# Lower Confidence Bounds for the Probabilities of Correct Selection

Authors: Radhey S. Singh; Narinder Kumar
Journal: Journal of Probability and Statistics
 (2011)

---

## Abstract

We extend the results of Gupta and Liang (1998), derived for location parameters, to obtain lower confidence bounds for the probability of correctly selecting thet best populations (PCSt) simultaneously for all t=1,…,k−1 for the general scale parameter models, where k is the number of populations involved in the selection problem. The application of the results to the exponential and normal probability models is discussed. The implementation of the simultaneous lower confidence bounds for PCSt is illustrated through real-life datasets.


---

## Body

## 1. Introduction

The populationΠi is characterized by an unknown scale parameter θi(>0), i=1,…,k. Let Ti be an appropriate statistic for θi, based on a random sample of size n from population Πi, having the probability density function (pdf) fθi(x)=(1/θi)f(x/θi) with the corresponding cumulative distribution function (cdf) Fθi(x)=F(x/θi), x>0, θi>0, i=1,…,k.F(·) is an arbitrary continuous cdf with pdf f(·). Let the ordered values of Ti’s and θi’s be denoted by T[1],…,T[k] and θ[1],…,θ[k], respectively. Let T(i) be the statistic having a scale parameter θ[i]. Let Π(i) denote the population associated with θ[i], the ith smallest of θi’s. Any other population or sample quantity associated with Π(i) will be denoted by the subscript (i) attached to it. Throughout, we assume that there is no prior knowledge about which of Π1,…,Πk is Π(i), i=1,…,k and that θ1,…,θk are unknown. Call the populations Π(k),Π(k-1),…,Π(k-t+1) as the t best populations.In practice, the interest is to select the populationsΠ(k),Π(k-1),…,Π(k-t+1), that is, the populations associated with the largest unknown parameters θ[k],θ[k-1],…,θ[k-t+1]. For this, the natural selection rule “select the populations corresponding to t largest Ti’s, that is, T[k],T[k-1],…,T[k-t+1] as the t best populations” is used. However, it is possible that selected populations according to the natural selection rule may not be the best. Therefore, a question which naturally arises is: what kind of confidence statement can be made about these selection results? Motivated by this, we make an effort to answer this question.LetCSt (a correct selection of the t best populations) denote the event that t best populations are actually selected. Then, the probability of correct selection of the t best populations (PCSt) is: (1.1a)PCSt(θ)=P{max⁡1≤i≤k-t⁡T(i)<min⁡k-t+1≤j≤k⁡T(j)}=∫∏i=1k-tF(yθ[i])d{1-∏j=k-t+1kF-(yθ[j])}(1.1b)=∫∏j=k-t+1kF̅(yθ[j])d∏i=ik-tF(yθ[i]),where F̅(·)=1-F(·) and θ=(θ1,…,θk).For thek populations differing in their location parameters μ1,…,μk, Gupta and Liang [1] provided a novel idea to construct simultaneous lower confidence bounds for the PCSt for all t=1,…,k-1. Their result was applied to the selection of the t best means of normal populations. For other references under location set up, one may refer to the papers cited therein.For other relevant references, one may refer to Gupta et al. [2], Gupta and Panchpakesan [3], Mukhopadhyay and Solanky [4], and the review papers by Gupta and Panchapakesan [5, 6], Khamnei and Kumar [7], and the references cited therein.In this article, we use the methodology and results of Gupta and Liang [1] to derive simultaneous lower confidence bounds for the PCSt for all                          t=1,…,k-1 under the general scale parameter models. Section 2 deals with obtaining such intervals. The application of the results to the exponential and normal probability models is discussed in Section 3. In the case of an exponential distribution, Type-II censored data is also considered. In Section 4, we have given some numerical examples, based on real life data sets, to illustrate the procedure of finding out simultaneous lower confidence bounds for the probability of correctly selecting the t best populations (PCSt).


