Multiple-Decision Procedures for Testing the Homogeneity of Mean for k Exponential Distributions

In multiple-decision procedures, a crucial objective is to determine the association between the probability of a correct decision (CD) and the sample size. A review of some methods is provided, including a subset selection formulation proposed by Huang and Panchapakesan, a multidecision procedure for testing the homogeneity of means by Huang and Lin, and a similar procedure for testing the homogeneity of variances by Lin and Huang. In this paper, we focus on the use of the Lin and Huang method for testing the null hypothesisH 0 of homogeneity of means for k exponential distributions. We discuss the decision rule R, evaluation of the critical value C, and the infimum of P(CD | R) for k independent random samples from k exponential distributions. In addition, we also observed that a lower bound for the probability of CD relative to the number of the common sample size is determined based on the desired probability of CD when the largest mean is sufficiently larger than the other means. We explain the results by using two examples.


Introduction
A multiple-decision problem can be defined as a situation where a person or a group of people must select the number of possible actions from a given finite set.Gupta and Huang [1] and Lin and Gupta [2] presented the selection procedures relevant to multiple-decision theory, including indifference zone selection and subset selection.They suggested that preferences among alternatives can be determined by maximizing the expected value of a numerical utility function or equivalently minimizing the expected value of a loss function.They indicated that the subset selection procedures have been studied and applied widely in determining the required sample size, which is the number of replications or batches used for selecting the optimal population among  populations and for selecting a subset.
Huang and Panchapakesan [3] suggested a modification of the subset selection formulation on the largest mean and the smallest variance.Huang and Lin [4] presented a multidecision procedure for testing the homogeneity of means when the sample sizes and unknown variance are unequal.Lin and Huang [5] used a similar procedure for testing the hypothesis  0 regarding the homogeneity of the variances.The purpose of this paper was to use the Lin and Huang method for testing the hypothesis  0 regarding the homogeneity of the means for  exponential distributions.When  0 , the hypothesis, is rejected, the main objective was to obtain a nonempty subset  of the  populations that will include the population related to the largest means (called the best population).In this case, a correct decision (CD) is said to occur if the selected subset  contains the best populations.
The paper is organized as follows.In Section 2, we introduce the definitions and notations of decision rule  for  exponential distributions.In Section 3, we discuss the evaluation of the critical value of our test and the infimum of the probability of a correct decision CD.In Section 4, the performance of the method is illustrated with two examples and the behavior of our procedure is analyzed.Finally, concluding remarks are provided in Section 5.

Related Concepts of the Decision Rule
In this section, we use the Lin and Huang [5] method to identify the decision rule  for  exponential distributions.
First, given , where 0 <  < 1, we want to find a  such that the condition where  is the critical value for the decision rule  and  is a given probability of Type I error at level .
Second, given Δ > 0 and  * , where 1/ <  * < 1, we want to find a nonempty subset  = {1 ≤  ≤  | τ ≥ } of the  populations that contains the best populations and it is necessary that inf
Lemma 3.According to the Lin and Huang [5] appendix, we can get where   We have Therefore, the critical value  is However inf which is the desired result.
where [] denotes the lowest integer greater than or equal to .

Examples
In this section, we provide two examples to explain the results of performing Theorems 4 and 5. Example 1.This example is from Nelson [7].In this example, the results of a life test experiment are described in which specimens of electrical insulating fluid were subjected to a constant voltage stress.The length of time until each specimen failed, or "broke down, " was observed.Table 1 gives results for five groups of specimens, tested at voltages ranging from 30 to 38 kilovolts (kV).We use the data on times to breakdown (in minutes) at each of the five voltage levels for our example.
The computed values are given in Table 2 based on the assumption that  = 0.01.
We obtained  = 2.8511.Because τ1 ≥  and τ2 ≥ , using the decision rule , we reject  0 : τ1 = τ2 = τ3 = τ4 = τ5 and select the subset containing populations 1 and 2. We identified these two populations as contributing substantially.We claim that the select subset contains the population with the largest mean.
Example 2. Based on the same assumption as Theorem 5, given the number of populations ,  = 3, 4, 5, and 6, as well as  = 0.05 and 0.01 and Δ = 1.5(0.1)2.5 and  * = 0.6, 0.8, 0.9, and 0.95, we can determine  by using (14), so that inf ∈Ω Δ {CD | } ≥  * .Several selected combinations of  in each case are tabulated in Tables 4, 5, 6, and 7 which show the populations  that have the minimal sample size  required to satisfy the  * .

Concluding Remarks
In this study, we considered the methods of the Lin and Huang theorems to propose a framework for analyzing and synthesizing multiple-decision procedures used for testing the homogeneity of means for  exponential distributions [5].

Table 1 :
Times to breakdown (in minutes) at each of the five voltage levels.