Alternative Tests for the Poisson Distribution

Transcription

1 Chiang Mai J Sci 015; 4() : Contibuted Pape Altenative Tests fo the Poisson Distibution Manad Khamkong*[a] and Pachitjianut Siipanich [b] [a] Depatment of Statistics, Chiang Mai Univesity, Chiang Mai, Thailand [b] School of Applied Statistics, National Institute of Development Administation, Bangkok, Thailand *Autho fo coespondence; manadk@cmuacth Received: Decembe 01 Accepted: 6 Febuay 014 ABSTRACT Poisson distibution is well used as a standad model fo analyzing count data Tests based on the index of dispesion (ID) concening the vaiance and the mean ae not able to disciminate between the Poisson distibution and some othes An altenative method fo the test that data comes fom Poisson distibution ae poposed based on skewness popeties, which is the same as coefficient of vaiation popeties Monte Calo studies show that the poposed tests ae poweful competito to the ID tests when altenative distibution is not Poisson but its vaiance is close to the mean Keywods: coefficient of vaiation, goodness of fit test, index of dispesion, skewness 1 INTRODUCTION In many applications, the vaiable of inteest is given in the fom of an event count o a nonnegative intege value which efes to the numbe of occuences of a paticula phenomenon ove a fixed set of time, distance, aea o space Some examples of such data ae the numbe of oad accident victims pe week, numbe of cases with a specific disease in epidemiology, etc Poisson distibution is a standad and good model fo analyzing count data and it seems to be the most common and fequently used as well Equality of mean and vaiance is an impotant chaacteistic of Poisson family of distibutions In pactice, a sample with equal dispesion is ae, ie, ovedispesion (sample vaiance is geate than sample mean), and unde-dispesion (sample vaiance is less than sample mean) usually occu even when a andom sample is dawn fom a Poisson distibution On the othe hand, equal dispesion is held by many othe distibutions such as beta-binomial distibution with paametes 5, a1 and b/ and discete unifom distibution with suppot in 0, 1,,,, eg 4 It is vey inteesting to be able to test whethe a sample is dawn fom a Poisson distibution Suppose X1, X,, X n is a andom sample of size n fom a discete distibution A poblem of testing fo Poissonity is given by H 0 : X is distibuted as Poisson ( ), (0, ), against H 1 : X is not distibuted as Poisson Seveal methods have been poposed to test fo goodness of fit of a Poisson distibution which can be divided into 4 goups The fist goup consists of tests developed fom some chaacteistics on diffeent odes of moment

2 Chiang Mai J Sci 015; 4() 775 of the Poisson family The vey fist test in this goup is based on the index of dispesion (ID, vaiance-to-mean atio) poposed by ( n 1) M Fishe et al [1] and is defined by D, X which is distibuted appoximately, unde the null hypothesis, as chi-squaed with n 1 1 n degees of feedom Note that X X n i i 1 1 n and M ( ) X X n 1 i Gat and Pettigew [] i 1 have poposed test statistics based on the conditional moment of the cumulants of Poisson distibution as follow: Z 4 Z Z nm X ( n 1), X ( nx 1) nm X ( n 1)( n ) 6 X ( nx 1) ( nx ) + ( n ) and n ( n 1m ) 4 n ( 1m ) + X ( n 1)( n )( n ) X ( nx 1) 108( nx ) 1( n + 1)( nx )( nx ) ( n ) nn ( )( n ) 1 n whee m ( X X) n i 1 i These test statistics ae asymptotically nomally distibuted as n Bohning's test [] is one of the well known tests which is impossible to ignoe The test statistic is defined on diffeence between vaiance and n 1 M mean of a sample as BN 1 X and is also asymptotically nomal distibuted as n Last but not the least, to be mentioned is the test based on equality of squaed skewness and kutosis poposed by Gupta et al [4], n m ( m m ) m G 4 Asymptotic distibution X + 6 X X of G was shown to be a standad nomal The second goup of tests fo Poissonity consists of tests based on an empiical distibution function as discussed by seveal authos and eviewed by Kalis and Xekalaki [5] The thid goup of tests is based on the pobability geneating function (PGF) such as the test of Kochelakota and Kochelakota [6] which elies on the diffeence between Poisson PGF and its empiical PGF and the value of t, and many othe developed tests as has been eviewed by Gütle and Henze [7], and the last goup consists of othe existing tests fo Poissonity such as Bown and Zhao [8] have poposed a test based on Anscombe s vaiance stabilizing tansfomation Simulation techniques have been used to compae between vaious tests fo Poissonity in the liteatues The Bohning's test seems to be pefeed [4] if a Poisson hypothesis is tested against eithe ove-dispesed o undedispesed altenative while it fails against equally dispesed altenatives In addition, Fishe s ID test obtained high powe in almost all cases except the case whee ID was closed to 1 [7] The above findings exhibit that these two well known tests, Fishe s ID and Bohning's test, pefom vey well when testing against a family of distibutions chaacteized by non equal dispesion but not vice vesa Hence, two altenative tests fo Poissonity based on the popeties of sample skewness and the coefficient of vaiation ae poposed in this pape so that Poisson distibution can be disciminated fom anothe distibution having equal o almost equal dispesion GOODNESS OF FIT TEST FOR POISSONITY 1 Useful Popeties of Poissonity Besides equal dispesion, anothe inteesting chaacteistic of the Poisson family is that the skewness and coefficient of vaiation (CV) ae equal to the ecipocal squae oot of its mean Skewness of a distibution of a andom vaiable X, in geneal, is measued by coefficient of skewness which is denoted by γ and is defined as follow 1 µ γ 1 (1) µ

