转换轴Descend 下降Descending counts 按频数的降序排列Descending means 按均值的降序列表Descending values 按变量值的降序排列Descriptive 描述性的Descriptive Statistics 描述统计Descriptives 描述统计Destination 目标Deviation 偏差，离差df (=degrees of freedom) 自由度Diagnostics 诊断Dialog 对话（框）Dichotomies 二分变量，二分模式Dichotomy 二分法，二分值，二分变量Dictionary 字典Directory 目录Discrete 离散的Discrete missing values 离散缺失值Dispersion 离散趋势Display 显示Display axis line 显示轴线Display chart with case labels 在图中显示个案标签Display clustered bar charts 显示聚类条图Display Data Info 显示数据的基本信息Display derived axis 显示转换轴Display frequencies tables （在输出结果中）显示频数表Display groups defined by missing values 对由缺失值定义的组也加以显示Display labels 显示标签Display legend 显示图例Display normal curve 显示正态曲线Display order 输出结果列表顺序Display summary tables 显示提要表Display the ReadMe file now? 是否现在显示软件说明文件Distribution 分布Division 分，除法Do not filter cases 不过滤个案Dollar 元，美元-->带美元符号的数值型变量Dot 圆点，句号-->带圆点的数值型变量Draft output 文本输出文件Drop-line 垂线图Durbin-Watson 系列相关检验，Durbin-Watson 系数EEdit 编辑-->文件编辑Enter 进入-->强行进入法Entry 进入Equal 等于Equal Variances Assumed 等方差假定Equal Variances Not Assumed 不假定等方差Equality 相等Estimate 估计-->估计值Eta η系数（用于测量一个定类变量与一个定比/定距变量之间的相关比率）Eta Squared η2Every 每一个Every N labels 每N个标签Exactly 精确地Exactly M cases from the first N cases 在前N个个案中随机选择M个个案Exclude 排除，拒绝Exclude cases analysis by analysis 剔除分析变量为缺失值的个案Exclude cases listwise 剔除任何含有缺失值的个案Exclude cases listwise within categories 排除分类变量中的缺失值Exclude cases listwise within dichotomies 排除二分变量中的缺失值Exclude cases pairwise 剔除参与相关系数计算的两个变量中有缺失值的个案Exclude cases test-by-test 排除对比中的缺失值Excluded Variables 被拒绝变量Existing 现有的Exit 退出Expect 期望Expected 期望的-->期望频次Expected Range 理论分布范围Expected Value 期望值，理论值Exponential 指数的-->指数分布Export model information to XML file 将模型的信息输出到XML 文件External file is keyed table 外部文件为关键表Extreme 极端FF F检验的值Factor 因素，因子-->影响因素变量File 文件-->文件操作File is already sorted 数据文件已排序Filter 过滤-->过滤变量Filter variable 过滤变量Filtered 过滤掉的-->生成过滤变量Find 查找Finish 完成First Case 第一个个案（最小值）First value 第一个观测值Fixed and random effects 确定性影响因素和随机影响因素Flag 旗帜，标记；做标记Flag significant correlations 标出达到显著性水平的相关系数Font 字体Footnote 脚注Format 格式Fraction 比率Frame 框，框架Freedom 自由Frequencies 频数分析，按频数作图Frequency 频率，频数Friedman Friedman检验Function 函数GGallery 美术馆-->图形转换功能Gamma γ等级相关系数General 一般General Linear Model 一般线性模型Get from data 由样本观测值确定Go to Case 查找个案Graph 图形Graphs 统计图Grid 网格Grid line 格线Grid lines 用竖线作刻度标志Group 群，组，分组Group Based on 根据……分组Group Statistics 分组统计分析Grouped Median 分组中位值Grouping Variable 分组变量HHelp 帮助Hidden 隐藏High 最大值Histogram 直方图Homogeneity 同质性-->齐次性Homogeneity of variance 方差齐次性Homogeneity-of-variance 方差齐次性检验Horizontal 水平的，横向的Horizontal Alignment 横向对齐方式Hypothesis 假设IIf condition is satisfied 如果满足一定条件Include 包含，包括Include constant in equation 在方程中包含常数项Increment 增量Independent 独立的-->自变量Independent List 自变量/分组变量列表Independent-Samples T Test 独立样本的T 检验Independent-Samples Test 独立样本检验Indicate 指明，标明Indicate case source as variable 生成一个表明个案来源的变量Individual 单个，个体Influence 影响Influence Statistics 影响因素统计量Info (=Information) 信息Information 信息Inner 里面的Inner Frame 内框-->给图形增加内框Input Variable 输入变量Insert 插入Inside 在（某范围）之内Install (动) 安装Installation (名) 安装Into Different Variables 用重新编码的变量生成一个新变量Into Same Variables 用重新编码的变量取代原变量JJustification 对齐方式KK Independent Samples K个独立样本检验Kappa κ系数Kendall's Coefficient of Concordance 肯德尔和谐系数Kendall's tau-b 肯得尔等级相关τ-b 系数Kendall's tau-c 肯得尔等级相关τ-c 系数Kendall's W Kendall W 检验Key 关键的Key Variables 关键变量Kolmogorov-Smirnov Z 柯尔莫哥洛夫-斯米诺夫Z 检验Kruskal-Wallis H Kruskal-Wallis H检验（用秩的平方和检验）Kurtosis 峰度，峰度系数LLabel 标签-->变量名标签Label Cases by 以……为个案标签Label Text 标签文字Lambda λ系数Largest value 最大值Last Case 最后一个个案（最大值）Last value 最后一个观测值Launch 开始Launch tutorial now? 现在开始看使用辅导Layer 层-->层变量Left 左-->居左，左对齐Left/bottom 左下Legend 图例Legend Title 图例标题Less 较少，较小Less than 小于Levene StatisticLevene’s Test for Equality of Variances 方差齐性检验License 许可，授权Likelihood 似然性Likelihood Ratio 似然比卡方Line 行，线-->线形图Line Represents 单线条所代表的统计量Line Style 线型Linear 线性Linear model 线性模型Linear Regression 线性回归分析，线性回归模型Lines Represent 多线条所代表的统计量List 列表Listwise 整个数据表Location 位置Log 对数-->对数尺度Lose 丢失Lost 丢失掉Low 最小值Lower 下限Lower Bound 下限Lowest through X 从最小值到XLSD (=Least-significance difference) 能达到显著性水平的最小差异MMajor 主要的Major Divisions 大分度Mann-Whitney U 曼-维特尼U检验（用秩和检验的方法进行）Margin 边距Margins 边距设置Marker 标记，标志Match 匹配Match cases on key variables in sorted files 按排序的关键变量匹配个案Match variables across response sets 输出与多选变量进行交叉分析的匹配变量的选择数和以选择总数为基础计算的边缘频率分布Matrix 矩阵-->矩阵散点图Maximum 最小值McNemar 麦克讷马Mean 均值Mean Difference 均值差Mean of Values 变量值的平均数Mean Rank 平均秩和Mean Square 平均平方和Means 平均数分析Means and standard deviations 均值与标准差Means plot 均值分布图Measure 测量-->测量层次Measurement level 测量层次Measures of AssociationMedian 中位数-->中位数检验法Median of Values 变量值的中位数Merge 合并Merge Files 合并(数据)文件Method 方法Minimum 最大值Minor 次要的Minor Divisions 小分度Missing 缺失，变量的缺失值Missing value 缺失值Missing Value Analysis 缺失值分析Mix 混合Mixed 混合的-->混合对齐Mode 众数Mode of Values 变量值的众数Model 模型Model fit 模型配置Model Summary 模型概要More 更多（选项）Moses 莫西Moses extreme reactions 莫西极值反应Most Extreme 最极值Mult (=Multiple) Response Sets 多选变量集Multiple 多个-->多线图Multiple Response 多选变量分析Multiple Variables 多变量选项NN (=Number) 个案数N of cases 各类别的频数Name 名称-->变量名Name & Label 名称与标签Name Variable 名称变量Negative 负Negative Difference 负差Negative Rank 负秩Network 网络New 新的-->新建(各种文件)New Value 新值New Working Data File 新的当前数据文件Next 下一个-->下一步No 否，无No missing values 无缺失值Nominal 定类变量Nominal by Interval 定类变量与定距变量间的相关系数None 无Nonlinear 非线性的-->非线性模型Nonparametric 非参数的Nonparametric Tests 非参数检验Normal 正态的-->正态分布Normal curve 正态曲线Normal probability plot 残差的正态概率图Notation 记号，记法Note 便条Notes 说明文字Number 数字，编号-->数字型变量Number Above 大于设定值的个案数Number Below 小于设定值的个案数Number of Cases 个案数目，频数Number of Runs 游程数Numeric 数值的-->标准数值型变量OObject 对象Observation 观察，观测-->样本观测值Observed 观测到的-->观测到的频次Observed Prop. (=proportion) 观察到的比例Odds 几率Off 关闭，未开启Offset from 距离OK 行了-->执行Old and New Values 新旧变量值的转换Old Value 旧值Omit 省略，略去On 开启One Sample T Test 单样本T 检验One-sample K-S(Kolmogorov-Smirnov) test 单样柯尔莫哥洛夫-斯米诺夫检验One-Sample Statistics 单样本统计量One-Sample Test 单样本检验One-tailed 单尾的-->单尾检验One-Way ANOV A 简单方差分析Open 打开Open an existing data source 打开现存的数据文件Open another type of file 打开另一种类型的数据文件Open File 打开文档Option 选择，先项Options 先项Order 顺序Order by 排序选项Ordinal 定序变量Organization 组织Organize 组织（动）Organize output by groups 按分组变量组织输出Organize output by variables 各变量单独输出Orientation 定向Other summary function 其它汇总函数Outer 外面的Outer Frame 外框-->给图形增加外框Outlier 远离中心者-->远离均值的值Outliers outside X standard deviations 离均值X个标准差之外的值Output 输出-->输出文件Output file specification 规定输出文件Output Variable 输出变量Output variables are strings 输出变量是字符型变量Outside 在（某范围）之外Overlay 重叠-->重叠散点图PPackage 包Pair 对，一对Paired 已配对的Paired Sample Correlations 配对样本相关系数Paired Sample Statistics 配对样本统计分析Paired Sample Test 配对样本检验Paired Variables 已配对样本Paired-Samples T Test 配对样本的t 检验Pairwise 成对地Parameter 参数Parametric 参数的Part and partial correlations 部分相关与偏相关系数Part correlation 部分相关系数Partial 部分，偏Partial correlation 偏相关系数Paste 粘贴PC (=personal computer) 个人电脑Pct (=Percent) 百分比Pearson 皮尔逊Pearson Chi-Square 皮尔逊卡方值Pearson Correlation 皮尔逊相关系数Percentage 百分比Percentage Above 大于设定值的个案的百分比Percentage Below 小于设定值的个案的百分比Percentages 按百分比作图Percentages Based on 百分比基于……Percentile 百分位数Percentile Values 百分位数选项栏Percents and totals based on respondents 百分比与总数基于回答者人数Personal 个人的-->个人安装Personal or Shared Installation 个人或共享安装Phi and Cramér's V 列联相关的V系数Pie chart 圆饼图，圆形图Pivot 支点，枢轴，在枢轴上的旋转运动-->表格转置（行列互换）Plot 绘图，图形Point 点Poisson 泊桑-->泊桑分布Positive 正Positive Difference 正差Positive Rank 正秩Post Hoc 发生于其后者-->确定均值有差别后的多重比较Predict 预测Predicted 预测的Predicted Values 预测值Prediction Intervals 预测区间Predictor 预测变量Preview 预览Print 打印Print Preview 打印预览Probability 概率Process 处理Processor 处理器Produce 产生，生成Provide 提供pt.