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Journal of Probability and Statistics
VolumeΒ 2011, Article IDΒ 765058, 11 pages
http://dx.doi.org/10.1155/2011/765058
Research Article

Lower Confidence Bounds for the Probabilities of Correct Selection

1Department of Mathematics and Statistics, University of Guelph, ON, Canada N1G 2W1
2Department of Statistics, Panjab University, Chandigarh 160014, India

Received 6 October 2010; Accepted 12 January 2011

Academic Editor: A.Β Thavaneswaran

Copyright Β© 2011 Radhey S. Singh and Narinder Kumar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 the 𝑑 best populations (PCS𝑑) simultaneously for all 𝑑=1,…,π‘˜βˆ’1 for the general scale parameter models, where π‘˜ 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 PCS𝑑 is illustrated through real-life datasets.

1. Introduction

The population Π𝑖 is characterized by an unknown scale parameter πœƒπ‘–(>0), 𝑖=1,…,π‘˜. Let 𝑇𝑖 be an appropriate statistic for πœƒπ‘–, based on a random sample of size 𝑛 from population Π𝑖, having the probability density function (pdf) π‘“πœƒπ‘–(π‘₯)=(1/πœƒπ‘–)𝑓(π‘₯/πœƒπ‘–) with the corresponding cumulative distribution function (cdf) πΉπœƒπ‘–(π‘₯)=𝐹(π‘₯/πœƒπ‘–), π‘₯>0, πœƒπ‘–>0, 𝑖=1,…,π‘˜.𝐹(β‹…) is an arbitrary continuous cdf with pdf 𝑓(β‹…). Let the ordered values of 𝑇𝑖’s and πœƒπ‘–β€™s be denoted by 𝑇[1],…,𝑇[π‘˜] and πœƒ[1],…,πœƒ[π‘˜], respectively. Let 𝑇(𝑖) be the statistic having a scale parameter πœƒ[𝑖]. Let Ξ (𝑖) denote the population associated with πœƒ[𝑖], the 𝑖th smallest of πœƒπ‘–β€™s. Any other population or sample quantity associated with Ξ (𝑖) will be denoted by the subscript (𝑖) attached to it. Throughout, we assume that there is no prior knowledge about which of Ξ 1,…,Ξ π‘˜ is Ξ (𝑖), 𝑖=1,…,π‘˜ and that πœƒ1,…,πœƒπ‘˜ are unknown. Call the populations Ξ (π‘˜),Ξ (π‘˜βˆ’1),…,Ξ (π‘˜βˆ’π‘‘+1) as the 𝑑 best populations.

In practice, the interest is to select the populations Ξ (π‘˜),Ξ (π‘˜βˆ’1),…,Ξ (π‘˜βˆ’π‘‘+1), that is, the populations associated with the largest unknown parameters πœƒ[π‘˜],πœƒ[π‘˜βˆ’1],…,πœƒ[π‘˜βˆ’π‘‘+1]. For this, the natural selection rule β€œselect the populations corresponding to 𝑑 largest 𝑇𝑖’s, that is, 𝑇[π‘˜],𝑇[π‘˜βˆ’1],…,𝑇[π‘˜βˆ’π‘‘+1] as the 𝑑 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.

Let CS𝑑 (a correct selection of the 𝑑 best populations) denote the event that 𝑑 best populations are actually selected. Then, the probability of correct selection of the 𝑑 best populations (PCS𝑑) is: PCS𝑑(πœƒ)=𝑃max1β‰€π‘–β‰€π‘˜βˆ’π‘‘π‘‡(𝑖)<minπ‘˜βˆ’π‘‘+1β‰€π‘—β‰€π‘˜π‘‡(𝑗)ξ‚Ό=ξ€œπ‘˜βˆ’π‘‘ξ‘π‘–=1πΉξ‚΅π‘¦πœƒ[𝑖]𝑑1βˆ’π‘˜ξ‘βˆ’π‘—=π‘˜βˆ’π‘‘+1πΉξ‚΅π‘¦πœƒ[𝑗]ξ‚Άξƒ°=ξ€œ(1.1a)π‘˜ξ‘π‘—=π‘˜βˆ’π‘‘+1πΉξ‚΅π‘¦πœƒ[𝑗]ξ‚Άπ‘‘π‘˜βˆ’π‘‘ξ‘π‘–=π‘–πΉξ‚΅π‘¦πœƒ[𝑖]ξ‚Ά,(1.1b)where 𝐹(β‹…)=1βˆ’πΉ(β‹…) and πœƒ=(πœƒ1,…,πœƒπ‘˜).

