The chapter discusses the application of order statistics in statistical inference, particularly in nonparametric problems. Order statistics are the values of a sample arranged in ascending or descending order. The chapter covers the following key points: 1. **Introduction**: It introduces the importance of order statistics in statistical inference, especially in problems where the distribution of the population is unknown or only known to be continuous. The chapter highlights the significance of order statistics in developing simple and broadly applicable statistical procedures. 2. **Notation and Preliminary Definitions**: It defines the notation and basic concepts related to continuous cumulative distribution functions (CDFs) and random samples. It introduces the concept of order statistics and their coverage, which is the fraction of the population contained in an interval determined by these statistics. 3. **Sampling Distributions of Coverages**: The chapter discusses the sampling distributions of coverages for one-dimensional samples. It provides formulas for the probability distributions of order statistics and their sums, which are crucial for constructing confidence intervals and tolerance limits. 4. **Examples of Direct Applications**: It presents practical applications of coverage distributions, such as confidence limits for medians, quartiles, and quantiles, and population tolerance limits. These applications demonstrate how order statistics can be used to make inferences about population parameters. 5. **Distribution of Single Order Statistics**: The chapter explores the exact and limiting distributions of single order statistics. It discusses the asymptotic normality of order statistics and provides formulas for their limiting distributions under different conditions. 6. **Joint Distributions of Several One-Dimensional Order Statistics**: It examines the joint distributions of multiple order statistics and their applications. This includes the distribution of the sample range and midrange, and the limiting distributions of two or more order statistics in large samples. 7. **Confidence Bands for the CDF**: The chapter discusses the construction of confidence bands for the CDF of a continuous distribution. It provides methods for determining these bands and their properties, which are essential for estimating the unknown CDF from a sample. 8. **Sampling Distributions of Coverages in Two or More Dimensions**: Finally, the chapter extends the concepts to two-dimensional samples, discussing methods for constructing regions with distribution-free coverages and the slicing technique for analyzing these regions. Overall, the chapter provides a comprehensive overview of the theoretical and practical aspects of order statistics in statistical inference, emphasizing their role in nonparametric methods and their applications in various statistical problems.The chapter discusses the application of order statistics in statistical inference, particularly in nonparametric problems. Order statistics are the values of a sample arranged in ascending or descending order. The chapter covers the following key points: 1. **Introduction**: It introduces the importance of order statistics in statistical inference, especially in problems where the distribution of the population is unknown or only known to be continuous. The chapter highlights the significance of order statistics in developing simple and broadly applicable statistical procedures. 2. **Notation and Preliminary Definitions**: It defines the notation and basic concepts related to continuous cumulative distribution functions (CDFs) and random samples. It introduces the concept of order statistics and their coverage, which is the fraction of the population contained in an interval determined by these statistics. 3. **Sampling Distributions of Coverages**: The chapter discusses the sampling distributions of coverages for one-dimensional samples. It provides formulas for the probability distributions of order statistics and their sums, which are crucial for constructing confidence intervals and tolerance limits. 4. **Examples of Direct Applications**: It presents practical applications of coverage distributions, such as confidence limits for medians, quartiles, and quantiles, and population tolerance limits. These applications demonstrate how order statistics can be used to make inferences about population parameters. 5. **Distribution of Single Order Statistics**: The chapter explores the exact and limiting distributions of single order statistics. It discusses the asymptotic normality of order statistics and provides formulas for their limiting distributions under different conditions. 6. **Joint Distributions of Several One-Dimensional Order Statistics**: It examines the joint distributions of multiple order statistics and their applications. This includes the distribution of the sample range and midrange, and the limiting distributions of two or more order statistics in large samples. 7. **Confidence Bands for the CDF**: The chapter discusses the construction of confidence bands for the CDF of a continuous distribution. It provides methods for determining these bands and their properties, which are essential for estimating the unknown CDF from a sample. 8. **Sampling Distributions of Coverages in Two or More Dimensions**: Finally, the chapter extends the concepts to two-dimensional samples, discussing methods for constructing regions with distribution-free coverages and the slicing technique for analyzing these regions. Overall, the chapter provides a comprehensive overview of the theoretical and practical aspects of order statistics in statistical inference, emphasizing their role in nonparametric methods and their applications in various statistical problems.