The subset selection approaches developed herein follow the seminal work by Gupta [7] . Let $T_{ij}$ be $n$ independent identically distributed(iid) random variables from population ${\Pi }_i$, for $i=1,2,\cdots ,k$ and $j=1,2,\cdots ,n$. Let ${\gamma }_{\left[1\right]}\le {\gamma }_{\left[2\right]}\le \cdots \le {\gamma }_{\left[k\right]}$ be the ordered thresholds, where ${\gamma }_{\left[k\right]}$ corresponds to the best population. Since $t\ge \gamma$, then the minimum order statistic is a typical estimator for $\gamma$. So, let $Y_{\left(i\right)}$ be the minimum of the sample from the population with threshold parameter ${\gamma }_{\left[i\right]}$,

$Y_{\left(i\right)}=\underset{1\le j\le n}{\text{min}}\left(X_{ij}\right),i=1,2,\cdots ,k.$(2.2)

The selection rule is then,

$R_1:\text{Select}{\Pi }_i\text{iff}{Y}_i\ge \underset{1\le j\le k}{\text{max}}\left(Y_j\right)-d,d\ge 0.$(2.3)

The distribution of the minimum order statistic for independent random variables $X_i,i=1,2,\cdots ,n$ following (2.1) can easily be derived.

If $Y=\text{min}\left(X_1,X_2,\cdots ,X_n\right)$, then

$\begin{array}{c}\mathrm{Pr}\left(Y\le y\right)=1-\mathrm{Pr}\left(Y>y\right)\\ =1-\mathrm{Pr}\left(X_i>y,i=1,2,\cdots ,n\right)\\ =1-\underset{i=1}{\overset{n}{{\displaystyle \prod }}}\mathrm{Pr}\left(X_i>y\right)\text{by}\text{independence}\\ =1-\underset{i=1}{\overset{n}{{\displaystyle \prod }}}\left[1-\mathrm{Pr}\left(X_i\le y\right)\right]\\ =1-\underset{i=1}{\overset{n}{{\displaystyle \prod }}}\left\{1-\left[1-{\text{e}}^{-\left(y-\gamma \right)}\right]\right\}\\ =1-{\text{e}}^{-n\left(y-\gamma \right)},y\ge \gamma .\end{array}$ (2.4)

From here, the pdf of $Y$ is $f_Y\left(y\right)=n{\text{e}}^{-n\left(y-\gamma \right)}$, $y\ge \gamma$. The probability of a (CS) now follows:

$\begin{array}{c}\mathrm{Pr}\left(\text{CS}\right)=\mathrm{Pr}\left(Y_{\left(k\right)}\ge \underset{1\le j\le k-1}{\max }\left(Y_{\left(j\right)}\right)-d\right)\\ =\mathrm{Pr}\left(\underset{1\le j\le k-1}{\max }\left(Y_{\left(j\right)}\right)\le Y_{\left(k\right)}+d\right)\\ ={\displaystyle {\int }_{{\gamma }_{\left[k\right]}}^{\infty }\underset{j=i}{\overset{k-1}{{\displaystyle \prod }}}\mathrm{Pr}\left(Y_{\left(j\right)}\le y+d\right)\text{d}{F}_{Y_{\left(k\right)}}\left(y\right)}\\ ={\displaystyle {\int }_{{\gamma }_{\left[k\right]}}^{\infty }\underset{j=i}{\overset{k-1}{{\displaystyle \prod }}}\mathrm{Pr}\left(Y_{\left(j\right)}\le y+d\right)\cdot n\cdot {\text{e}}^{-n\left(y-{\gamma }_{\left[k\right]}\right)}\text{d}y}.\end{array}$

Now, let $x=y-{\gamma }_{\left[k\right]}$, and so $y=x+{\gamma }_{\left[k\right]}$. Then since ${\gamma }_{\left[k\right]}-{\gamma }_{\left[j\right]}\ge 0$, $j=1,2,\cdots ,k-1$