For the π‘˜ populations differing in their location parameters πœ‡1,…,πœ‡π‘˜, Gupta and Liang [1] provided a novel idea to construct simultaneous lower confidence bounds for the PCS𝑑 for all 𝑑=1,…,π‘˜βˆ’1. Their result was applied to the selection of the 𝑑 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 𝑑=1,…,π‘˜βˆ’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 𝑑 best populations (PCS𝑑).

2. Simultaneous Lower Confidence Bounds for PCS𝑑

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 PCS𝑑(πœƒ) can be expressed as PCS𝑑(πœƒ)=π‘˜ξ“π‘—=π‘˜βˆ’π‘‘+1𝑃𝑑𝑗(πœƒ),(2.1) where for each 𝑗=π‘˜βˆ’π‘‘+1,…,π‘˜,π‘ƒπ‘‘π‘—ξ€œ(πœƒ)=π‘˜βˆ’π‘‘ξ‘π‘–=1𝐹𝑦Δ𝑑𝑗𝑖(1)π‘—βˆ’1ξ‘π‘š=π‘˜βˆ’π‘‘+1πΉξ€·π‘¦Ξ”π‘‘π‘—π‘šξ€Έ(2)π‘˜ξ‘π‘™=𝑗+1𝐹𝑦Δ𝑑𝑗𝑙(3)𝑑𝐹(𝑦),(2.2) where Δ𝑑𝑗𝑖(1)=πœƒ[𝑗]/πœƒ[𝑖]β‰₯1 for 1β‰€π‘–β‰€π‘˜βˆ’π‘‘<𝑗; Ξ”π‘‘π‘—π‘š(2)=πœƒ[𝑗]/πœƒ[π‘š]β‰₯1 for π‘˜βˆ’π‘‘+1β‰€π‘š<𝑗 and Δ𝑑𝑗𝑙(3)=πœƒ[𝑗]/πœƒ[𝑙]≀1 for π‘˜βˆ’π‘‘+1≀𝑗<π‘™β‰€π‘˜. Here, βˆπ‘‘π‘ β‰‘1 if 𝑑<𝑠. Note that for each 𝑗(π‘˜βˆ’π‘‘+1β‰€π‘—β‰€π‘˜), 𝑃𝑑𝑗(πœƒ) is increasing in Δ𝑑𝑗𝑖(1), and decreasing in Ξ”π‘‘π‘—π‘š(2) and Δ𝑑𝑗𝑙(3), respectively. Thus, if we develop simultaneous lower confidence bounds for Δ𝑑𝑗𝑖(1), 1β‰€π‘–β‰€π‘˜βˆ’π‘‘ and upper confidence bounds for Ξ”π‘‘π‘—π‘š(2) and Δ𝑑𝑗𝑙(3), π‘˜βˆ’π‘‘+1β‰€π‘šβ‰€π‘—β‰€π‘™β‰€π‘˜, π‘šβ‰ π‘—, 𝑙≠𝑗 for all 𝑑=1,…,π‘˜βˆ’1, then, simultaneous lower confidence bounds for PCS𝑑(πœƒ) for all 𝑑=1,…,π‘˜βˆ’1 can be established.

