Extending the rank likelihood for semiparametric copula estimation
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组织左转弯车流的可行方法【中英文实用版】Title: Feasible Methods for Organizing Left-Turning TrafficTitle: 组织左转车流的可行方法Introduction:Efficiently managing left-turning traffic is crucial for maintaining smooth traffic flow and ensuring the safety of motorists.There are several feasible methods that can be implemented to organize left-turning traffic effectively.引言:有效地管理左转车流对于保持流畅的交通流和确保驾驶员的安全至关重要。
有几种可行的方法可以实施以有效地组织左转车流。
Method 1: Left-Turn LanesOne of the most common methods for organizing left-turning traffic is the use of left-turn lanes.These dedicated lanes allow motorists to queue up and make their left-hand turns at a designated time, reducing confusion and improving traffic flow.方法一:左转车道组织左转车流最常见的方法之一是使用左转车道。
这些专用的车道允许驾驶员排队并在指定时间进行左转,减少混乱并改善交通流。
方法二:左转绿灯延时Another effective method is to implement left-turn green lightextensions.By extending the duration of the green light for left-turning vehicles, motorists have more time to complete their turns, reducing the likelihood of gridlock and improving overall traffic efficiency.方法二:左转绿灯延时另一种有效的方法是实施左转绿灯延时。
金属封边工艺流程Metal edge sealing is an important process in the manufacturing of metal products. It involves the application of a protective coating to the edges of metal parts to prevent corrosion, ensure structural integrity, and improve aesthetics. 金属封边是金属制品制造中重要的工艺流程。
它涉及在金属零件的边缘上涂抹一层保护性涂层,以防止腐蚀,确保结构完整性,并提高美观度。
From a technical perspective, the metal edge sealing process involves several steps. First, the edges of the metal parts are cleaned and prepared for coating. This may involve sanding, degreasing, and other surface preparation techniques. Then, a sealant or protective coating is applied to the edges using specialized equipment such as spray guns or automated sealing machines. The coated edges are then cured or dried to ensure adhesion and durability. 从技术角度来看,金属封边工艺流程涉及几个步骤。
首先,金属零件的边缘被清洁并准备涂层。
这可能涉及砂磨、脱脂和其他表面处理技术。
手写修改合同英语作文Title: Amendment of Contract。
In response to your request for the amendment of the contract, I am writing to provide the necessary modifications. Please find below the proposed changes along with the rationale for each amendment:1. Amendment to Clause 2: Scope of Services。
Original Clause: The scope of services shall include A, B, and C.Proposed Amendment: The scope of services shall be expanded to include D, in addition to A, B, and C.Rationale: The addition of service D is essential to meet the evolving needs of both parties and to ensure comprehensive coverage of all necessary tasks.2. Amendment to Clause 4: Payment Terms。
Original Clause: Payment shall be made within 30 days of receipt of invoice.Proposed Amendment: Payment shall be made within 45 days of receipt of invoice.Rationale: Extending the payment term to 45 days will provide both parties with a more reasonable timeframe for processing payments, reducing the likelihood of any financial strain.3. Amendment to Clause 7: Termination。
合同有效期延期顺延函回函格式I understand that you are looking for a letter format for extending the validity period of a contract. This is a common situation in business, and it's important to handle it professionally and effectively. When writing a letter to request an extension of a contract, it's crucial to be clear, courteous, and persuasive. You want to convey the reasons for the extension and demonstrate that it is in the best interest of both parties.Firstly, it's important to start the letter by addressing the recipient in a polite and formal manner. This sets the tone for the rest of the letter and shows respect for the recipient. You can start by stating the purpose of the letter, which is to request an extension of the validity period of the contract. Be sure to provide the details of the original contract, such as the date it was signed, the parties involved, and the original validity period. This helps to provide context and clarity for the request.Next, you should clearly state the reasons for requesting the extension. Whether it's due to unforeseen circumstances, delays in project completion, or any other valid reason, it's important to be honest and transparent. Providing a detailed explanation of the situation will help the recipient understand the need for the extension and may increase the likelihood of a positive response.In addition to explaining the reasons for the extension, it's also important to propose a new validity period forthe contract. This shows that you have thought about the situation and have a clear plan for moving forward. Be sure to suggest a reasonable and realistic new validity period that takes into account the circumstances that led to the extension request.It's also important to express your willingness to negotiate and accommodate the needs of the recipient. This demonstrates a collaborative and cooperative approach,which can help to build a positive and productive working relationship. You can also offer to provide any additionalinformation or documentation to support your request, if necessary.Finally, it's important to conclude the letter by expressing gratitude for the recipient's consideration of the request. A courteous and appreciative tone can leave a positive impression and may influence the recipient's response. You can also offer to discuss the matter further or provide any additional information if needed.In conclusion, when writing a letter to request an extension of the validity period of a contract, it's important to be clear, courteous, and persuasive. By providing the details of the original contract, explaining the reasons for the extension, proposing a new validity period, and expressing willingness to negotiate, you can increase the likelihood of a positive response. A polite and appreciative tone can leave a positive impression and help to build a productive working relationship.。
谈谈你心中的直播带货主题作文英文回答:The advent of livestream e-commerce has revolutionized the retail landscape, offering a unique and engaging way for businesses to connect with consumers. This innovative shopping format has gained immense popularity in recent years, with both retailers and consumers embracing its advantages.Livestream e-commerce seamlessly merges the worlds of online shopping and live entertainment, providing an immersive and interactive shopping experience. Unlike traditional e-commerce platforms, where customers browse static product listings, livestream e-commerce allows them to interact with products in real-time. Viewers can ask questions, engage with the host, and even make purchases directly through the livestream.The interactive nature of livestream e-commerce makesit particularly well-suited for product demonstrations. Hosts can showcase the features and benefits of products in a dynamic and engaging way, providing viewers with a more comprehensive understanding of the offerings. Additionally, livestreams allow for real-time feedback, enabling businesses to gather valuable insights into customer preferences and adjust their offerings accordingly.Another significant advantage of livestream e-commerce is its ability to build trust and establish personal connections between sellers and buyers. By interacting with the host on a live stream, viewers get to know the people behind the products, forming a deeper understanding of the brand and its values. This personal touch helps fostertrust and increases the likelihood of purchases.Furthermore, livestream e-commerce offers businesses the opportunity to reach a wider audience. By partnering with popular livestreaming platforms or influencers, businesses can tap into a vast network of potential customers. Live streaming events can also be recorded and shared on social media, further extending the reach of thecontent.While livestream e-commerce holds immense potential for businesses, it also presents certain challenges. One of the key considerations is the need for high-quality production values. Live streams require a stable internet connection, clear audio, and engaging visuals to capture the attention of viewers. Additionally, businesses must invest intraining their staff to effectively present products and engage with customers during live events.Another challenge is the issue of product returns and exchanges. Unlike traditional e-commerce, where customers can easily return or exchange products, livestream e-commerce often involves purchases made on the spot. Businesses must establish clear policies and provide adequate support to handle any post-purchase issues.Despite these challenges, livestream e-commerce remains a promising avenue for businesses looking to connect with consumers in a more engaging and effective way. By leveraging the power of live video, businesses can creatememorable shopping experiences, build stronger relationships with customers, and drive sales growth.中文回答:直播带货这一新颖的电商模式近年来受到了广泛关注,它将网络购物与现场娱乐无缝结合,为企业和消费者提供了独特的互动购物体验。
清洁工具类英语作文Title: The Importance of Cleaning Tools in Everyday Life。