(=point) 点-->象素点QQuartile 四分位数RR squared change R2 变化Random 随机的Random Number 随机数Random Number Seed 随机数初始值Random sample of cases 随机选择个案样本Range 范围，全距，极差Range of Grouping Variable 分组变量值的范围Range of missing values 缺失值范围Rank 秩Rank Assigned to Ties 对同值的秩的分配方法选项Rank Cases 对个案排秩Rank Types 秩变量类型Ratio 比率Read 读Read File 读取文件Read Text Data 读入文本格式的数据ReadMe 软件说明文件Ready 准备好了Ready to Install Files 准备安装界面Recent 最近Recently Used Data 最近用过的数据（文件）Recently Used Files 最近用过的文件Recode 重新编码Redo 恢复-->恢复上一步被撤销的操作Reference 参考，参照Reference Line 参照线Refresh 刷新Regression 回归Regression Standardized Residual 回归分析的标准化残差Remote 远程Remote server 远程服务器Removal 剔除Remove 取消-->删除Rename 重命名Replace 替换Replace Missing Values 替换缺失值Replace with mean 用平均值代替（缺失值）Replace working data file 替代当前的数据文件Report 报告Represent 代表Request 要求Reset 重设Residual 残余的，残余物-->残差Residual Statistics 残差统计表Response 回答Right 右-->居右，右对齐Right/top 右上Row 行-->单元格中个案数占行总数的百分比Row Order 行序（行的排列顺序）Run 执行Run Pending Transforms 运行等待中的转换Runs 游程Runs Test 游程检验SS.E. mean 均值的标准误S.E. of mean predictions 对均值标准误的预测Sample 样本Sample size 样本大小Satisfy 满足Save 保存Save As 另存为Save number of case in break group as variable 存储各分组变量的个案数Save standardized values as variables 将标准分存为新变量Save to New File 存为新文档Scale 尺度-->定距/定比变量Scatter 驱散-->散点Scatterplot 散点图Scientific 科学的Scientific Notation 科学计数法Script 程序Seed 种子-->初始值Select 选择Selected Label 选定的标签Selection Variable 选择变量Sequential 序列的Sequential ranks to unique values 相同值的秩次取第一个出现的秩次值Serial 系列的Serial Number 系列号Series 系列Series Displayed as 图形转换选项Server 服务器Set 集，集合Set Definition 多选变量集的定义Set Markers by 以……为标志Setting 设置Setup 安装Setup Complete 完成安装界面Setup Type 安装类型Shade 阴影Shading 阴影设置Share 共享Show 显示Sig. (=significance) 显著性，显著性水平Sign 符号-->符号检验法Significance level 显著性水平Simple 简单的-->简单条形图，单线图，简单散点图Skewness 偏度，偏度系数Slice 片，块-->扇形块Smallest value 最小值Social Science 社会科学Software 软件Software License Agreement 软件授权使用协议Somers' d Somers 等级相关d 系数Sort 分类，排序Sort Cases 对个案进行归类、排序Sort order 排序规则Sort the file by grouping variables 按分组变量对数据文件进行排序Source 来源Specific 专门的，特定的Specific Values 专门值Specification 规定，规格Split 分割Split Files 分割文件SPSS (Statistical Package for Social Sciences) 社会科学统计软件包SPSS Data Editor 数据窗口编辑界面SPSS/PC(for DOS) DOS 版个人电脑用SPSSStacked 分段条的-->分段条形图Standard deviation 标准差Standard Error of the Estimate 估计值的标准误Standardized 标准化的-->标准化残差Standardized Coefficient 标准化(回归)系数Standardized Residual Plots 标准化残差图Standardized value 标准分Statistic 统计量Statistical 统计的Statistics 统计学，统计量(复数)Statistics for First Layer 第一层分组的统计量Status 状态Std. Error Difference 标准误之差Std.(=Standard) Deviation 标准差Step 步Stepping 分步引入或剔除变量Stepping Method Criteria 变量引入模型或从模型中剔除的判断标准Stepwise 逐步-->逐步进入法Stop 停止，中断String 字符串-->字符型变量Studentized 将残差学生化Studentized deleted 将残差学生化并删除Style 风格，形式-->线型Subtitle 副标题Sum 总和Sum of Ranks 秩和Sum of Squares 总平方和Sum of Squares and Cross-products 离差平方和与叉积和Sum of Values 变量值的和Summaries for groups of cases 个案组概要（变量值模式）Summaries of separate variables 不同变量的概要（变量模式）Summary 概要-->统计概要Summary Function for Selected Va r i abl e s 选定变量的概要函数Suppress 压制，禁止Suppress table 隐藏表格Suppress tables with more than N categories 分组数大于N时禁止频数表在结果中输出Switch 切换Syntax 语句-->语句文件System 系统System-missing 系统缺失值System-or User-missing 系统或用户缺失值Tt t 统计量T test T 检验Table 表格Tables for 为（某变量）列表-->多选变量分析中要分析的变量Tail 尾Template 模板Tendency 趋势Test 检验Test distribution is Normal 待检分布为正态分布Test for linearity 检验线性相关Test Homogeneity of Variances 方差齐次性检验Test of Significance 显著性检验Test Pair List 待检变量对列表Test Prop.(=proportion) 待检比例Test Proportion 待检概率Test Type 检验类型，检验方法Test Value 待检值Test Variable 分析变量，待检变量Test Variable List 待检变量列表Tests for Several Independent Samples 多个独立样本检验Tick 标记Tick mark 标记Tick marks 用点作刻度标志Tick marks for skipped labels 给被越过的标签作标记Tie 同值-->相等的秩，无差（相等）Time 时间-->时间型变量Time Series 时间系列Title 标题Title Justification 标题对齐方式，标题位置Top 顶部-->上边框Total 总数-->单元格中个案数占个案总数的百分比Transform 转换-->数据转换Transpose 转置Transpose Rows and Columns 行列互换Transposition 转置（名）t-test for Equality of Means 均值相等的t 检验Tutorial 辅导-->使用辅导Two Independent Samples 两个独立样本的检验Two-Related-Samples Tests 两个相关样本检验Two-tailed 双尾的-->双尾检验Type 类型-->变量类型Typical 典型安装UUncertainty 不确定性Uncertainty coefficient 不定系数Undo 撤销-->撤销上一步操作Uniform 均匀的-->均匀分布Unique 唯一的Unpaired 不匹配Unpaired variable 不匹配变量Unselected 未被选择的Unselected Case Are 未被选中的个案的处理方法选项Unstandardized 非标准化的-->非标准化残差Unstandardized Coefficient 非标准化(回归)系数Untransposed 未被转置的Unweighted 不加权的-->不加权的个案数Unweighted missing 不加权的缺失值数Upper 上限Upper Bound 上限Use chart specifications from 用……的图形规格Use F value 以F值为标准Use filter variable 使用过滤变量Use probability of F 以F的概率为标准Use specified range 使用特定的范围Use specified values 使用给定的变量值User 用户User Information 用户信息界面Utilities 实用程序Utility 利用，用途VValid 有效的Valid Cases 有效个案Valid Percent 有效百分比Va lue 值Value label 变量值标签Va lue s 变量值标签Values are group midpoints 取值为各组中点Values are grouped midpoints 变量值为各组中点Values of individual cases 个案值（观测值模式）Va r i abl e 变量Variable label 变量名标签Variable list 变量列表，按变量顺序列表Variable Name and Label 变量名称与标签Variable View 变量(编辑)窗口Variables Are Coded As 变量编码方式Variables Entered/Removed 引入或剔除的变量Variables in Set （多选变量）集中的变量Variance 方差Vertical 垂直的，纵向的Vertical Alignment 纵向对齐方式View 观看-->窗口外观控制Viewer 浏览器WWeigh Cases 给变量加权Weight 重量，轻重-->线的粗细Weight Estimation 权重估计Weighted 加权的-->加权的个案数Weighted missing 加权的缺失值数Welch Welch 检验Welcome 欢迎(界面)Width 宽度Wilcoxon Wilcoxon 检验(用符号秩检验)Wilcoxon Signed Rank Test Wilcoxon符号秩检验Wilcoxon W 威尔科克松W 检验Window 窗口-->窗口控制Windows 视窗With normal curve 加上正态曲线Within Groups 组内值Working 当前正在用的Working Data File is keyed table 当前数据文件为关键表Working Date File 当前数据文件XX Axis X 轴X through highest 从X到最大值X through Y 从X到YYY Axis Y轴Yes 是ZZ Z 检验值Z Approximation Z 近似估计(整理自网络)。
概率论与数理统计专业词汇中英对照表概率论与数理统计专业词汇中英对照表导语：概率论与数理统计是数学的一个有特色且又十分活跃的分支，下面是YJBYS店铺收集整理的概率论与数理统计专业词汇中英对照表，欢迎参考!Aabsolute value 绝对值accept 接受acceptable region 接受域additivity 可加性adjusted 调整的alternative hypothesis 对立假设analysis 分析analysis of covariance 协方差分析analysis of variance 方差分析arithmetic mean 算术平均值association 相关性assumption 假设assumption checking 假设检验availability 有效度average 均值Bbalanced 平衡的band 带宽bar chart 条形图beta-distribution 贝塔分布between groups 组间的bias 偏倚binomial distribution 二项分布binomial test 二项检验Ccalculate 计算case 个案category 类别center of gravity 重心central tendency 中心趋势chi-square distribution 卡方分布chi-square test 卡方检验classify 分类cluster analysis 聚类分析coefficient 系数coefficient of correlation 相关系数collinearity 共线性column 列compare 比较comparison 对照components 构成，分量compound 复合的confidence interval 置信区间consistency 一致性constant 常数continuous variable 连续变量control charts 控制图correlation 相关covariance 协方差covariance matrix 协方差矩阵critical point 临界点critical value 临界值crosstab 列联表cubic 三次的，立方的cubic term 三次项cumulative distribution function 累加分布函数curve estimation 曲线估计Ddata 数据default 默认的definition 定义deleted residual 剔除残差density function 密度函数dependent variable 因变量description 描述design of experiment 试验设计deviations 差异df.(degree of freedom) 自由度diagnostic 诊断dimension 维discrete variable 离散变量discriminant function 判别函数discriminatory analysis 判别分析distance 距离distribution 分布D-optimal design D-优化设计Eeaqual 相等effects of interaction 交互效应efficiency 有效性eigenvalue 特征值equal size 等含量equation 方程error 误差estimate 估计estimation of parameters 参数估计estimations 估计量evaluate 衡量exact value 精确值expectation 期望expected value 期望值exponential 指数的exponential distributon 指数分布extreme value 极值Ffactor 因素，因子factor analysis 因子分析factor score 因子得分factorial designs 析因设计factorial experiment 析因试验fit 拟合fitted line 拟合线fitted value 拟合值fixed model 固定模型fixed variable 固定变量fractional factorial design 部分析因设计frequency 频数F-test F检验full factorial design 完全析因设计function 函数Ggamma distribution 伽玛分布geometric mean 几何均值group 组Hharmomic mean 调和均值heterogeneity 不齐性histogram 直方图homogeneity 齐性homogeneity of variance 方差齐性hypothesis 假设hypothesis test 假设检验Iindependence 独立independent variable 自变量independent-samples 独立样本index 指数index of correlation 相关指数interaction 交互作用interclass correlation 组内相关interval estimate 区间估计intraclass correlation 组间相关inverse 倒数的`terate 迭代Kkernal 核Kolmogorov-Smirnov test 柯尔莫哥洛夫-斯米诺夫检验kurtosis 峰度Llarge sample problem 大样本问题layer 层least-significant difference 最小显著差数least-square estimation 最小二乘估计least-square method 最小二乘法level 水平level of significance 显著性水平leverage value 中心化杠杆值life 寿命life test 寿命试验likelihood function 似然函数likelihood ratio test 似然比检验linear 线性的linear estimator 线性估计linear model 线性模型linear regression 线性回归linear relation 线性关系linear term 线性项logarithmic 对数的logarithms 对数logistic 逻辑的lost function 损失函数。
