DISCRIMINATING BETWEEN NORMAL AND GUMBEL DISTRIBUTIONS Authors: Abdelaziz Qaffou Departmet of Applied Mathematics, Faculty of Scieces ad Techiques, Sulta Moulay Slimae Uiversity, Bei Mellal. Morocco aziz.qaffou@gmail.com) Abdelhak Zoglat Departmet of Mathematics, Faculty of Scieces, Mohammed V Uiversity, Rabat. Morocco azoglat@gmail.com) Abstract: The ormal ad Gumbel distributios are much alike i practical egieerig, i flood frequecy ad they are similar i appearace, especially for small samples. So the aim of this paper is to discrimiate betwee these two distributios. Cosiderig the logarithm of the ratio of the maximized likelihood RML) as test statistic, its asymptotic distributio is foud uder both ormal ad Gumbel distributios, which ca be used to compute the probability of correct selectio PCS). Fially, Mote Carlo MC) simulatios are performed to examie how the asymptotic results work for fiite sample sizes. Key-Words: Normal ad Gumbel distributios; probability of correct selectio; ratio of maximized likelihood, Mote Carlo simulatio. AMS Subject Classificatio: 62H10, 62H12, 62H15, 65C05. The opiios expressed i this text are those of the authors ad do ot ecessarily reflect the views of ay orgaizatio.
2 Abdelaziz Qaffou ad Abdelhak Zoglat
Discrimiatig Betwee Normal ad Gumbel Distributios 3 1. INTRODUCTION I egieerig practice, risk criteria ad ecoomic cosideratios are importat parts of a project desig. These criteria are crucial, for example, i the desig of a urba sewer etwork, the sizig of a hydraulic structure, or the coceptio of a storage capacity system. The adequate kowledge of desig evets e.g., desig flood magitudes) is ofte helpful for the proper sizig of a project to avoid the high iitial ivestmets associated with the oversizig of the project ad the large future failure costs resultig from its udersizig. To estimate these desig evets, statistical frequecy aalysis of hydrological data is ofte used; it cosists of fittig a probability distributio to a set of recorded hydrological values e.g., aual maximum flood series) ad obtaiig estimated results cocerig the uderlyig populatio. Estimates are ofte eeded for such quatities as the magitude of a extreme evet quatile) x T, correspodig to a retur period T. Evidetly, the reliability of the estimates depeds largely o the quality of the data as well as the legth of the period of record. The aim of this paper is to discrimiate betwee ormal ad Gumbel distributios. These two distributios are widely applied i egieerig ad ofte used as a model for hydrologic data sets. Some of its recet applicatio areas iclude flood frequecy aalysis, etwork ad software reliability egieerig, uclear egieerig ad epidemic modelig. There are may practical applicatios where Gumbel ad ormal distributios are similar i appearace ad the two distributios caot be distiguished from oe aother. Normal ad Gumbel distributios belog to the locatio scale family. Discrimiatig betwee ay two geeral probability distributio fuctios from the locatio scale family was widely ivestigated i the literature. See, for istace, 1], 2], 4], 5], 6], 7], 15] ad 9] who studied the discrimiatio problem i geeral betwee the two models. Besides, 16], 19] ad 22] studied the discrimiatio problem betwee logormal ad gamma distributios. 3] ad 10] studied the discrimiatio problem betwee Weibull ad gamma distributios. Recetly, Gupta ad Kudu cosidered the discrimiatio problem betwee Weibull ad geeralized expoetial distributios, betwee gamma ad geeralized expoetial distributios ad betwee logormal ad geeralized expoetial distributios see, 12], 13], 18]). Amog the discrimiatio problems, the oe for Weibull ad logormal distributios is particularly importat ad has received much attetio; this is because the two distributios are the most popular oes for aalyzig the lifetime of electroic products. 8] adopted the ratio of maximized likelihood RML) i discrimiatig betwee the two distributios for complete data ad provided the percetile poits for some sample sizes by simulatio. Recetly, 17] cosidered the discrimiatio problem for complete data usig the RML procedure.