Cleaning tools play a crucial role in maintaining cleanliness and hygiene in our daily lives. From sweeping floors to scrubbing dishes, these tools enable us to keep our surroundings clean and free from harmful bacteria and germs. In this essay, we will explore the significance of cleaning tools and how they contribute to a healthier environment.First and foremost, cleaning tools such as brooms, mops, and vacuum cleaners are essential for keeping our living spaces tidy. Sweeping away dust and debris, mopping up spills, and vacuuming carpets and upholstery help eliminate dirt and allergens that can accumulate over time. By regularly using these tools, we can prevent the buildup of dust and grime, creating a more pleasant and comfortable environment for ourselves and others.Furthermore, cleaning tools are indispensable in maintaining personal hygiene. Items such as toothbrushes, sponges, and scrub brushes are used daily to cleanourselves and our belongings. Proper oral hygiene, for example, requires the use of a toothbrush and toothpaste to remove food particles and plaque from our teeth and gums. Similarly, scrubbing brushes and sponges are used to clean dishes, utensils, and household surfaces, preventing the spread of harmful bacteria and viruses.In addition to promoting cleanliness, cleaning toolsalso contribute to our overall well-being by reducing the risk of illness and infection. Bacteria and germs thrive in dirty environments, increasing the likelihood ofcontracting diseases such as colds, flu, and food poisoning. By regularly cleaning and disinfecting our surroundings, we can minimize the spread of pathogens and protect ourselves and our loved ones from illness.Moreover, cleaning tools play a vital role inpreserving the lifespan of our belongings. Regularmaintenance and cleaning help prevent wear and tear, extending the longevity of household items such as furniture, appliances, and clothing. For example, using alint roller to remove lint and pet hair from clothing helps keep fabrics looking fresh and prevents them from deteriorating over time. Similarly, regularly cleaning kitchen appliances such as ovens and refrigerators helps ensure optimal performance and prevents the buildup of grease and food residues.In conclusion, cleaning tools are essential for maintaining cleanliness, promoting personal hygiene, and safeguarding our health and well-being. By regularly using these tools to clean our surroundings and belongings, wecan create a cleaner, healthier, and more pleasant environment for ourselves and future generations. Therefore, it is important to recognize the significance of cleaning tools and incorporate them into our daily routines.。
关于锻炼好处的英文作文Title: The Benefits of Exercise。
Exercise is an integral aspect of maintaining a healthy lifestyle, with its advantages extending beyond physical well-being to encompass mental and emotional health. Inthis essay, we will explore the myriad benefits of regular physical activity.Firstly, engaging in exercise promotes physical fitness and overall health. Regular workouts contribute to weight management by burning calories and building muscle mass. This, in turn, reduces the risk of obesity and associated health conditions such as diabetes, heart disease, and hypertension. Moreover, exercise strengthens the cardiovascular system, improving heart and lung function, and enhances flexibility and joint mobility, reducing the likelihood of injury.Furthermore, exercise plays a crucial role in promotingmental health and well-being. Physical activity stimulates the release of endorphins, neurotransmitters that act as natural mood lifters, thereby alleviating symptoms of stress, anxiety, and depression. Regular exercise has been shown to enhance cognitive function, including memory, concentration, and problem-solving skills. It fostersbetter sleep patterns, leading to improved overall quality of life.Additionally, participating in physical activities fosters social connections and community engagement. Whether through team sports, group fitness classes, or outdoor recreational pursuits, exercise provides opportunities for individuals to connect with others who share similar interests and goals. These socialinteractions not only enhance motivation and accountability but also contribute to a sense of belonging and support network, which are crucial for mental and emotional well-being.Moreover, exercise serves as a powerful tool for stress management. In today's fast-paced world, stress has becomeubiquitous, leading to detrimental effects on both physical and mental health. Engaging in physical activity helps to reduce the production of stress hormones such as cortisol while promoting relaxation and a sense of calm. Whetherit's a brisk walk in nature, a yoga session, or a high-intensity workout, exercise provides a healthy outlet for releasing pent-up tension and improving overall resilience to stressors.Furthermore, regular exercise has long-term benefitsfor longevity and quality of life. Studies haveconsistently shown that individuals who lead active lifestyles tend to live longer and enjoy a higher quality of life in their later years. By maintaining muscle strength, bone density, and overall functional capacity, exercise helps to preserve independence and mobility as we age. It also reduces the risk of chronic diseases associated with aging, such as osteoporosis and sarcopenia, thereby promoting healthy aging and longevity.In conclusion, exercise offers a multitude of benefits that extend far beyond physical fitness. From promotingmental health and emotional well-being to fostering social connections and enhancing overall quality of life, regular physical activity is essential for leading a healthy and fulfilling life. By incorporating exercise into our daily routine, we can reap the rewards of improved physical, mental, and emotional health both now and in the years to come.。
酒店免费延迟退房英语作文延迟退房服务是酒店行业中的一项常见服务,提供给客人在原定退房时间之后继续留宿的机会,通常是免费的。
这种服务可以给客人更多的灵活性和舒适度,同时也展现了酒店的服务质量和关怀。
下面是一篇参考网上下载最多的范文,供您参考:Title: The Value of Complimentary Late Checkout Service in Hotels。
In the bustling world of hospitality, hotels strive to go above and beyond to ensure their guests have a memorable and comfortable stay. One such service that has become increasingly popular is complimentary late checkout. This service allows guests to extend their stay beyond the standard checkout time without incurring additional charges. In this essay, we will explore the significance of complimentary late checkout service in hotels and itsimpact on guest satisfaction.First and foremost, complimentary late checkout offersguests added flexibility and convenience. Whether they have a late flight, a busy schedule, or simply wish to enjoy a few extra hours of relaxation, late checkout provides them with the opportunity to tailor their stay to their needs. This flexibility is especially appreciated by business travelers who may have meetings or appointments extending into the afternoon.Moreover, offering complimentary late checkout is a gesture of goodwill and hospitality from the hotel. It demonstrates to guests that their comfort and satisfaction are top priorities. By providing this extra service at no additional cost, hotels can enhance their reputation and foster positive word-of-mouth marketing. Guests are more likely to recommend a hotel that goes the extra mile to accommodate their needs.Additionally, complimentary late checkout cancontribute to a smoother departure process for both guests and hotel staff. Rather than rushing to vacate their room by a specific time, guests can take their time to pack, unwind, and prepare for their departure at their own pace.This can lead to a more relaxed and enjoyable experiencefor guests, reducing the likelihood of stress orfrustration.Furthermore, hotels can benefit from offering complimentary late checkout through increased guest loyalty and repeat business. When guests feel valued and appreciated, they are more likely to return to the samehotel for future stays. Additionally, they may be more inclined to book directly through the hotel's website or loyalty program, rather than through third-party booking platforms.In conclusion, complimentary late checkout serviceplays a significant role in enhancing the overall guest experience in hotels. It provides added flexibility, demonstrates hospitality, streamlines the departure process, and fosters guest loyalty. By offering this service, hotels can differentiate themselves from competitors and leave a lasting impression on their guests. It's not just about providing a place to stay; it's about creating memorable moments and exceeding expectations.This essay illustrates the importance of complimentary late checkout service in hotels and its positive impact on guest satisfaction and loyalty. By recognizing the value of this service and implementing it effectively, hotels can elevate the guest experience and distinguish themselves in a competitive market.。
保险行业英语单词Insurance Industry English VocabularyIntroduction:The insurance industry is a vital sector in the modern world, providing financial protection and coverage for individuals and businesses alike. In order to fully understand and navigate the complexities of this industry, it is essential to familiarize oneself with the key English vocabulary related to insurance. This article aims to present a comprehensive list of essential insurance industry English words that will enable readers to enhance their understanding of this field.I. Types of Insurance:1. Life Insurance - Provides financial protection to beneficiaries in the event of the insured person's death.2. Health Insurance - Offers coverage for medical expenses and healthcare services.3. Property Insurance - Protects against damage or loss of property due to various perils such as fire, theft, or natural disasters.4. Auto Insurance - Provides coverage for damages and liabilities related to vehicles.5. Liability Insurance - Protects against legal liabilities arising from injury or damage caused to others.II. Insurance Policies:1. Premium - The amount paid to the insurance company to obtain coverage.2. Deductible - The initial amount that policyholders must pay out-of-pocket before the insurance coverage kicks in.3. Coverage Limit - The maximum amount an insurance policy will pay for a covered loss.4. Exclusion - Specific conditions or circumstances that are not covered by the insurance policy.5. Claims - Requests made by policyholders to receive benefits or compensation for