Goodness-Of-Fit For The Generalized Exponential Distribution. Abstract
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1 Goodess-Of-Fit For The Geeralized Expoetial Distributio By Amal S. Hassa stitute of Statistical Studies & Research Cairo Uiversity Abstract Recetly a ew distributio called geeralized expoetial or expoetiated expoetial distributio was itroduced ad studied quite extesively by the authors (see Gupta ad Kudu, 999, 00a, 00b, 00, 003). A class of goodess-of-fit tests for the geeralized expoetial distributio with estimated parameter is proposed. The tests are based o the empirical distributio fuctio. These test statistics are available whe the hypothesized distributio is completely specified. Whe the parameters of the geeralized expoetial distributio are ot kow ad must be estimated from the sample data, the stadard Tables for these test statistics are ot valid. This article uses Mote Carlo ad Pearso system techiques to create Tables of critical values for such situatios. Moreover, the power of the proposed test statistics is ivestigated for a umber of alterative distributios. The results of the power studies showed that the test statistic proposed by Liao ad Shimokawa (999) is the most powerful goodess-of-fit test amog the competitors. Smirov statistic, Watso statistic, Critical values, Geeralized expoetial distributio, Power test.. troductio Recetly a ew distributio, amed geeralized expoetial distributio or expoetiated expoetial distributio was itroduced ad studied quite extesively by Gupta ad Kudu (999, 00a, 00b, 00, 003). The geeralized expoetial has the distributio fuctio with the desity fuctio Key words: Aderso-Darlig test statistic, Cramer vo Mises test statistic, Kolmogorov- F( x,, ) = (- e - λ x α α λ ) ; α, λ, x > 0 f ( x, α, λ) = α λ(- e - λ x ) α e λx. α, λ, x > 0 Where α is the shape parameter ad λ is the scale parameter. Whe the shape parameter α equals oe it reduces to a oe-parameter expoetial distributio, that is, geeralized expoetial is a geeralizatio of a oe-parameter expoetial distributio. Geeralized expoetial distributio with shape parameter α ad scale parameter λ will be deoted by (.) (.)
2 GE(α, λ ). t is observed i Gupta ad Kudu (999) that the two-parameter GE(α, λ ) ca be used quite effectively i aalyzig may lifetime skewed data, ad the properties of the two-parameter GE(α, λ ) distributio are quite close to the correspodig properties of the two-parameter gamma distributio. Gupta ad Kudu (00a) estimate the ukow parametersα ad λ usig differet methods of estimatio. They compare maximum likelihood estimators with momet estimators, least square estimators, weighted least square estimators, estimators based o percetiles, ad estimators based o the liear combiatio of order statistics i terms of their bias ad mea square error. They cocluded from their simulatio that the percetile estimators have smaller bias i almost all cases for estimatigα ad λ followed by the least square estimators ad weighted least square estimators. For the mea square error, the maximum likelihood estimators have smaller mea square error compared to other estimators. Goodess-of-fit tests are desiged to measure the compatibility of a radom sample with a theoretical probability distributio fuctio. Several goodess-of-fit tests are available i the literature such as those of Kolmogorov-Smirov (K-S) statistic, Cramer-vo-Mises (C- M) statistic, Aderso-Darlig (A-D) statistic, Watso test statistic, ad L test statistic which itroduced by Liao ad Shimokawa (999). These test statistics are geerally measure, i differet ways the distace betwee a cotiuous distributio fuctio F(x) ad the empirical distributio fuctio F (x). They are also called empirical distributio fuctio test statistics. However, these tests require cotiuous uderlyig distributios with kow parameters. Moreover, goodess-of-fit tests are ot distributio free whe the parameters must be estimated from the sample data. the last two decades, may authors (for example, Lawless, (98); Liao ad Shimokawa, (999b); Littell et al (979); Park et al (994); Stephes, (974) ) have reported that the A-D ad C-M test