Discrete Dynamics in Nature and Society, Article ID 70074, 5 pages http://dx.doi.org/0.55/204/70074 Research Article Multiple-Decision Procedures for Testing the Homogeneity of Mean for Exponential Distributions Han-Ching Chen, Her Pei Shan, and Nae-Sheng Wang Department of Statistics, Feng Chia University, No. 00, Wenhua Road, Xitun District, Taichung City 407, Taiwan Correspondence should be addressed to Han-Ching Chen; hang.ching@msa.hinet.net Received 25 June 204; Accepted 4 August 204; Published 9 August 204 Academic Editor: Yunqiang Yin Copyright 204 Han-Ching Chen et al. 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 wor is properly cited. 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 Panchapaesan, 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 hypothesis H 0 of homogeneity of means for exponential distributions. We discuss the decision rule R,evaluationofthe critical value C, and the infimum of P(CD R)for independent random samples from 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 [] 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 Panchapaesan [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 unnown variance are unequal. Lin and Huang [5] used a similar procedure for testing the hypothesis H 0 regarding the homogeneity of the variances. The purpose of this paper was to use the Lin and HuangmethodfortestingthehypothesisH 0 regarding the homogeneity of the means for exponential distributions. When H 0, the hypothesis, is rejected, the main objective was to obtain a nonempty subset E 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 E contains the best populations. The paper is organized as follows. In Section 2, we introduce the definitions and notations of decision rule R 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 remars are provided in Section5. 2. Related Concepts of the Decision Rule In this section, we use the Lin and Huang [5] methodto identify the decision rule R for exponential distributions.
2 Discrete Dynamics in Nature and Society Let X i,x i2,...,x ini, i =,2,...,,beindependent random samples from exponential distribution Γ(, θ i ), θ i > 0, i =,2,...,. We define the τ i = θ i /( j= θ j) / as the distance between θ i and all other θ j s. Then the MLE of τ i is θ τ i = i ( θ, j= j ) / () where θ i = n i X n ij = X i. i=,2,...,. (2) i j= For testing H 0 :θ =θ 2 =...,θ, the test statistic that arises naturally is max i τ i. We now present the steps of decision rule R for exponential distribution as follows. First, given α,where0<α<,wewanttofindac such that the condition P{max τ i C H 0 } α, (3) i where C is the critical value for the decision rule R and α is a givenprobabilityoftypeierroratlevelα. Second, given >0and P,where/<P <,we want to find a nonempty subset E={ j τ j C} of the populations that contains the best populations and it is necessary that inf θ Ω P{CD R} P,whereP(CD R) = P Ω (max i τ i C and τ () C) and Ω = {θ = (θ,...,θ ) τ [] },whereτ [] τ [2] τ [] denote the ordered τ i and τ i = θ i /( θ j= j ) / and θ () is associated with the population having the largest θ []. 