covered losses.6. Policyholder - The individual or entity that holds an insurance policy.7. Insured - The person or property covered by the insurance policy.8. Underwriter - The person or company responsible for evaluating and pricing insurance risks.9. Renewal - The process of extending an insurance policy beyond its initial term.10. Endorsement - A written amendment or modification to an existing insurance policy.III. Insurance Terms and Concepts:1. Actuary - A professional who uses mathematical and statistical methods to assess risks and determine insurance premiums.2. Risk Assessment - The evaluation of potential risks to estimate the likelihood of loss or damage.3. Loss Ratio - The ratio of incurred losses and expenses to earned premiums, used as a measure of an insurer's profitability.4. Premium Rate - The price of insurance per unit of coverage, typically expressed as a percentage of the insured amount.5. Claims Adjuster - An individual responsible for investigating and assessing claims submitted to an insurance company.6. Insurable Interest - The legal or financial interest a policyholder must have in the subject matter insured for a policy to be valid.7. Reinsurance - The practice of insurance companies transferring a portion of their risks to other insurers to reduce their exposure.8. Policyholder Surplus - The excess of an insurance company's assets over liabilities, serving as a measure of its financial strength.9. Salvage - Property or assets that have been damaged but still retain value and can be sold by the insurance company to recover losses.10. Subrogation - The legal process through which an insurance company seeks reimbursement from a third party for losses paid to the insured.Conclusion:As the insurance industry continues to evolve and expand, mastering the relevant English vocabulary is crucial for professionals and individuals seeking to navigate this field effectively. This article has provided acomprehensive list of essential insurance industry English words, covering various types of insurance, policy terms, and key concepts. By familiarizing oneself with these vocabulary terms and their meanings, readers can enhance their understanding and fluency in discussing insurance-related matters.。
my ……英语作文Title: Importance of Safety Education: A Classroom MeetingSafety education plays a pivotal role in our lives, extending far beyond mere school requirements. It equips us with essential knowledge and skills to safeguard ourselves and others from potential hazards in diverse settings. Today, we convened to delve into the significance of safety education and its profound impact on our well-being.To begin with, safety education heightens our awareness of potential dangers. Whether at home, school, or outdoors, understanding safety protocols enables us to identify risks promptly and take preemptive measures. For instance, familiarity with fire safety procedures can avert accidents and save lives during emergencies.Furthermore, safety education instills responsible behavior. Learning traffic regulations, for instance, transforms us into conscientious pedestrians and drivers, thereby diminishing the likelihood of road accidents.Similarly, mastering the proper handling of tools and equipment minimizes workplace injuries and fosters a secure learning environment.Moreover, safety education empowers us to respond effectively in crises. Fundamental first aid training, such as CPR techniques, equips us to administer immediate assistance to those in distress. This knowledge proves invaluable in potentially saving lives before professional aid arrives.Additionally, safety education cultivates a culture of mutual concern and respect. Prioritizing safety demonstrates our esteem for ourselves and others by creating secure surroundings. It encourages collaboration and teamwork, as each person contributes to upholding a safe environment for the community.Ultimately, safety education imparts lifelong skills that extend well beyond the classroom. Whether at home, work, or engaging in leisure activities, applying safety principles ensures our well-being in diverse situations.By continually updating our knowledge and staying abreast of new safety practices, we contribute to a safer and healthier society.In conclusion, safety education is indispensable for personal growth and community welfare. By recognizing potential risks, practicing responsible conduct, and knowing how to react in emergencies, we empower ourselves to lead safer lives. Let us pledge to implement today’s lessons to foster a safety-oriented culture within our school and beyond.Together, through our commitment to safety education, we can effect positive change and ensure that every individual feels secure and protected in their environment. Thank you for your attention and dedication to understanding this critical topic.。
关于介绍人的英语作文Title: The Role of Referral in Professional Networking。
In the contemporary landscape of professional networking, the role of referrals stands as a pivotal force driving opportunities and connections. A referral, in its essence, is a recommendation or endorsement provided by one individual to another, typically within a professional context. This essay delves into the significance of referrals, elucidating their impact on career advancement, fostering trust, and expanding one's professional network.First and foremost, referrals serve as a potentcatalyst for career advancement. In today's competitive job market, securing employment opportunities can be arduous, often reliant on more than just qualifications and experience. A well-timed referral from a trusted contactcan elevate a candidate's chances of landing a job, often bypassing the initial screening stages and granting direct access to decision-makers. Employers frequently prioritizereferred candidates due to the implicit trust imbued by the recommender's endorsement, thus increasing the likelihood of success for the referred individual.Moreover, referrals play a vital role in fosteringtrust and credibility within professional circles. In an era inundated with information and choices, trust has emerged as a scarce yet invaluable commodity. When an individual receives a referral from someone they trust or respect, they are more inclined to view the recommended individual favorably, attributing a degree of credibility to their skills and character. This trust not only expedites the establishment of professional relationships but also cultivates a positive reputation for both the referrer and the referred, enhancing their standing within their respective industries.Furthermore, referrals facilitate the expansion ofone's professional network by leveraging existing connections. Human networks are inherently interconnected, with each individual serving as a node capable of bridging disparate connections. When one extends or receives areferral, they effectively tap into the expansive network of the referrer, gaining access to individuals and opportunities that may have otherwise remained elusive. As referrals propagate through networks, they facilitate the organic growth of relationships and open doors to new avenues of collaboration and advancement.However, it is imperative to acknowledge that the efficacy of referrals is contingent upon the quality and relevance of the connections involved. While a referral from a trusted colleague or mentor can be invaluable, indiscriminate referrals devoid of merit can potentially undermine one's credibility and dilute the potency of their network. Thus, it is incumbent upon individuals to exercise discretion and discernment when extending or soliciting referrals, ensuring that they align with the recipient's needs and aspirations.In conclusion, referrals constitute a cornerstone of professional networking, wielding considerable influence in shaping career trajectories and fostering meaningful connections. By harnessing the power of referrals,individuals can expedite their career advancement,cultivate trust and credibility, and expand their professional networks with precision and efficacy. As the adage goes, "It's not just what you know, but who you know," and referrals serve as the linchpin that bridges the gap between knowledge and opportunity in the dynamic landscape of professional endeavors.。
表示提出要求的高级英语作文句子Asking for something is a common occurrence in our daily lives. Whether it's a simple request for a favor or a more formal appeal for assistance, the way we phrase our requests can greatly impact how they are received. In this article, we will explore some advanced English sentences that can be used to express requests in a polite, professional, and effective manner.1. I would be grateful if you could kindly provide me with the necessary information regarding the upcoming conference.2. Could you please send me the latest updates on the project by the end of the day?3. I am in need of your expertise to help me analyze the data and make recommendations for our marketing strategy.4. It would be greatly appreciated if you could spare a few moments of your time to meet with me and discuss the upcoming budget proposal.5. I kindly request your assistance in coordinating the logistics for the upcoming event.6. Would it be possible for you to review my presentation slides and provide feedback before the meeting tomorrow?7. I humbly ask for your guidance in navigating the complexities of the new software system.8. Could you please consider extending the deadline for submission of the final report due to unforeseen circumstances?9. I would like to request a budget increase to accommodate additional expenses for the upcoming project.10. I respectfully request your support in securing funding for the new research initiative.By using these advanced English sentences to express your requests, you can effectively communicate your needs while maintaining a professional and respectful tone. Remember to always show gratitude and courtesy when asking for something, as this will increase the likelihood of a positive response from the recipient.。