## 2. Simultaneous Lower Confidence Bounds forPCSt

Most of the results in this Section are as a simple consequence of the results obtained by Gupta and Liang [1].From (1.1a), the PCSt(θ) can be expressed as (2.1)PCSt(θ)=∑j=k-t+1kPtj(θ),
where for each j=k-t+1,…,k,(2.2)Ptj(θ)=∫∏i=1k-tF(yΔtji(1))∏m=k-t+1j-1F̅(yΔtjm(2))∏l=j+1kF̅(yΔtjl(3))dF(y),
where Δtji(1)=θ[j]/θ[i]≥1 for 1≤i≤k-t<j; Δtjm(2)=θ[j]/θ[m]≥1 for k-t+1≤m<j and Δtjl(3)=θ[j]/θ[l]≤1 for k-t+1≤j<l≤k. Here, ∏st≡1 if t<s. Note that for each j(k-t+1≤j≤k), Ptj(θ) is increasing in Δtji(1), and decreasing in Δtjm(2) and Δtjl(3), respectively. Thus, if we develop simultaneous lower confidence bounds for Δtji(1), 1≤i≤k-t and upper confidence bounds for Δtjm(2) and Δtjl(3), k-t+1≤m≤j≤l≤k, m≠j, l≠j for all t=1,…,k-1, then, simultaneous lower confidence bounds for PCSt(θ) for all t=1,…,k-1 can be established.Also, from (1.1b), the PCSt(θ) can be expressed as (2.3)PCSt(θ)=∑i=1k-tQti(θ),
where for each i=1,…,k-t,(2.4)Qti(θ)=∫∏m=1i-1F(zδtim(1))∏l=i+1k-tF(zδtil(2))∏j=k-t+1kF̅(zδtij(3))dF(z)
and δtim(1)=θ[i]/θ[m]≥1 for 1≤m<i≤k-t; δtil(2)=θ[i]/θ[l]≤1 for  1≤i<l≤k-t; and δtij(3)=θ[i]/θ[j]≤1 for i≤k-t<j≤k. Note that for each i=1,…,k-t,Qti(θ) is increasing in δtim(1), δtil(2), and decreasing in δtij(3), respectively. Thus, if simultaneous lower confidence bounds for δtim(1) and  δtil(2), 1≤m≤i≤l≤k-t, m≠i, l≠i and upper confidence bounds for δtil(3), i≤k-t<j≤k can be obtained, and, thereafter, by using  (2.3) and (2.4), we can obtain simultaneous lower confidence bounds for the PCSt(θ) for all t=1,…,k-1.We use the results of Gupta and Liang [1] to construct simultaneous lower confidence bounds for all Δtji(1), δtim(1), δtil(2), and upper confidence bounds for all Δtjm(2), Δtjl(3), and δtil(3) for all t=1,…,k-1.For eachP*(0<P*<1), let c(k,n,P*) be the value such that  (2.5)Pθ-{[max1≤i≤k(Ti/θi)min1≤j≤k(Tj/θj)]≤c(k,n,P*)}=P*.
Note that since Ti has a distribution function F(y/θi), i=1,…,k, the value of c=c(k,n,P*) is independent of the parameter θ. Let(2.6)E={max⁡1≤i≤k(Ti/θi)min⁡1≤j≤k(Tj/θj)≤c},E1={(T[i]cT[j])+≤θ[i]θ[j]≤(cT[i]T[j]),∀1≤j<i≤k},E2={(T[i]cT[j])≤θ[i]θ[j]≤(cT[i]T[j])-,∀1≤i<j≤k},