statistics are more powerful tha the K-S test. Liao ad Shimokawa (999) cocluded that the L test statistic is the most powerful goodess-of-fit test amog the correspodig K-S, C-M ad A-D test statistics for testig the type- extreme-value ad - parameter Weibull distributios with estimated parameters. Hassa (999) cocluded that the A-D test statistic is more powerful tha the K- S ad C-M test statistics for testig the geeralized gamma distributio. Aother class of goodess of fit tests based o the empirical Laplace trasform was discussed by may authors ( for example Barighaus ad Heze (99), Heze ad Meitais (00a), Heze ad Meitais (00b) Meitais ad liopoulosl (003) ). this article, extesive Tables of goodess-of-fit critical values for the geeralized expoetial distributio are developed through simulatio for the K-S, C-M, A-D, Watso statistic, ad L test statistic. We cocetrate o the most practical case i which the parameters are ot kow. This problem is studied through three differet cases, whe oe of the two parameters is ukow ad whe both parameters are ukow. Usig a Mathcad (00), critical values for these test statistics will be obtaied usig two differet techiques. The first method is based o the Mote Carlo simulatio, while the secod method used Pearso system to obtai the samplig distributios of the proposed test statistics, from the
3 resultig samplig distributios critical values for the test statistics are obtaied. additio, power comparisos of test statistics are ivestigated. The paper is orgaized as follows. Sectio deals with the estimatio of ukow parameters uder three cases. Sectio 3 discusses the problem of obtaiig the critical values for the test statistics by usig two differet methods. Sectio 4 gives power comparisos amog the K-S, C-M, A-D, Watso, ad L test statistics. Fially coclusios are show i Sectio.. Estimatio Of The Ukow Parameters This Sectio is cocered with the maximum likelihood estimatio of the ukow parameters α ad λ for the GE (α, λ ). This problem is studied through three cases. Case, the maximum likelihood estimators i which both parameters α ad λ are ukow. Let X, X,,X be a radom sample from a geeralized expoetial distributio with ukow parameters α ad λ. The maximum likelihood estimator of λ, say obtaied as a solutio of the equatio ˆλ ca be ˆ λ xi x ie ] ˆ ˆ ˆ -λx i -λ xi i= (- e ) i= [ λ i= l(- e + ][ ) x = 0. The exact solutio for equatio (.) requires iterative techique. Oce the maximum i (.) likelihood estimator obtaied as ˆλ is obtaied the maximum likelihood estimator ofα, say ˆα, ca be ˆ α =. (.) - ˆ λ xi l(- e ) i= Case, the maximum likelihood estimator of λ whe the shape parameter α is kow. For kow α Gupta ad Kudu (00) obtaied the maximum likelihood estimator of λ as a fixed poit solutio of equatio v ( ) v λ = λ = i=, where - (-αe -λxi (- e x ) ) i ( ). x i λ λ (.3) Case 3, the maximum likelihood estimator ofα, whe the scale parameter λ is kow. Without loss of geerality Gupta ad Kudu (00) take λ =. f λ is kow they obtaied the maximum likelihood estimator ofα, as ˆ α =. (.4) -x l(- e i ) i= 3
4 3. Critical Values Calculatios A goodess-of-fit test is used to test the ull hypothesis H 0 : the radom sample X, X,, X comes from distributio (.). this Sectio, the Kolmogorov-Smirov statistic D, Cramer-vo-Mises statisticw, Aderso-Darlig statistic A, Watso test statisticu, ad L test statistic which itroduced by Liao ad Shimokawa (999) will be described. The A-D statistic is a modificatio of C-M statistic givig more weight to observatios i the tail of the distributio, which is useful i detectig outliers (see Aderso ad Darlig (94), Stephes (977)). The Watso statistic is a modificatio of the C-M test statistic; it is also measure the discrepacy betwee the empirical distributio fuctio ad the hypothesized distributio fuctio. L test statistic measures the average of all weighted distaces over the etire rage of x, which combies the characteristic of the K-S, C-M ad A-D statistics (see, Liao ad Shimokawa (999)). The aim i this Sectio is to obtai Tables of goodess-of-fit critical values for all test statistics usig two differet methods. The first method by usig Mote Carlo simulatio. The secod method by obtaiig the samplig distributios for the proposed test statistics usig Pearso system techique. From the resultig samplig distributios the critical values for the test statistics will be obtaied. The two methods are carried out via Mathcad (00) package. 