卖身边的物品英语作文英文回答:Introduction.The decision to sell personal items can be prompted by various reasons, ranging from financial constraints to a desire for decluttering or a change in lifestyle. As we accumulate possessions throughout our lives, it becomes essential to periodically assess their value and significance to determine if they still meet our current needs and aspirations.Benefits of Selling Personal Items.Selling unwanted items offers numerous benefits, both tangible and intangible. From a practical standpoint, it can provide additional income to supplement expenses or fund future goals. It also helps declutter living spaces, reducing stress and promoting a sense of order andspaciousness. Additionally, selling personal items can be an act of environmental responsibility, as it reduces waste by extending the lifespan of products.Emotional Considerations.While the practical benefits of selling personal items are undeniable, it is crucial to acknowledge the emotional considerations that may arise. For many individuals, possessions carry sentimental value and serve as reminders of past experiences or relationships. It is important to approach the process of decluttering with sensitivity and mindfulness, carefully evaluating the emotional attachment to each item before making a decision.Methods of Sale.The advent of online marketplaces has revolutionized the way we sell personal items. E-commerce platforms such as eBay, Amazon, and Craigslist provide a convenient and global reach, enabling sellers to connect with potential buyers from diverse locations. However, traditional methodssuch as garage sales, flea markets, and consignment shops remain viable options for those who prefer a more local approach.Strategies for Maximizing Value.To maximize the value of your personal items, it is essential to present them in their best possible condition. Thoroughly clean and repair items before listing them for sale. High-quality photographs and detailed descriptions will entice potential buyers and increase the likelihood of a successful transaction. Researching similar items on the market will provide you with valuable insights into pricing strategies and market demand.Legal and Ethical Considerations.When selling personal items, it is paramount to adhere to legal and ethical guidelines. Ensure that you have the legal right to sell the items and that they are not counterfeit or stolen. Clearly disclose any defects or flaws in the items to potential buyers, and avoid engagingin deceptive marketing practices.Conclusion.The decision to sell personal items should be based on careful consideration of both practical and emotional factors. By approaching the process with sensitivity, embracing the benefits of decluttering, and employing effective selling strategies, you can maximize the value of your unwanted items while promoting a more sustainable and organized lifestyle.中文回答:引言。
Unit 1 Human ReflectionsBefore Reading Activities1. Fill in the blanks with the words you hear.(1) crush; (2) lifetime; (3) deceive; (4) smile; (5) right; (6) grateful; (7) arrives;(8) assurance; (9) return; (10) contentⅡ. Language FocusVocabularyPart A1. stranded;2. emeritus;3. erode;4. wondrous;5. yearning;6. shackled;7. salvation;8. nibble;9. imperfection; 10. eruptPart B1. B2. C3. A4. D5. A6. B7. C8. A9. D 10. BBank-Cloze(1) bored; (2) affiliate; (3) conducted; (4) valid; (5) ranging(6) fulfilling; (7) revealed; (8) tended; (9) Pressure; (10) toughTranslationPart A1. For some, marriage is the grave of love, while for others, marriage is an effective salvation for those who lead a solitary life.2. Blessed with a great historical responsibility, the Conference is destined to have far-reaching impact on the development of the organization.3. All of these show people’s yearning for a bet ter life, so they have been carried forward generation after generation.4. If Congress approved the bill now, the president warned, the fragile peace process that he is trying to keep could fall apart.5. The couple must be tolerant of the little imperfec tions in each other’s character, otherwise their marriage may end up in divorce.Part B爱情是一部电话机,渴望它响起时,它却总是悄无声息;不经心留意时,它又叮铃铃地响起。
家务劳动清单英语作文Household Chores: A Comprehensive Guide to Maintaining a Tidy HomeMaintaining a clean and organized home is a fundamental aspect of daily life, yet it often goes unnoticed or underappreciated. Household chores, while sometimes seen as mundane tasks, play a crucial role in creating a comfortable living environment and promoting a sense of well-being. In this essay, we will explore the various household chores that are essential for keeping a home in pristine condition and the benefits they provide.One of the most basic and essential household chores is cleaning. This encompasses a wide range of tasks, from dusting and vacuuming to mopping and scrubbing. Regular cleaning helps to remove dirt, dust, and grime, preventing the buildup of allergens and creating a healthier indoor environment. It also helps to preserve the condition of furniture, flooring, and other household items, extending their lifespan and maintaining a pleasant aesthetic.Laundry is another crucial household chore that requires attention. This includes washing, drying, and folding clothes, as well as ironingand mending garments as needed. Keeping on top of laundry not only ensures that everyone in the household has clean clothes to wear but also helps to prevent the accumulation of dirty laundry, which can lead to unpleasant odors and unsanitary conditions.Cooking and meal preparation are also integral parts of household chores. Preparing nutritious and delicious meals for oneself and family members is a way to promote health and well-being. This includes planning menus, grocery shopping, cooking, and cleaning up after meals. By taking an active role in meal preparation, individuals can ensure that their household is well-fed and that the food they consume is of high quality.Another important household chore is the maintenance of outdoor spaces, such as yards, gardens, and patios. This may involve tasks like mowing the lawn, trimming hedges, weeding, and cleaning outdoor furniture. Maintaining the outdoor areas of a home not only enhances the overall aesthetic but also provides a pleasant and inviting space for relaxation and recreation.Household organization and decluttering are also essential chores that contribute to a well-functioning home. This includes tasks like sorting through and organizing possessions, disposing of or donating items that are no longer needed, and maintaining a tidy and efficient storage system. By keeping the home organized,individuals can reduce stress, improve productivity, and create a more calming and harmonious living environment.In addition to the physical tasks involved in household chores, there are also administrative and managerial responsibilities that come with running a household. This includes tasks like paying bills, managing household finances, scheduling maintenance and repairs, and coordinating the activities of family members. Effectively managing these administrative tasks helps to ensure the smooth operation of the household and reduces the likelihood of unexpected disruptions or emergencies.While household chores may sometimes be seen as burdensome or monotonous, they play a vital role in maintaining a healthy, functional, and comfortable living environment. By regularly engaging in these tasks, individuals can enjoy a sense of accomplishment, improved physical and mental well-being, and a greater appreciation for the effort required to keep a household running smoothly.Moreover, the division of household chores can foster a sense of shared responsibility and cooperation within a household. When all members of a family or household contribute to the upkeep of the home, it can strengthen relationships, promote a sense of community, and instill a greater appreciation for the contributions ofothers.In conclusion, household chores are an essential component of daily life, and their importance should not be underestimated. By embracing these tasks and developing a routine that works for one's individual needs and preferences, individuals can create a harmonious and well-maintained home that supports their overall well-being and quality of life.。
处理好剩下的食物英语作文标题,Effective Strategies for Handling Leftover Food。
