The Weibull distribution is one of the most widely employed models in reliability and survival analysis due to its flexibility in modeling various hazard rate behaviors. It is frequently used to characterize the lifetime distributions of components and systems in engineering applications, including mechanical parts, electronic devices, and structural materials. Beyond engineering, it is also extensively applied in biomedical and epidemiological studies to model time-to-event data, such as the latency period of diseases or time to failure in biological systems. See, for example, Lawless [1] and Nelson [2] .

The exponential distribution rises as a special case of the Weibull distribution when $\beta$, the shape parameter, equals 1. The exponential distribution, characterized by a constant failure rate, is especially useful for modeling electronic components and systems with memoryless lifetimes. In contrast, the Weibull distribution with the shape parameter, allows it to model increasing, decreasing, or constant failure rates, making it suitable for a wide range of mechanical, structural, and industrial applications. This adaptability makes the Weibull distribution a cornerstone in life data analysis, failure prediction, and maintenance scheduling. Both distributions support parameter estimation, hazard rate modeling, and reliability function derivation, providing critical insights into product life cycles, risk assessment, and quality control. (Meeker, et al. [3] )

Two authoritative resources on subset selection procedures are the comprehensive works by Gupta and Panchapakesan [4] , and Gibbons et al. [5] . In particular, subset selection methods for populations following the exponential distribution have been extensively studied in the literature. For instance, Ng [6] presents procedures for identifying desirable exponential populations under both known and unknown scale parameter scenarios. The definition of a “good” population in this context follows the criteria outlined in Lam [7] .

In this Section, the exponential threshold model considered by McDonald and Hodaj [8] is further assessed within the framework of subset selection rules. Specifically, the performance of their selection rules R₁ and R₂ will be compared. Let ${\pi }_i$, $i=1,\cdots ,k$, be k (≥2) independent populations with random draws from ${\pi }_i$ following an exponential probability distribution with scale parameter equal to 1 and threshold parameter equal to ${\Upsilon }_i$. Without loss of generality, the scale parameter can be any known value. If not equal to 1, then simply divide all the sample values by that common known scale parameter and proceed with the modified sample as coming from populations with unit scale parameter. Let $X_{ij}$, $j=1,\cdots ,n$, denote an independent random sample of size n from the i^th population. Let $Y_i=\min \left(X_{ij},j=1,\cdots ,n\right)$, and ${\bar{X}}_i$ equal the sample mean of $X_{ij},j=1,\cdots ,n$.

The goal of the subset selection rules is to select a subset of the k populations so as to include the “best” population, i.e., the population that is associated with the largest threshold parameter, $\Upsilon$, with a user prescribed probability (P*) no less than 1/k. That is, the probability of a Correct Selection (CS) is at least equal to a user specified value of P* no matter what the underlying configuration of the population threshold parameters may be. If two or more populations possess the largest threshold parameter, one of these is tagged at random and denoted as the “best”.

The two subset selection rules considered are:

R₁: Select ${\pi }_i$ iff $Y_i\ge \max \left(Y_j,j=1,\cdots ,k\right)-d$, $d\ge 0$, and (1.1)

R₂: Select ${\pi }_i$ iff ${\bar{X}}_i\ge \max \left({\bar{X}}_j,j=1,\cdots ,k\right)-b$, $b\ge 0$. (1.2)

The nonnegative constants, d and b, are chosen to satisfy the P* condition, i.e.,

$\min \mathrm{Pr}\left(\text{CS}\right)\ge P*$, (1.3)

where the minimum is taken over all possible configurations of ${\Upsilon }_i,i=1,\cdots ,k$. Computational methods for these constants are given in [8] .

We first consider the case of two populations and a large sample size. This will give a basis for comparing the accuracy of a relevant simulation methodology to be introduced in Section 3. By the Central Limit Theorem, the distribution of the sample mean, ${\bar{X}}_i$, is approximately normal with mean and variance ${\Upsilon }_i+1$ and 1/n, respectively. Now consider selection rule R₂ given in (1.2). Let ${\bar{X}}_{\left(i\right)}$ denote the sample mean drawn from the population associated with ${\Upsilon }_{\left[i\right]}$, where ${\Upsilon }_{\left[1\right]}=1\le {\Upsilon }_{\left[2\right]}=1+\delta$, $\delta \ge 0$. Then as n grows large, and noting ${\bar{X}}_{\left(2\right)}\ge {\bar{X}}_{\left(2\right)}-b$, it follows that Pr(CS) approaches