$\begin{array}{c}\mathrm{Pr}\left(\text{CS}\right)=n{\displaystyle {\int }_0^{\infty }\underset{j=i}{\overset{k-1}{{\displaystyle \prod }}}\mathrm{Pr}\left(Y_{\left(j\right)}-{\gamma }_{\left[j\right]}\le x+{\gamma }_{\left[k\right]}-{\gamma }_{\left[j\right]}+d\right){\text{e}}^{-nx}\text{d}x}\\ \ge n{\displaystyle {\int }_0^{\infty }\underset{j=i}{\overset{k-1}{{\displaystyle \prod }}}\mathrm{Pr}\left(Y_{\left(j\right)}-{\gamma }_{\left[j\right]}\le x+d\right){\text{e}}^{-nx}\text{d}x}.\end{array}$ (2.5)

The parameter configuration ${\gamma }_1={\gamma }_2=\cdots ={\gamma }_k$ is referred to as the Least Favorable Configuration (LFC) since it yields the minimum value of $\text{Pr}\left(\text{CS}\right)$. Setting the expression in (2.5) equal to a specified value $P^{\text{*}}$ ( $k^{-1}<P^{\text{*}}<1$), the value of $d$ can now be determined. Now the selection rule $R_1$ will ensure that $\text{Pr}\left(\text{CS}\right)\ge P^{\text{*}}$ no matter the configuration of ${{\gamma }^{\prime }}_{i}s$.

We assume that the scale parameter is common for all the populations ${\Pi }_i$ for $i=1,2,\cdots ,k$ and it is known, but it’s not 1. Does this change the selection rule for subset selection? Let $T~\text{exp}\left(\eta ,\gamma \right)$. The random variable $\frac{T}{\eta }$ is still exponentially distributed with scale parameter $\eta =1$,

$\begin{array}{c}\text{Pr}\left(\frac{T}{\eta }\le t\right)=\text{Pr}\left(T\le \eta t\right)\\ =1-\text{exp}\left[-\frac{\eta t-\gamma }{\eta }\right]\\ =1-\text{exp}\left[-\left(t-\frac{\gamma }{\eta }\right)\right]\\ =1-\text{exp}\left[-\left(t-{\gamma }^{\prime }\right)\right].\end{array}$ (2.6)

Or otherwise,

$F\left(\frac{t}{\eta }|\gamma \right)=1-\text{exp}\left[-\left(t-{\gamma }^{\prime }\right)\right],t\ge {\gamma }^{\prime },{\gamma }^{\prime }=\frac{\gamma }{\eta }.$ (2.7)

Hence, we get back to the previous case where scale parameter $\eta =1$ and the same selection rule, $R_1$, can be applied.

If $\beta =1$, then $E\left(T\right)=\eta \text{Γ}\left(2\right)+\gamma =\eta +\gamma$ and $V\left(T\right)={\eta }^2\left[\text{Γ}\left(3\right)-{\text{Γ}}^2\left(2\right)\right]={\eta }^2$. Suppose ${\Pi }_i,i=1,2,\cdots ,k$ are $k$ independent populations with ${\Pi }_i$ distributed as Weibull $\left({\gamma }_i,\eta ,\beta =1\right)$, and $\eta$ is a common value known. Let $X_{ij}$ be an independent random sample from ${\Pi }_i$, $j=1,2,\cdots ,n$. Our goal is to select a subset of the populations such that the population having the largest $\gamma$ value is contained in the subset with a specified probability $P^{\text{*}}$ ( $k^{-1}<P^{\text{*}}<1$). Denote the ordered $\gamma$-values by

${\gamma }_{\left[1\right]}\le {\gamma }_{\left[2\right]}\le \cdots \le {\gamma }_{\left[k\right]}.$

Since $\gamma$ is a threshold, as covered in Section 2 it is reasonable to consider a selection rule based on the minimum sample values from the population, such as $R_1$ given in 0.15.