In today's society, food waste has become a significant issue that affects both the environment and economy. According to research, approximately one-third of all food produced worldwide is wasted each year. Leftover food plays a significant role in this waste, contributing to environmental degradation and economic inefficiency. Therefore, it is crucial to adopt effective strategies for handling leftover food to minimize waste and promote sustainability.One of the most effective strategies for managing leftover food is meal planning. Meal planning involves carefully planning meals for the week, considering portion sizes and ingredients needed. By planning meals in advance, individuals can purchase only the necessary ingredients, reducing the likelihood of excess food that may go to waste. Additionally, meal planning allows for better utilizationof leftovers, as meals can be designed to incorporate leftover ingredients from previous meals.Another essential aspect of handling leftover food is proper storage. Proper storage helps to preserve food freshness and prevent spoilage, extending the shelf life of leftovers. Leftovers should be stored in airtight containers or resealable bags to maintain freshness and prevent odors from permeating the refrigerator. It is also important to label leftovers with the date they were prepared to ensure they are consumed before they spoil. By storing leftovers correctly, individuals can reduce food waste and save money by maximizing the use of leftover ingredients.Furthermore, creative cooking techniques can help transform leftovers into new and exciting dishes. Instead of simply reheating leftovers, individuals can experiment with different recipes and cooking methods to repurpose leftover ingredients. For example, leftover vegetables can be incorporated into soups, stir-fries, or omelets, while leftover proteins such as chicken or beef can be used tomake sandwiches, salads, or casseroles. By thinking creatively and being resourceful in the kitchen,individuals can minimize food waste while enjoyingdelicious meals made from leftover ingredients.In addition to individual actions, communityinitiatives and organizations play a vital role in addressing the issue of leftover food. Food rescue programs, for example, work to collect excess food from restaurants, grocery stores, and other sources and redistribute it to those in need. By rescuing surplus food that wouldotherwise go to waste, these programs help to alleviate hunger and reduce food waste simultaneously. Furthermore, educational campaigns and awareness-raising initiatives can help raise public awareness about the importance ofreducing food waste and provide individuals with the knowledge and tools they need to take action.In conclusion, handling leftover food effectively is essential for reducing food waste, promoting sustainability, and ensuring food security for all. By adopting strategies such as meal planning, proper storage, creative cookingtechniques, and community initiatives, individuals can play a significant role in minimizing food waste andcontributing to a more sustainable food system. Together, we can work towards a future where leftover food is valued and utilized rather than wasted.。
容器播种技术流程Container seeding technology refers to a method of sowing seeds in containers, such as trays or pots, instead of directly in the ground. This technique has gained popularity for its numerous benefits, such as better control over germination, improved plant growth, and reduced risk of seedling loss. By starting seeds in containers, gardeners can create optimal growing conditions, such as proper soil moisture and temperature, which can lead to healthier plants in the long run. Additionally, container seeding allows for easier monitoring of seedlings and reduces the likelihood of pests and diseases affecting young plants.容器播种技术是指在容器中播种种子的方法,例如盘子或盆中,而不是直接在地面上播种。
这种技术因其诸多好处而受到了欢迎,如更好地控制发芽、改善植物生长,并降低幼苗丢失的风险。
通过在容器中种植种子,园艺爱好者可以创造最佳的生长条件,如适当的土壤湿度和温度,这有助于植物长期健康生长。
此外,容器播种可以更容易地监测幼苗,并减少害虫和疾病影响年轻植物的可能性。
One of the main advantages of container seeding technology is the ability to extend the growing season. By starting seeds indoors or ina greenhouse, gardeners can get a head start on the growing season and have plants ready for transplanting once the weather warms up. This can be particularly beneficial in regions with short growing seasons or unpredictable weather patterns, as it allows for a longer growing period and a greater chance of successful harvests. Container seeding also provides flexibility in terms of planting schedules, as seeds can be started at any time of the year, regardless of outdoor conditions.容器播种技术的主要优势之一是延长生长季节的能力。
The Importance of UnderstandingPercentages in Daily LifePercentages, a mathematical concept that often appears in our daily lives, plays a crucial role in understanding and analyzing data. From grades in school to sales in stores, percentages help us compare and interpret information more effectively. In this article, we will explore the significance of percentages in our daily lives and how understanding them can benefit us.In the academic world, percentages are often used to evaluate students' performance. For example, when a student receives a score of 85% on a test, it quickly gives them and their teachers an understanding of how well they performed relative to the maximum score. This percentage-based evaluation system not only helps students identify their strengths and weaknesses but also encourages them to improve their performance.In the business world, percentages are equally important. Business owners often rely on percentage-based data to make informed decisions. For instance, if a store reports a 20% increase in sales from the previous month,the owner can use this information to plan future strategies, such as stocking more popular items or extending the store's hours.Moreover, percentages are also crucial in understanding statistics and probability. When faced with data about the likelihood of certain events happening, percentages help us assess the risks and make informed decisions. For example, if a doctor informs a patient that they have a 90% chance of success with a certain surgery, the patient can use this information to decide whether to proceed with the surgery or seek a second opinion.In conclusion, percentages are an integral part of our daily lives, and understanding them is essential for success in various aspects of life. From academics to business and even in our daily decision-making, percentages provide us with a valuable tool to compare, analyze, and interpret data. Bymastering this concept, we can make more informed decisions and achieve better outcomes in various scenarios.**百分比在日常生活中的重要性**百分比是一个经常出现在我们日常生活中的数学概念,它在理解和分析数据方面起着至关重要的作用。
江苏省常州市第一中学2024-2025学年高三上学期10月月考英语试题一、听力选择题1.Who is most probably Esther?A.A director.B.A reporter.C.A ballet dancer.2.What did Sarah emphasize just now?A.Teamwork.B.Responsibility.C.Creative thinking. 3.What does the woman think of the weather?A.It’s very warm.B.It’s pretty hot.C.It’s super cold.4.What does Greg ask Chloe to do?A.Draw up a notice.B.Send an application.C.View an advertisement. 5.What is the conversation mainly about?A.A charity event.B.A volunteer.C.A restaurant.听下面一段较长对话,回答以下小题。
6.What will the man probably be doing at 10:00 a.m. tomorrow?A.Having a meal with Ms. Carey.B.Talking about marketing strategies.C.Attending a budget review meeting.7.When is tomorrow’s conference call scheduled to begin?A.At 5:30 p.m.B.At 4:30 p.m.C.At 3:00 p.m.听下面一段较长对话,回答以下小题。
8.What is the probable relationship between the speakers?A.Friends.B.Fellow passengers.C.Air hostess and passenger. 9.Why does the man go to France?A.For sightseeing.B.For shopping.C.For business. 10.Where does the woman come from?A.The Netherlands.B.Britain.C.Thailand.听下面一段较长对话,回答以下小题。
关于品牌延伸的英语作文Title: Exploring Brand Extension Strategies。
Brand extension is a strategic marketing initiative where a company uses its existing brand name to launch new products or enter new markets. It is a complex process that requires careful planning, market research, and understanding of consumer behavior. In this essay, we will delve into various aspects of brand extension strategies, discussing their benefits, challenges, and key considerations.First and foremost, brand extension offers several advantages to companies. One of the primary benefits is leveraging the existing brand equity. Established brands already enjoy recognition, trust, and loyalty from consumers, which can significantly reduce the time and resources required to establish a new brand from scratch. By extending the brand into related or complementary product categories, companies can capitalize on thepositive associations consumers have with the original brand, thereby increasing the likelihood of success for the new offerings.Moreover, brand extension allows companies to diversify their product portfolio and capture new market segments.For example, a clothing brand known for its casual wear may extend its brand to include accessories such as bags, shoes, or sunglasses, thereby catering to a broader range of consumer preferences and increasing revenue streams. This diversification also helps mitigate risks associated with relying too heavily on a single product or market.However, despite its potential benefits, brandextension also presents certain challenges and risks. Oneof the main concerns is diluting the original brand equity. If the new products fail to meet consumer expectations ordo not align with the core values of the brand, it can tarnish the brand's reputation and erode trust amongexisting customers. Therefore, it is essential for companies to ensure that any brand extension maintains consistency with the original brand identity while offeringsomething unique and valuable to consumers.Another challenge is cannibalization, where the introduction of new products under the same brand cannibalizes sales of existing products. Companies need to carefully assess the potential impact of brand extension on their existing product lines and develop strategies to minimize cannibalization. This may involve targeting different market segments or positioning the new products as complementary rather than substitutes to existing offerings.When considering brand extension, companies must also conduct thorough market research to understand consumer needs, preferences, and purchase behavior. This involves identifying market gaps, assessing competitive landscape, and evaluating the feasibility of extending the brand into new categories. Additionally, companies should payattention to cultural and regional differences that may influence consumer perceptions and acceptance of the brand extension.Furthermore, effective communication is crucial in successfully launching a brand extension. Companies need to clearly communicate the rationale behind the extension, highlighting how it builds upon the existing brand's strengths and benefits consumers. This may involve leveraging various marketing channels such as advertising, social media, and public relations to generate awareness and generate excitement around the new products.In conclusion, brand extension is a valuable strategy for companies looking to expand their product offerings and reach new markets. By leveraging the existing brand equity, diversifying the product portfolio, and capturing new consumer segments, brand extension can drive growth and enhance competitiveness. However, it is essential for companies to approach brand extension with caution, considering the potential risks and challenges involved, and executing it thoughtfully with a deep understanding of consumer behavior and market dynamics.。