Also, from (1.1b), the PCS𝑑(πœƒ) can be expressed as PCS𝑑(πœƒ)=π‘˜βˆ’π‘‘ξ“π‘–=1𝑄𝑑𝑖(πœƒ),(2.3) where for each 𝑖=1,…,π‘˜βˆ’π‘‘,π‘„π‘‘π‘–ξ€œ(πœƒ)=π‘–βˆ’1ξ‘π‘š=1πΉξ€·π‘§π›Ώπ‘‘π‘–π‘šξ€Έ(1)π‘˜βˆ’π‘‘ξ‘π‘™=𝑖+1𝐹𝑧𝛿𝑑𝑖𝑙(2)π‘˜ξ‘π‘—=π‘˜βˆ’π‘‘+1𝐹𝑧𝛿𝑑𝑖𝑗(3)𝑑𝐹(𝑧)(2.4) and π›Ώπ‘‘π‘–π‘š(1)=πœƒ[𝑖]/πœƒ[π‘š]β‰₯1 for 1β‰€π‘š<π‘–β‰€π‘˜βˆ’π‘‘; 𝛿𝑑𝑖𝑙(2)=πœƒ[𝑖]/πœƒ[𝑙]≀1 for 1≀𝑖<π‘™β‰€π‘˜βˆ’π‘‘; and 𝛿𝑑𝑖𝑗(3)=πœƒ[𝑖]/πœƒ[𝑗]≀1 for π‘–β‰€π‘˜βˆ’π‘‘<π‘—β‰€π‘˜. Note that for each 𝑖=1,…,π‘˜βˆ’π‘‘,𝑄𝑑𝑖(πœƒ) is increasing in π›Ώπ‘‘π‘–π‘š(1), 𝛿𝑑𝑖𝑙(2), and decreasing in 𝛿𝑑𝑖𝑗(3), respectively. Thus, if simultaneous lower confidence bounds for π›Ώπ‘‘π‘–π‘š(1) and 𝛿𝑑𝑖𝑙(2), 1β‰€π‘šβ‰€π‘–β‰€π‘™β‰€π‘˜βˆ’π‘‘, π‘šβ‰ π‘–, 𝑙≠𝑖 and upper confidence bounds for 𝛿𝑑𝑖𝑙(3), π‘–β‰€π‘˜βˆ’π‘‘<π‘—β‰€π‘˜ can be obtained, and, thereafter, by using (2.3) and (2.4), we can obtain simultaneous lower confidence bounds for the PCS𝑑(πœƒ) for all 𝑑=1,…,π‘˜βˆ’1.

We use the results of Gupta and Liang [1] to construct simultaneous lower confidence bounds for all Δ𝑑𝑗𝑖(1), π›Ώπ‘‘π‘–π‘š(1), 𝛿𝑑𝑖𝑙(2), and upper confidence bounds for all Ξ”π‘‘π‘—π‘š(2), Δ𝑑𝑗𝑙(3), and 𝛿𝑑𝑖𝑙(3) for all 𝑑=1,…,π‘˜βˆ’1.

For each π‘ƒβˆ—(0<π‘ƒβˆ—<1), let 𝑐(π‘˜,𝑛,π‘ƒβˆ—) be the value such that π‘ƒπœƒβˆ’ξƒ―ξƒ¬max1β‰€π‘–β‰€π‘˜ξ€·π‘‡π‘–/πœƒπ‘–ξ€Έmin1β‰€π‘—β‰€π‘˜ξ€·π‘‡π‘—/πœƒπ‘—ξ€Έξƒ­ξ€·β‰€π‘π‘˜,𝑛,π‘ƒβˆ—ξ€Έξƒ°=π‘ƒβˆ—.(2.5) Note that since 𝑇𝑖 has a distribution function 𝐹(𝑦/πœƒπ‘–), 𝑖=1,…,π‘˜, the value of 𝑐=𝑐(π‘˜,𝑛,π‘ƒβˆ—) is independent of the parameter πœƒ. Let𝐸=max1β‰€π‘–β‰€π‘˜ξ€·π‘‡π‘–/πœƒπ‘–ξ€Έmin1β‰€π‘—β‰€π‘˜ξ€·π‘‡π‘—/πœƒπ‘—ξ€Έξƒ°,𝐸≀𝑐1=𝑇[𝑖]𝑐𝑇[𝑗]ξ‚Ά+β‰€πœƒ[𝑖]πœƒ[𝑗]≀𝑐𝑇[𝑖]𝑇[𝑗]ξ‚Άξ‚Ό,𝐸,βˆ€1≀𝑗<π‘–β‰€π‘˜2=𝑇[𝑖]𝑐𝑇[𝑗]ξ‚Άβ‰€πœƒ[𝑖]πœƒ[𝑗]≀𝑐𝑇[𝑖]𝑇[𝑗]ξ‚Άβˆ’ξ‚Ό,,βˆ€1≀𝑖<π‘—β‰€π‘˜(2.6) where 𝑦+=max(1,𝑦) and π‘¦βˆ’=min(1,𝑦).