Package‘sEparaTe’August18,2023Title Maximum Likelihood Estimation and Likelihood Ratio TestFunctions for Separable Variance-Covariance StructuresVersion0.3.2Maintainer Timothy Schwinghamer<***************************.CA>Description Maximum likelihood estimation of the parametersof matrix and3rd-order tensor normal distributions with unstructuredfactor variance covariance matrices,two procedures,and for unbiasedmodified likelihood ratio testing of simple and double separabilityfor variance-covariance structures,two procedures.References:Dutilleul P.(1999)<doi:10.1080/00949659908811970>,Manceur AM,Dutilleul P.(2013)<doi:10.1016/j.cam.2012.09.017>,and Manceur AM,DutilleulP.(2013)<doi:10.1016/j.spl.2012.10.020>.Depends R(>=4.3.0)License MIT+file LICENSEEncoding UTF-8LazyData trueRoxygenNote7.2.0NeedsCompilation noAuthor Ameur Manceur[aut],Timothy Schwinghamer[aut,cre],Pierre Dutilleul[aut,cph]Repository CRANDate/Publication2023-08-1807:50:02UTCR topics documented:data2d (2)data3d (2)lrt2d_svc (3)lrt3d_svc (5)mle2d_svc (7)12data3d mle3d_svc (8)sEparaTe (10)Index11 data2d Two dimensional data setDescriptionAn i.i.d.random sample of size7from a2x3matrix normal distribution,for a small numerical example of the use of the functions mle2d_svc and lrt2d_svc from the sEparaTe packageUsagedata2dFormatA frame(excluding the headings)with42lines of data and4variables:K an integer ranging from1to7,the size of an i.i.d.random sample from a2x3matrix normal distributionId1an integer ranging from1to2,the number of rows of the matrix normal distributionId2an integer ranging from1to3,the number of columns of the matrix normal distribution value2d the sample data for the observed variabledata3d Three dimensional data setDescriptionAn i.i.d.random sample of size13from a2x3x2tensor normal distribution,for a small numerical example of the use of the functions mle3d_svc and lrt3d_svc from the sEparaTe packageUsagedata3dFormatA frame(excluding the headings)with156lines of data and5variables:K an integer ranging from1to13,the size of an i.i.d.random sample from a2x3x2tensor matrix normal distributionId3an integer ranging from1to2,the number of rows of the3rd-order tensor normal distribution Id4an integer ranging from1to3,the number of columns of the3rd-order tensor normal distribu-tionId5an integer ranging from1to2,the number of edges of the3rd-order tensor normal distribution value3d the sample data for the observed variablelrt2d_svc Unbiased modified likelihood ratio test for simple separability of avariance-covariance matrix.DescriptionA likelihood ratio test(LRT)for simple separability of a variance-covariance matrix,modified tobe unbiased infinite samples.The modification is a penalty-based homothetic transformation of the LRT statistic.The penalty value is optimized for a given mean model,which is left unstruc-tured here.In the required function,the Id1and Id2variables correspond to the row and column subscripts,respectively;“value2d”refers to the observed variable.Usagelrt2d_svc(value2d,Id1,Id2,subject,data_2d,eps,maxiter,startmat,sign.level,n.simul)Argumentsvalue2d from the formula value2d~Id1+Id2Id1from the formula value2d~Id1+Id2Id2from the formula value2d~Id1+Id2subject the replicate,also called the subject or individual,thefirst column in the matrix (2d)datafiledata_2d the name of the matrix dataeps the threshold in the stopping criterion for the iterative mle algorithm(estimation) maxiter the maximum number of iterations for the mle algorithm(estimation)startmat the value of the second factor variance-covariance matrix used for initialization,i.e.,to start the mle algorithm(estimation)and obtain the initial estimate of thefirst factor variance-covariance matrixsign.level the significance level,or rejection rate in the testing of the null hypothesis of simple separability for a variance-covariance structure,when the unbiased mod-ified LRT is used,i.e.,the critical value in the chi-square test is derived bysimulations from the sampling distribution of the LRT statistic n.simul the number of simulations used to build the sampling distribution of the LRT statistic under the null hypothesis,using the same characteristics as the i.i.d.random sample from a matrix normal distributionOutput“Convergence”,TRUE or FALSE“chi.df”,the theoretical number of degrees of freedom of the asymptotic chi-square distribution that would apply to the unmodified LRT statistic for simple separability of a variance-covariance structure“Lambda”,the observed value of the unmodified LRT statistic“critical.value”,the critical value at the specified significance level for the chi-square distribution with“chi.df”degrees of freedom“mbda”will indicate whether or not the null hypothesis of separability was rejected, based on the theoretical LRT statistic“Simulation.critical.value”,the critical value at the specified significance level that is derived from the sampling distribution of the unbiased modified LRT statistic“mbda.simulation”,the decision(acceptance/rejection)regarding the null hypothesis of simple separability,made using the theoretical(biased unmodified)LRT“Penalty”,the optimized penalty value used in the homothetic transformation between the biased unmodified and unbiased modified LRT statistics“U1hat”,the estimated variance-covariance matrix for the rows“Standardized_U1hat”,the standardized estimated variance-covariance matrix for the rows;the standardization is performed by dividing each entry of U1hat by entry(1,1)of U1hat“U2hat”,the estimated variance-covariance matrix for the columns“Standardized_U2hat”,the standardized estimated variance-covariance matrix for the columns;the standardization is performed by multiplying each entry of U2hat by entry(1,1)of U1hat“Shat”,the sample variance-covariance matrix computed from the vectorized data matrices ReferencesManceur AM,Dutilleul P.2013.Unbiased modified likelihood ratio tests for simple and double separability of a variance-covariance structure.Statistics and Probability Letters83:631-636.Examplesoutput<-lrt2d_svc(data2d$value2d,data2d$Id1,data2d$Id2,data2d$K,data_2d=data2d,n.simul=100)outputlrt3d_svc An unbiased modified likelihood ratio test for double separability of avariance-covariance structure.DescriptionA likelihood ratio test(LRT)for double separability of a variance-covariance structure,modified tobe unbiased infinite samples.The modification is a penalty-based homothetic transformation of the LRT statistic.The penalty value is optimized for a given mean model,which is left unstructured here.In the required function,the Id3,Id4and Id5variables correspond to the row,column and edge subscripts,respectively;“value3d”refers to the observed variable.Usagelrt3d_svc(value3d,Id3,Id4,Id5,subject,data_3d,eps,maxiter,startmatU2,startmatU3,sign.level,n.simul)Argumentsvalue3d from the formula value3d~Id3+Id4+Id5Id3from the formula value3d~Id3+Id4+Id5Id4from the formula value3d~Id3+Id4+Id5Id5from the formula value3d~Id3+Id4+Id5subject the replicate,also called individualdata_3d the name of the tensor dataeps the threshold in the stopping criterion for the iterative mle algorithm(estimation) maxiter the maximum number of iterations for the mle algorithm(estimation)startmatU2the value of the second factor variance-covariance matrix used for initialization startmatU3the value of the third factor variance-covariance matrix used for initialization,i.e.,startmatU3together with startmatU2are used to start the mle algorithm(estimation)and obtain the initial estimate of thefirst factor variance-covariancematrix U1sign.level the significance level,or rejection rate in the testing of the null hypothesis of simple separability for a variance-covariance structure,when the unbiased mod-ified LRT is used,i.e.,the critical value in the chi-square test is derived bysimulations from the sampling distribution of the LRT statistic n.simul the number of simulations used to build the sampling distribution of the LRT statistic under the null hypothesis,using the same characteristics as the i.i.d.random sample from a tensor normal distributionOutput“Convergence”,TRUE or FALSE“chi.df”,the theoretical number of degrees of freedom of the asymptotic chi-square distribution that would apply to the unmodified LRT statistic for double separability of a variance-covariance structure“Lambda”,the observed value of the unmodified LRT statistic“critical.value”,the critical value at the specified significance level for the chi-square distribution with“chi.df”degrees of freedom“mbda”,the decision(acceptance/rejection)regarding the null hypothesis of double sep-arability,made using the theoretical(biased unmodified)LRT“Simulation.critical.value”,the critical value at the specified significance level that is derived from the sampling distribution of the unbiased modified