Finite Sample Theory of Order Statistics and Extremes
Finite Sample Theory of Order Statistics and Extremes
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阶次统计和极值的有限样本理论
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发表时间：
2011
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通讯作者：
A. Dasgupta
中科院分区：
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作者：
A. Dasgupta
样本的有序值称为样本的顺序统计量，最小和最大的称为极值。顺序统计量和极值是我们在概率和统计学中研究的一组随机变量的最重要的函数之一。研究序列的高点和低点是很自然的兴趣，而其他顺序统计量有助于理解分布中概率的集中度，或者等价地，分布所代表的总体的多样性。顺序统计量在统计推断中也很有用，其中参数的估计通常基于顺序统计量的一些合适的函数。特别是，中位数非常重要。对于来自固定分布的固定数量n个观测值的顺序统计量，有一个发展良好的理论，也有一个渐近理论，其中n趋于无穷大。我们在本章中讨论固定n的情况。当观测来自离散分布时，顺序统计量的分布理论在符号上和代数上都是复杂的，因为可能有几个实际上相等的观测。样本值之间的这些联系使分布理论变得很麻烦。因此，我们集中讨论连续的情形。
The ordered values of a sample of observations are called the order statistics of the sample, and the smallest and the largest are called the extremes. Order statistics and extremes are among the most important functions of a set of random variables that we study in probability and statistics. There is natural interest in studying the highs and lows of a sequence, and the other order statistics help in understanding the concentration of probability in a distribution, or equivalently, the diversity in the population represented by the distribution. Order statistics are also useful in statistical inference, where estimates of parameters are often based on some suitable functions of the order statistics. In particular, the median is of very special importance. There is a well-developed theory of the order statistics of a fixed number n of observations from a fixed distribution, as also an asymptotic theory where n goes to infinity. We discuss the case of fixed n in this chapter. A distribution theory for order statistics when the observations are from a discrete distribution is complex, both notationally and algebraically, because of the fact that there could be several observations which are actually equal. These ties among the sample values make the distribution theory cumbersome. We therefore concentrate on the continuous case.