where y+=max(1,y) and y-=min(1,y).Lemma 2.1.
(a)E⊂E1∩E2 and, therefore,
(b)Pθ{E1∩E2}≥Pθ{E}=P* for all θ.Proof.
Part (a) follows on the lines of Lemma  2.1 of Gupta and Liang [1] by noting that θ[i]/θ[j]≥1 as j<i and θ[i]/θ[j]≤1 for i<j, we have E⊂E1 andE⊂E2. Therefore, E⊂E1∩E2.
Part (b) follows immediately from part (a) and (2.5).
For eacht=1,…,k-1 and j=k-t+1,…,k, let
(2.7)Δ̂tji(1)=(T[j]cT[i])+for1≤i≤k-t;Δ̂tjm(2)=(cT[j]T[m])fork-t+1≤m<j;Δ̂tjl(3)=(cT[j]T[l])-forj<l≤k.
Also, for eacht=1,…,k-1 and i=1,…,k-t, let
(2.8)δ̂tim(1)=(T[i]cT[m])+for1≤m≤i-1;δ̂til(2)=(T[i]cT[l])fori+1≤l≤k-t;δ̂tij(3)=(cT[i]T[j])-fork-t+1≤j≤k.The following Lemma is a direct result of Lemma2.1.Lemma 2.2.
With probability at leastP*, the following (A1) and (A2) hold simultaneously.
(A1) For eacht=1,…,k-1 and each j=k-t+1,…,k,
(2.9)Δtji(1)≥Δ̂tji(1),∀i=1,…,k-t;Δtjm(2)≤Δ̂tjm(2),∀k-t+1≤m<j;Δtjl(3)≤Δ̂tjl(3),∀j<l≤k.
(A2) For eacht=1,…,k-1 and each i=1,…,k-t,
(2.10)δtim(1)≥δ̂tim(1),∀1≤m≤i-1;δtil(2)≥δ̂til(2),∀i+1≤l≤k-t;δtij(3)≤δ̂tij(3),∀k-t+1≤j≤k.
Now, for each t=1,…,k-1 and each j=k-t+1,…,k, define
(2.11)P̂tj=∫∏i=1k-tF(yΔ̂tji(1))∏m=k-t+1j-1F̅(yΔ̂tjm(2))∏l=j+1kF̅(yΔ̂tjl(3))dF(y),
and for each t=1,…,k-1, define
(2.12)P̂t=∑j=k-t+1kP̂tj.
Also, for each t=1,…,k-1 and each i=1,…,k-t, define
(2.13)Q̂ti=∫∏m=1i-1F(zδ̂tim(1))∏l=i+1k-tF(zδ̂til(2))∏j=k-t+1kF̅(zδ̂tij(3))dF(z),(2.14)Q̂t=∑i=1k-tQ̂ti.
Define
(2.15)PtL=max⁡(P̂t,Q̂t).
The authors propose PtL=max(P̂t,Q̂t) as an estimator of a lower confidence bound of the PCSt(θ) for each t=1,…,k-1. The authors have the following theorem.Theorem 2.3.
Pθ-{PCSt(θ)≥PtL for all t=1,…,k-1}≥P* for all θ.Proof.
Note thatPtj(θ) is increasing in Δtji(1) and decreasing in Δtjm(2) andΔtjl(3). Also, Qti(θ) is increasing in δtim(1), δtil(2) and decreasing in δtij(3). Then, by using (2.2), (2.4), (2.11), (2.13), and Lemma 2.2, we have
(2.16)Pθ{Ptj(θ)≥P̂tj,∀j=k-t+1,...,k,andQti(θ)≥Q̂ti,∀i=1,…,k-t,t=1,…,k-1}≥P*.
Then, by (2.1), (2.3), (2.12), (2.14), and (2.16), we have
(2.17)P*≤P{PCSt(θ)≥P̂t,PCSt(θ)≥Q̂t,∀t=1,…,k-1}=Pθ(PCSt(θ)≥PtL∀t=1,…,k-1}.
This proves the theorem.