LRT statistic“mbda.simulation”,the decision(acceptance/rejection)regarding the null hypothesis of double separability,made using the unbiased modified LRT“Penalty”,the optimized penalty value used in the homothetic transformation between the biased unmodified and unbiased modified LRT statistics“U1hat”,the estimated variance-covariance matrix for the rows“Standardized_U1hat”,the standardized estimated variance-covariance matrix for the rows;the standardization is performed by dividing each entry of U1hat by entry(1,1)of U1hat“U2hat”,the estimated variance-covariance matrix for the columns“Standardized_U2hat”,the standardized estimated variance-covariance matrix for the columns;the standardization is performed by multiplying each entry of U2hat by entry(1,1)of U1hat“U3hat”,the estimated variance-covariance matrix for the edges“Shat”,the sample variance-covariance matrix computed from the vectorized data tensorsReferencesManceur AM,Dutilleul P.2013.Unbiased modified likelihood ratio tests for simple and double separability of a variance-covariance structure.Statistics and Probability Letters83:631-636.Examplesoutput<-lrt3d_svc(data3d$value3d,data3d$Id3,data3d$Id4,data3d$Id5,data3d$K,data_3d=data3d,n.simul=100)outputmle2d_svc Maximum likelihood estimation of the parameters of a matrix normaldistributionDescriptionMaximum likelihood estimation for the parameters of a matrix normal distribution X,which is char-acterized by a simply separable variance-covariance structure.In the general case,which is the case considered here,two unstructured factor variance-covariance matrices determine the covariability of random matrix entries,depending on the row(one factor matrix)and the column(the other factor matrix)where two X-entries are.In the required function,the Id1and Id2variables correspond to the row and column subscripts,respectively;“value2d”indicates the observed variable.Usagemle2d_svc(value2d,Id1,Id2,subject,data_2d,eps,maxiter,startmat)Argumentsvalue2d from the formula value2d~Id1+Id2Id1from the formula value2d~Id1+Id2Id2from the formula value2d~Id1+Id2subject the replicate,also called individualdata_2d the name of the matrix dataeps the threshold in the stopping criterion for the iterative mle algorithmmaxiter the maximum number of iterations for the iterative mle algorithmstartmat the value of the second factor variance-covariance matrix used for initializa-tion,i.e.,to start the algorithm and obtain the initial estimate of thefirst factorvariance-covariance matrixOutput“Convergence”,TRUE or FALSE“Iter”,will indicate the number of iterations needed for the mle algorithm to converge“Xmeanhat”,the estimated mean matrix(i.e.,the sample mean)“First”,the row subscript,or the second column in the datafile“U1hat”,the estimated variance-covariance matrix for the rows“Standardized.U1hat”,the standardized estimated variance-covariance matrix for the rows;the stan-dardization is performed by dividing each entry of U1hat by entry(1,1)of U1hat“Second”,the column subscript,or the third column in the datafile“U2hat”,the estimated variance-covariance matrix for the columns“Standardized.U2hat”,the standardized estimated variance-covariance matrix for the columns;the standardization is performed by multiplying each entry of U2hat by entry(1,1)of U1hat“Shat”,is the sample variance-covariance matrix computed from of the vectorized data matrices ReferencesDutilleul P.1990.Apport en analyse spectrale d’un periodogramme modifie et modelisation des series chronologiques avec repetitions en vue de leur comparaison en frequence.D.Sc.Dissertation, Universite catholique de Louvain,Departement de mathematique.Dutilleul P.1999.The mle algorithm for the matrix normal distribution.Journal of Statistical Computation and Simulation64:105-123.Examplesoutput<-mle2d_svc(data2d$value2d,data2d$Id1,data2d$Id2,data2d$K,data_2d=data2d) outputmle3d_svc Maximum likelihood estimation of the parameters of a3rd-order ten-sor normal distributionDescriptionMaximum likelihood estimation for the parameters of a3rd-order tensor normal distribution X, which is characterized by a doubly separable variance-covariance structure.In the general case, which is the case considered here,three unstructured factor variance-covariance matrices determine the covariability of random tensor entries,depending on the row(one factor matrix),the column (another factor matrix)and the edge(remaining factor matrix)where two X-entries are.In the required function,the Id3,Id4and Id5variables correspond to the row,column and edge subscripts, respectively;“value3d”indicates the observed variable.Usagemle3d_svc(value3d,Id3,Id4,Id5,subject,data_3d,eps,maxiter,startmatU2,startmatU3)Argumentsvalue3d from the formula value3d~Id3+Id4+Id5Id3from the formula value3d~Id3+Id4+Id5Id4from the formula value3d~Id3+Id4+Id5Id5from the formula value3d~Id3+Id4+Id5subject the replicate,also called individualdata_3d the name of the tensor dataeps the threshold in the stopping criterion for the iterative mle algorithmmaxiter the maximum number of iterations for the iterative mle algorithmstartmatU2the value of the second factor variance covariance matrix used for initialization startmatU3the value of the third factor variance covariance matrix used for initialization,i.e.,startmatU3together with startmatU2are used to start the algorithm andobtain the initial estimate of thefirst factor variance covariance matrix U1Output“Convergence”,TRUE or FALSE“Iter”,the number of iterations needed for the mle algorithm to converge“Xmeanhat”,the estimated mean tensor(i.e.,the sample mean)“First”,the row subscript,or the second column in the datafile“U1hat”,the estimated variance-covariance matrix for the rows“Standardized.U1hat”,the standardized estimated variance-covariance matrix for the rows;the stan-dardization is performed by dividing each entry of U1hat by entry(1,1)of U1hat“Second”,the column subscript,or the third column in the datafile“U2hat”,the estimated variance-covariance matrix for the columns“Standardized.U2hat”,the standardized estimated variance-covariance matrix for the columns;the standardization is performed by multiplying each entry of U2hat by entry(1,1)of U1hat“Third”,the edge subscript,or the fourth column in the datafile“U3hat”,the estimated variance-covariance matrix for the edges“Shat”,the sample variance-covariance matrix computed from the vectorized data tensorsReferenceManceur AM,Dutilleul P.2013.Maximum likelihood estimation for the tensor normal distribution: Algorithm,minimum sample size,and empirical bias and dispersion.Journal of Computational and Applied Mathematics239:37-49.10sEparaTeExamplesoutput<-mle3d_svc(data3d$value3d,data3d$Id3,data3d$Id4,data3d$Id5,data3d$K,data_3d=data3d) outputsEparaTe MLE and LRT functions for separable variance-covariance structuresDescriptionA package for maximum likelihood estimation(MLE)of the parameters of matrix and3rd-ordertensor normal distributions with unstructured factor variance-covariance matrices(two procedures),and for unbiased modified likelihood ratio testing(LRT)of simple and double separability forvariance-covariance structures(two procedures).Functionsmle2d_svc,for maximum likelihood estimation of the parameters of a matrix normal distributionmle3d_svc,for maximum likelihood estimation of the parameters of a3rd-order tensor normaldistributionlrt2d_svc,for the unbiased modified likelihood ratio test of simple separability for a variance-covariance structurelrt3d_svc,for the unbiased modified likelihood ratio test of double separability for a variance-covariance structureDatadata2d,a two-dimensional data setdata3d,a three-dimensional data setReferencesDutilleul P.1999.The mle algorithm for the matrix normal distribution.Journal of StatisticalComputation and Simulation64:105-123.Manceur AM,Dutilleul P.2013.Maximum likelihood estimation for the tensor normal distribution:Algorithm,minimum sample size,and empirical bias and dispersion.Journal of Computational andApplied Mathematics239:37-49.Manceur AM,Dutilleul P.2013.Unbiased modified likelihood ratio tests for simple and doubleseparability of a variance covariance structure.Statistics and Probability Letters83:631-636.Index∗datasetsdata2d,2data3d,2data2d,2data3d,2lrt2d_svc,3lrt3d_svc,5mle2d_svc,7mle3d_svc,8sEparaTe,1011。
