Subset Selection Rules Research Articles

An essentially complete class of multiple decision procedures

Abstract Let π 1 ,…, π k represent k (⩾2) independent populations. The quality of the i th population π i is characterized by a real-valued parameter θ i , usually unknown. We define the best population in terms of a measure of separation between θ i 's. A selection of a subset containing the best population is called a correct selection (CS). We restrict attention to rules for which the size of the selected subset is controlled at a given point and the infimum of the probability of correct selection over the parameter space is maximized. The main theorem deals with construction of an essentially complete class of selection rules of the above type. Some classical subset selection rules are shown to belong to this class.

Nonparametric Selection Procedures Applied to State Traffic Fatality Rates

This article reviews the practical aspects of several nonparametric subset selection rules useful in block design problems, and discusses advantages and disadvantages of these methods. The populations are assumed stochastically ordered by the parameter of interest. Rules based on ranked observations are given for selecting a subset of populations which contains, with a specified confidence level, the population characterized by the smallest (or largest) parameter value. These procedures are applied to state traffic fatality rates recorded yearly (1960-76). New England states and Middle Atlantic states comprise most of the subset asserted, with a 90% confidence level, to contain the state with the smallest fatality rate; whereas, Southern states, Southwestern states and Rocky Mountain states generally comprise the subset for the state with the largest fatality rate.

Nonparametric Selection Procedures Applied to State Traffic Fatality Rates

This article reviews the practical aspects of several nonparametric subset selection rules useful in block design problems, and discusses advantages and disadvantages of these methods. The populations are assumed stochastically ordered by the parameter of interest. Rules based on ranked observations are given for selecting a subset of populations which contains, with a specified confidence level, the population characterized by the smallest (or largest) parameter value. These procedures are applied to state traffic fatality rates recorded yearly (1960-76). New England states and Middle Atlantic states comprise most of the subset asserted, with a 90% confidence level, to contain the state with the smallest fatality rate; whereas, Southern states, Southwestern states and Rocky Mountain states generally comprise the subset for the state with the largest fatality rate.

Nonparametric selection procedures applied to state traffic fatality rates

This article reviews the practical aspects of several nonparametric subset selection rules useful in block design problems, and discusses advantages and disadvantages of these methods. The populations are assumed stochastically ordered by the parameter of interest. Rules based on ranked observations are given for selecting a subset of populations which contains, with a specified confidence level, the population characterized by the smallest (or largest) parameter value. These procedures are applied to state traffic fatality rates recorded yearly (1960-76). New England states and Middle Atlantic states comprise most of the subset asserted, with a 90% confidence level, to contain the state with the smallest fatality rate; whereas, Southern states, Southwestern states and Rocky Mountain states generally comprise the subset for the state with the largest fatality rate. Note that while this example is not based on simulation data, such data would be analyzed in exactly the same fashion.

On the Distribution of the Maximum and Minimum of Ratios of Order Statistics

Let $X_i(i = 0, 1, \cdots, p)$ be $(p + 1)$ independent and identically distributed nonnegative random variables each representing the $j$th order statistic in a random sample of size $n$ from a continuous distribution $G(x)$ of a nonnegative random variable. Let $G_{j,n}(x)$ be the cumulative distribution function of $X_i(i = 0, 1, \cdots, p)$. Consider the ratios $Y_i = X_i/X_0 (i = 1, 2, \cdots, p)$. The random variables $Y_i (i = 1, 2, \cdots, p)$ are correlated and the distribution of the maximum and the minimum is of interest in problems of selection and ranking for restricted families of distribution. The distribution-free subset selection rules using the percentage points of these order statistics are investigated in a companion paper by Barlow and Gupta (1969). In the present paper, we discuss the distribution of these statistics, in general, for any $G(x)$ and then derive specific results for $G(x) = 1 - e^{-x/\theta}, x > 0, \theta > 0$. Section 2 deals with the distribution of the maximum while Section 3 discusses the distribution of the minimum. Some asymptotic results are given in Section 4, while Section 5 describes the tables of the percentage points of the two statistics.