4 Abdelaziz Qaffou ad Abdelhak Zoglat I the preset work, to discrimiate betwee ormal ad Gumbel distributios, we cosider the ratio of maximized likelihood RML) as test statistic. Based o the result of 21], the asymptotic distributio of the logarithm of the RML is foududerboth ormal adgumbel distributios, which ca beused to compute the probability of correct selectio PCS). For small sample size, maximum likelihood estimators MLE) of Gumbel parameters are biased; heceforth, we will use a correctio for the bias itroduced by 11] ad 14]. The rest of the paper is orgaized as follows. Sectio 2 is dedicated to the mathematical otatios that we use i this paper. I Sectio 3 we describe the logarithm of RML as test statistic, their asymptotic distributios uder both ormal ad Gumbel distributios are obtaied. Mote Carlo simulatios are preseted i Sectio 4 to examie how the asymptotic results work for fiite samples. Fially, we coclude the paper i Sectio 5. 2. NOTATION To facilitate the aalysis that follows, we use the followig otatios. A ormal distributio with mea µ ad variace σ 2, deoted by Nµ,σ 2 ), has a probability desity fuctio pdf) give by f N x,µ,σ 2 ) = 1 σ 2π exp x µ)2 2σ 2, x R. The maximum likelihood estimators of µ ad σ 2 are respectively give by 2.1) ˆµ = 1 X i := X ad ˆσ 2 = 1 X i X) 2. A Gumbel distributio with locatio parameter α ad scale parameter β, deoted by Gα,β), has a pdf give by f G x,α,β) = 1 β exp x α exp x α ], x R, β β ad the maximum likelihood estimators of its parameters satisfy the followig equatios 2.2) = X X i exp X i exp X i ad ˆα = l 1 exp X i ]. These estimators, obtaied as umerical solutios to the above equatios, are kow to be biased whe the sample size is small. 11] proposed a correctio for
Discrimiatig Betwee Normal ad Gumbel Distributios 5 that bias: c = 1 0.8/ ad ˆα c = 1 c l exp X i c ] 0.7 c. Usig a rather theoretical aalysis, 14] made more accurate correctios leadig to the followig estimators: c = 1+ 0.7716 ) ] 1 ad ˆα c = c l exp X i 0.3698 c c. It is to beoted that i istaces whe a o-egative radom variable is eeded, it is the discrimiatio betwee the logormal ad the Weibull distributios that might be of iterest, buti such a case the results of the preset study remai applicable because the ormal ad the logormal also the Gumbel ad the Weibull distributios) are liked by a simple logarithmic trasformatio. The discrimiatio betwee logormal ad Weibull has bee proposed by 17]. 3. THE TEST STATISTIC AND ITS ASYMPTOTIC DISTRIBU- TION Assume that the radom sample X 1,...,X is kow to come from either a ormal distributio, X Nµ,σ 2 ), or a Gumbel distributio, X Gα,β). The log-likelihood ratio statistic, T, is defied as the logarithm of the ratio of two maximized likelihood fuctios: ) L N ˆµ,ˆσ 2 ) T = l L G ˆα, ) where L N µ,σ 2 ) ad L G α,β) the likelihood fuctios uder a ormal distributio ad a Gumbel distributio, respectively. The decisio rule for discrimiatig betwee the ormal ad the Gumbel distributios is to choose the ormal if T > 0, ad to reject the ormal i favor of the Gumbel, otherwise. Because both of these two distributios are of the locatio scale type, oe importat property of the T statistic is that it is idepedet of the parameters from both distributios see, 8]). Let us look at the expressios of T i terms of the correspodig MLEs.
6 Abdelaziz Qaffou ad Abdelhak Zoglat Note that 3.1) T = ll N ˆµ,ˆσ 2 ) ll G ˆα, ) = lˆσ l 2π 1 2ˆσ 2 l Xi ˆα = lˆσ l 2π 1 2ˆσ 2 + 1 X i ˆα + ] X i ˆµ) 2 +exp X i ˆα ] ] X i ˆµ) 2 +l exp X i exp ˆα. Usig 2.2), we get 3.2) exp ˆα = exp X i. If we replace the MLE fidig i the equatios 2.1) ad the last equatio 3.2) i 3.1), we obtai T = l ˆσ +ˆµ ˆα + 2 1 l2π). We deote T c, the ew test statistic which itroduces a correctio for bias of maximum likelihood estimators proposed by 14]. Therefore, T c ca be writte as: T c = l ˆσc +ˆµ ˆα c + c 2 1 l2π). Note that T ad T c are asymptotically equivalet, the we state the followig lemma: Lemma 3.1. distributio. The test statistics T ad T c have the same asymptotic Proof: We have T c = l ˆσc + ˆµ ˆα c + 1 1 l2π) ad c 2 3.3) l ˆσc = l ˆσ +l 1+ 0.7716 ) = l ˆσ +o1). I additio, c = +o1) ad ˆα c = ˆα+o p 1) lead to 3.4) ˆµ ˆα c c = ˆµ ˆα +o p 1).