3 776 Chiang Mai J Sci 015; 4() whee µ i epesents the i th cental moment of X If the coefficient of skewness is positive, the distibution is skewed to the ight that is the distibution has long ight tail If it is negative, then the distibution is skewed to the left Fo the family of Poisson distibutions with paamete > 0, the nd cental moment o the vaiance of X is µ and the coefficient of 1 / skewness is equal to > 0 Then, fom / / / equation (1), µµ 1 µ 1 and thus µ µ 0 () A CV is a measue of vaiability and is known to be independent of scale o, in othe wods, it isn t affected by the units of measuement In paticula, the CV of the Poisson family with paamete, is µ 1 / CV () µ Hee again µ fo the Poisson family equation () can be ewitten as µ µ 0, and hence, subtacting the latest equation by equation () attains µ ( µ µ ) 0 + (4) Equation () and (4) ae the fist two conclusion of popeties of Poisson family of distibutions It is possible to veify that the unbiased estimatos of µ µ and µ ( µ + µ ) ae, espectively, S M M and S M ( M X) 1 +, whee ( / ) n X 1 n i 1X, M ( 1/ ( n 1)) n ( ) i i 1X X and i M ( n/ ( n 1)( n )) n ( X X ) Consequently, i 1 i n E M, E M µ 1+ n ( n 1) 1 18n 6n E ( M µ ) + +, n n 1 ( n 1)( n ) ( )( ) ( )( ) E X µ M µ E X µ M µ n E M ( ), 1 6n µ µ + n n 1 and E ( M )( M ) Accodingly, the follows thus simply veified Theoem 1 Let X1, X,, X n be independent and identically distibuted andom vaiables fom a Poisson distibution with mean Then n (i) ES [ ] 0 and VS [ ] 4+ 1 n 1 n (ii) ES [ ] 0 and [ ] n VS 1+ n 1 n (5) (6) X ( nx 1) nx 6 (iii) T1 4+ n and X ( nx 1) nx 6 T 1+, n ae consistent estimatos, espectively of VS [ 1] and VS [ ] Poposed Tests fo Poissonity and Thei Asymptotic Distibutions The poposed tests fo Poissonity ae based on the statistics S 1 and S and thei asymptotic distibutions ae veified in Theoem Theoem Let X1, X,, X n be a andom sample of size n selected fom a Poisson distibution with mean and let SK and CSK M M X ( nx 1) nx n M ( M + X) X ( nx 1) nx n D (7) (8) Then SK N( 0, 1) and CSK N( 0, 1) as n Poof Since S 1 M M and S M ( M + X) has continuous functions in the D

4 Chiang Mai J Sci 015; 4() 777 neighbohood of ( µ, µ ) and ( µµ,, µ ), espectively By the Theoem 816 of Lehmann and Casella [9], we have whee σ D M M N0σ (, 1) and D M ( M + X) N( 0, σ ), n ( n 1) ( n ) n σ 1 + ( n 1) ( n ) and Hence, by Theoem 1 and conclusions of the theoem follow fom Slutsky s theoem fo fixed as n Accoding to Theoem, the two appopiate test statistics fo Poisson distibution testing, SK and CSK, as defined in equation (7) and (8), espectively, ae then poposed If a distibution fom which a sample is dawn is a Poisson distibution, then SK (and CSK) should take on a small value (closed to zeo), and vice vesa Theefoe, the null hypothesis of Poissnity would be ejected at level α when SK > Z α and CSK > Z / α, / whee Z α is the uppe α / quantile of the / standad nomal distibution EMPIRICAL RESULTS This section pesents some selected empiical esults fo testing Poissonity The two poposed tests, SK and CSK, espectively, will be compaed to five othe tests, ie, Fishe s test (D), Bohning's test (BN), the two tests of Gat and Pettigew (Z and Z 4 ) and Gupta s test (G) The powe of these tests ae evaluated by a Monte Calo simulation using 10,000 eplications fo a sample of modeate size (n 50 and 100) and the significant level to be consideed is α 005, povided by the statistical package R [10] Many altenative discete distibutions with ove-, unde- o equal-dispesion ae taken into account whee the involved paametes ae chosen so that the means of all distibutions above ae closest to the mean of the Poisson distibution unde the null hypothesis, (, 05] [11] Some popeties of these distibutions ae exhibited in Table 1 To investigate the sampling distibution of the two poposed tests, we used the quantile plot (Q-Q plot) in an analogous way as Baksh et al [1] Figue 1, as shown, fo the sampling distibution investigating of the two poposed tests (SK and CSK, espectively) against the BN test, seeing that the two poposed tests ae also convege to standad nomal distibution as n tends to infinity In Table, fo the null hypothesis of Poissonity is tue, the test statistics D, BN, Z, SK and CSK ae quite accuate in the sense that the empiical Type I eo ate in all cases is which coincides to the desied significant level ( α 005 ) While the empiical Type I eo ates of the tests which ely on the high cental moments (Z 4 and G) neve each the nominal level The estimated powes of all tests at the level 005 ae always high powe as the sample size inceases Unde-dispesed altenative with binomial distibution, the empiical powe of the D test is geate than those of the othe investigated tests, though it seems to be not much high when n 50 While the est of the tests (ie, Z, Z 4, G, SK and CSK), ae found to be vey poo Ove-dispesed altenatives include the negative binomial (NB) and zeo-inflated Poisson (ZIP) All two tests elated to the index of dispesion (ID) namely D and BN seem to be outstanding in tems of empiical powe fo NB and ZIP distibutions The two poposed tests, SK and CSK, attain a lowe powe than those two tests based on ID fo NB distibutions and the powe of CSK inceases as n inceases fo ZIP distibution Mixed-dispesion altenatives with discete unifom (DU), beta-binomial (BB)