$\begin{array}{c}\mathrm{Pr}\left(\text{CS}\right)=\mathrm{Pr}\left({\bar{X}}_{\left(2\right)}\ge {\bar{X}}_{\left(1\right)}-b\right)\\ =\mathrm{Pr}\left({\bar{X}}_{\left(1\right)}-{\bar{X}}_{\left(2\right)}\le b\right)\\ =\Phi \left[\left(b+\delta \right)/\text{sqrt}\left(2/n\right)\right],\end{array}$ (1.4)

where Ф(⋅) is the cumulative distribution function (cdf) of a normal variable with mean 0 and variance 1. The move from the second to the third equality in (1.4) results from the normality of a linear combination of independent normal variates, e.g., ${\bar{X}}_{\left(1\right)}-{\bar{X}}_{\left(2\right)}$, Navidi [ [9] , Chapt. 4]. The mean and variance of ${\bar{X}}_{\left(1\right)}-{\bar{X}}_{\left(2\right)}$ are −δ and 2/n, respectively. Since δ ≥ 0, it follows from (1.4) that $\mathrm{Pr}\left(\text{CS}\right)\ge \Phi \left[b/\text{sqrt}\left(2/n\right)\right]$. Thus, for k = 2 and a given value of n and P*, the b-value is given by

$b=\text{sqrt}\left(2/n\right)*{\Phi }^{-1}\left(P*\right),$ (1.5)

where Ф⁻¹(⋅) is the inverse function of the cdf Ф(⋅).

An incorrect selection (ICS) occurs when the population associated with ${\Upsilon }_{\left[1\right]}$ is included in the selected subset. Thus, following the derivation of (1.4)

$\begin{array}{c}\mathrm{Pr}\left(\text{ICS}\right)=\mathrm{Pr}\left({\bar{X}}_{\left(1\right)}\ge {\bar{X}}_{\left(2\right)}-b\right)\\ =\mathrm{Pr}\left({\bar{X}}_{\left(2\right)}-{\bar{X}}_{\left(1\right)}\le b\right)\\ =\Phi \left[\left(b-\delta \right)/\text{sqrt}\left(2/n\right)\right].\end{array}$ (1.6)

Let $S_{\left[i\right]}=1$ if the population associated with ${\Upsilon }_{\left[i\right]}$ is included in the selected subset, otherwise $S_{\left[i\right]}=0$, $i=1,2$. Then the expected size of the selected subset (ESS) is

$\begin{array}{c}\text{ESS}=E\left(S_{[1]}\right)+E\left(S_{[2]}\right)\\ =\mathrm{Pr}\left(\text{ICS}\right)+\mathrm{Pr}\left(\text{CS}\right).\end{array}$ (1.7)

Appendix A provides R-code for calculating the Operating Characteristics (OCs), i.e., Pr(CS), Pr(ICS), and ESS, for given values of k = 2, n, P*, and b. Table 1 provides such output for a reasonably large value of n = 25. Gupta and Panchapakesan [ [4] , Sec. 11.2] and Gibbons et al. [ [5] , Sec. 3.2] address the expected subset size properties within the context of criteria for evaluating the performance of a subset selection procedure.

Now consider the case of k (≥2) populations with a random sample of size n drawn from each of the populations. Each of the populations follow an exponential distribution with scale parameter equal to 1 and threshold parameter equal to 0 for k-1 populations and equal to δ (≥0) for the remaining population. Using subset selection rule R₂ (1.2) a subset of the populations is chosen to contain the “best” population, i.e., the one associated with the threshold value δ, with a probability no less than a prescribed P* (1/k < P* < 1).

The R-code in Appendix B is used to simulate the operating characteristics Pr(CS), Pr(ICS), and ESS for specified values of k, n, P*, δ and N. The process of generating a data set as specified is repeated N times. For each of the repeats, the subset selection is made, and the populations chosen given a score of 1. The averages of the population scores estimate the probabilities that the individual populations are chosen by R₂. From these probability estimates the OCs are then computed, the ESS being the sum of the estimated selection probabilities of the individual populations.

To assess the accuracy of the simulation approach, the values similar to those given in Table 1 are calculated and displayed in Table 2 . Note the b-value is determined by the simulation of 200,000 draws and differs very slightly from those given in Section 2. The entries in Table 2 are quite close to those in Table 1 . For example, the ESS entry in Table 2 for δ = 0.5 is 0.145% less than the corresponding entry from Table 1 . Overall, the mean absolute percentage difference for ESS Table 2 entries compared to those from Table 1 is 0.115%. Thus, for k = 2 the simulation results are in very close agreement with the exact values obtained from the Central Limit Theorem presented in Section 2.