Lemma 2.1. (a)πΈβŠ‚πΈ1∩𝐸2 and, therefore,
(b)π‘ƒπœƒ{𝐸1∩𝐸2}β‰₯π‘ƒπœƒ{𝐸}=π‘ƒβˆ— for all πœƒ.

Proof. Part (a) follows on the lines of Lemma  2.1 of Gupta and Liang [1] by noting that πœƒ[𝑖]/πœƒ[𝑗]β‰₯1 as 𝑗<𝑖 and πœƒ[𝑖]/πœƒ[𝑗]≀1 for 𝑖<𝑗, we have πΈβŠ‚πΈ1 andπΈβŠ‚πΈ2. Therefore, πΈβŠ‚πΈ1∩𝐸2.
Part (b) follows immediately from part (a) and (2.5).
For each 𝑑=1,…,π‘˜βˆ’1 and 𝑗=π‘˜βˆ’π‘‘+1,…,π‘˜, let Δ𝑑𝑗𝑖𝑇(1)=[𝑗]𝑐𝑇[𝑖]ξ‚Ά+Δfor1β‰€π‘–β‰€π‘˜βˆ’π‘‘;π‘‘π‘—π‘šξ‚΅(2)=𝑐𝑇[𝑗]𝑇[π‘š]Δforπ‘˜βˆ’π‘‘+1β‰€π‘š<𝑗;𝑑𝑗𝑙(3)=𝑐𝑇[𝑗]𝑇[𝑙]ξ‚Άβˆ’for𝑗<π‘™β‰€π‘˜.(2.7)
Also, for each 𝑑=1,…,π‘˜βˆ’1 and 𝑖=1,…,π‘˜βˆ’π‘‘, let Μ‚π›Ώπ‘‘π‘–π‘šξ‚΅π‘‡(1)=[𝑖]𝑐𝑇[π‘š]ξ‚Ά+̂𝛿for1β‰€π‘šβ‰€π‘–βˆ’1;𝑑𝑖𝑙𝑇(2)=[𝑖]𝑐𝑇[𝑙]̂𝛿for𝑖+1β‰€π‘™β‰€π‘˜βˆ’π‘‘;𝑑𝑖𝑗(3)=𝑐𝑇[𝑖]𝑇[𝑗]ξ‚Άβˆ’forπ‘˜βˆ’π‘‘+1β‰€π‘—β‰€π‘˜.(2.8)

The following Lemma is a direct result of Lemma 2.1.

