## Retrospective Theses and Dissertations

#### Title

Stochastic comparisons of order statistics

Dissertation

1988

#### Degree Name

Doctor of Philosophy

Statistics

Herbert David

#### Abstract

Let X[subscript]1,...,X[subscript] n and Y[subscript]1,...,Y[subscript] n be independent random samples from distributions with respective c.d.f.s F(x) and G(x). If X is stochastically smaller than Y (X ≤[over] st Y), i.e., F(x) ≥ G(x) ∀ x, then it is well known that X[subscript] r:n≤[over] st Y[subscript] r:n. If X[subscript]1,...,X[subscript] n (Y[subscript]1,...,Y[subscript] n) is a random sample from a life distribution F(G), then the order statistics may be interpreted as the successive failure times of the n components of a system. From this interpretation, X[subscript] r:n is the failure time of a k-out-of-n system of identical components, where k = n - r + 1. Since P/X[subscript] r:n > t is the reliability of a k-out-of-n system at time t, we can compare the reliabilities of the X and Y systems. Stochastic ordering is a very strong kind of ordering. Consequently, many other weaker orderings have been studied in the statistical literature. In this dissertation, the properties of order statistics under these orderings is investigated extensively. As an example of situations treated, consider that X and Y are symmetric about 0 and that X is more peaked than Y, i.e., F(x) ≥ G(x) for x ≥ 0. It is easy to see that neither of X[subscript] r:n, Y[subscript] r:n is stochastically larger than the other. However, it can be shown that E(X[subscript] r:n) E(Y[subscript] r:n) according as r >[over]<1[over] 2(n + 1). The situation when the X's (Y's) are independent but not identically distributed is also considered;Order statistics are dependent because of the inequality relations among them. It is well known that cov(X[subscript] r:n,X[subscript] s:n) ≥ 0 under the i.i.d. assumption. However, cov(X[subscript] r:n,X[subscript] s:n) can be negative if X[subscript]1,X[subscript]2,...,X[subscript] n are sufficiently negatively dependent. More generally, the following question arises: Is it necessarily true that in random samples the covariance of two order statistics X[subscript] i:n,X[subscript] j:n decreases as i and j draw apart? Tukey (1958) was the first to study this and related questions. Here Tukey's pioneering results are corrected, extended, and illustrated numerically for selected distributions;Reference. Tukey, J. W. 1958. A problem of Berkson, and minimum variance orderly estimators. Ann. Math. Statist. 29:588-592.

#### DOI

https://doi.org/10.31274/rtd-180813-13239

#### Publisher

Digital Repository @ Iowa State University, http://lib.dr.iastate.edu/

Song-Ho Kim

en

AAI8909160

application/pdf

78 pages

COinS