3. Method A Mote Carlo Simulatio is used to create critical values for the proposed test statistics for a geeralized expoetial distributio with ukow parameters. The followig steps are used i calculatig critical values for the proposed test statistics: Step (): A radom sample X, X,,X from geeralized expoetial was geerated. Firstly a radom sample U (), U (),, U () of order statistics from a uiform (0,) distributio was geerated, the the obtaied as follows x i-th order statistic from the GE( - λ α, λ ) with α =0. ad λ = will be α () i = ( )l[- (U () i ) ], i=,,, (3.) Step (): This radom sample was used to estimate the ukow parameters by method of maximum likelihood metioed i Sectio. Step (3): The resultig maximum likelihood estimators of the ukow parameters uder each case were the used to determie the hypothesized cumulative distributio fuctio for the geeralized expoetial distributio. Step (4): Selected sample size as = () 0 ad 00. The appropriate test statistics was calculated for the give values of, as follows 4
5 . The K-S test statistic D is D i = max{ max[ F0 (x ˆ, ˆ)],max[F ˆ, ˆ) λ i i i - () i, α λ 0(x () i, α ]} (3.) Where F0(x i, ˆ, λ ) is a cumulative distributio fuctio of GE(α, λ ) distributio, αˆ ad λˆ are α ˆ the estimated parameters usig maximum likelihood estimators of α ad λ,. The C-M statistic W is represeted by the followig formula W = + i= i - [F0 (x () i, ˆ, α ˆ) λ - ]. (3.3) A 3. The A-D statistic is = (i -)[lf (x, ˆ, α ˆ) λ + l{- F (x +, ˆ, α ˆ)}] λ A 0 () i 0 ( -i ) (3.4) i= 4. The Watso statistic U is U = W. Liao ad Shimokawa L, statistic is L F0 (x () i, ˆ, ˆ) {[ ] - }. i= α λ i max[ F (x (3.) 0 () i 0 () i { }. (3.6) = F (x, ˆ, α ˆ),F λ (x, ˆ, α ˆ)[ λ F (x () i 0 () i i, ˆ, α ˆ) λ ], ˆ, α ˆ)] i= 0 λ Step : This procedure was repeated 0000 times, thus geeratig 0000 idepedet values of the appropriate test statistics. These 0000 values were the raked, ad the values of these test statistics at seve sigificace levels, i.e., γ = 0.0, 0.0, 0.0, 0.0, 0., 0.0, ad 0. are calculated. These provided the critical values for that particular test uder each of the three cases ad sample size used. Tables -3 list the critical values for the statistics D, W, each case, ad 3, usig Mote Carlo method. A, U ad L ad for 3. Method B Pearso s system techique is used to obtai the samplig distributios of the proposed test statistics. The Pearso system of distributios was origiated by Karl Pearso (89). The criterio for fixig the distributio family is β ( β + 3) = 4(4β 3) (β 3β 6) K (3.7)
6 Where β = ad. M 3 M 3 β are the measures of skewess ad kurtosis = M 4 M respectively ad M i is the ith momet about mea. Pearso classified the differet members of system accordig to their shapes ito a umber of types. So for differet values of K, there exist differet types of distributios. The followig steps are used i calculatig critical values for the test statistics usig Pearso s techique: Step : Repeat the above steps from -3 i method A, the mea, variace, skewess, kurtosis ad Pearso coefficiet are calculated for each test statistic ad sample size uder each case. Step : The resultig values of equatio (3.7) yielded the types of distributios that appear i Tables 4-6. Step 3: For ay particular distributio, the costats ad the parameters of distributios are calculated. The method of momets are used to estimate the parameters of these differet types. These provided the critical values for the above test statistics at sigificace levels, γ = 0.0, 0.0, 0.0, 0.0, 0., 0.0 ad 0., for differet sample sizes. As a result of computer simulatio, the followig fuctios are obtaied. Each fuctio is defiig a specific type of Pearso s curves. particular, the type Pearso s curves has the desity fuctio m >-, where l, l, m ad m are the parameters of the family of distribut ios ad k is a costat. While, type V Pearso