Appendix C provides an R-code to compute the operating characteristics of selection rule R₁ (1.1) and the appropriate constant d = d (k, n, P*). This code is structured very similar to that of Appendix B with the exception that the simulation of the sample means statistics are replaced by the sample minimum value statistics. A comparison of the OCs of the two selection procedures is given in Table 3 (to four dp) and plotted in Figure 1 for k = 10, n = 25, P* = 0.95. These values are based on 200,000 simulations for a slippage configuration of exponential populations with common scale parameter equal to 1 and nine populations with threshold parameters equal to 0 and one population, the “best”, with threshold parameter δ = 0(0.1)1. While the Pr(CS) for the two selection procedures are very close, the ESS for selection rule R₁ based on the sample minimum values is substantially less than that for R₂ based on the sample means.

A second parametric configuration for OC comparisons is the equi-spaced configuration. In this particular setup for k = 10 populations, the threshold parameters are fixed at ${\Upsilon }_i=\left(i-1\right)\cdot \delta$, $i=1,\cdots ,k$. Thus, the difference between any two adjacently ordered population threshold parameters is δ. R-codes in Appendix D and Appendix E provide the OCs for selection rules R₂ and R₁, respectively. Table 4 provides the output for k = 10, n = 25, P* = 0.95 for N = 200,000 simulations. The quantity $\mathrm{Pr}\left({\pi }_i\right)$ is the estimated probability of selecting the i^th population. Population π₁₀ is the “best” and its selection probability is denoted by Pr(CS). The ESS is the sum of the ten estimated selection probabilities. As in the case with the slippage configuration, the ESS for R₁ is substantially less than that for R₂ for all positive values of δ.

The ESS values for the two parameter configurations are displayed in Figure 1 (see Appendix F and Appendix G). The values for the slippage (equi-spaced) configuration are given in the left (right) side. Clearly the selection procedure R₁ outperforms R₂ with respect to these metrics. This is somewhat explained by the moments of the sample minimum value and the sample mean given in Table 5 . Since both R₁ and R₂ can be expressed in terms of unbiased estimators of ϒ, the sample minimum is more efficient as its variance is a factor of n⁻¹ times that of the sample mean.

McDonald and Hodaj [8] generate a data set for k = 10 exponential populations with a random sample of size 25 from each population. This data set is produced using the R-code in their Appendix B. For each of the population draws, the minimum value and sample means are calculated and rules R₁ and R₂ applied to select subsets to contain the “best” with P* = 0.75, 0.90, 0.95, 0.975, and 0.99. The findings are reported in their Table 4 , and show the means procedure R₂ chooses fewer populations for four of the P* values and an equal number for one value of P*. These findings seem somewhat at odds with what has been reported in this article. However, the data set leading to these findings in [8] is based on 10 exponential populations all with threshold parameters equal to 0 and rate parameters (=1/scale) equal to 1/i, i = 1(1)10. The expected values of the ten populations are 1(1)10. Thus, the selection rules were applied to exponential populations differing in expected values but not threshold values, and those results are not meaningful for the model under consideration in this article. The five lines in the R-code from Appendix B [8] beginning with “gamma < −seq (from = 1, to =10, by = 1)” and ending with “M[,i] < −rexp (n,lambda[i])” should be replaced by the four lines beginning with “gamma < −seq (from = 1, to =2.8, by = 0.2)” and ending with “M[,i] < −rexp(n,1)+gamma[i]” from Appendix H given here. This issue will now be further addressed.

Using the R-code in Appendix H random samples of size 25 are generated from 10 exponential populations having ϒ-values equal to 1(0.2)2.8. Table 6 gives the minimum and mean values of these samples. Table 7 gives the constants required to implement the two selection rules for five values of P*, and Table 8 indicates which of the ten populations are selected by R₁ and R₂ for each of the five P* values. For P* = 0.75 each of the selection rules only select the “best” population, i.e., π₁₀. For all values of P*, R₁ only selects the “best” population, whereas R₂ progressively chooses two or three populations as P* increases from 0.90 to 0.99. For this one set of data R₁ outperforms R₂ insofar as choosing an equal or smaller number of populations for any of the P* values investigated. This aligns with the earlier results herein obtained with respect to expected subset sizes, and displayed in Table 3 and Table 4 .

Table 3 and Table 4 provide results for moderately large values of k = 10 and n = 25. Table 9 and Table 10 are very similar to those of 3 and 4 only using “small” values for the number of populations and common sample sizes, k = 5 and n = 10. Results for Table 9 and Table 10 are obtained from the R-codes in (Appendices B-E). The conclusions from these computations are very similar to those derived earlier. The selection rules based on the sample minimums, R₁, are notably better than those based on the sample means, R₂, with respect to the Pr(CS) and ESS.