Since the population means are ordered as the population $\gamma$-values, it is also reasonable to consider a selection rule based on the population sample means, ${\bar{X}}_i$. That is,

$R_2:\text{Choose}{\Pi }_i\text{iff}{\bar{X}}_i\ge \underset{1\le j\le k}{\text{max}}\left({\bar{X}}_j\right)-b,$(2.8)

where ${\bar{X}}_i=\left({\displaystyle {\sum }_{j=1}^n{X}_{ij}}\right)/n$ and $b=b\left(k,n,P^{\text{*}}\right)$ is a nonnegative number chosen to satisfy the $P^{\text{*}}$ condition,

$\underset{\Omega }{\min }\mathrm{Pr}\left(\text{CS}\right)\ge P^{*},$ (2.9)

where $\Omega =\left({\gamma }_1,{\gamma }_2,\cdots ,{\gamma }_k\right)$.

Let ${\bar{X}}_{\left(i\right)}$ be the sample mean drawn from the population possessing the mean value $\eta +{\gamma }_{\left[k\right]}$. Using the Central Limit Theorem, the distribution of ${\bar{X}}_{\left(i\right)}$ is approximately normal with mean $\eta +{\gamma }_{\left[k\right]}$ and variance $V\left(T\right)/n={\eta }^2/n$. So for large $n$,

$\begin{array}{c}\mathrm{Pr}\left(\text{CS}\right)=\mathrm{Pr}\left({\bar{X}}_{\left(k\right)}\ge \underset{1\le j\le k-1}{\max }{\bar{X}}_{\left(j\right)}-b\right)\\ =\mathrm{Pr}\left(Z_k\ge Z_j+\left({\gamma }_{\left[j\right]}-{\gamma }_{\left[k\right]}\right)-\left(\sqrt{n}/\eta \right)b,j=1,2,\cdots ,k-1\right)\\ \ge {\displaystyle {\int }_{-\infty }^{\infty }{\Phi }^{k-1}\left(x+cb\right)\varphi \left(x\right)\text{d}x},\end{array}$ (2.10)

where $Z_i$, $i=1,2,\cdots ,k$, are independent standardized normal random variables and $c=\sqrt{n}/\eta$. Thus, $b=b\left(k,n,P^{\text{*}}\right)$ is determined by setting Equation (2.10) equal to $P^{\text{*}}$ and solving for $b$.

For small values of $n$, the selection procedure can be based on the sum of the random variables, $n{\bar{X}}_i$, $i=1,2,\cdots ,k$, and utilizing the property that a sum of $n$ independent exponential random variables follows the gamma distribution with parameters $\left(n,\eta \right)$ assuming the threshold parameter, $\gamma$, is equal to 0 (Casella and Berger [8] ).

To assess the sensitivity of the subsets selected using selection rules $R_1$ and $R_2$ to the assumption of a common known scale parameter, $\eta$, set a bound on possible values of the parameter. That is, say the assessment of the analyst is $L<\eta <U$. Now discretize the interval $\left(L,U\right)$ into $m$ discrete values, i.e., $L=v_1<v_2<\cdots <v_m=U$. Follow the procedures in Sections (2.1.1) and (2.2) with $\eta$ assumed to be equal to $v_i$ to obtain the selected subsets. Repeat the process using the remaining values of $v=\left(v_1,\cdots ,v_m\right)$. Compare the $m$ selected subsets and assess the sensitivity to the selections based on the uncertainty of the scale parameter. To assess the sensitivity to the assumption that the shape parameter, $\beta$, is equal to one is not possible since the LFC has not been determined for values of $\beta$ not equal to one.

In this section, an example for each of the selection rules that were developed previously, rules (2.3) and (2.8) are given. The R-code for these rules is given in the Appendix section.

Using the selection rule for the largest threshold based on minimum order statistics from the $k$ populations, the $d$-values for different levels of the probability $P^{\text{*}}$ are computed.

The example will be for $k=10$ populations. From each population, a sample of size $n=25$ is drawn, with an equi-spaced configuration for the threshold parameters, i.e., ${{\gamma }^{\prime }}_{i}s=1\to 10$, by 1. Simulations are set for $N=10000$. The $d$-values obtained are given in Table 1 . The R-code for generating the $d$-values is found in Appendix A. The $P^{\text{*}}$ values based on simulations can be checked using the R-code in Appendix D.