3. Assessment of the Critical Value C and the Infimum of P(CD R) In this section, we want to estimate the critical value C and the infimum of P(CD R)for exponential distribution. Lemma. Let X i2,...,x ini, i =,2,...,,be independent random samples from exponential distribution Γ(, θ i ), i=,2,...,.themleofθ i is θ i = n i n i j= n i θi Γ(n i,θi), Lemma 2. The MLE of τ i is X ij = X i. i=,2,...,, θ i θ i Γ(n i, n i ). (4) where { c j = { { j=i, j =i. Thus, ln τ i ln τ i is a linear combination of independent loggamma random variables with coefficients ( (/)) for ln( θ i /θ i ) and / for ln( θ j /θ j ), j=,2,...,. Lemma 3. AccordingtotheLinandHuang[5] appendix,we can get Pr (ln τ i ln τ i <C) +exp { π (C b i )/ 3a i }, (7) where a i = ( )2 2 + n i 2, b n i = +. j 2n i 2n (8) j j= j=i Theorem 4. Under the same assumption of Lemma, for testing H 0 : θ = θ 2 =...,θ,giventhesamplessizesn,...,n and 0 < α <,thecriticalvaluec for the decision rulersatisfiesthepr {max i τ i C H 0 } α which is approximately C = exp(( 3a [] /π) ln((/α) ) + b [] ), where a [] = max i a i, b [] = max i b i and (a i,b i ) are given by (8). Further,given>0,onethenhasinf θ Ω P{CD R} /(+exp{π(ln C b [] )/ 3a [] }) = P,whereb [] = min i b i and a [] = min i a i. Proof. Under H 0,wehaveτ i =θ i /( j= θ j) / =for each i=,2,...,. Therefore, the ln τ i =0for each i=,2,...,.and Pr {max i τ i C H 0 } j= = Pr {max ln τ i ln C, for some i=,..., H 0 } i = Pr {ln τ i ln τ i ln C ln τ i, for some i, i=,2,..., H 0 } = i= i= i= Pr {ln τ i ln τ i ln C ln τ i } Pr {ln τ i ln τ i ln C} (since the ln τ i =0) ( +exp { π (ln C b i )/ 3a i } ) (by Lemma 3) (6) θ τ i = i ( θ, for θ j= j ) / i = X i. i=,2,...,, ln τ i ln τ i = j= ln θ j c j, θ j (5) i= = [ +exp {(π ln C b i )/ 3a i } ] +exp {π (ln C b [] )/ 3a [] } α (say). (9)
Discrete Dynamics in Nature and Society 3 Table : Times to breadown (in minutes) at each of the five voltage levels. Voltage level (V) n Breadown times 30 7.05, 22.66, 2.02, 75.88, 39.07, 44.2, 20.46, 43.40, 94.90, 47.30, 7.74 32 5 0.40, 82.85, 9.88, 89.29, 25.0, 2.75, 0.79, 5.93, 3.9, 0.27, 0.69, 00.58, 27.80, 3.95, 53.24 34 9 0.96, 4.5, 0.9, 0.78, 8.0, 3.75, 7.35, 6.50, 8.27, 33.9, 32.52, 3.6, 4.85, 2.78, 4.67,.3, 2.06, 36.7, 72.89 36 5.97, 0.59, 2.58,.69, 2.7, 25.50, 0.35, 0.99, 3.99, 3.67, 2.07, 0.96, 5.35, 2.90, 3.77 38 8 0.47, 0.73,.40, 0.74, 0.39,.3, 0.09, 2.38 Table 2: Computed values. 2 3 4 5 n i 5 9 5 8 a i 0.0706 0.056 0.0477 0.056 0.09 b i 0.0053 0.0069 0.039 0.0069 0.0223 θ i 75.98 4.74 4.35895 4.606 0.962 τ i 6.6858 3.623.2635 0.4053 0.0806 We have +exp { π(ln C b []) 3a [] }= α, ln C b [] = 3 a [] π Therefore, the critical value C is However C=exp ( 3a [] π ln ( α ). (0) ln ( α )+b []). () Theorem 5. Under the same assumption of Lemma and assuming n = n 2 = = n = n,givenlevelα, where 0<α<,andn,thecriticalvalueC is C=exp ( 3 π ( ) ln ( )). (3) n α Furthermore, given P,where/<P <and >0,under thedecisionrulethatr satisfies Pr{max i ln τ i ln C H 0 } αand inf θ Ω P{CD R} P,wehavethecommon sample size n as follows: n=[ [ ( ) ( ln {( α) P } ln {α ( P 2 )} ) ( 3/π) ln ] +, ] (4) where [x] denotes the lowest integer greater than or equal to x. Proof. By Theorem 4, n i =n, i=,2,...,,wehave a i = n, b i =0 for each i=,2,...,, (5) and the critical value C is inf P {CD R} = inf {Pr ( τ () C)} = inf {Pr (ln τ () ln c)} inf θ Ω d p(ln τ () ln τ [] ln C ln ) = +exp { π (ln C ln b () )/ 3a () } +exp {π (ln C ln b () )/ 3a () } (by Lemma 3) +exp {π (ln C ln b [] )/ 3a [] } P, which is the desired result. (2) C=exp ( 3a () π ln ( α )+b ()) = exp ( 3 π n ln ( α )+b ()) = exp ( 3 π n ln ( α )), (6) which is the desired result. Given P,where/ < P < and > 0,usingthe propriety of Theorem 4,wehave inf P {CD R} +exp {π (ln C ln ) / 3a} and we have π(ln C ln )/ 3a ln(/p ). Let P, (7)