## 3. Applications to Exponential and Normal Distributions

### 3.1. Exponential Distribution

(i)  Complete Data
LetXij, j=1,…,n denote a random sample of size n from the two-parameter exponential population Πi having pdf f(x)=(1/θi)exp{-(x-μi)/θi}, i=1,…,k. Let Mi=min1≤j≤nXij and Yi=∑j=1n(Xij-Mi). Here, (Mi,Yi) is a sufficient statistic for (μi,θi), i=1,…,k.Yi/θi has a standardized gamma distribution with shape parameter θ=n-1, i=1,…,k. Then, based on statistics Y1,…,Yk by applying the natural selection rule for each t=1,…,k-1, the associated PCSt is
(3.1)PCSt(θ)=∑j=k-t+1kPtj(θ)=∑i=1k-tQti(θ),
where
(3.2)Ptj(θ)=∫∏i=1k-tF(yΔtji(1))∏m=k-t+1j-1F̅(yΔtjm(2))∏l=j+1kF̅(yΔtjl(3))dF(y),Qti(θ)=∫∏m=1i-1F(zδtim(1))∏l=i+1k-tF(zδtil(2))∏j=k-t+1kF̅(zδtij(3))dF(z),
and F(·) is the distribution function of the standardized gamma distribution with shape parameter θ=n-1.For eachP*(0<P*<1), let c=c(k,P*,n) be the P* quantile of the distribution of the random variable Z defined as Z={max1≤i≤k(Yi/θi)}/{min1≤i≤k(Yi/θi)}, the extreme quotient of independent and identically distributed random variables Yi.Givenk,n,P* the value of c can be obtained from the tables of Hartley’s ratio Z with 2(n-1) degrees of freedom refer to Pearson and Hartley [8].For eacht=1,…,k-1 and each j=k-t+1,…,k, let(3.3)P̂tj=∫∏i=1k-tF(yΔ̂tji(1))∏m=k-t+1j-1F̅(yΔ̂tjm(2))∏l=j+1kF-(yΔ̂tjl(3))dF(y),
and for each t=1,…,k-1 and each i=1,…,k-t, let (3.4)Q̂ti=∫∏m=1i-1F(zδ̂tim(1))∏l=i+1k-tF(zδ̂til(2))∏j=k-t+1kF̅(zδ̂tij(3))dF(z),
where Δ̂tji(1), Δ̂tjm(2), and Δ̂tjl(3) are defined as (2.7) and δ̂tim(1), δ̂til(2), and δ̂tij(3) are defined in (2.8) with c chosen from Pearson and Hartley’s tables.For eacht=1,…,k-1, let (3.5)P̂t=∑j=k-t+1kP̂tj,Q̂t=∑i=1k-tQ̂ti.
Then, by Theorem 2.3, we can conclude the following.Theorem 3.1.
Pθ{PCSt(θ)≥max(P̂t,Q̂t) for all t=1,…,k-1}≥P*  for all θ.(ii) Type-II Censored Data
From each populationΠi, i=1,…,k, we take a sample of n items. Let Xi[1],…,Xi[n] denote the order statistic representing the failure times of n items from population Πi, i=1,…,k. Let r be a fixed integer such that 1≤r≤n. Under Type-II censoring, the first r failures from each population Πi are to be observed. The observations from population Πi cease after observing Xi[r]. The (n-r) items whose failure times are not observable beyond Xi[r] become the censored observations. Type-II censoring was investigated by Epstein and Sobel [9]. The sufficient statistic for θi, when location parameters are known, is
(3.6)Ui=∑j=1rXi[j]+(n-r)Xi[r],i=1,…,k.Ui is called the total time on test (TTOT) statistic. It is easy to verify that Ui/θi has standardized gamma distribution with shape parameter r,i=1,…,k. Again, the results of complete data can be applied simply by taking ϑ=r.


### 3.2. Normal Distribution

LetΠi denote the normal population with mean μi and variance θi (both unknown), i=1,…,k. The sufficient statistic for θi based on a random sample Xi1,…,Xin of size n from Πi is Yi*=(1/(n-1))∑j=1n(Xij-X̅i)2, where X̅i=(1/n)∑j=1nXij, i=1,…,k. It can be verified that {(n-1)Yi*}/(2θi) is a standardized gamma variate with shape parameter (n-1)/2, i=1,…,k. Once again, the above results of exponential distribution can be used with ϑ=(n-1)/2.To illustrate the implementation of the simultaneous lower confidence bounds for the probability of correctly selecting thet best populations (PCSt), we consider the following examples.