Using the R-code in Appendix B random samples of size 25 are generated from 10 exponential populations having $\gamma$-values equal to 1 up to 10 with step size 1. Table 2 gives the minimum and mean values of these samples, respectively. The populations selected using rule $R_1$ are given in Table 4 for the five values of $P^{\text{*}}$ given in Table 1 .

For the selection rule based on the sample means, the $b$-value is calculated using the R-code in Appendix C, which for $k=10$, $n=25$ is given in Table 3 :

The selected populations using $R_2$ with the example simulated exponential dataset, $\gamma =1\to 10$, step size 1, are given in Table 4 . The sample means procedure yields subsets with smaller size in most of the $P^{\text{*}}$ values. For this particular simulation $R_2$ would be preferred over $R_1$ since the selected subset size is favorable. Another similarly simulated data set could well result in $R_1$ being preferred over $R_2$. Simulation studies, yet to be published, show that the expected subset size of the selected populations is smaller for $R_1$ than for $R_2$ with equi-spaced threshold parameters (or slippage parameter) greater than 0.

Following the Indifference Zone approach formulated by Bechhofer [9] , the population yielding the largest minimum order statistics, $Y_{\left[k\right]}$ would be asserted to be the “best” population. The statistical goal is to have the probability of this assertion to be correct with a specified probability, $P^{\text{*}}$, over the parameter space ${\gamma }_{\left[k\right]}-{\gamma }_{\left[k-1\right]}\ge d^{\text{*}}$, where $d^{\text{*}}$ is a user specified meaningful difference between the best population and all the remainder ones. The minimum of the sample size $n=n\left(k,P^{\text{*}},d^{\text{*}}\right)$ needs to be determined so that the $\text{Pr}\left(\text{CS}\right)\ge P^{\text{*}}$ for the parameter configuration ${\gamma }_{\left[k\right]}-{\gamma }_{\left[k-1\right]}\ge d^{\text{*}}$. Other parameter configurations comprise the “indifference zone” in the sense that the $\text{Pr}\left(\text{CS}\right)$ is not necessarily applicable.

For this decision rule,

$\begin{array}{c}\mathrm{Pr}\left(\text{CS}\right)=\mathrm{Pr}\left(Y_{\left(k\right)}\ge \max Y_{\left(j\right)},j=1,2,\cdots ,k-1\right)\\ =\mathrm{Pr}\left(Y_{\left(j\right)}\le Y_{\left(k\right)},j=1,2,\cdots ,k-1\right)\\ =\mathrm{Pr}\left(Y_{\left(j\right)}-{\gamma }_{\left[j\right]}\le Y_{\left(k\right)}-{\gamma }_{\left[k\right]}+{\gamma }_{\left[k\right]}-{\gamma }_{\left[j\right]},j=1,2,\cdots ,k-1\right)\\ \ge \mathrm{Pr}\left(Y_{\left(j\right)}-{\gamma }_{\left[j\right]}\le Y_{\left(k\right)}-{\gamma }_{\left[k\right]}+d^{*},j=1,2,\cdots ,k-1\right)\\ =n{\displaystyle {\int }_0^{\infty }{\left[1-{\text{e}}^{-n\left(x+d^{\text{*}}\right)}\right]}^{k-1}{\text{e}}^{-nx}\text{d}x}.\end{array}$ (2.11)

The LFC is ${\gamma }_{\left[1\right]}=\cdots ={\gamma }_{\left[k-1\right]}={\gamma }_{\left[k\right]}-d^{\text{*}}$. The above integral expression is calculated with the R-code found in Appendix 5 and through iteration we can determine the minimum sample size required to meet the $\text{Pr}\left(\text{CS}\right)$ requirement.

For example, for $k=5$ populations and requirement $P^{\text{*}}=0.90$ with $d^{\text{*}}=0.25$, Table 5 gives evaluations of (2.11). Thus, a sample size of 12 would be the minimum size requirement to meet the experimental goal.