On concomitants of order statistics

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.