Concomitants of order statistics
Concomitants of order statistics
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DOI：
10.31274/rtd-180813-3286
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发表时间：
1976
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影响因子：
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通讯作者：
Shie-Shien Yang
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文献类型：
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作者：
Shie-Shien Yang
设（Xi, Yi）， 1≤i≤n，是绝对连续随机向量（X，Y）中大小为n的样本。设Xi:n为x样本的i阶统计量，Y[i:n]为它的伴随量。本文研究了与Y[i:n]相关的三个问题。第一个问题是关于序统计量伴随量在相关样本中的分布。我们推导了COS在一组特定的相关样本下的有限样本和渐近分布，其中X形成一个等相关的多元正态样本。本文扩展了文献中关于COS分布理论的现有结果，这些理论通常假设独立同分布（i.i.d）或独立样本。我们研究的第二个问题是来自i.i.d样本的伴随子子集的有序统计量的分布。具体地，我们研究了v:m和Wt:n−m的有限样本和渐近分布，其中v:m是伴随子集{Y[i:n]， i = n−m + 1，…的第五阶统计量。， n}，而Wt:n−m是伴随子子集{Y[j:n]， j = 1，…的第n阶统计量。, n−m}。我们证明，通过适当的归一化，v:m和Wt:n - m都以n - 1/2阶的收敛速度收敛到正态分布。我们提出了对这些阶统计量的边际分布的高阶扩展，即使对于中等样本量，也比正态近似准确得多。然后导出了（v:m,Wt:n−m）的有限样本和渐近联合分布。我们应用这些结果并确定在常用的选择程序中感兴趣的事件的概率。我们还应用结果来研究ii在基因-疾病关联研究的两阶段设计中识别疾病易感基因的能力。我们考虑的第三个问题是在解释变量(X)处于其分布的特定分位数的情况下，估计响应变量(Y)的条件均值。我们提出了两个基于序统计量伴随量的估计量。第一个是核平滑估计量，第二个可以被认为是自举估计量。我们研究了这些估计量的渐近性质，并用仿真比较了它们的有限样本性质。
Let (Xi, Yi), 1 ≤ i ≤ n, be a sample of size n from an absolutely continuous random vector (X,Y ). Let Xi:n be the ith order statistic of the X-sample and Y[i:n] be its concomitant. We study three problems related to the Y[i:n]’s in this dissertation. The first problem is about the distribution of concomitants of order statistics (COS) in dependent samples. We derive the finite-sample and asymptotic distribution of COS under a specific setting of dependent samples where the X’s form an equally correlated multivariate normal sample. This work extends the available results on the distribution theory of COS in the literature, which usually assumes independent and identically distributed (i.i.d) or independent samples. The second problem we examine is about the distribution of order statistics of subsets of concomitants from i.i.d samples. Specifically, we study the finite-sample and asymptotic distributions of Vs:m and Wt:n−m, where Vs:m is the sth order statistic of the concomitants subset {Y[i:n], i = n−m + 1, . . . , n}, and Wt:n−m is the tth order statistic of the concomitants subset {Y[j:n], j = 1, . . . , n−m}. We show that with appropriate normalization, both Vs:m and Wt:n−m converge in law to normal distributions with a rate of convergence of order n−1/2. We propose a higher order expansion to the marginal distributions of these order statistics that is substantially more accurate than the normal approximation even for moderate sample sizes. Then we derive the finite-sample and asymptotic joint distribution of (Vs:m,Wt:n−m). We apply these results and determine the probability of an event of interest in commonly used selection procedures. We also apply the results to study the power of ii identifying the disease-susceptible gene in two-stage designs for gene-disease association studies. The third problem we consider is about estimating the conditional mean of the response variable (Y ) given that the explanatory variable (X) is at a specific quantile of its distribution. We propose two estimators based on concomitants of order statistics. The first one is a kernel smoothing estimator, and the second one can be thought of as a bootstrap estimator. We study the asymptotic properties of these estimators and compare their finite sample behavior using simulation.