Discriminating between Generalized Exponential and Gamma Distributions

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1 Joural of Probability ad Statistical Sciece 4, 4-47, Aug 6 Discrimiatig betwee Geeralized Expoetial ad Gamma Distributios Orawa Supapueg Kamo Budsaba Adrei I Volodi Praee Nilkor Thammasat Uiversity Uiversity of Regia Silpakor Uiversity ABSTRACT Geeralized Expoetial ad Gamma distributios are the most popular i aalyzig skewed lifetime data They have may similar properties Nevertheless they have some differet properties, especially whe the lifetime data aalysis emphasizes the tail of the probabilities We ca observe that it will be more efficiet if we ca select the correct distributio for a give data Therefore i this article, we ivestigate the asymptotic method for distiguishig these two distributios It is observed that the asymptotic distributio is idepedet of a uisace parameter We perform some umerical experimets to observed that the asymptotic method works for differet sample sizes Keywords Expoetial distributio; Geeralized gamma distributio; Geeralized ivariat property; Lifetime distributio; Separate hypothesis; Statistical testig Itroductio Nowadays we ca observe a dramatic icrease of productio i differet areas of activities Regardless of what is produced, it is crucial to pay more attetio to the reliability of products Reliability of a device or a product, it is a importat idicatio of its quality The sigificat matter about reliability theory is the cocept of lifetime distributios There are may differet lifetime distributios because every product will provide differet iformatio about its lifetime so that we should be careful ad critical i selectig a lifetime distributio to describe lifetime data from a represetative sample of uits Received May 5, revised July 5, i fial form August 6 Orawa Supapueg aorstat@gmailcom ad Kamo Budsaba are affiliated to the Departmet of Mathematics ad Statistics at Thammasat Uiversity, Ragsit Ceter, Pathum Thai, Thailad Adrei Volodi is a Professor i the Departmet of Mathematical ad Statistics at Uiversity of Regia, Regia, Saskatchewa, Caada; adrei@uregiaca Praee Nilkor was affiliated to Silpakor Uiversity, Bagkok 7, Thailad The authors gratefully ackowledge the fiacial support provided by Thammasat Uiversity Research Fud uder the TU Research Scholar, Cotract No TN45/558 6 Susa Rivers Cultural Istitute, Hsichu City, Taiwa, Republic of Chia JPSS: ISSN 76-8

2 4 JPSS Vol 4 No August 6 pp 4-47 Methods This research desired to develope statistical tests for discrimiatig betwee models Gamma ad Geeralized Expoetial as the followig: Case : H : X Gamma H : X Geeralized Expoetial Case : H : X Geeralized Expoetial H : X Gamma which these distributios are the special case of the Geeralized Gamma distributio Therefore we will cosider a radom sample, X, X,, X, from Geeralized Gamma distributio ad the specified the parameters to each distributio Stacy [] proposed the Geeralized Gamma distributio ggd with probability desity x θ f θ I λ, β = + β +λ θ x λ exp x I x >, θ >, λ, β > Γ θ +λ which is a popular model of reliability theory i the study of the lifetime distributio The Geeralized Gamma distributio is very complicated fuctio It will be easier if we use ivariat property statistic uder the trasformatio x = θx ad also by the simple idea that we do ot test this parameter The ivariat tests are a fuctio of the maximal ivariat T = {Y, Y,, Y } where Y i = x i x, i =,,, [], Chapter 6 So oce agai, it is simple to take θ = ad also there are o iformatio lose Therefore we use the Geeralized Gamma distributio with θ = where the probability desity fuctio as the follow f xi λ, β = Γ + β +λ x λ exp Note that we assig β = for the Gamma distributio x I x >, λ, β > λ = for the Geeralized Expoetial distributio I [] ad [4] Volodi discussed the statistic T = c T + c T This research will focus o the statistic: T = T /T where T = l X i l X i Which is T = l, T = l i= ly i l i= Y i + X i i= Y i + i= Y ily i i= Y i+ X i l X i X i

3 Discrimiatig betwee Geeralized Expoetial ad Gamma Distributios O Supapueg et al 4 Statistics T ad T deped o three simple statistics: T = X i, T = l X i, T = X i lx i The i this otatio, statistics T ad T ca be writte as The mai test statistic takes the form T = T l T, T = l T T T T = T lt lt T T The test statistic T = gt, T, T is a fuctio of the three mai statistics The Asymptotic distributio of T is foud with the help of the Delta method, which is a procedure of the stochastic represetatio of T with the accuracy O P / We expad fuctio g i Taylor series by the powers of T i µ i, i =,,, where µ i = ET i ad the mathematical expectatio is take uder the assumptio that oe of hypotheses is true ull or alterative These two distributios are reduced to the ordial Expoetial distributio E whe parameters λ = τ = β = Because of that we ca iterpret the Expoetial distributio E as a boudary that separates ull hypotheses ad the alterative Based o this observatio we ca defie the critical costat for all tests For Case used critical costat C, for the test statistic T > C, is the α percetile of the statistic whe the sample is take from the Expoetial distributio E Ad for the case used the critical costat C, for the test statistic T < C, is the α percetile of the statistic whe the sample is take from the E Theoretical Results I order to fid the mea, variace ad covariace of the statistics T i, i =,, for the Geeralized Gamma distributio It is eccessary to calculate E[X k lx j ] for k, j =,, Therefore our task is to calculate these all momets Lemma Let radom variable X has the Geeralized Gamma distributio with θ = the Γ j λ+k+ µ k,j = E[X k lx j ] =, + β j Γ where Γ j a = dj dx j Γx x=a +λ