Discrimiatig Betwee Normal ad Gumbel Distributios 7 From 3.3) ad 3.4) we obtai Tc = T + o p1). The Tc ad T have the same limit distributio, thus for ǫ > 0 ad sufficietly large, we have P Tc < t ] T P < t ] < ǫ. Immediately P T c < t] P T < t] < ǫ, for ǫ > 0 ad is sufficietly large. Fially, if lim PT < t] exists, the lim PT c < t] = lim PT < t]. + + + 3.1. Asymptotic distributio of T c uder the ormal distributio Suppose data are comig from a ormal distributio Nµ,σ 2 ). Based o 18], the followig theorem ca be stated: Theorem 3.1. AssumethatthesampleX 1,...,X follows Nµ,σ 2 ), the the test statistic T c is asymptotically ormally distributed with mea E N T) ad variace Var N T). Proof: The proof of this theorem is based o the Lemma 3.1, the followig Lemma 3.2 ad the Cetral Limit Theorem CLT). Lemma 3.2. Deote T LN µ,σ 2 ) ) = l L G α, β), where α ad β are give by the followig equatio ad may deped o µ ad σ, E N lf G X, α, β)] = max α,β E Nlf G X,α,β)], the ˆα α a.s, β a.s ad T E NT) is asymptotically equivalet to T E N T). The proof of this lemma is similar to that of Theorem 1 preseted by White i 21], the the proof of Theorem 3.1 is established by provig that T E N T) is asymptotically ormal based o the cetral limit theorem. As for the eeded quatities α ad β i Lemma 3.2, E N T) ad variace Var N T) i Theorem 3.1, they are derived by first referrig to Lemma 3.2 ad performig the followig calculatio: X α E N lf G X,α,β)] = lβ E N ) E N exp X α )) β β = lβ µ α β exp µ α ) + σ2 β 2β 2.
8 Abdelaziz Qaffou ad Abdelhak Zoglat We maximize with respect to α ad β, we get α = µ σ 2 ad β = σ. By the secod poit of Lemma 3.2, E N T) ad Var N T) are calculated. LN µ,σ E N T) E N l 2 )] ) L G α, β) for sufficietly large, we obtai = E N lf N X,µ,σ 2 ) lf G X, α, β)] = E N lf N X,µ,σ 2 )] E N lf G X, α, β)] = E N lσ l 2π 1 ) ] X µ 2 2 σ E N l β X α exp X α ] β β = lσ l 2π 12 lσ 32 ) ) = 1 l ) 2π, E N T) lim = 0.081016. + I additio, Var N lf N X,µ,σ 2 )] = Var N 1 2σ X µ) 2 )] = 1 2 2 ad takig ito accout that e 1 2 z 2 e z φz)dz = 2 ad e 1 2 ze z φz)dz = 1 where φ.) is the stadard ormal probability desity fuctio, the we have Var N lf G X, α, β) ] = Var N X α exp X α ] β β X µ = Var N )+Var N e 1 2 exp X µ )] σ σ X µ +2e 1 2 CovN ;exp X µ ] σ σ = e 2 ad Cov N lf N X,µ,σ 2 ),lf G X, α, β) ] X = 1 ) ] µ 2 2 Cov N, X µ σ σ + 1 2 e 1 2 CovN X µ σ ) 2,exp X µ σ ) ] = 1 2,
Discrimiatig Betwee Normal ad Gumbel Distributios 9 thus, Var N T) Var N lf N X,µ,σ 2 )+Var N lf G X, α, β)] 2Cov N lf N X,µ,σ 2 );lf G X, α, β)] The Fially, lim + E N T) e 5 2. Var N T) lim = 0.218282. + ad lim + Var N T) are idepedet of µ ad σ, the the asymptotic distributio of T is idepedet of µ ad σ. The from Theorem 3.1, thetest statistic T c isasymptotically ormallydistributedwithmea0.081016 ad variace 0.218282. 