Lemma 2.2. With probability at least π‘ƒβˆ—, the following (A1) and (A2) hold simultaneously.
(A1) For each 𝑑=1,…,π‘˜βˆ’1 and each 𝑗=π‘˜βˆ’π‘‘+1,…,π‘˜, Δ𝑑𝑗𝑖(Δ1)β‰₯𝑑𝑗𝑖(Ξ”1),βˆ€π‘–=1,…,π‘˜βˆ’π‘‘;π‘‘π‘—π‘šξΞ”(2)β‰€π‘‘π‘—π‘šΞ”(2),βˆ€π‘˜βˆ’π‘‘+1β‰€π‘š<𝑗;𝑑𝑗𝑙Δ(3)≀𝑑𝑗𝑙(3),βˆ€π‘—<π‘™β‰€π‘˜.(2.9)
(A2) For each 𝑑=1,…,π‘˜βˆ’1 and each 𝑖=1,…,π‘˜βˆ’π‘‘, π›Ώπ‘‘π‘–π‘šΜ‚π›Ώ(1)β‰₯π‘‘π‘–π‘šπ›Ώ(1),βˆ€1β‰€π‘šβ‰€π‘–βˆ’1;𝑑𝑖𝑙̂𝛿(2)β‰₯𝑑𝑖𝑙𝛿(2),βˆ€π‘–+1β‰€π‘™β‰€π‘˜βˆ’π‘‘;𝑑𝑖𝑗̂𝛿(3)≀𝑑𝑖𝑗(3),βˆ€π‘˜βˆ’π‘‘+1β‰€π‘—β‰€π‘˜.(2.10) Now, for each 𝑑=1,…,π‘˜βˆ’1 and each 𝑗=π‘˜βˆ’π‘‘+1,…,π‘˜, define 𝑃𝑑𝑗=ξ€œπ‘˜βˆ’π‘‘ξ‘π‘–=1𝐹𝑦Δ𝑑𝑗𝑖(1)π‘—βˆ’1ξ‘π‘š=π‘˜βˆ’π‘‘+1πΉξ‚€π‘¦ξΞ”π‘‘π‘—π‘šξ‚(2)π‘˜ξ‘π‘™=𝑗+1𝐹𝑦Δ𝑑𝑗𝑙(3)𝑑𝐹(𝑦),(2.11) and for each 𝑑=1,…,π‘˜βˆ’1, define 𝑃𝑑=π‘˜ξ“π‘—=π‘˜βˆ’π‘‘+1𝑃𝑑𝑗.(2.12) Also, for each 𝑑=1,…,π‘˜βˆ’1 and each 𝑖=1,…,π‘˜βˆ’π‘‘, define 𝑄𝑑𝑖=ξ€œπ‘–βˆ’1ξ‘π‘š=1πΉξ€·π‘§Μ‚π›Ώπ‘‘π‘–π‘šξ€Έ(1)π‘˜βˆ’π‘‘ξ‘π‘™=𝑖+1𝐹𝑧̂𝛿𝑑𝑖𝑙(2)π‘˜ξ‘π‘—=π‘˜βˆ’π‘‘+1𝐹𝑧̂𝛿𝑑𝑖𝑗𝑄(3)𝑑𝐹(𝑧),(2.13)𝑑=π‘˜βˆ’π‘‘ξ“π‘–=1𝑄𝑑𝑖.(2.14) Define 𝑃𝑑𝐿𝑃=max𝑑,𝑄𝑑.(2.15) The authors propose 𝑃𝑑𝐿𝑃=max(𝑑,𝑄𝑑) as an estimator of a lower confidence bound of the PCS𝑑(πœƒ) for each 𝑑=1,…,π‘˜βˆ’1. The authors have the following theorem.

Theorem 2.3. π‘ƒπœƒβˆ’{PCS𝑑(πœƒ)β‰₯𝑃𝑑𝐿 for all 𝑑=1,…,π‘˜βˆ’1}β‰₯π‘ƒβˆ— for all πœƒ.

Proof. Note that 𝑃𝑑𝑗(πœƒ) is increasing in Δ𝑑𝑗𝑖(1) and decreasing in Ξ”π‘‘π‘—π‘š(2) andΔ𝑑𝑗𝑙(3). Also, 𝑄𝑑𝑖(πœƒ) is increasing in π›Ώπ‘‘π‘–π‘š(1), 𝛿𝑑𝑖𝑙(2) and decreasing in 𝛿𝑑𝑖𝑗(3). Then, by using (2.2), (2.4), (2.11), (2.13), and Lemma 2.2, we have π‘ƒπœƒξ‚†π‘ƒπ‘‘π‘—ξπ‘ƒ(πœƒ)β‰₯𝑑𝑗,βˆ€π‘—=π‘˜βˆ’π‘‘+1,...,π‘˜,and𝑄𝑑𝑖𝑄(πœƒ)β‰₯𝑑𝑖,βˆ€π‘–=1,…,π‘˜βˆ’π‘‘,𝑑=1,…,π‘˜βˆ’1β‰₯π‘ƒβˆ—.(2.16) Then, by (2.1), (2.3), (2.12), (2.14), and (2.16), we have π‘ƒβˆ—ξ‚†β‰€π‘ƒPCS𝑑𝑃(πœƒ)β‰₯𝑑,PCS𝑑𝑄(πœƒ)β‰₯𝑑,βˆ€π‘‘=1,…,π‘˜βˆ’1=π‘ƒπœƒξ€·PCS𝑑(πœƒ)β‰₯𝑃𝑑𝐿.βˆ€π‘‘=1,…,π‘˜βˆ’1(2.17) This proves the theorem.

