Selection Procedures for Restricted Families of Probability Distributions

Abstract

Let $\Pi_1, \Pi_2, \cdots, \Pi_k$ be $k$ populations. The random variable $X_i$ associated with $\Pi_i$ has a continuous distribution $F_i, i = 1, 2, \cdots, k$. We are primarily interested in selecting a subset such that the probability is at least $P^\ast$ that the selected subset includes the population with the largest (smallest) quantile of a given order $\alpha (0 < \alpha < 1)$. We assume each $F_i$ has a unique $\alpha$-quantile, $\xi_{\alpha i}$. Let $F_{\lbrack i\rbrack}(x) = F_{\lbrack x\rbrack}$ denote the cumulative distribution function of the population with the $i$th smallest $\alpha$-quantile. In the following, we consider families of distributions ordered in a certain sense with respect to a specified continuous distribution $G$ and propose and study a selection procedure which is different from the non-parametric procedure of Rizvi and Sobel (1967). We assume (a) $F_{\lbrack i\rbrack} (x) \geqq F_{\lbrack k\rbrack}(x), i = 1, 2, \cdots, k$ and all $x$. (b) $\mathbf{\exists}$ a continuous distribution $G \ni F_{\lbrack i\rbrack} \underset{\sim}{\prec} G, \mathbf{\forall}i = 1, 2, \cdots, k$, where $\underset{\sim}{\prec}$ denotes a partial ordering relation on the space of distributions. A relation $\underset{\sim}{\prec}$ on the space of distributions is a partial ordering if \begin{equation*}\begin{split}F \underset{\sim}{\prec} F\quad \mathbf{\forall} \text{distributions} F \\ F \underset{\sim}{\prec} G,\quad G \underset{\sim}{\prec} H \text{implies} F \underset{\sim}{\prec} H.\\ \end{split}\end{equation*} Note that $F \underset{\sim}{\prec} G$ and $G \underset{\sim}{\prec} F$ do not necessarily imply $F \equiv G$. Various special cases in addition to stochastic ordering are: (i) $F \prec_\ast G \operatorname{iff} F(0) = G(0) = 0$ and $G^{-1}F(x)/x$ is nondecreasing in $x \geqq 0$ on the support of $F$. (ii) $F \prec_c G \operatorname{iff} G^{-1}F(x)$ is convex on the support of $F$. (iii) $F \prec_r G \operatorname{iff} F(0) = G(0) = \frac{1}{2}$ and $G^{-1}F(x)/x$ is increasing (decreasing) for $x$ positive (negative) on the support of $F$. (iv) $F \prec_s G \operatorname{iff} F(0) = G(0) = \frac{1}{2}$ and $G^{-1}F$ is concave-convex about the origin, on the support of $F$; i.e., $\{x\mid 0 < F(x) < 1\}$. If $G(x) = 1 - e^{-x}$ for $x \geqq 0$, then (i) defines the class of IFRA distributions studied by Birnbaum, Esary and Marshall (1966) while (ii) defines the class of IFR distributions studied by Barlow, Marshall and Proschan (1963). For any distribution $G, F \prec_\ast G \operatorname{iff} F(x)$ crosses $G(\theta x)$ at most once and from below if at all as a function of $x$ for all $\theta > 0$. If $G(x) = 1 - \exp (-x^\lambda)$ for $x \geqq 0$ and $\lambda > 0$, then $F \prec_\ast G$ implies that $F$ is "sharper" than the family of Weibull distributions with shape parameter $\lambda$. Implications of orderings defined by (iii) were studied by Lawrence (1966). Van Zwet (1964) studies orderings defined by both (ii) and (iv). Clearly $\prec_c$ ordering implies $\prec_\ast$ ordering and $\prec_s$ ordering implies $\prec_r$ ordering. If $\mathbf{X}_i = (X_{i1}, X_{i2}, \cdots, X_{in})$ is the observed sample from the $i$th population, then we restrict ourselves to the class of statistics $T_i = T(\mathbf{X}_i)$ that preserve both ordering relations (a) and (b), i.e., $(a') P_{F_\lbrack i\rbrack}\{T(\mathbf{X}) \leqq x\} \geqq P_{F_\lbrack k\rbrack}\{T(\mathbf{X}) \leqq x\}$ for all $x$ and $i = 1, 2, \cdots, k$. $(b') F_{T(X_i)} \precsim G_{T(\mathbf{Y})}, i = 1, 2, \cdots, k$, where $F_{T(\mathbf{X}_i)}$ represents the cdf of $T(\mathbf{X}_i)$ under $F_{\lbrack i\rbrack}$ and $G_{T(\mathbf{Y})}$ is the cdf of $T(\mathbf{Y})$ under $G, \mathbf{Y} = (Y_1, Y_2, \cdots, Y_n)$ being a random sample from $G$. In Section 2 of this paper, we propose and study procedures $R (R')$ for selecting the population with the largest (smallest) $\alpha$-quantile for distributions which are $\prec_\ast$ ordered with respect to a specified distribution $G$. The infimum of the probability of a correct selection is obtained in Theorem 2.1 and asymptotic evaluation is given in Theorem 2.2. Section 3 deals with quantile selection procedures for the class of IFRA distributions. In Section 4, we study the efficiency of procedure $R$ with respect to a procedure studied by Rizvi and Sobel (1967) under scale type slippage configurations. Asymptotic relative efficiency of $R$ with respect to a selection procedure for the gamma populations proposed by Gupta (1963) is also investigated. Section 5 deals with selection procedures for the median for distributions that are $\prec_r$ ordered with respect to a specified $G$. In Section 6 we propose a selection procedure with respect to the means for distributions that are $\prec_c$ ordered with respect to $G(x) = 1 - e^{-x}$. Application to the selection of gamma populations is also given in Section 6.