a rX iv:mat h /61413v2[mat h.ST]11Mar27Extending the rank likelihood for semiparametric copula estimation Peter D.Hoff∗February 2,2008Abstract Quantitative studies in many fields involve the analysis of multivariate data of diverse types,including measurements that we may consider binary,ordinal and continuous.One approach to the analysis of such mixed data is to use a copula model,in which the associations among the variables are parameterized separately from their univariate marginal distributions.The purpose of this article is to provide a simple,general method of semiparametric inference for copula models via a type of rank likelihood function for the association parameters.The proposed method of inference can be viewed as a generalization of marginal likelihood estimation,in which inference for a parameter of interest is based on a summary statistic whose sampling distribution is not a function of any nuisance parameters.In the context of copula estimation,the extended rank likelihood is a function of the association parameters only and its applicability does not depend on any assumptions about the marginal distributions of the data,thus making it appropriate for the analysis of mixed continuous and discrete data with arbitrary marginal distributions.Estimation and inference for parameters of the Gaussian copula are available via a straightforward Markov chain Monte Carlo algorithm based on Gibbs sampling.Specification of prior distributions or a parametric form for the univariate marginal distributions of the data is not necessary.Some key words :Bayesian inference,latent variable model,marginal likelihood,Markov chain Monte Carlo,multivariate estimation,polychoric correlation,rank likelihood,sufficiency.1IntroductionStudies involving multivariate data often include measurements of diverse types.For example,a survey or observational study may record the sex,education level and income of its participants,thus including measurements that we may consider binary,ordinal and continuous.Such studiesFigure1:Univariate histograms of the GSS data.are generally concerned with statistical associations among the variables,but not necessarily the scale on which the variables are measured.One approach to data analysis in these situations is to obtain rank-based measures of bivariate association,such as the rank correlation or“Spearman’s rho”.Such procedures are scale-free,but involve ad-hoc methods for dealing with ties and pro-vide inference that is generally limited to hypothesis tests of bivariate association.These issues make such procedures problematic for the analysis of much of social science survey data,in which the variables are often discrete and the hypotheses of interest generally concern multivariate and conditional associations.For example,Figure1shows histograms of six demographic variables of male respondents to the1994General Social Survey.The variables INC,DEG and CHILD refer to the income,highest degree and number of children of a survey respondent,and PINC,PDEG and PCHILD refer to similar variables of the respondent’s parents(further details on the variables are given in Section4).All of these variables are ordered categorical variables,even though some of them have many levels.Additionally,our interests in these variables involve measures of con-ditional association:An assessment of the relationship between income and number of children2PredictorResponse INC DEG PCHILD PDEG INC NA7.03(<.01) 4.07(<.01) 1.40(.12)NA.04(<.01).02(.08)Part of the above problem can be resolved by jointly modeling the variables of interest.A number of latent-variable methods have been recently developed to accommodate non-Gaussian multivariate data.These methods generally proceed by modeling each component of a vector of observations with a parametric exponential family model,in which the parameters for each compo-nent involve an unobserved latent variable.For example,Chib and Winkelmann[2001]present a model for a vector of correlated count data in which each component is a Poisson random variable with a mean depending on a component-specific latent variable.Dependence among the count vari-ables is induced by modeling the vector of latent variables with a multivariate normal distribution. Similar approaches are proposed by Dunson[2000]and described in Chapter8of Congdon[2003]. The model of Chib and Winkelmann can be viewed as a copula model,in which the association parameters are modeled separately from the marginal distributions of the observed data.Such a modeling approach can be applied to a wide variety of multivariate analysis problems:An old mathematical result known as Sklar’s Theorem says that every multivariate probability distribution can be represented by its univariate marginal distributions and a copula,which is a type of joint distribution withfixed marginals.Pitt et al.[2006]develop an estimation procedure for multivariate normal copula models in which the marginal distributions belong to specified parametric families.Unfortunately,the marginal distributions of survey data such as age,number of children,income and education level generally do not belong to standard families.For such data a semiparametric estimation strategy may be appropriate,in which the associations among the variables are represented with a simple parametric model but the marginal distributions are estimated nonparametrically.In the case where all the variables are continuous,Genest et al.[1995]suggest a“pseudo-likelihood”approach to estimation, in which the observed data is transformed via the empirical marginal distributions to obtain pseudo-data that can be used to estimate the association parameters.Klaassen and Wellner[1997]study a similar type of estimation in the case of the Gaussian copula.Such estimators are well-behaved for continuous data but can fail for discrete data,making them somewhat inappropriate for the analysis of mixed continuous and discrete data.For ordinal discrete data with a known number of categories,the dependence induced by the Gaussian copula model is called polychoric correlation. Olsson[1979]describes a two-stage estimation procedure for the parameters in the copula,and this and other estimation strategies appear in a number of software packages including SAS PROC FREQ and the LISREL module PRELIS.Kottas et al.[2005]describe a nonparametric estimation procedure in which the copula is based on a mixture of normal distributions.However,such pro-cedures do not accommodate continuous data,and may even be problematic for discrete data with a large number of categories,as inference in this case requires the simultaneous estimation of the large number of parameters specifying the marginal distributions.As an alternative to these procedures,this article presents an approach to copula estimation in4which the marginal distributions are arbitrary and of unspecified types,thus accommodating both discrete and continuous data.This is achieved by the use of a likelihood function that depends on the association parameters only,and does not make assumptions about the form of the univariate marginal distributions.Inference based on such a likelihood is therefore appropriate for the joint analysis of continuous and ordinal discrete data.For continuous data,the likelihood function we propose is derived from the marginal probability of the ranks,and can be seen as a multivariate version of a“rank likelihood”[Pettitt,1982,Heller and Qin,2001]which does not depend on the univariate marginal distributions.Unfortunately,for discrete data the probability of the observed ranks is not free of these nuisance parameters.To solve this problem,we derive a likelihood that is equivalent to the distribution of the ranks for continuous data but is also free of the nuisance parameters for discrete data.This likelihood function is derived from the probability that the latent variables of the copula model satisfy the partial ordering induced by the observed data.We call this function an extended rank likelihood,as it generalizes the concept of rank likelihood.This likelihood can also be seen as a generalization of a marginal likelihood,which is based on a statistic whose sampling distribution depends only on the parameter of interest and not on any nuisance parameters.In what follows we work with the Gaussian copula model,although the basic ideas can be extended to other parametric families of copulas.In the next section we review the general Gaussian copula model,and discuss how inference for discrete data using existing semiparametric methods is problematic.Section3derives the extended rank likelihood as a general approach to semiparametric copula estimation and discusses parameter estimation in the context of Bayesian inference using a relatively simple Gibbs sampling scheme.The primary goal of this paper is to provide a simple method of inference for the multivariate relationships