## 3.1. Exponential Distribution

(i)  Complete Data
LetXij, j=1,…,n denote a random sample of size n from the two-parameter exponential population Πi having pdf f(x)=(1/θi)exp{-(x-μi)/θi}, i=1,…,k. Let Mi=min1≤j≤nXij and Yi=∑j=1n(Xij-Mi). Here, (Mi,Yi) is a sufficient statistic for (μi,θi), i=1,…,k.Yi/θi has a standardized gamma distribution with shape parameter θ=n-1, i=1,…,k. Then, based on statistics Y1,…,Yk by applying the natural selection rule for each t=1,…,k-1, the associated PCSt is
(3.1)PCSt(θ)=∑j=k-t+1kPtj(θ)=∑i=1k-tQti(θ),
where
(3.2)Ptj(θ)=∫∏i=1k-tF(yΔtji(1))∏m=k-t+1j-1F̅(yΔtjm(2))∏l=j+1kF̅(yΔtjl(3))dF(y),Qti(θ)=∫∏m=1i-1F(zδtim(1))∏l=i+1k-tF(zδtil(2))∏j=k-t+1kF̅(zδtij(3))dF(z),
and F(·) is the distribution function of the standardized gamma distribution with shape parameter θ=n-1.For eachP*(0<P*<1), let c=c(k,P*,n) be the P* quantile of the distribution of the random variable Z defined as Z={max1≤i≤k(Yi/θi)}/{min1≤i≤k(Yi/θi)}, the extreme quotient of independent and identically distributed random variables Yi.Givenk,n,P* the value of c can be obtained from the tables of Hartley’s ratio Z with 2(n-1) degrees of freedom refer to Pearson and Hartley [8].For eacht=1,…,k-1 and each j=k-t+1,…,k, let(3.3)P̂tj=∫∏i=1k-tF(yΔ̂tji(1))∏m=k-t+1j-1F̅(yΔ̂tjm(2))∏l=j+1kF-(yΔ̂tjl(3))dF(y),
and for each t=1,…,k-1 and each i=1,…,k-t, let (3.4)Q̂ti=∫∏m=1i-1F(zδ̂tim(1))∏l=i+1k-tF(zδ̂til(2))∏j=k-t+1kF̅(zδ̂tij(3))dF(z),
where Δ̂tji(1), Δ̂tjm(2), and Δ̂tjl(3) are defined as (2.7) and δ̂tim(1), δ̂til(2), and δ̂tij(3) are defined in (2.8) with c chosen from Pearson and Hartley’s tables.For eacht=1,…,k-1, let (3.5)P̂t=∑j=k-t+1kP̂tj,Q̂t=∑i=1k-tQ̂ti.
Then, by Theorem 2.3, we can conclude the following.Theorem 3.1.
Pθ{PCSt(θ)≥max(P̂t,Q̂t) for all t=1,…,k-1}≥P*  for all θ.(ii) Type-II Censored Data
From each populationΠi, i=1,…,k, we take a sample of n items. Let Xi[1],…,Xi[n] denote the order statistic representing the failure times of n items from population Πi, i=1,…,k. Let r be a fixed integer such that 1≤r≤n. Under Type-II censoring, the first r failures from each population Πi are to be observed. The observations from population Πi cease after observing Xi[r]. The (n-r) items whose failure times are not observable beyond Xi[r] become the censored observations. Type-II censoring was investigated by Epstein and Sobel [9]. The sufficient statistic for θi, when location parameters are known, is
(3.6)Ui=∑j=1rXi[j]+(n-r)Xi[r],i=1,…,k.Ui is called the total time on test (TTOT) statistic. It is easy to verify that Ui/θi has standardized gamma distribution with shape parameter r,i=1,…,k. Again, the results of complete data can be applied simply by taking ϑ=r.


## 3.2. Normal Distribution

LetΠi denote the normal population with mean μi and variance θi (both unknown), i=1,…,k. The sufficient statistic for θi based on a random sample Xi1,…,Xin of size n from Πi is Yi*=(1/(n-1))∑j=1n(Xij-X̅i)2, where X̅i=(1/n)∑j=1nXij, i=1,…,k. It can be verified that {(n-1)Yi*}/(2θi) is a standardized gamma variate with shape parameter (n-1)/2, i=1,…,k. Once again, the above results of exponential distribution can be used with ϑ=(n-1)/2.To illustrate the implementation of the simultaneous lower confidence bounds for the probability of correctly selecting thet best populations (PCSt), we consider the following examples.