本人收集整理的统计专业英语词汇:AAbsolute value 绝对值accept接受acceptable reegion接受域additivity可加性adjusted调整的 alternative hypothesis对立假设analysis分析analysis of covariance协方差分析analysis of variance方差分析arithmetic mean 算术平均数assoviation 相关性assumption 假设assumption checking 假设检验availability有效度average 均值Bbalanced平衡的 band带宽bar chart条形图bathtub curve浴盆曲线beta-distribution贝塔分布between groups组间的bias偏倚binomial distribution二项分布binomial test二项检验Ccalculate计算 case个案category类别center of gravity重心central tendency中心趋势 chi-aquare distribution卡方分布chi-aquare test卡方检验classify分类cluster analysis聚类分析coefficient系数coefficient of correlation相关分析collinearity共线性column列compare比较comparison对照components构成，分量compound复合的 compsite hypothesis复合假设concomitant cariable共变量confidence interval置信区间consistency一致性constant常数continuous variable连续变量 control charts控制图correlation相关 correspondence analysis对应分析covariance协方差 covariance matrix协方差矩阵critical point临界点critical value临界植crosstab列联表cubic三次的，立方的cubic term三次项curve estimation曲线估计Ddata数据default默认的definition定义deleted residual剔除残差density function密度函数dependent variable因变量description描述deviations差异df.(degree of *******) 自由度diagnostic诊断dimension维discrete variable离散变量discriminant function判别函数discriminatory analysis半别分析distance距离distribution分布Eequal相等effects of interaction交互效应efficiency有效性eigenvalue特征值equal size等含量 equation方程error误差estimate估计estimation of parameters参数估计estimations估计量evaluate衡量 exact value精确值expectation 期望expected value期望值exponential指数的exponential distributon指数分布extreme value极值Ff-text f检验facto*因素，因子**ctor analysis因素分析factor score因子得分factorial designs析因分析 factorial experiment析因试验failure rate失效率 fit拟合fitted line拟合线 fitted value拟合值fixed model固定模型fixed variable固定变量frequency频数function函数Ggamma distribution 伽玛分布geometric mean 几何均值group 组 growth curve 生长曲线Hharmomic mean 调和均值hererogeneity 不齐性histogram 直方图homogeneity 齐性homogeneity of variance 方差齐性hypothesis 假设hypothesis test 假设检验IIndependence 独立 independent-samples 独立样本independent variable 自变量 index 指数index of correlation 相关指数interaction 交互作用interclass correlation 组内相关interval estimate 区间估计intraclass correlation 组间 inverse 倒数的iterate 迭代KKernel 核kolmogorov-smirnov test 柯尔莫哥洛夫-斯米诺夫检验kurtosis 峰度Llarge sample problem大样本问题layer 层least significant difference 最小显著差数least-square estimation 最小二乘估计least-square method最小二乘法level 水平level of significance显著性水平leverage value 中心化杠杆值life 寿命 life test 寿命试验likelihood function似然函数 likelihood ratio test 似然比检验linear线性的linear estimator 线性估计linear model 线性模型linear regression 线性回归linear relation线性关系linear term 线性项logarithmic 对数的logarithms 对数logistic 逻辑的lost function损失函数Mm-estimator m估计main effect 主效应maintainability维护度matrix 矩阵maximum 最大值maximum likelihood estimation 极大似然估计mean fine between failure 平均无故障工作时间mean squared deviation(MSD) 均方差mean sum of square 均方和measure 衡量media 中位数minimum 最小值missing values 缺失值mixed model 混合模型mode 众数model 模型monte carle method 蒙特卡罗法moving average 移动平均值multicollinearity 多元共线性multiple comparison 多重比较multiple correlation 多重相关multiple correlation coefficient 多元相关系数multiple regression analysis 多元回归分析multiple regression equatiom 多元回归方程multiple response 多响应multivariate analysis 多元分析Nnegative relationship 负相关nonadditively 不可加性nonlinear 非线性nonlinear regression 非线性回归nonparametric tests 非参数检验normal distribution 正太分布null hypothesis 零假设number of cases 个案数Oone-sample 单样本one-tailed test 单侧检验one-way anova 单方向分析one-way classification 单向分类optimal 优化的optimum allocation 最优配制order 排序order statistics 次序统计量origin 原点orthogonal 正交的outliers 异常值Pp-p plot pp概率图paried-sample成对样本paired observation 成对观测数据parameter 参数partial correlation 偏相关partial correlation coefficient 偏相关系数partial regression coefficient 偏回归系数percent 百分比 percentiles 百分位数pie chart 饼图point estimate 点估计poisson distribution 泊松分布polynomial curve 多项式曲线polynomial regression 多项式回归polynomials 多项式positive relationship 正相关power 幂predict 预测predicted value 预测值prediction intervals 预测区间principal component analysis 主成分分析proability 概率probit analysis 概率分析proportion 比例QQ-Q polt QQ概率图Quadratic 二次的Quadratic term 二次项Quality control 质量控制Quantitative 数量的，度量的Quartiles 四分位数RRandom 随机的Random number 随机数Random sampling 随机取样Random seed 随机数种子Random variable 随机变量Randomizatiom 随机化Range 极差Rank 秩rank correlation 秩相关rank statistic 秩统计量rector 向量regression analysis 回归分析regression coefficient 回归系数regression line 回归线reject 拒绝rejection region 拒绝域relationship 关系reliability 可靠性reliability analysis 可靠性分析reliability test 可靠性试验repeated 重复的report 报告，报表residual 残差residual sum of squares剩余平方和response 响应risk function 风险函数robustness 稳健性root mean square 标准差row 行run 游程run test 游程检验Ss-curve s型曲线sample 样本sample size 样本容量sample space 样本空间sampling 取样sampling inspection 抽样检验scatter chart 散点图separately 单独地sets 集合sign test 符号检验significance 显著性significance level 显著性水平significance testing 显著性检验significant 显著的，有效的significant digits 有效数字skewed distribution 偏态分布skewness 偏度small sample problem 小样本问题smooth 平滑sort 排序soruces of variation 方差来源space 空间spread 扩展square 平方standard deviation 标准离差standard error of mean 均值的标准差standardization 标准化standardize 标准化statistic 统计量statistical quality control 统计质量管理std.residual 标准残差stem-and-leaf plot 茎叶图stepwise regression analysis 逐步回归stimulus 刺激strong assumption 强假设stud.deleted residual 学生化剔除残差stud.residual学生化残差subsamples次级样本sufficient statistic充分统计量sum 和sum of square 平方和summary 概括Tt-distibution t分布t-test t检验table 表test criterion 检验判据test for linearity线性检验test of goodness of fit拟合优度检验test pf independence 齐性检验test rules 独立性检验test statistics 检验法则testing function 检验统计量test 检验time series 时间序列tolerance limits 容许限total 总共transformation 转换treatment 处理trimmed mean 截尾均值true value 真值two-tailed test 双尾检验Uunbalanced 不平衡的unbiased estimation 无偏估计unbiasedness 不偏性unequal size 不等含量uniform distribution 均匀分布Vvalue of estimator 估计值variable 变量variance 方差variance components 方差分量variance ratio 方差比various 不同的Wweight 加权weighted average 加权平均值wishart distribution 维夏分布within groups 组内的z-score z分数。
COMPARING AND RANKING COV ARIANCE STRUCTURES OF M-GARCHVOLATILITY MODELSS´e bastien Laurent1,Jeroen V.K.Rombouts2,Annastiina Silvennoinen3and Francesco Violante4October4,2006AbstractA large number of different parameterizations have been introduced to model conditionalvariance dynamics in a multivariate framework.A major problem with these models is thelarge number of parameters that have to be estimated and the many constraints,often difficultto make explicit,that have to be imposed to ensure semi-positive definiteness of the covari-ance matrices.This paper compares and ranks different multivariate GARCH type modelsin terms of their ability to estimate the in-sample conditional variance-covariance structureand the accuracy of their out-of-sample variance forecasts.The models are compared usingHansens Superior Predictive Ability test redefined in a multivariate framework by providingthree alternative metrics to evaluate the distance between variance matrices.In this paper wealso test the preliminary version of our MGARCH Package for Ox,extension of Laurent andPeterss G@RCH Package.Up to date,the package includes:BEKK models(Full,Diagonaland Scalar),CCC-DCC models and RiskMetrics.Keywords:Volatility,Multivariate GARCH,Covariance Models,SPA test,Financial econo-metrics.JEL Classification:C10,C32,C51C52,C53,G101CeReFim,Universit´e de Namur,and CORE,Universit´e Catholique de Louvain.2HEC Montr´e al,CIRANO,CIRPEE and CREF.3Stockholm School of Economics.4FUNDP Namur and CORE,Universit´e Catholique de Louvain.The authors would like to thank Kilian Mie for useful suggestions and comments.Correspondence to S´e bastien Laurent,FUNDP,Rempart de la Vi´e rge,8,B-5000Namur,Belgium.Telephone:+32 81724869.Fax:+3281724840.E-mail:urent@fundp.ac.beor to Francesco Violante,FUNDP,Rempart de la Vi´e rge,8,B-5000Namur,Belgium.Telephone:+3281724810. Fax:+3281724840.E-mail:violante@core.ucl.ac.beContents1Introduction12Multivariate GARCH:an Overview22.1BEKK-GARCH(1,1) (2)2.2RiskMetrics (3)2.3CCC-GARCH(1,1) (3)2.4DCC(1,1)-GARCH(1,1) (3)3Model Comparison:distance metrics43.1Frobenius Metric (4)3.2Eigenvalue Metric (4)3.3Foerstner and Moonen Metric (5)3.4Cosinus Mass Metric (5)4Realized Volatility7 5Model Comparison:Hansen SPA Test8 6Data and Empirical Results10 7Conclusions131IntroductionSince