3.2. Asymptotic distributio of T c uder the Gumbel distributio Now we tur to the case where the sample comes from a Gumbel distributio Gα, β). As before, based o Kudu, Gupta, ad Maglick 18], the followig theorem ca be stated: Theorem 3.2. We suppose that the sample X 1,...,X follows Gα,β), the thetest statistic T c is asymptotically ormally distributedwith mea E G T) ad variace Var G T). Oce agai, the proof of this theorem is straightforward from the cetral limit theorem ad the followig lemma. Lemma 3.3. Deote T LN µ, σ 2 ) ) = l, where µ ad σ are give by L G α,β) the followig equatio ad may deped o α ad β: E N lf G X, µ, σ 2 )] = max µ,σ E Nlf G X,µ,σ 2 )] the ˆµ µ a.s, ˆσ σ a.s ad T E GT) is asymptotically equivalet to T E G T ). It is ow possible to evaluate µ ad σ by referrig to Lemma 3.3 ad performig the followig calculatio: E G lf N X,µ,σ 2 )] = E G 1 ] X µ)2 l2π lσ 2 2σ 2.
10 Abdelaziz Qaffou ad Abdelhak Zoglat Sice X follows Gα,β), it is immediate that E G X) = α+βγ ad Var G X) = π 2 6 β2, where γ 0.5772...the Euler costat). Therefore, E G lf N X,µ,σ 2 )] = 1 2 l2π lσ 1 2σ 2E GX 2 2µX +µ 2 ) = 1 l2π lσ 2 1 π 2 β 2 ] 2σ 2 +α+βγ) 2 2µα+βγ)+µ 2. 6 Maximizig with respect to µ ad σ yields µ = α + βγ ad σ = π 6 β. The quatities E G T) ad Var G T) ca be derived usig agai Lemma 3.3, E G T) E G lfn X, µ, σ 2 ) lf G X,α,β) ] E G l σ l 2π 1 ) ] X µ 2 2 σ +E G lβ + X α +exp X α ] β β E G l πβ 6 1 2 l2π 1 2 +E G lβ + X α β 32 lπ + 12 ) l3 +E G 3 X α π 2 β X α+βγ) πβ 6 ] +exp X α β ) 2 γ + X α +e β 2 α X β we put Z = X α, the Z follows G0,1) ad we obtai β E G T) 3 2 lπ + 2 l3+e G 3 ] π 2Z γ)2 +Z +exp Z 3 2 lπ + 3 l3 2 π 2E GZ γ) 2 ]+E G Z]+E G exp Z] 32 lπ + 12 l3 3π π 2 ) 2 +γ +1 6 for sufficietly large, we obtai E G T) lim = 0.090573. +
Discrimiatig Betwee Normal ad Gumbel Distributios 11 Similarly, Var G T) Var G lfn X, µ, σ 2 ) lf G X,α,β) ] Var G l σ l 2π 1 ) X µ 2 +lβ + X α 2 σ β +exp X α ] β Var G 3 ) ] X α 2 π 2 γ + X α +exp X α β β β Var G 3 ] π 2Z γ)2 +Z +exp Z, the Sice both lim + E G T) Var G T) lim = 0.283408. + ad lim + Var G T) doot depedo α ad β, the asymptotic distributio of T is idepedet of α ad β. The from Theorem 3.2, the test statistic T c is asymptotically ormally distributed with mea 0.090573 ad variace 0.283408. 4. PCS AND MC SIMULATION It is assumed that the data have bee geerated from oe of the two distributios: Nµ,σ 2 ) or Gα,β). The the discrimiatio procedure based o a radom sample X = X 1,...,X is as follows. Choose ormal distributio if T c > 0 ad Gumbel distributio if T c < 0. If the data were origially comig from Nµ,σ 2 ), the PCS N ca be writte as follows: PCS N = PT c > 0 data follow a ormal distributio). Similarly, if the data were origially comig from Gα,β), the PCS N ca be writte as follows: PCS G = PT c < 0 data follow Gumbel distributio). Sice for ormal distributio E N T) PCS N = PT c > 0] Φ VarN T) ) 0.081016 = Φ 0.218282 = Φ0.1734 ) )
12 Abdelaziz Qaffou ad Abdelhak Zoglat where Φ is the distributio fuctio of the stadard ormal distributio. I the same maer, we have for Gumbel distributio PCS G = PT c < 0] = 1 PT c > 0] ) E G T) 1 Φ VarG T) ) 0.090573 = 1 Φ 0.283408 ) 0.090573 = Φ 0.283408 = Φ0.1701 ). We use Mote-Carlo simulatios to examie how the asymptotic results work for small sizes. All computatios are performed usig the statistical freeware R 20]. We compute the PCS based o simulatios ad those based o the asymptotic ormality results. Sice the distributio of T c is idepedet of the locatio ad scale parameters, we take the locatio ad scale parameters to be