3. Applications to Exponential and Normal Distributions

3.1. Exponential Distribution

(i)  Complete Data
Let 𝑋𝑖𝑗, 𝑗=1,…,𝑛 denote a random sample of size 𝑛 from the two-parameter exponential population Π𝑖 having pdf 𝑓(π‘₯)=(1/πœƒπ‘–)exp{βˆ’(π‘₯βˆ’πœ‡π‘–)/πœƒπ‘–}, 𝑖=1,…,π‘˜. Let 𝑀𝑖=min1≀𝑗≀𝑛𝑋𝑖𝑗 and π‘Œπ‘–=βˆ‘π‘›π‘—=1(π‘‹π‘–π‘—βˆ’π‘€π‘–). Here, (𝑀𝑖,π‘Œπ‘–) is a sufficient statistic for (πœ‡π‘–,πœƒπ‘–), 𝑖=1,…,π‘˜.π‘Œπ‘–/πœƒπ‘– has a standardized gamma distribution with shape parameter πœƒ=π‘›βˆ’1, 𝑖=1,…,π‘˜. Then, based on statistics π‘Œ1,…,π‘Œπ‘˜ by applying the natural selection rule for each 𝑑=1,…,π‘˜βˆ’1, the associated PCSt is PCS𝑑(πœƒ)=π‘˜ξ“π‘—=π‘˜βˆ’π‘‘+1𝑃𝑑𝑗=(πœƒ)π‘˜βˆ’π‘‘ξ“π‘–=1𝑄𝑑𝑖(πœƒ),(3.1) where π‘ƒπ‘‘π‘—ξ€œ(πœƒ)=π‘˜βˆ’π‘‘ξ‘π‘–=1𝐹𝑦Δ𝑑𝑗𝑖(1)π‘—βˆ’1ξ‘π‘š=π‘˜βˆ’π‘‘+1πΉξ€·π‘¦Ξ”π‘‘π‘—π‘šξ€Έ(2)π‘˜ξ‘π‘™=𝑗+1𝐹𝑦Δ𝑑𝑗𝑙𝑄(3)𝑑𝐹(𝑦),π‘‘π‘–ξ€œ(πœƒ)=π‘–βˆ’1ξ‘π‘š=1πΉξ€·π‘§π›Ώπ‘‘π‘–π‘šξ€Έ(1)π‘˜βˆ’π‘‘ξ‘π‘™=𝑖+1𝐹𝑧𝛿𝑑𝑖𝑙(2)π‘˜ξ‘π‘—=π‘˜βˆ’π‘‘+1𝐹𝑧𝛿𝑑𝑖𝑗(3)𝑑𝐹(𝑧),(3.2) and 𝐹(β‹…) is the distribution function of the standardized gamma distribution with shape parameter πœƒ=π‘›βˆ’1.

For each π‘ƒβˆ—(0<π‘ƒβˆ—<1), let 𝑐=𝑐(π‘˜,π‘ƒβˆ—,𝑛) be the π‘ƒβˆ— quantile of the distribution of the random variable Z defined as 𝑍={max1β‰€π‘–β‰€π‘˜(π‘Œπ‘–/πœƒπ‘–)}/{min1β‰€π‘–β‰€π‘˜(π‘Œπ‘–/πœƒπ‘–)}, the extreme quotient of independent and identically distributed random variables π‘Œπ‘–.

Given π‘˜,𝑛,π‘ƒβˆ— the value of 𝑐 can be obtained from the tables of Hartley’s ratio 𝑍 with 2(π‘›βˆ’1) degrees of freedom refer to Pearson and Hartley [8].