## 4. Examples

Example 4.1.
Hill et al. [10] considered data on survival days of patients with inoperable lung cancer, who were subjected to a test chemotherapeutic agent. The patients are divided into the following four categories depending on the histological type of their tumor: squamous, small, adeno, and large denoted by π1,, π2, π3, and π4, respectively, in this article. The data are a part of a larger data set collected by the Veterans Administrative Lung Cancer Study Group in the USA.
We consider a random sample of eleven survival times from each group, and they are given in Table1.
Using the standard results of reliability (refer to Lawless [11]), one can check the validity of the two-parameter exponential model for Table 1. In this example, the populations with larger survival times (i.e., larger Yi’s) are desirable.
For Table1 data set:
(4.1)Y1=3841,Y2=383,Y3=361,Y4=1374.
Hence, according to natural selection rule, the populations π1,π2, and π4 are selected as the t (t=1,2,3) best populations, that is, for t=1, population π1 which has largest survival time is the best; for t=2, populations π1 and π4 which have the two largest survival times are the best; and for t=3, populations π1,, π2, and π4 which have the three largest survival times are the best. However, it i,s possible that selected populations according to the natural selection rule may not be the best. Therefore, we wish to find out a confidence statement that can be made about the probability of correctly selecting the t best populations (PCSt) simultaneously for all t=1,2,3.
Here,k=4, n=11, and, by taking P*=0.95, we get, from the tables of Pearson and Hartley [8], c=c(k,n,P*)=3.29.
Then,P̂t and Q̂t computed for the above data set using (3.5) are given in Table 2.  From Table 2, we have, with at least a 95% confidence coefficient, that simultaneously PCS1(θ)≥0.551725, PCS2(θ)≥0.33380, and PCS3(θ)≥0.174162.Table 1π1:99911224299111115873893825357π2:2521138722072499899π3:24183151905273836487π4:5216419531543340133111231378Table 2T123P̂t0.4071250.1439430.088946Q̂t0.5517250.333800.174162Max(P̂t,Q̂t)0.5517250.333800.174162Example 4.2.
Nelson [12] considered the data which represent times to breakdown in minutes of an insulating fluid subjected to high voltage stress. The times in their observed order were divided into three groups. After analyzing the data, it was shown to follow an exponential distribution. We consider the following data based on a random sample of size 11 each from the three groups and the observations are in Table 4.
For the above data set:(4.2)Y1=20.82,Y2=21.17,Y3=20.67.
Hence, according to natural selection rule, the populations π1, π2 are selected as the           t (t=1,2) best populations, that is, for t=1, population π1 which has largest survival time is the best; and for t=2, populations π1 and π2 which have the two largest survival times are the best. However, it is possible that selected populations according to the natural selection rule may not be the best. Therefore, we wish to find out a confidence statement that can be made about the probability of correctly selecting the t best populations (PCSt) simultaneously for all t=1,2.
Here,k=3, n=11, and, by taking P*=0.95, we get, from the tables of Pearson and Hartley [8], c=c(k,n,P*)=2.95.
Then,P̂t and Q̂t computed for the above data set using (3.5) are given in Table 3.
From Table3, we have, with at least a 95% confidence coefficient, that simultaneously PCS1(θ)≥0.424471 and PCS2(θ)≥0.248274.Table 3T12P̂t0.4244710.164871Q̂t0.1638550.248274max(P̂t,Q̂t)0.4244710.248274Table 4π1:1.540.661.701.822.750.660.550.1810.610.630.71π2:1.992.151.080.930.820.492.803.820.023.723.57π3:3.170.801.131.082.121.561.342.107.213.835.13Example 4.3.
Proschan [13] considered the data on intervals between failures (in hours) of the air-conditioning system of a fleet of 13 Boeing 720 jet air planes. After analyzing the data, he found that the failure distributions of the air-conditioning system for each of the planes was well approximated as exponential. We consider the following data based on four random samples of size seven each, and the observations in the samples are mentioned in Table 5.
For the above data set:(4.3)Y1=1046,Y2=96,Y3=226,Y4=139.
Hence, according to natural selection rule, the populations π1,π3, and π4 are selected as the t (t=1,2,3) best populations.
Here,k=4, n=7 and, by taking P*=0.99, we get, from the tables of Pearson and Hartley [8], c=c(k,n,P*)=6.90.
Proceeding on the lines similar to Examples4.1 and 4.2, we have, with at least a 99% confidence coefficient, that simultaneously PCS1(θ)≥0.360517, PCS2(θ)≥0.217558, and PCS3(θ)≥0.154598.Table 5π1:41310016944720111867π2:10142044292623π3:1148054631824π4:2234622302314



---

*Source: 765058-2011-02-24.xml*