概率论与数理统计(英文) 第九章
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174 9. Nonparametric Statistics
Statistical analyses that do not depend upon the
knowledge of the distribution and parameters of
the population are called nonparametric or
distribution-free methods.
不依赖于总体的分布及其参数的统计方法称为非参数方法或非分布方法。
9.1 Sign Test 符号检验
1
The simplest of all nonparametric methods is the sign test, which
is usually used to test the significance of the difference between two
means in a paired experiment.
最简单的非参数检验是符号检验
检验两个总体均值差的显著程度
It is particularly suitable when the various pairs are observed under
different conditions, a case in which the assumption of normality
may not hold. However, because of its simplicity, the sign test is
often used even though the populations are normally distributed. As
is implied by its name in this test only the sign of the difference 175 between the paired variates is used.
若两个总体的均值相等，那么符号‘＋’、‘－’的概率一样。
D = sign of (X1－X2 )
If p denotes the probability of a difference D being positive and
q the probability of its being negative, we have as hypothesis p=1/2.
appropriate test statistic is X , X~B(n, p)， X --- N(‘+”)
To test the hypothesis
012:0H
112:0H
we will reject 0H in favor of 1H only if the proportion of plus
signs is sufficiently less than 1/2, that is , when the value x of our
random variable is small. Hence, if the computed P-value
12()PPXxwhenp
is less than or equal to the significance level , we reject 0H in
favor of 1H.
To test the hypothesis
012:0H
112:0H 176 we reject 0H in favor 1H when the proportion of plus signs is
significantly less than or significantly greater than 1/2. This, of
course, is equivalent to x being sufficiently small or sufficiently
large, respectively. Therefore, if /2xn and the computed P-value
122()PPXxwhenp
is less than or equal to , or if /2xn and the computed P-value
122()PPXxwhenp
is less than or equal to , we reject 0H in favor 1H.
Example 9.1.1
A taxi company is trying to decide whether the use of radial tires
instead of regular belted tires improves fuel economy. Sixteen cars
are equipped with radial tires and driven over a prescribed test
course.
Without changing drivers, the same cars are then equipped
with the regular belted tires and driven once again over the test
course. The gasoline consumption, in kilometers per liter, was
recorded as follows:
Car Radial tires Belted tires D
1 4.2 4.1 ＋
2 4.7 4.9 －
3 6.6 6.2 ＋
4 7.0 6.9 ＋ 177 5 6.7 6.8 －
6 4.5 4.4 ＋
7 5.7 5.7
8 6.0 5.8 ＋
9 7.4 6.9 ＋
10 4.9 4.9
11 6.1 6.0 ＋
12 5.2 4.9 ＋
13 5.7 5.3 ＋
14 6.9 6.5 ＋
15 6.8 7.1 －
16 4.9 4.8 ＋
Can we conclude using the 5% level of significance that cars
equipped with radial tires have better fuel economy than those
equipped with regular belted tires?
Solution Let 1 and 2 represent the mean kilometers per liter for
cars equipped with radial and belted tires, respectively,
1. 012:0H. 112:0H
2. Test statistics: Binomial variable X with p=1/2.
3. 0.05
4. Calculations: After replacing each positive difference by a “＋” 178 symbol and each negative difference by a “－” symbol, and then
discarding the two zero differences, we obtain the sequence
＋ － ＋ ＋ － ＋ ＋ ＋ ＋ ＋ ＋ ＋ －＋
for which 14n and 11x. Using the normal-curve approximation,
7np, 14/2npq,
we find
10.57(11)(1.87)0.030714/2XnpPPXPPZnpq
5. Decision: Since 0.03070.05P, we reject 0H and conclude that,
on the average, radial tires do improve fuel economy.
符号检验的利弊
n 必须比较大
因为对于 n =5的样本，会出现永远不拒绝“总体均值相等“的假设。（ 极端情形2(1/2)5=0.0625，全部为正号情形 ）
对双边检验，n至少6以上，越大越好。
179 9-2 Rank-Sum Test 秩和检验
the Wilcoxon rank-sum test or Wilcoxon
two
–sample test
is an appropriate alternative to the two-sample
t-test described
a sample (x1,x2, …,xn1) from X’s population
a sample (y1,y2, …,yn2) from Y’s population
Hypothesis: 12 总体均值是否相等
n= n1 +n2
In Wilcoxon’s test:
These n variates
are
graded
(or ranked)
according to in creasing size
n个变量从小到大排列并编号
x1, y1, x2, x3, y2, …
1 2 3 4 5
w1=1+3+4+…