thefirst seminal paper of Kraft and Engle(1982)an increasing interest in modelling con-ditional covariances has developed a large number of multivariate volatility models,see Bawens, Laurent and Rombouts(2006)for an extensive survey.In principle all these models suffer from the increase of the cross-sectional dimension due to the large number of parameters to be simultaneously estimated which increases at least quadratically1, even when parameter restrictions,tipically parameter pooling,are imposed.Yet a further compli-cation rises from the necessity to impose non-linear parameter constraints to ensure semi-positive definite conditional covariance matrices.The aim of this paper is to compare and rank,in terms of ability to estimate the sample covari-ance structure and forecast accuracy,some of the most well known and widely used multivariate GARCH models.The specifications cosidered include the BEKK(Engle and Kroner1995),CCC (Bollerslev1990),DCC(Engle2001),RiskMetrics(JPMorgan1996).After checking the consistency of the procedure-i.e.estimation,forecast,comparison-using ad hoc simulated data we compare different model specifications using daily returns for two stocks traded on the NYSE for the period1980-1999,while the out-of-sample realized volatility has been computed using5-minute data from1995to1999.Models are evaluated using a loss function,which is applied both to the sample conditional covariance estimation and to the sequence of one step ahead covariance forecasts.The latent conditional covariance in the loss function is substituted,depending on the context,either by simulated data or by realized volatility.The sequence of conditional covariance forecasts has been obtained by means of a moving window of dimension k within which at each iteration the one step ahead forecast has been computed replacing the expectations for the squared innovation with its observed value.The coefficients are updated every k observations by restimating the model including the last k out of sample observations into the dataset.Since the objective of this paper is to compare covariance matrices the choice of an appropriate loss function has to be compatible to the multivariate framework.In this paper we investigate the problem providing three different notions of distance between matrices.The goodness offit of each model to the latent conditional covariance is then tested using Hansen(2001)Superior Predictive Ability Test(SPA).This test has the advantage that it properly accounts for the full set of models and is therefore likely to detect a superior model when such exist.In fact,SPA test evaluates whether a particular model(benchmark)when compared to the true data generating process(DGP)-simulated data or realized volatility-is significantly outperformed by other models,while taking into account the whole set of models that are being compared.In this paper we also introduce thefirst version of a new OxMetrix application to estimate and forecast multivariate GARCH models.The objective is to extend the work done by Laurent 1Engle and Sheppard(2005)and Peters(2006)with their G@RCH package dedicated to univariate ARCH-type models.Other than including procedures to estimate and forecast BEKK,CCC,Engle’s DCC and RiskMetrics specifications for the conditional covariance,it provides also some features to evaluate distance metrics,such as Frobenius norm,Eigenvalue metric(spectral norm),Foerstner and Moonen metric and Cosinus Mass metric.The paper is organized as follows.Section2provide a brief overview of several GARCH specification considered in this paper,while the loss functions are described in Section3.Section 4and5,contain some details on realized volatility and of the Hansen(2001)SPA test.Empirical results are presented in Section6.Finally,Section7contains some concluding remarks.2Multivariate GARCH:an OverviewConsider a vector price process,p t of dimension(N×1),let define the compounded daily return by r t=log(p t)−log(p t−1).The process r t is a vector stocastic process conditioned on the sigmafield, t−1generated by the information avaliable at t−1.We denoteµt≡E(r t| t−1)the conditional mean vector and H t≡E(r t r t| t−1)the conditional variance matrix.The specification used for the mean equation is:r t=µt+ t(1)z t(2)t=H1/2tis a(N×N)positive definite matrix and z t is i.i.d random vector with E(z t)=0 where H1/2tand V ar(z t)=I N.In this paper we consider six specifications for H t,namely the BEKK by Engle and Kroner, 1995,toghether with the diagonal and the scalar variants,the RiskMetrics by JPMorgan,1996, the Constant Conditional Correlation(CCC)by Bollerslev,1990and the Dynamic Conditional Correlation(DCC)by Engle,2001.In the following Section we will briefly describe the models formulation2.2.1BEKK-GARCH(1,1)The BEKK-GARCH(1,1)is defined as follows:H t=C∗ C∗+A∗ t−1 t−1A∗+G∗ H t−1G∗,(3) where C∗,A∗and G∗are(N×N)matrices and C∗is upper triangular.The number of parameters to be estimated is N(5N+1)/2.2See Bawens,Laurent and Rombouts(2006)for further detailTo reduce this number we can impose the diagonality of the parameter matrices A ∗and G ∗-i.e.Diagonal BEKK specification -or impose that A ∗and G ∗are equal to a scalar times the identity matrix -i.e.Scalar BEKK.2.2RiskMetricsThe RiskMetrics is an integrated Scalar-BEKK specification -i.e.A ∗A +G ∗G =I k ;where I is the identity matrix and k is the number of series -where the variance equation parameters are fixed and chosen ex-ante (does not require numerical optimization).The value assigned to theparameters is a 2ij =0.06and g 2ij =0.94∀i =j and 0elsewhere.2.3CCC-GARCH(1,1)The CCC is defined as follows:H t =D t RD t =ρijh iit h jjt(4)whereD t =diag (h 1/211t ...h 1/2NNt )(5)h iit can be defined as any univariate GARCH model,andR =(ρij )(6)is a symmetric positive definite matrix with ρii =1,∀i .R is the matrix containing the constant conditional correlations ρij .In this paper we specify the univariate process as a GARCH(1,1)specification for each conditional variance in D t :h iit =ωi +αi 2i,t −1+βi h ii,t −1(7)2.4DCC(1,1)-GARCH(1,1)The DCC(1,1)model by Engle (2002)can be specified as:R t =diag (q −1/211,t ...q −1/2NN,t )Q t diag (q −1/211,t ...q −1/2NN,t )(8)where the (N ×N )symmetric positive definite matrix Q t =(q ij,t )is given by:Q t =(1−θp −θq )Q +θp u t −1u t −1+θq Q t −1(9)with u t = i,t /(h ii,t ).Q is the (N ×N )unconditional variance matrix of u t ,and θp and θq are nonnegative scalar parameters with θp +θq <1.3Model Comparison:distance metricsIt is not obvious which loss function is more appropriate to evaluate the distance between two matrices as required by the problem we are facing of comparing covariance matrices-i.e.the latent (Σt)and the estimated(H t)covariance matrices sequences.In this paper we use the following four loss functions which,for their properties,seems to be appropriate for our purposes.3.1Frobenius MetricThe idea behind Frobenius norm is very similar to that of the Euclidean norm on R n.The Frobenius distance between two matrices(Σt and H t)is defined as:d2t=i,j|σt,ij−h t,ij|2∀i,j(10)=T race[(Σt−H t)(Σt−H t)∗]= rows(Σt−H t)i=1SV i(Σt−H t)where(.)∗denotes the conjugate transpose of(Σt−H t)and SV(.)are the singular values of (Σt−H t)which are defined as the eigenvalues of[(Σt−H t)(Σt−H t)∗]1/2.Since(Σt−H t)∗has only real entries and it is symetric then it is defined as self-adjoint or Hermitian matrix,that is (Σt−H t) =(Σt−H t)∗.Therefore we can compute the Frobenius norm as the square root of the sum of element-wise squared differences,that is:d t=i,j(σt,ij−h t,ij)2∀i,j(11)Given the second equality in Eq.10this distance metric is also called trace norm which is defined as the sum of all the singular values of(Σt−H t).3.2Eigenvalue MetricThe Eigenvalue metric,or spectral norm,is based on the concept of a standard principal component analysis.AssumingΣt and H t to be two covariance matrices,therefore symetric and with real entries,then the difference matrix at time t is Hermitian and the distance is given by:d t=λ1max(Σt,H t)(12)whereλ1max(Σt,H t)represents the largest eigenvalue of the matrix(Σt−H t)times its transpose -i.e.