For each 𝑑=1,…,π‘˜βˆ’1 and each 𝑗=π‘˜βˆ’π‘‘+1,…,π‘˜, let𝑃𝑑𝑗=ξ€œπ‘˜βˆ’π‘‘ξ‘π‘–=1𝐹𝑦Δ𝑑𝑗𝑖(1)π‘—βˆ’1ξ‘π‘š=π‘˜βˆ’π‘‘+1πΉξ‚€π‘¦ξΞ”π‘‘π‘—π‘šξ‚(2)π‘˜ξ‘βˆ’π‘™=𝑗+1𝐹𝑦Δ𝑑𝑗𝑙(3)𝑑𝐹(𝑦),(3.3) and for each 𝑑=1,…,π‘˜βˆ’1 and each 𝑖=1,…,π‘˜βˆ’π‘‘, let 𝑄𝑑𝑖=ξ€œπ‘–βˆ’1ξ‘π‘š=1πΉξ€·π‘§Μ‚π›Ώπ‘‘π‘–π‘šξ€Έ(1)π‘˜βˆ’π‘‘ξ‘π‘™=𝑖+1𝐹𝑧̂𝛿𝑑𝑖𝑙(2)π‘˜ξ‘π‘—=π‘˜βˆ’π‘‘+1𝐹𝑧̂𝛿𝑑𝑖𝑗(3)𝑑𝐹(𝑧),(3.4) where Δ𝑑𝑗𝑖(1), ξΞ”π‘‘π‘—π‘š(2), and Δ𝑑𝑗𝑙(3) are defined as (2.7) and Μ‚π›Ώπ‘‘π‘–π‘š(1), ̂𝛿𝑑𝑖𝑙(2), and ̂𝛿𝑑𝑖𝑗(3) are defined in (2.8) with 𝑐 chosen from Pearson and Hartley’s tables.

For each 𝑑=1,…,π‘˜βˆ’1, let 𝑃𝑑=π‘˜ξ“π‘—=π‘˜βˆ’π‘‘+1𝑃𝑑𝑗,𝑄𝑑=π‘˜βˆ’π‘‘ξ“π‘–=1𝑄𝑑𝑖.(3.5) Then, by Theorem 2.3, we can conclude the following.

Theorem 3.1. π‘ƒπœƒ{PCS𝑑𝑃(πœƒ)β‰₯max(𝑑,𝑄𝑑) for all 𝑑=1,…,π‘˜βˆ’1}β‰₯π‘ƒβˆ— for all πœƒ.

(ii) Type-II Censored Data
From each population Π𝑖, 𝑖=1,…,π‘˜, 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,…,π‘˜. Let π‘Ÿ be a fixed integer such that 1β‰€π‘Ÿβ‰€π‘›. Under Type-II censoring, the first π‘Ÿ failures from each population Π𝑖 are to be observed. The observations from population Π𝑖 cease after observing 𝑋𝑖[π‘Ÿ]. The (π‘›βˆ’π‘Ÿ) items whose failure times are not observable beyond 𝑋𝑖[π‘Ÿ] become the censored observations. Type-II censoring was investigated by Epstein and Sobel [9]. The sufficient statistic for πœƒπ‘–, when location parameters are known, is π‘ˆπ‘–=π‘Ÿξ“π‘—=1𝑋𝑖[𝑗]+(π‘›βˆ’π‘Ÿ)𝑋𝑖[π‘Ÿ],𝑖=1,…,π‘˜.(3.6)π‘ˆπ‘– is called the total time on test (TTOT) statistic. It is easy to verify that π‘ˆπ‘–/πœƒπ‘– has standardized gamma distribution with shape parameter π‘Ÿ,𝑖=1,…,π‘˜. Again, the results of complete data can be applied simply by taking πœ—=π‘Ÿ.