4 44 JPSS Vol 4 No August 6 pp 4-47 Proof We have µ k,j = Γ +λ x λ+k lx j exp{ x }dx Cosider trasform variable, let t = x The we get x = t ad dx = t β dt Hece µ k,j = Γ +λ t λ+k β l t j e t dt = + β j Γ +λ t λ+k β l t j e t dt The itegral ta l t j e t dt = dj da j ta e t dt = dj Γ a + = Γ j a + Hece da j we obtai µ k,j = Γ j [ λ+k+ j / + β Γ +λ ] For this we use the followig supportig fuctio ad formulas The derivative of the lγx is called Digamma Euler fuctio which deoted as ψx = Γ x/γx ad Trigamma Euler fuctio is i the from ψ x = Γ x/γx Γ x/γx Hece, Γ x = Γxψx ad Γ x = Γx[ψ x + ψx ] Sice gt is differetiable fuctio So we expad T = gt = T lt /lt T /T by the firstorder Taylor series is T = gt = gµ + where T = T, T, T ad µ= µ, µ, µ g i µt i µ i i= Lemma Suppose X has Geeralized Gamma Distributio The asymptotic mea ad variace of the test statistic is as followig: µ T = E[T ] = gµ = µ lµ lµ µ, µ [ ] Var[T ] = E[gT gµ ] = E g i µt i µ i i= = where g µ = µ g i µ σi + g i µg j µη ij, i= µ +lµ µ lµ µ ij µ µ, g µ = lµ µ, g µ = µ µ µ µ lµ µ lµ µ µ We derived the asymptotic probability distributio of the test statistic by cetral limit theorem Let T be a sequece of radom variables such that T µ T d N, σ T Therefore the asymptotic probability distributio of the test statistic is For X G[λ]: T µg d N, σ G

5 Discrimiatig betwee Geeralized Expoetial ad Gamma Distributios O Supapueg et al 45 which µ G = ψ + λ l+ λ l+ λ ψ + λ, σg = i=g Gi µ σgi + g Gi µg Gj µη Gij, ij where g ψ + λ ψ + λψ+ λ + ψ + λl+ λ ψ + λ G µ = + λl+ λ ψ + λ, For X GE[β]: which ψ+λ g G µ = l+ λ ψ + λ, g G µ = +λ l+λ +λ l+ λ ψ + λ µ E = where g E µ = ψ lγ lγ T µe lγ + lγ σ E = i=g Ei µ σ Ei + Γ Γ ψ g E µ = g E µ = lγ ij ψ lγ ψ Γ Γ Γ Γ d N, σ E ψ g Ei µg Ej µη Eij, + lγ, lγ lγ lγ lγ ψ, lγ ψ ψ lγ lγ ψ Γ Γ, 4 Simulatio Results The objective of this study is a developmet of statistical tests for discrimiatig betwee the popular lifetime distributio which cocludes 4 cases see method I order to obtai umerical iformatio to demostrate the effectiveess of tests, computer simulatios will be performed The measuremet capability of the tests based o both size of the test ad empirical power

6 46 JPSS Vol 4 No August 6 pp 4-47 I this work used the umber of simulatio N =, for each of the three parameters λ, τ ad β icludig form 9 to with icremet start from 9, 8, 7, 6, etc util However the result were preset i this paper oly case with = 5 as the figure Size of The Test Empirical Power parameter parameter Figure Iterpretatio from simulatio studies for Geeralized Expoetial vs Gamma test with = 5 Commet i figure : Size of the test decreases startig about assiged sigificat level ad cotiuous early Empirical power starts at for the smallest parameter ad rapidly icreases to the assiged sigificat level whe parameter icreases to After that it icreases to 8 whe parameter icreases to 5 Afterwards it slightly icreases approximate 98 for parameter icreases to 6 Subsequetly it is steady about 98 Discussio Computer simulatio was used to assess the proposed tests 4 sample size was geerated ad the statistics τ calculated for every idividual test Type I error ad power of each test calculated ad it was observed from the simulatio results that all costructed four tests are powerful ubiased test whe parameter is greater tha zero All tests are much more powerful whe parameter ad sample size are more valuable Refereces [] Lehma, E L ad Romao, J P 5 Testig Statistical Hypotheses, rd ed,

7 Discrimiatig betwee Geeralized Expoetial ad Gamma Distributios O Supapueg et al 47 Spriger, New York [] Stacy, E W 96 A geeralizatio of the gamma distributio, The Aals of Mathematical Statistics,, 87-9 [] Volodi, I N 98 O discrimiatig betwee types coected with the geeralized gamma distributio, Selected Traslatios i Mathematical Statistics ad Probability, 5, -7 [4] Volodi, I N 974 The discrimiatio of the gamma ad weibull distributios Russia, TeoriyaVeroyatostei ieeprimeeiya, 9, 98-44

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