(Σt−H t)(Σt−H t) .SinceΣt and H t are covariance matrices,therefore symetric and semi-positive definite,standard principal component analysis implies that the largest eigenvalue is non-negative and captures the difference in form ofΣt and H t completely,providing a metric for the largest amount of error between these two matrices.Alternatively,the spectral norm can be defined as the lagest singular value of the semi positive definite matrix (Σt −H t ),i.e.:d t =λ1max(Σt −H t )(Σt −H t )(13)where once again λ1max (.)represents the largest eigenvalue of (.).From here to the end of this paper and in our empirical analysis we will always refer to the definition provided in Eq.12.3.3Foerstner and Moonen MetricThis metric has been introduced by Foerstner and Moonen (1999)as a generalization of the idea introduced in Section 3.2.The distance between two symetric semi-positive definite matrices Σt and H t is defined as the sum of the squared logarithms of the eigenvalues,that is:d t = n i =1ln 2(λ1i (Σt ,H t ))(14)with eigenvalues λ1i (Σt ,H t )from the system |λΣt −H t |=0.The square root of summing squares closely resemble the Euclidean metric.Again,since the covariance matrices are symetric and semi-positive definite,the eigenvalues are ensured to be positive 3.3.4Cosinus Mass MetricThis measure is quite uncommon with respect to metrics commonly used in the econometrics literature but its simplicity appeals in our framework.Since Σt and H t are symetric semi-positive definite matrices the distance can be defined as:d t =cos −1 (LT Σt )(LT H t )[(LT Σt ) (LT Σt )]·[(LT H t ) (LT H t )](15)where LT Σt =vech (Σt )and LT H t =vech (H t )and vech (·)denotes the operator that stacks thelower triangular portion of a (k ×k )matrix as a (k (k +1)/2×1)vector.Intuitively,the distance is measured by the angle between the two vectors defined by the the stacked lower triangulars of Σt and H t ,and hence the smaller is the angle,the more Σt and H t are closer.Given these four definitions we can define the fit of a multivariate GARCH-type model by the average distance between the matrices Σt and H t over t :ModelF itness =1T Tt =1d t(16)To provide an example we generate two random variables with mean zero,conditional variance that follows individually an univariate GARCH (1,1)process and conditional correlation generated by the DCC(1,1)process by Engle (2002).The paramaters of the univariate models and of the unconditional correlation equation are:3Furtherdetails on the properties of this metric are in Foerstner and Moonen (1999)Table1:DGP specification for GARCH(1,1)-DCC(1,1)processSample size:20000Number of series:2GARCH(1,1)DCC(1,1)Equation forσ21Equation forσ22Correlation Equationω10.2ω20.3¯ρ0.6α10.1α20.2θp0.2β10.7β20.5θq0.7We will now estimate the simulated process using a CCC and a DCC model respectively and then we will compute the distance between the sequences of both in-sample conditional variance (H t)and conditional correlation(R t)sequences and the underlyingΣt andΨt using the four distance metrics illustrated in the beginning of this Section.Table2and3contain the estimation results.Table2:Estimation results for GARCH(1,1)-CCCSample size:20000Number of series:2GARCH(1,1)Equation forσ21Equation forσ22Coefficient Std.Error t-value Coefficient Std.Error t-value ω10.1788020.0242987.359ω20.2899790.01958614.81α10.0897940.008086111.10α20.1877510.009884318.99β10.7334490.02976924.64β20.5254220.02462721.33CCCCoefficient Std.Error t-valueρ0.462270.005046991.596The results in Table4confirm that DCC model betterfits the true DGP outperforming the CCC whether we consider the Frobenius norm,Eigenvalue metric,Forstner and Moonen or Cosinus Mass metric.The difference in the scale is obviously due to the different intuition behind each metric.If on one side this provides a simple way to rank different models,based on sample per-formances,whether we compare in sample estimation or out of sample forecasts,in the next sections we will describe a tool to test whether the differences between these loss functions-i.e. X t=(Σt−H t,0)−(Σt−H t,i)∀i-significantly differs from zero and therefore if one model statistically outperforms the all the others.Table3:Estimation results for GARCH(1,1)-DCC(1,1)Sample size:20000Number of series:2GARCH(1,1)Equation forσ21Equation forσ22Coefficient Std.Error t-value Coefficient Std.Error t-value ω10.1788020.0242987.359ω20.2899790.01958614.81α10.0897940.008086111.10α20.1877510.009884318.99β10.7334490.02976924.64β20.5254220.02462721.33DCC(1,1)Coefficient Std.Error t-valueρ0.518910.01148145.197θp0.173230.004844935.755θq0.743300.007657897.064Table4:Distance MetricsSample size:20000-Number of series:2Frobenius MetricΣt−H t,DCC0.13564Ψt−R t,DCC0.13953Σt−H t,CCC0.37177Ψt−R t,CCC0.36814Eigenvalue MetricΣt−H t,DCC0.10710Ψt−R t,DCC0.098662Σt−H t,CCC0.27548Ψt−R t,CCC0.26032Forstner MetricΣt−H t,DCC0.23337Ψt−R t,DCC0.22773Σt−H t,CCC0.56888Ψt−R t,CCC0.56565Cos Mass MetricΣt−H t,DCC0.062785Ψt−R t,DCC0.062947Σt−H t,CCC0.16320Ψt−R t,CCC0.166414Realized VolatilityIn our empirical analysis we substitute the realized variance matrix to the latent conditional covariance when evaluating the goodness of the endogeneously generated volatility implied by the different M-GARCH-type models.The daily realized variance is computed from the intraday returs and is defined as:r t+∆,∆≡p t+∆−p t(17) for t=∆,2∆,...,T and∆=1/m,where m is the number od daily intervals.The corresponding daily return,for t=1,2,...,T over the time interval of length∆=1/m,isdefined by:r t+1≡mi=1r t+i∆,∆≡r t+∆,∆+r t+2∆,∆+...+r t+1,∆(18)Then,starting fron the(T m×n)matrix of intraday returns,we define the daily realizedvolatility as:V t+1≡1/∆i=1r t+i∆,∆r t+i∆,∆=R t+1R t+1(19)where the(m×n)matrix R t+1is defined by R t+1=(r t+∆,∆,r t+2∆,∆,...,r t+1,∆).The matrix V t+1represents the empirical counterpart to the daily quadratic return variation.In fact as the sampling frequency of the intraday returns increases,∆→0-i.e.m→∞,V t+1converges almost surely to the quadratic variation(QV):∆→0,V t+1(∆)as−→t+1tΣu du≡QV t(20) Therefore the realized volatility V t becomes a less noisy measure for the latent volatility as the intraday frequency∆reduces4.As shown in Andersen et al.(2002),to ensure positive semi definiteness of the realized covariance matrices it suffices that the columns in the matrix of returns R t are linearly independent.However this condition,which could eventually be preserved even for high dimensional problems,will be violated when the cross-sectional dimension increases with respect to sampling frequency of the intraday return.The condition n<m constitutes the dimensional upper bound to ensure semi positive definitenes of the realized covariance matrices since if the rank(R t)<n,V t=R t R t will not be full rank and will fail to be semi positive definite.5Model Comparison:Hansen SPA TestA way to test model differences across models is the asimptotic test by Diebold and Mariano (1995)or the non parametric test by Pesaran and Timmermann(1992).However the limit of these tests is that they only implement pairwise comparisons across models.Since our aim is to test whether a model,when compared to the true DGP,is outperformed by all the other models, we need to test their performances together.An alternative method(Andersen et al.,2002),in the tradition of Mincer-Zarnowitz(1969),is to evaluate volatility forecasts,and thus their relative performances,by projecting the realized volatility on a constant and the various model forecasts. The Mincer-Zarnowitz regression takes the form:vech(V t+1)=b0+b1vech(H t+1,i)+b2vech(H t+1,j)+u t+1(21) where b0is expected to be equal to(0,...,0)and b1=(1,...,1)and b2=(0,...,0)-or alternatively b1=(0,...,0)and b2=(1,...,1)-defines the best model forecast for the realized volatility.The R2of the regression represents the goodness offit and therefore the term of comparison between different models.However,the R2of a Mincer-Zarnowitz regression is not an ideal criterion for comparing volatility models,because it does not penalize a biased forecast5.4Note that the optimal choice for the intraday frequency needs to take into account the distortion due to microstructure effects5See Hansen and Lunde(2005)In this paper,we use the Test for Superior Predictive