between variables,such as INC,CHILD,DEG described above,whose univariate marginal distributions cannot be well approximated with simple parametric models.In Section 4we present an analysis of these and other demographic characteristics of males in the1994 U.S.workforce and their parents.In particular,we are interested in the statistical associations among income,education and number of children of the survey respondents,and how they relate to similar characteristics of the parents of the survey respondents.The data come from the1994 General Social Survey,and include a number of discrete and non-Gaussian random variables.In addition to estimating a Gaussian copula model for these data,we estimate and describe the conditional dependencies among the variables on the Gaussian scale,as well as provide predictive and conditional distributions on the original scale of the data.Section5considers notions of statistical sufficiency relevant to the rank likelihood,and a dis-cussion follows in Section6.52Semiparametric copula estimationLet y1and y2be two random variables with continuous CDF’s F1and F2.The transformed variables u1=F1(y1)and u2=F2(y2)both have uniform marginal distributions.The term “copula modeling”generally refers to a model that parametrizes the joint distribution of u1and u2 separately from the marginal distributions F1and F2.A semiparametric copula model includes a parametric model for the joint distribution of u1and u2,but lacks any parametric restrictions on F1or F2.Any continuous multivariate distribution can be used to form a copula model via an inverse-CDF transformation.For example,the bivariate normal distribution can be used to generate dependent data with arbitrary marginals F1and F2as follows:1.sample z1z2 ∼bivariate normal 00 , 1ρρ1 ;2.set y1=F−11[Φ(z1)],y2=F−12[Φ(z2)],where F−1(u)=inf{y:F(y)≥u}denotes the pseudo-inverse of a CDF F.The correspondence to the usual copula formulation can be seen by noting thatΦ(z)=u is uniformly distributed.Suppose(y1,1,y1,2),...,(y n,1,y n,2)are samples from a population that we wish to model with a Gaussian copula.If the marginal distributions F1and F2were continuous and known,then the values z i,j=Φ−1[F j(y i,j)]could be treated as observed data andρcould be estimated directly from the z’s,perhaps using the unbiased estimatorˆρ=1n+1ˆFj(y i,j)]≡Φ−1[˜F j(y i,j)],where the rescalingis to avoid infinities.For continuous data,the estimator˜ρ=1What could go wrong with such an estimator in situations involving discrete data?In general, these pseudo-data estimators of copula parameters will be problematic for discrete data because transformations of such data do not really change the data distribution,they just change the sample space.Consider the simple case of a continuous variable y1and a binary variable y2such that Pr(y2=0)=Pr(y2=1)=1/2.Letting˜z i,j=Φ−1[˜F j(y i,j)],the distribution of˜z1,1,...,˜z n,1 will have an approximately standard normal distribution,but˜z i,2will be approximately equal to eitherΦ−1(1n+1)orΦ−1(n√n+1).As n increases so does the expectation of˜ρ,and it is not a consistent estimator.One problem here is that all of the ˜z i,2’s such that y i,2=1are being pushed to the extreme standard normal quantileΦ−1(n2Φ−1(n2/π.3Estimation using the extended rank likelihoodIn this section we derive a likelihood function that depends on the association parameters and not on the unknown marginal distributions.For continuous data this function is equivalent to the distribution of the multivariate ranks.This is not the case of discrete data,for which the distribution of the ranks depends on the univariate marginal distributions.In this case the derived likelihood function contains less total information than one based on the ranks,but it is free of any parameters describing the marginal distributions.3.1Extended rank likelihoodGeneralizing from the previous section,the Gaussian copula sampling model can be expressed as follows:z1,...,z n|C∼i.i.d.multivariate normal(0,C),(3)y i,j=F−1j[Φ(z i,j)],where C is a p×p correlation matrix and each F−1j denotes the(pseudo)inverse of an unknown univariate CDF,not necessarily continuous.Our goal is to make inference on C,and not on the potentially high-dimensional parameters F1,...,F p.If the z’s were observed we could use them to directly estimate C.The z’s are not observed of course,but the y’s do provide a limited amount of information about them,even absentany knowledge of the F’s:Since the F’s are non-decreasing,observing y i1,j <y i2,jimplies that7z i 1,j <z i2,j.More generally,observing Y=(y1,...,y n)T tells us that Z=(z1,...,z n)T must liein the set{Z∈R n×p:max{z k,j:y k,j<y i,j}<z i,j<min{z k,j:y i,j<y k,j}}.We can take the occurrence of this event as our data.Letting D be thefixed subset of R n×p generated by the observed value of Y,we can calculate the following“likelihood”:Pr(Z∈D|C,F1,...,F p)= D p(Z|C)d Z=Pr(Z∈D|C).(4) As a function of the parameters,this likelihood depends only on the parameter of interest C and not the nuisance parameters F1,...,F p.Estimation of C can proceed by maximizing Pr(Z∈D|C) as a function of C,or by obtaining a posterior distribution Pr(C|Z∈D)∝p(C)×Pr(Z∈D|C).The likelihood function(4)can be seen as a type of marginal likelihood function for estimation in the presence of a nuisance parameter:Consider a generic statistical problem in which the density for data y depends on a parameter of interestθand a nuisance parameterψ.If there exists a statistic t(y)whose distribution depends onθonly,then the density of y may be decomposed asp(y|θ,ψ)=p(t(y),y|θ,ψ)=p(t(y)|θ)×p(y|t(y),θ,ψ).In this situation,estimation ofθcan be based on the marginal likelihood p(t(y)|θ),eliminating the need to estimate the nuisance parameterψ(see,for example,Section8.3of Severini[2000]).The likelihood function Pr(Z∈D|C)in our copula estimation problem can be derived analogously,by decomposing the probability of the observed data asp(Y|C,F1,...,F p)=p(Z∈D,Y|C,F1,...,F p)(5)=Pr(Z∈D|C)×p(Y|Z∈D,C,F1,...,F p).(6) Equation(5)holds because the event Z∈D occurs whenever Y is observed.This derivation can be made rigorous by deriving the density p(Y|C,F1,...,F p)from the limit of Pr(∩i,j(y i,j−ǫ,y i,j]|C,F1,...,F p)asǫ→0.As in the case of marginal likelihood,our approach is to estimate C using only Pr(Z∈D|C),the part of the observed data likelihood(6)that depends on the parameter of interest C and not on the nuisance parameters F1,...,F p.Since our likelihood function is based on the marginal probability of an event that is a superset of observing the ranks,we refer to it as an extended rank likelihood.3.2Estimation of the copula parametersBayesian inference for C can be achieved via construction of a Markov chain having a stationary distribution equal to p(C|Z∈D)∝p(C)×p(Z∈D|C).In the case of the Gaussian cop-ula with a semi-conjugate prior distribution,the Markov chain can be constructed quite easily8using Gibbs sampling.This prior distribution for C is defined as follows:Let V have an inverse-Wishart(ν0,ν0V0)prior distribution,parameterized so that E[V−1]=V−10,and let C be equal in distribution to the the correlation matrix with entries V[i,j]/],Φ[z u−µi,jσjV[i,i]V[j,j].Iteration of this algorithm generates a Markov chain in C whose stationary distribution is p(C|Z∈D).This algorithm is easily modified to accommodate data that are missing-at-random:If y i,j is missing,the full conditional distribution of z i,j is the unconstrained normal distribution with mean µi,j and varianceσ2j given above.The reader may have noticed that the samples of Z are based on the covariance matrix V and not the correlation matrix C.To see why this does not matter for estimation of C,compare our original model,V∼inverse-Wishart(ν0,ν0V0){C[i,j]}={V[i,j]/V[j,j],and let C=Cov(˜z)y i,j=G j(˜z i,j).9The z’s in thefirst formulation are equal in distribution to the˜z’s in the second,and so posterior inference for C is equivalent under either model.The Gibbs sampling scheme outlined above is based on a Markov chain in V and z1,...,z n based on the second formulation.Note that in this formulation the observed data implies the same ordering D on both the˜z’s and the z’s.Additionally, posterior estimation of C is invariant to changes in the prior distribution on V that do not alter the induced prior on C.For example,if V0and V′0are two different covariance matrices with the same correlations,then the posterior distribution of C under V∼inverse-Wishart(ν0,ν0V0)will be equal to that under V∼inverse-Wishart(ν0,ν0V′0).4Income,education and intergenerational mobilityThe U.S.census reports a strong positive relationship between income and educational attain-ment[Day and Newburger,2002].However,in many studies both of these variables have been shown to be associated with a number of family background variables such as parental income, parental educational attainment and number of siblings[Ermisch and Francesconi,2001,Blake, 1985].Additionally,some researchers have suggested that having children reduces opportunities for educational attainment[Moore and Waite,1977],while others have found evidence that eco-nomic status of males is positively associated with their fertility[Hopcroft,2006].Results such as these are generally based on univariate regression models in which one variable from a sample survey is selected as a“response”or“dependent”variable and the others as“control”or“inde-pendent”variables.However,all of the variables in these studies are randomly sampled and all are potentially dependent on one another.In this section we describe the multivariate dependencies among income,education and number of children using the Gaussian copula model and the semiparametric estimation procedure described in Section3.Specifically,we analyze survey data on1002males in the