s curve has the desity x m x m f (x) = k(+ ) (- ), l < x < l (3.8) l l x d - x f (x) = k (+ ) exp[- ta ( )], < x < a a where k is a costat, d, a ad ψ are the parameters ψ (3.9) Pearso s curves that fitted to the test statistics is type V ad it has the desity fuctio where k3 is a costat, e, h ad p are the parameters of distributios. Pearso s system techique. t is clear from these Tables that: (3.0). Whe the two parameters are ukow ad oe of the two parameters is ukow,. e -h f (x) = k3(x - p) x, p x < of distributios. The last type of Tables 4-6 list the critical values for the test statistics ad the distributio type usig the samplig distributios for K-S are type for all sample size. The samplig distributios for C-M, A-D, ad Watso statistic are type V for all sample size ad uder each case. 3. Whe the two parameters are ukow the samplig distributios for L test are of type V for small (=) ad large (=00) sample sizes. While the samplig distributios for L test are of type V for all sample sizes expect = ad =00. 6
7 4. Whe oe of the two parameters is ukow, the samplig distributios of L test statistic are of types V ad V for all sample sizes. 4. Power Study The power of a goodess-of-fit test is defied as the probability that a statistic will lead to the rejectio of the ull hypothesis, H 0, whe it is false, i.e. whe a sample is ot from the hypothesised populatio but a alterative populatio (Ma et al (974)). Let the complemet of the ull hypothesis be the alterative hypothesis H a. The power of a gooessof-fit test at the sigificace level γ is deoted by committig a type error, faillig to reject a false ull hypothesis. β, where β is the probability of A power compariso was made amog K-S statistic, C-M statistic, A-D statistic, Watso statistic, ad L test statistic for the geeralized expoetial distributio with ukow shape ad scale parameters. The power was determied by geeratig 0000 radom sample of size =, ad 30 from each of seve alteratives for each test. Here =, ad = 30 represet small, moderate alterative distributios are listed below:. A stadard ormal distributio.. The Weibull distributio with desity x t- t ad fairly large sample sizes respectively. All the t exp(-x ) t- 3. The gamma distributio with desity ( (t)) x exp(-x), deoted by W (t). - Γ, deoted by Γ (t). 4. The expoetial distributio with desity t exp(-tx), deoted by exp (t).. The chi-square distributio with desity Γ( 6. The uiform distributio o the iterval [0, ]. For each test, the appropriate test ) x exp(- x ), deoted by χ statistic was calculated ad compared to its respective critical values ad couted the umber of rejectios of the ull hypothesis. The power results for the tests at the sigificace level. γ = 0.0 are preseted i Table 7.. Coclusios For differet sigificace levels ad sample sizes, the chage of critical values for all test statistics uder case are greater tha that the correspodig uder case ad case 3. As becomes larger ad γ lower, the critical values for test statistics decrease mootoically, for all test statistics i each case. Power studies usig several differet distributioal forms show that L statistic is geerally superior to other test statistic. For sample size equal 30, The A-D test statistic is more powerful tha The K-S, C-M, ad Watso test statistic. The Watso statistic is ot appearig to be powerful across this group of differet distributios. The power of the test statistic icreases as the sample size icreases. 7
8 Table Critical Poits Of Test Statistics Usig Method A C ase : Bo th λ Ad α Ukow Sample Size Test Sigificace leve l γ Statistics D W A U L D W A U L D W A U L D W A U L D W A U L D W A U L D W A U L D W A U L D W A U L D W A U L D W A U L
9 Table Critical Poits Of Test Statistics Usig Method A Case : Scale Parameter λ Ukow Sample Size Test Sigificace leve l γ Statistics D W A U L D W A U L D W A U L D W A U L D W A U L D W A U L D W A U L D W A U L D W A U L D W A U L D W A U L
10 Sample Size Table 3 Critical P oits Of Test Statistics Usig Method A Case 3 : Shape Parame ter α Ukow Test Sigificace leve l γ Statistics D W A U L D W A U L D W A U L D W A U L D W A U L D W A U L D W A U L D W A U L D W A U L D W A U L D W A U L