Throughout this article, when using the selection rule R₂ based on the sample mean values, the Central Limit Theorem or computer simulation was invoked to calculate the implementation constant, b, and evaluate the OCs of the procedure. In this Section, the exact distribution of the sum of independent exponential random variables is used and the resultant OCs compared to those in Table 2 and Table 3 .

As before, the setup is k populations each having an independent random sample of size n, $X_{ij}$, $i=1,\cdots ,k$, $j=1,\cdots ,n$, and $X_{ij}$ following an exponential distribution with unknown threshold parameter ${\Upsilon }_i$ and a common known scale parameter $\eta$. Without loss of generality, $\eta$ is assumed to be 1. Denote the sum of the sample values from the i^th population, ${\pi }_i$, by $S_i$. The i^th sample mean is then simply $S_i/n$. The random variables $X_{ij}-{\Upsilon }_i$ follow an exponential distribution with a zero threshold value and unit scale parameter. Then

$n\left({\bar{X}}_i-{\Upsilon }_i\right)=\underset{j=1}{\overset{n}{{\displaystyle \sum }}}\left(X_{ij}-{\Upsilon }_i\right)$ (5.1)

follows a gamma distribution (scale version) with probability density

$g\left(x|n,\eta =1\right)=x^{n-1}\exp \left(-x\right)/\left(n-1\right)!,x\ge 0.$ (5.2)

The subset selection rule R₂ (1.2) can now be rewritten as

R₃: Select ${\pi }_i$ iff $S_i\ge \max \left(S_j,j=1,\cdots ,k\right)-nb$, $b\ge 0$. (5.3)

Following the setup from Section 3 for slippage configurations, all but one of the populations have a zero threshold parameter, and the “best” population has a threshold parameter δ ≥ 0. The OCs can be derived as in earlier sections

$\mathrm{Pr}\left(\text{CS}\right)={\displaystyle {\int }_0^{\infty }{G}^{k-1}}\left[x+n\left(b+\delta \right)\right]\cdot g\left(x\right)\text{d}x,$ (5.4)

where g(⋅) is the probability density (5.2) and G(⋅) is the corresponding cumulative distribution function. The value b is chosen so as to satisfy the P* condition (1.3). Continuing,

$\mathrm{Pr}\left(\text{ICS}\right)={\displaystyle {\int }_0^{\infty }{G}^{k-2}}\left(x+nb\right)\cdot G\left[x+n\left(b-\delta \right)\right]\cdot g\left(x\right)\text{d}x,$ (5.5)

and

$\text{ESS}=\text{Pr}\left(\text{CS}\right)+\left(k-1\right)\text{Pr}\left(\text{ICS}\right)$ (5.6)

The OCs for R₃ using sample sums with associated gamma distributions can be calculated using the R-code given in Appendix I with input for k, n, b, and δ. R-codes given in Appendix C and Appendix D can be used to calculate the b-value for selection rule R₂ (1.2) for slippage configurations and equi-spaced configurations respectively.

Three methods have been presented for implementing the selection rule R₂. The first is a simulation approach using the R-code in Appendix B. The second is employing the Central Limit Theorem (CLT) and treating the sample means as normally distributed random variables as in the R-code of Appendix J. The final approach uses the distribution of the sample sums as gamma random variables as in Appendix I. The OCs for these approaches are given in Table 11 . The entries for the simulation approach (blue) agree with those given in Table 3 . While the results for the three approaches are somewhat close, the results for simulation (blue) and gamma (purple) approaches are in very close agreement. The ESS based on the CLT yields slightly lower values than the other two entries.

For k = 2 and n sufficiently large so that the sample means follow, approximately, a normal distribution, the exact OCs for R₂ can be calculated. Results so obtained are in very close agreement with results based on simulations, thus providing support for the simulation approach for practical applications. The OCs for R₁ are substantially better than those for R₂: higher Pr(CS) and lower ESS for δ > 0. Comparisons herein made of the OCs for R₁ with those for R₂, k = 25 and n = 10, strongly favor R₁ for both slippage and equi-spaced configurations. The same conclusion followed when similar analyses were done with k = 5 and n = 10. While

an earlier result reported in [8] seems to be somewhat at odds with that reported here, it is, in fact, not. Further clarification of the finding in [8] is herein given and shown not to be based on the exponential models herein considered.

Using the fact that sums of exponential random variables follow a gamma probability distribution permits exact calculations of the OCs for the selection rule R₁. A comparison of these exact results with those based on simulations and those based on the normal distribution shows the three approaches yield quite comparable estimates. Overall, the selection rule based on the sample minimums, R₁, has superior OCs over those based on the sample means, R₂, in the cases herein examined.