zero ad oe respectively i all cases. We cosider differet sample sizes, amely = 20, 30, 40, 50, 60 ad 100. First we cosider the case whe the data comes from ormal distributio. I this case we geerate a radom sample of size from N0,1), we compute T c ad check whether T c is positive or egative. We replicate the process 10 000 times ad obtai a estimate of PCS. Similarly, we obtai the results whe the data comes from Gumbel distributio. The results are reported i Table 1. Sample size ) MC Asymptotic results 10 0.62 0.70) 0.71 0.70) 20 0.75 0.79) 0.78 0.77) 30 0.81 0.84) 0.84 0.82) 40 0.86 0.88) 0.86 0.85) 50 0.90 0.91) 0.89 0.88) 60 0.91 0.92) 0.91 0.90) 70 0.93 0.94) 0.93 0.92) 80 0.94 0.95) 0.94 0.94) 90 0.95 0.96) 0.95 0.95) 100 0.96 0.97) 0.96 0.95) Table 1: PCS s based o Mote Carlo simulatios MC) with 10 000 replicatios ad those based o the asymptotic results AR) whe the data come from the ormal Gumbel) distributio respectively. The compariso betwee the MC simulatio ad the asymptotic results shows that the asymptotic approximatio works quite well eve for small samples. Results also reveal that it is easy to discrimiate betwee ormal ad Gumbel
Discrimiatig Betwee Normal ad Gumbel Distributios 13 distributios eve for a small sample as 20. For example, the compariso of the results of Table 1 with those of Kudu ad Maglick 17] shows that the selectio betwee the ormal ad Gumbel distributios gives a asymptotic approximatio more accurate eve for a small sample size whe the data comes from Gumbel distributio. Table 1 shows that the miimum sample size eeded to choose betwee ormal ad Gumbel distributios is less tha 50; it is also clear that the power of the test varies betwee 0.62 ad 0.96 as the sample size varies betwee 10 ad 100. 5. CONCLUSION The ormal ad Gumbel distributios are ofte cosidered as competig models whe the variable of iterest takes values from to +. I this work we cosider the statistic based o the RML ad obtai asymptotic distributios of the test statistics uder ull hypothesis. Usig MC simulatios we compare the probability of correct selectio with these asymptotic result ad it is observed that eve whe the sample size is as small as 20, these asymptotic results work quite well for a wide rage of the parameter space. Therefore, these asymptotic results ca be used to estimate the PCS. Our method ca be used for discrimiatig betwee ay two members of the differet locatio ad scale families. ACKNOWLEDGMENTS The authors would like to thak the Reviewer for his valuable commets which had improved the earlier versio of the paper ad oe Associate Editor for some very costructive suggestios. REFERENCES 1] Atkiso, A. 1969). A test for discrimiatig betwee models, Biometrika, 56, 337 347. 2] Atkiso, A. 1970). A method for discrimiatig betwee models with discussios), Joural of Royal Statistical Society, Series B, 32, 323 353. 3] Bai, L.J., ad Egelhardt, M. 1980). Probability of correct selectio of Weibull versus Gamma based o likelihood ratio, Statit.-Theor.Meth, 9, 375 381. 4] Chambers, E.A., Cox, D.R. 1967). Discrimiatig betwee alterative biary respose models, Biometrika, 54, 573 578.
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