3.2. Normal Distribution

Let Π𝑖 denote the normal population with mean πœ‡π‘– and variance πœƒπ‘– (both unknown), 𝑖=1,…,π‘˜. The sufficient statistic for πœƒπ‘– based on a random sample 𝑋𝑖1,…,𝑋𝑖𝑛 of size 𝑛 from Π𝑖 is π‘Œπ‘–βˆ—βˆ‘=(1/(π‘›βˆ’1))𝑛𝑗=1(π‘‹π‘–π‘—βˆ’π‘‹π‘–)2, where π‘‹π‘–βˆ‘=(1/𝑛)𝑛𝑗=1𝑋𝑖𝑗, 𝑖=1,…,π‘˜. It can be verified that {(π‘›βˆ’1)π‘Œπ‘–βˆ—}/(2πœƒπ‘–) is a standardized gamma variate with shape parameter (π‘›βˆ’1)/2, 𝑖=1,…,π‘˜. Once again, the above results of exponential distribution can be used with πœ—=(π‘›βˆ’1)/2.

To illustrate the implementation of the simultaneous lower confidence bounds for the probability of correctly selecting the 𝑑 best populations (PCS𝑑), 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 πœ‹1,, πœ‹2, πœ‹3, and πœ‹4, 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 Table 1.
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 Table 1 data set: π‘Œ1=3841,π‘Œ2=383,π‘Œ3=361,π‘Œ4=1374.(4.1) Hence, according to natural selection rule, the populations πœ‹1,πœ‹2, and πœ‹4 are selected as the 𝑑 (𝑑=1,2,3) best populations, that is, for 𝑑=1, population πœ‹1 which has largest survival time is the best; for 𝑑=2, populations πœ‹1 and πœ‹4 which have the two largest survival times are the best; and for 𝑑=3, populations πœ‹1,, πœ‹2, and πœ‹4 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 𝑑 best populations (PCS𝑑) simultaneously for all 𝑑=1,2,3.
Here, π‘˜=4, 𝑛=11, and, by taking π‘ƒβˆ—=0.95, we get, from the tables of Pearson and Hartley [8], 𝑐=𝑐(π‘˜,𝑛,π‘ƒβˆ—)=3.29.
Then, 𝑃𝑑 and 𝑄𝑑 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(πœƒ)β‰₯0.551725, PCS2(πœƒ)β‰₯0.33380, and PCS3(πœƒ)β‰₯0.174162.

tab1
Table 1
tab2
Table 2

Example 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: π‘Œ1=20.82,π‘Œ2=21.17,π‘Œ3=20.67.(4.2) Hence, according to natural selection rule, the populations πœ‹1, πœ‹2 are selected as the 𝑑 (𝑑=1,2) best populations, that is, for 𝑑=1, population πœ‹1 which has largest survival time is the best; and for 𝑑=2, populations πœ‹1 and πœ‹2 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 𝑑 best populations (PCS𝑑) simultaneously for all 𝑑=1,2.
Here, π‘˜=3, 𝑛=11, and, by taking π‘ƒβˆ—=0.95, we get, from the tables of Pearson and Hartley [8], 𝑐=𝑐(π‘˜,𝑛,π‘ƒβˆ—)=2.95.
Then, 𝑃𝑑 and 𝑄𝑑 computed for the above data set using (3.5) are given in Table 3.
From Table 3, we have, with at least a 95% confidence coefficient, that simultaneously PCS1(πœƒ)β‰₯0.424471 and PCS2(πœƒ)β‰₯0.248274.

tab3
Table 3
tab4
Table 4

Example 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: π‘Œ1=1046,π‘Œ2=96,π‘Œ3=226,π‘Œ4=139.(4.3) Hence, according to natural selection rule, the populations πœ‹1,πœ‹3, and πœ‹4 are selected as the 𝑑 (𝑑=1,2,3) best populations.
Here, π‘˜=4, 𝑛=7 and, by taking π‘ƒβˆ—=0.99, we get, from the tables of Pearson and Hartley [8], 𝑐=𝑐(π‘˜,𝑛,π‘ƒβˆ—)=6.90.
Proceeding on the lines similar to Examples 4.1 and 4.2, we have, with at least a 99% confidence coefficient, that simultaneously PCS1(πœƒ)β‰₯0.360517, PCS2(πœƒ)β‰₯0.217558, and PCS3(πœƒ)β‰₯0.154598.

tab5
Table 5

Acknowledgments

The authors thank the editor, the associate editor, and an anonymous referee for their helpful comments which led to the improvement of this paper.

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