Ability by Hansen (2001)which is based on the Reality Check Test by White (2000)6and the work of Diebold and Mariano (1995)and West (1996).This test controls for the whole set of models to be compared.Since our aim is to compare the performances of two or more multivariate models,while Hansen’s SPA Test is originally designed for the univariate framework 7,first we have to introduce new notions of distance metrics -i.e.the distance metrics difined in Section 3-to fit the test to the multivariate setting 8.However,since the test in itself is based on the average amount of distance between pairs of points in time,which is a scalar regardless of the dimension,it re-mains substantially the same but for some algebrical adaptation to the higher dimensional setting.Let n =0,1,...,N denote the number of estimated models for which we compute one-step-ahead covariance forecasts and let denote H n,1,...,H n,T the sequence of covariance matrices forecasts for model n which will be compared to the latent covariance matrix Σt by mean of a predefined loss function L n,t ≡L (Σt ,H n,t ),t =1,...,T .Yet,let denote with n =0the model that is chosen as the benchmark and that is compared to the set of competing models n =1,...,N .By using SPA Test we can analyse whether any of the competing models outperforms the benchmark model in terms of predictive ability.The relative performance of model n (n =1,...,N ),with respect to the benchmark model (n =0)at time t ,is defined as:X t,n =L 0,t −L n,t ;n =1,...,N ;t =1,...,T (22)Hansen (2001)showed that under reasonable assumptions the moment λn ≡E [X t,n ]is well-defined for n =1,...,N .Therefore,a λn >0implies that model n outperforms the benchmark and allows for a formulation of the null hypothesis of the SPA test.More clearly,under the null hypothesis we will verify if λn ≤0for n =1,...,N .If one of the competing models outperforms the benchmark,the null hypothesis will be rejected.Focusing just on the model which yields the highest λn gives us the following null hypothesis:H 0:λs max ≡max n λn ωnn ≤0(23)where λs max denotes the best standardized relative performance and ωnn the asymptotic vari-ance of λn .Since the sample average ¯X t,n =t −1 T i =1X i,n is a consistent estimate for λn the corresponding test statistic is:T sm t =√t ¯X s t,max (24)where ¯X s t,max =max n ¯X t,n ˆωnnand ˆω2nn is a consistent estimator of ω2nn ≡lim n →∞V ar (√n ¯X t,n )and is 6Hansen’sSpa Test demostrates better power properties.See Hansen (2001)for further details 7In the multivariate framework arrays of covariance matrices rather than vectors of variances are compared 8The choice of an appropriate loss function in the univariate case usually boils down to some measure of squared errors.The test in fact provides two predefined loss functions:Mean Squared Error (MSE)and Mean Absolute Deviation (MAD)estimated using the bootstrap method9.Hence,T smtrepresents the largest t-statistic(of relative performance)where the superscript sm stands for’standardized maximum’.Under regularity conditions,stated above:T sm t =max n¯Xt,nˆωnnp−→max nλnωnn(25)which is lower then0if and only ifλn≤0for some n.More clearly,the question is whether¯X s t,max is large enough in order to reject the null hypothesis ofλs max≤0.Therefore,the SPA test,using the bootstrap technique in order to estimate the distribution of¯X s t,max under the null hypothesis, allows to determine critical values as thresholds where¯X s t,max becomes too large to be consistent with the null hypothesis.Rather than giving a detailed description of the bootstrapping10we will focus on the general idea behind this method.Given the previous assumptions-i.e.λn≡E[X t,n] is well-defined for n=1,...,N-it holds that:√¯X−λ)d−→N(0,Ω)(26) whereλ=(λ1,...,λN) ,¯X=(¯X1,...,¯X N) andΩ=E[n(¯X−λ)(¯X−λ) ].The covariance matrixΩwill be estimated by bootstrapping methods.More precisely,the test uses the observed sample in order to generate k random resamples-i.e.iid bootstrap method-from the distribution of¯X.For each of these resamples an estimate ofω2nn is computed and therefore the distribution of¯X st,maxapproximated.Finally,critical values and p-values can be calculated from the estimated distribution of¯X s t,max.In other words,the SPA test is a method which yields the probability distribution against which one compares the best performance from those of some competing models.It generates the probability distribution of the model which performs best relative to the benchmark.This distribution is certainly not a standard one which we can look up in a text book but rather strongly depends on the models which enter our model comparison.As a result,the distribution is different in every case and one has tofind it by means of statistical methods,such as the bootstrap.6Data and Empirical ResultsOur empirical analysis is based on General Electric(GE)and IBM stock returns for the period January1,1981-December31,1999.Returns and realized variances are computed from equally spacedfive-minute prices from the Trade and Quote(TAQ)database-i.e.78intraday observations. The choice of5-min.intervals strikes a satisfactory trade offbetween the accuracy of the continuous time underlying process and the noise due to market microstructure effects.Missing values are obtained by linearly interpolating the closest previous and thefirst following 5-min.price.Thefive-minute returns are computed as thefirst difference of the logarithmic prices premultiplied by100.The dataset has been cleaned from weekends and holydays and similarly, 9See Hansen(2001)and Hansen,Kim and Lunde(2003)for further details10See Hansen(2001)for technical detailswe do not consider early closing days.The period from January1,1995to the end of the sample will be used for the out of sample evaluation to compare model forecasts.Wefinally end up with3652in sample and1187out of sample observations.Out of sample forecasts have been computed by means of a moving window of dimension k=100within which,at each iteration k,the one step ahead forecast has been computed replacing the expectations for the t+1 t+1 in the variance equation forecast by its observed value.The coefficients are updated every100 observations restimating the model by including the last100out of sample observations into the dataset.Estimation results for the six competing models are avaliable on request.The dynamic behind the one step ahead covariance matrices forecast is reported in Figure1.Table5contains the p-values of the out of sample one-step ahead forecasts comparisons under the null hypotesis that the benchmarck model,the BEKK-GARCH(1,1),is the best forecasting model.The consistent p-value-i.e.p−value c-is produced by the SPA test and is asimptotically valid and unbiased11. This p-value informs whether the benchmark model is outperformed,in terms of predictive ability, by one or more competing models.More precisely,a high p-value informs that there is no evidence that one or more competing models outperform the benchmark.The SPA test provides also an upper-i.e.p−value u-and a lower-i.e.p−value l-bound for the consistent p-value.Tables5, 6and7report the p-values-i.e.the consistent and its bounds-the sample performance of the model under the null(benchmark),sample informations and the bootstrap parameters12.Table5:SPA test p-values-ABenchmark model:BEKK-GARCH(1,1)Number of models:n=5Sample size:n=1100Test Statistic:TestStatScaledMax()Bootstrap parameters:B=1000(resamples)q=0.5(dependence)Loss function p−value l p−value c p−value u Sample performanceFrobenius Metric0.480000.716000.93500−3.48049Eigenvalue Metric0.477000.738000.95500−3.35674Forstner Metric0.590000.590000.99800−0.84536Cos Mass Metric0.486000.486001.00000−0.27176The results in Table5show that there is no evidence that BEKK-GARCH(1,1)is outper-formed by any of the other models.Moreover BEKK-GARCH(1,1)has always the best sample performance measured as the average distance between the out of sample realized variance and the BEKK-GARCH(1,1)one step ahead variance forecasts.Further evidence is given by the results reported in Table6where the SPA test is performed11See Hansen,Kim and Lunde(2003)for technical details12See Hansen,Kim and Lunde(2003)for details。