bor force(meaning not retired,in school or in an institution),obtained from the1994General Social Survey.Data and details for the survey are available at /GSS/.The relevant variables for this analysis include the income,education,and number of children of the survey respondent,as well as similar variables for the respondent’s parents.Age of the survey respondent is additionally included,as it is typically strongly related to income and number of children.The measurement scales for these variables are as follows:10Figure 2:MCMC samples of 11of the correlation coefficients,plotted every 50th scan.INC:income of the respondent in 1000s of dollars,binned into 21ordered categories DEG:highest degree ever obtained (None,HS,Associates,Bachelors,Graduate)CHILD:number of children ever had PINC:financial status of respondent’s parents when respondent was 16(on a 5-point scale)PDEG:maximum of mother’s and father’s highest degree PCHILD:number of siblings of the respondent plus one AGE:age of the respondent in yearsMissing data rates among each of the non-income variables was less than 4%.The missing data rates for INC and PINC were 10%and 48%respectively.However,the question PINC was asked on only half of the surveys,and so missing values for this variable can reasonably be considered as missing at random.4.1Estimation of CUsing an inverse-Wishart (p +2,(p +2)×I )prior distribution for V ,the Gibbs sampling scheme outlined in Section 3was iterated 25,000times with parameter values saved every 10scans,resulting in 2500samples of C for posterior analysis.Mixing of the Markov chain was quite good:Figure 2shows MCMC samples of 11elements of C ,corresponding to the odd order statistics of E [C |Z ∈D ].Convergence to stationarity appears to occur quickly,almost certainly within the first 5000scans.11INC CHILD DEG PDEGPCHILD PINC ++−−+++−−Figure 4:Reduced conditional dependence graph for the GSS data.These coefficients represent conditional dependencies among the underlying processes that give rise to the observed data.On this scale,the full conditional distribution of INC depends most strongly on DEG,and to a lesser extent on CHILD and AGE.Interestingly,the conditional relationship between INC and PINC has a non-negligible (>5%)probability of being less than or equal to zero.Figure 4summarizes these results with a graph indicating the conditional dependencies among the z -variables corresponding to the six variables of interest (implicitly conditioning on AGE).An edge is present between two nodes if the 95%credible interval for the associated regression parameter does not contain zero.This graph suggests that although INC and PINC are positively associated,this association is mediated by the intergenerational relationships of DEG,PDEG,CHILD and PCHILD.4.3Conditional distributions for the INC,DEG,PINC relationshipThe results in Figure 3suggest that,although INC and PINC are positively correlated,PINC is a relatively weak predictor of INC compared to DEG.However,PINC is a strong predictor of PDEG,and PDEG is a strong predictor of DEG,suggesting an indirect effect of PINC on INC.These conclusions about INC,DEG and PINC are made in terms of associations among the z -variables,although it is often desirable to report results on the scale of the original data.With this in mind,we now describe the relationship between INC,DEG and PINC on the original data scale,using an estimated predictive distribution Pr(INC,DEG,PINC),which we decompose as Pr(INC |DEG ,PINC)×Pr(DEG |PINC)×Pr(PINC).A predictive distribution for y can be obtained in a few different ways.Perhaps the simplest method is to combine the posterior distribution of C with the empirical univariate marginal dis-tributions ˆF1,...,ˆF p of the observed data (an alternative method is presented in the Discussion).Using this method,a predictive sample of y can be obtained as follows:131.sample C∼p(C|Z∈D);2.sample z∼multivariate normal(0,C);3.set y j=ˆF−1j(z j).Although this somewhat ad-hoc approach disregards uncertainty in the estimation of F1,...,F p(for prediction of y,not for estimation of C),it provides a predictive joint distribution that matches the observed data in terms of the univariate marginal distributions but has a simple,smooth Gaus-sian copula representing multivariate dependence.From these predictive samples we can obtain Monte Carlo estimates of various quantities of interest,including a consistent set of conditional distributions on the original scale of the data.Thefirst column of Figure5plots the predictive distribution of DEG conditional on PINC=x for x∈{1,2,3,4,5}.As on the z-scale,large values of PINC correspond to large values of DEG.The estimated conditional probability of someone notfinishing high-school given PINC=5 is5%,whereas for PINC=1it is22%,giving an odds ratio of odds(DEG=None|PINC=1)/ odds(DEG=None|PINC=5)=5.35.Similarly,the corresponding odds ratio for having a grad-uate degree is odds(DEG=Grad|PINC=5)/odds(DEG=Grad|PINC=1)=6.5.For comparison, the empirical conditional distributions are provided on the same plot.In general thefit is good, with most of the discrepancies occurring in categories of PINC with small sample sizes(n=28for PINC=1,and n=8for PINC=5).Note that if we were to estimate the above odds ratios using the empirical conditional distributions we would obtain ratios equal to infinity.In situations such as these where the sample size is low,we may prefer to estimate conditional distributions with a model that can share information across the categories of a variable,rather than use an empirical estimator having a high sampling variability.The second column of Figure5displays estimated quantiles of Pr(INC|DEG,PINC)for each combination of DEG and PINC.Specifically,each row corresponds to a single value of DEG,and each boxplot within a row corresponds to a single value of PINC.The boxplot provides5,25,50, 75and95%quantiles of Pr(INC|DEG,PINC).Note that the boxplots within a row indicate very small increases in INCOME with increasing values of PINC,while differences across rows indicate much larger increases with DEG(changes in the quantiles do not happen continuously due to the binned nature of the raw data).For high-school graduates(DEG=1),the estimated conditional mean incomes across levels of PINC are{23,25,26,28,29}in thousands of dollars.For college graduates(DEG=2),the estimated means are{41,41,43,44,47}.For these mean calculations,the income in a binned income category was taken as the average of the endpoints of the bin.For comparison,the actual values of INC for each combination of DEG and PINC are plotted on the corresponding boxplots(data are jittered to allow ties to be distinguished).As before, the main discrepancies occur for combinations of DEG and PINC for which there are few data.14Also,the predictive distributions based on the copula model are much smoother than the empirical versions:The empirical conditional means of INC for DEG=1and DEG=3are{23,27,24,27,8} and{41,44,35,58,75}respectively,across increasing levels of PINC.However,several of these empirical means are calculated from as few as3or4samples.5Notions of sufficiencyThe extended rank likelihood described above can be viewed as a generalization of marginal likeli-hood,a standard technique for dealing with nuisance parameters(see Section8.3of Severini[2000] for a review).One benefit of using such a likelihood is a gain in robustness,as inference no longer depends on assumptions about the relationship of the data to the nuisance parameters.Another benefit is a general simplification of the estimation problem,as the need to estimate a potentially high-dimensional set of parameters is eliminated.These benefits come at the cost of potentially losing information about the parameters of interest by only using part of the available data.Ide-ally,the statistic that generates the marginal likelihood is“partially sufficient”in the sense that it contains all relevant information in the data about the parameter of interest.Various definitions of partial sufficiency have been developed:Fraser[1956]defined S-sufficiency via properties of the marginal and conditional distributions of the statistic and the data.The concept of G-sufficiency was introduced in Barnard[1963]as a general principle for making inference about a parameter of interest when the inference problem remains invariant under a group of transformations.R´e mon [1984]developed a generalization of these notions based on profile likelihoods called L-sufficiency, which has been refined and studied by Barndorff-Nielsen[1988,1999].The general recommendation of these authors is to base inference for a parameter of interest on the sampling distribution of a statistic that is sufficient in some sense.If F1,...,F p are all continuous then there are no ties among the data,and knowledge of Z∈D provides a complete ordering of{y1,j,...,y n,j}for each j.This information is equivalent to the information contained in the ranks,and so Pr(Z∈D|C)is equivalent to the sampling distribution of the multivariate ranks.Following the notation of R´e mon[1984]we now show that the ranks r(Y)are a G-sufficient statistic in the sense of Barnard[1963]:Let C∈C describe the copula and F={F1,...,F p}∈F the marginal distributions,and so the parameter space isΩ=C×F and the model space is P={Pr(·|ω):ω∈Ω},where Pr(·|ω)is a probability measure on R p for each ω∈Ω.Furthermore,let G be the group of collections of p continuous strictly increasing functions, so that G={G=(G1,...,G p):G j is a continuous and strictly increasing function on R}.To each G∈G there corresponds a one-to-one function on P mapping P(·|ω)to P(G−1(·)|ω)and the model space is closed under the action of G.As a result,G induces a group¯G={f G:G∈G}on Ωdefined by P(·|f Gω)=P(G−1(·)|ω).15。