4 Discrete Dynamics in Nature and Society Table 3: Infimum of P{CD R}. 2 2.2 2.4 2.6 2.8 3.0 3.2 P 0.320 0.5095 0.685 0.8062 0.8850 0.938 0.9589 Table 4: Sample size n for = 3, 4, 5, 6, α = 0.0 (0.05), and P = 0.60..5.6.7.8.9 2.0 2. 2.2 2.3 2.4 2.5 3 69 (38) 52 (28) 4 (23) 33 (9) 28 (6) 24 (4) 22 (2) 9 () 7 (0) 6 (9) 5 (9) 4 84 (48) 63 (36) 50 (28) 4 (23) 35 (20) 30 (7) 26 (5) 23 (4) 2 (2) 9 () 8 () 5 96(56) 72(42) 57(33) 47(27) 39(23) 34(20) 30(8) 27(6) 24(4) 22(3) 20(2) 6 06 (62) 79 (47) 62 (37) 5 (30) 43 (26) 37 (22) 33 (20) 29 (8) 26 (6) 24 (4) 22 (3) Table 5: Sample size n for = 3, 4, 5, 6, α = 0.0 (0.05), and P = 0.80..5.6.7.8.9 2.0 2. 2.2 2.3 2.4 2.5 3 92(55) 69(42) 54(33) 45(27) 38(23) 32(20) 29(8) 25(6) 23(4) 2(3) 9(2) 4 2 (69) 84 (5) 66 (4) 54 (33) 46 (28) 39 (24) 34 (22) 3 (9) 28 (7) 25 (6) 23 (5) 5 26 (79) 94 (59) 74 (47) 6 (38) 5 (32) 44 (28) 39 (25) 35 (22) 3 (20) 28 (8) 26 (7) 6 38(87) 03(65) 8(52) 66(42) 56(36) 48(3) 42(26) 38(24) 34(22) 3(20) 28(8) Using (5) and C=exp(( 3a/π) ( )/n ln((/α) )),wethenhavetheminimalsamplesizen as follows: n [ [ ( ) ( ln {( α) P } ln {α ( P 2 )} ) ( 3/π) ln ] +. ] (8) Remar 6. The θ i defined in this study fulfills Lawless Corollary 4... (Type II censored test property) [6]. When the observations are Type II censored data, we can tae θ i = 2T i /θ i,wheret= r i= t (i) +(n r)t (r) and t (r) are the first r ordered observation of a random sample of size n from the exponential distribution. In this case, the T= r i= w i,where w i =nt (i), i=,2...,r,remainunchanged. 4. Examples In this section, we provide two examples to explain the results of performing Theorems 4 and 5. Example. 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 broe down, was observed. Table gives results for five groups of specimens, tested at voltages ranging from 30 to 38 ilovolts (V). We use the data on times to breadown (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 α = 0.0. We obtained C = 2.85. Because τ Cand τ 2 C, using the decision rule R,werejectH 0 : τ = τ 2 = τ 3 = τ 4 = τ 5 and select the subset containing populations and 2. We identified these two populations as contributing substantially. We claim that the select subset contains the population with the largest mean. For selected values of, = 2(0.2)3.2, inf θ Ω P{CD R} = P is tabulated in Table 3. The probability of a correct decision is at least 0.320 when is 2. This probability increases to 0.9589 when is 3.2. Example 2. Based on the same assumption as Theorem 5, given the number of populations, = 3, 4, 5, and6,as well as α = 0.05 and 0.0 and =.5(0.)2.5 and P = 0.6, 0.8, 0.9, and 0.95, we can determine n by using (4), so that inf θ Ω P{CD R} P. Several selected combinations of n in each case are tabulated in Tables 4, 5, 6, and7 which show the populations that have the minimal sample size n required to satisfy the P. 5. Concluding Remars In this study, we considered the methods of the Lin and Huang theorems to propose a framewor for analyzing and synthesizing multiple-decision procedures used for testing the homogeneity of means for exponential distributions [5].