11 Sample Test Size St atistics Table4 Critical Po its Of Test sta tistics U sig Metho d B D W A U L D W A U L D W A U L D W A U L D W A U L D W A U L D W A U L D W A U L D W A U L D W A U L D W A U L Case : Both λ Ad α Ukow Distributio Type V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V Sigificace level γ
12 Table Critical Poi ts Of Test Stat istics U sig Meth od B Case : Scale Parameter λ Ukow Sample Size Tes t Statistics D W A U L D W A U L D W A U L D W A U L D W A U L D W A U L D W A U L D W A U L D W A U L D W A U L D W A U L Distributio Type V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V Sigificace level γ V
13 Sample Size Table6 Critical Poits Of Test statistics U sig Method B Case 3: Shape Parameter α Ukow Test Statistics D W A U L D W A U L D W A U L D W A U L D W A U L D W A U L D W A U L D W A U L D W A U L D W A U L D W A U L Distributio T ype V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V Sigificace level γ
14 Ta ble 7 Power Of Tests For Geeralized Expoetial Distributio Leve l Of Sigificace γ = 0. 0 Sample Size 30 Test Alteratives Statistics Normal E E xp() xp(3) W() Γ ( 3 ) χ Uiform D W A U L D W A U L D W A U.6 L Etries are probability of rejectig H 0 whe the radom sample is actually from the alteratives distributios. stated Refereces. Aderso, T. W. ad Darlig, D. A. (94). " A test of goodess-of-fit". J. Amer. Statist. Assoc Barighaus, L. ad Heze, N. (99). " A class of cosistet tests for expoetiality based o the empirical Laplace trasform." A. st. Statist. Math., 43, Gupta, R. D. ad Kudu, D. (999). Geeralized expoetial distributios. Australia ad New Zealad Joural of Statistics, 4() Gupta, R. D. ad Kudu, D. (00a). Expoetiated expoetial distributio: statistical ifereces, a aterative to gamma ad Weibull distributios. Biometrical Joural Gupta, R. D. ad Kudu, D. (00b). Geeralized expoetial distributios: differet methods of estimatio. Joural of Statistical Computatio ad Simulatio, 69, Gupta, R. D. ad Kudu, D. (00). Geeralized expoetial distributios: statistical ifereces. Joural of Statistical Theory ad Applicatios
15 7. Gupta, R. D. ad Kudu, D. (003). Closeess of gamma ad geeralized expoetial distributio. Commicatio i Statistics Theory ad Methods 3, Hassa, A. S. (999). " Testig ad estimatio problems cocerig the geeralized life testig model". Ph.D thesis, Cairo uiversity, Egypt. 9. Heze, N. ad Meitais, S. (00a). " Test of fit for expoetiality based o empirical Laplace trasform." Statistics, 36, Heze, N. ad Meitais, S. (00b). " Goodess-of-fit tests based o a ew characterizatio of expoetial distributio." Commicatio i Statistics Theory ad Methods 3, Lawless, J. F. (98)." Statistical models ad methods for lifetime data. Joh Wiley & Sos.. Liao, M. ad Shimokawa, T. (999)." A ew goodess-of-fit test for Type- extreme-value ad -parameter Weibull distributios with estimated parameters.joural of Statistical Computatio ad Simulatio, 64, Liao, M. ad Shimokawa, T. (999b)." Goodess-of-fit test for Type- extreme-value ad -parameter Weibull distributios..eee Trasactios o reliability, 48(), Littell, R. C., McClave, J. T. ad Offe, W. W. (979). " Goodess-of-fit tests for two parameters Weibull distributio. Commum. Statist. Simula. Computa., B8, Ma, N. R., Schafer, R. E. ad Sigpurwalla, N. D. (974) Methods for statistical aalysis of reliability ad life data. Joh Wiley & Sos. 6. Mathcad 00 professioal. ( ). Mathsoft, c. 7. Meitais, S.G. ad liopoulos, G. (003) Tests of fit for the Rayleigh distributio based o the empirical Laplace trasform." A. st. Statist. Math., Park, W. J. ad Seoh, Musup (994). " More goodess-of-fit tests for the power law process. EEE Tras. O Reliability, 43, Pearso, K. (89)." Cotributios to the mathematical theory of evolutio.. Skew variatios i homogeeous material. Philosophical Trasactios of the Royal Society of Lodo, Series A, 86, Stephes, M. A. (974)." EDF statistics for goodess-of-fit ad some comparisos". J. Amer. Statist. Assoc. 69, Stephes, M. A. (977)." Goodess-of-fit for the exterme-value distributio". Biometrika, 64,
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