Discrete Dynamics in Nature and Society 5 Table 6: Sample size n for = 3, 4, 5, 6, α = 0.0 (0.05), and P = 0.90..5.6.7.8.9 2.0 2. 2.2 2.3 2.4 2.5 3 4 (72) 85 (54) 67 (43) 55 (35) 46 (30) 40 (26) 35 (23) 3 (20) 28 (8) 26 (7) 24 (5) 4 37 (89) 03 (67) 8 (53) 66 (43) 56 (36) 48 (3) 42 (28) 37 (25) 34 (22) 3 (20) 28 (9) 5 54 (0) 5 (76) 9 (60) 74 (49) 63 (4) 54 (36) 47 (3) 42 (28) 38 (25) 34 (23) 3 (2) 6 68 () 25 (83) 99 (66) 8 (54) 68 (45) 58 (39) 5 (34) 45 (30) 4 (27) 37 (25) 34 (23) Table 7: Sample size n for = 3, 4, 5, 6, α = 0.0 (0.05), and P = 0.95..5.6.7.8.9 2.0 2. 2.2 2.3 2.4 2.5 3 36(90) 02(68) 80(53) 66(44) 55(37) 48(32) 42(28) 37(25) 33(23) 30(2) 28(9) 4 63(0) 22(82) 96(65) 78(53) 66(45) 57(39) 50(34) 44(30) 40(27) 36(25) 33(23) 5 83(24) 36(93) 07(73) 88(60) 74(5) 64(44) 56(38) 49(34) 44(3) 40(28) 37(26) 6 98(36) 48(02) 6(80) 95(65) 80(55) 69(47) 60(42) 53(37) 48(33) 44(30) 40(28) We provided two examples and present the main results to explain Theorems 4 and 5 which can select the subset containing the population with the largest mean and effectively determine common sample size n to satisfy the requirement of P.Thispaperpresentstheuseofonetechniquetoboth select the optimal system among systems and construct an optimal rule for selecting a subset of independent random samples. We suggest employing the methods to facilitate the development of traditional statistical analyses used in the methodologies, techniques, and software applied in performing multiple-decision procedures for testing the homogeneity of means for exponential distributions problems, such as life testing and reliability engineering. [5] C. Lin and D. Huang, On some multiple decision procedures for normal variances, Communications in Statistics: Simulation and Computation,vol.36,no. 3,pp.265 275,2007. [6] J. F. Lawless, Statistical Models and Methods for Lifetime Data, New Yor, Wiley-Interscience, 2nd edition, 2003. [7] W. B. Nelson, Graphical analysis of accelerated life test data with the inverse power law model, IEEE Transactions on Reliability,vol.2,pp.2,972. Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper. References [] S. S. Gupta and D. Y. Huang, A note on optimal subset selection procedures, The Annals of Statistics,vol.8,no. 5, pp.64 67, 980. [2]X.LinandS.S.Gupta,Multiple decision theory: Raning and selection problems [Ph.D. thesis], Purdue University, 999. [3] D. Y. Huang and S. Panchapaesan, A modified subset selection formulation with special reference to one-way and two-way layout experiments, Communications in Statistics A: Theory and Methods,vol.5,no.7,pp.62 633,976. [4] D. Huang and C. Lin, Multiple decision procedures for testing homogeneity of normal means with unequal unnown variances, in Advances in Statistical Decision Theory and Applications, S. Panchapaesan and N. Balarishnan, Eds., Statistics for Industry and Technology, pp. 253 262, Birhäuser, Boston, Mass, USA, 997.
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