5 778 Chiang Mai J Sci 015; 4() Table 1 Some chaacteistics of selected discete distibutions Type of discete Distibution Mean Vaiance ID Skewness 1 Unde-dispesed Binomial: Bin, Ove-dispesed Negative binomial: NB, ( + ) Zeo-inflated Poisson: ZIP, p 1 p p p p (1 p)(1 p+ p) (1 p) p p (1 p)(1 p+ p) Equi- dispesed Poisson: Poi( Mixed-dispesed Beta binomial: ( ) a BB a,, ( 1) ( 1) ( )( a+ ) a + a + Discete unifom: ( 1) ( a + ) a + DU( ) + ( 1) Zeo-inflated Genealized Poisson: ( 1 θ ) ZIGP, θ, p ( 1 p) ( ) 1 p+ 1 θ p 1 p+ ( 1 θ) p ( 1 p)( 1 θ ) ( 1 p)( 1 θ ) 4 (1+ θ )(1 p) + (1 θ ) (1 p)p + (1 θ ) (p 1)p ( ) (1 θ ) (1 p) 1 p + (1 θ ) p and zeo-inflated genealized Poisson (ZIGP) distibutions The CSK test appeas to pefom vey well unde all mixed altenatives when ID is close to o highe than 1 While the two tests elated to ID (ie, D and BN) have vey low powe The Z is high powe fo all cases of the BB distibutions when n inceases 4 CONCLUSION AND DISCUSSION In geneal, tests based on the index of dispesion (ID) concened the fist two moments of distibution, mean and vaiance, ae outpefomed against many altenatives when ID is fa fom 1 [4,8] Howeve, the tests based on the ID sometimes cannot be disciminate between Poisson and othe distibutions such as discete unifom and beta-binomial distibution, etc The findings in this pape agee with this conclusion In this pape, two poposed altenative tests ae poposed based on the popeties of sample skewness, which is the same as the coefficient of vaiation, ae called SK and CSK Simulation esults shown that the estimated values of size of the SK and CSK tests at 5% level ae closed to the significant level in all cases and ae simila to the two test elated to ID (ie, D and BN) The powe of all two tests elated to ID (D and BN) ae poweful unde seveal cicumstances except when ID is close to 1 Howeve, when the ID is close to 1, the SK and CSK tests ae moe poweful than

6 Chiang Mai J Sci 015; 4() 779 Table Simulated values of powe of vaious tests fo Poissonity at significance α 005 against altenatives when n 50 and 100 with 10,000 eplications n Distibutions mean Vaiance ID Test Statistic D BN Z Z 4 G SK CSK Poi(1) Poi() Poi(5) Bin(0, 015) NB(0, 087) ZIP(15, 0048) DU() DU(4) DU(5) BB(5, 11, 067) BB(5, 100, 067) BB(5, 084, 067) ZIGP(8, 085, 004) ZIGP(8, 085, 0084) ZIGP(8, 085, 01) Poi(1) Poi() Poi(5) Bin(0, 015) NB(0, 087) ZIP(15, 0048) DU() DU(4) DU(5) BB(5, 11, 067) BB(5, 100, 067) BB(5, 084, 067) ZIGP(8, 085, 004) ZIGP(8, 085, 0084) ZIGP(8, 085, 01)

7 780 Chiang Mai J Sci 015; 4() Figue 1 Compaing of the Q-Q plots of simulated values of SK, CSK and BN against standad nomal quantiles

8 Chiang Mai J Sci 015; 4() 781 Figue 1 (Continued) the two tests elated to ID In some situation the CSK test offes a moe poweful test than the SK test, and in othe cases the powe of these two tests ae not significantly diffeent Theefoe, it can be ecommended that the CSK test is moe likely to be outpefomed fo detecting a non-poisson distibution whee the vaiance of the altenative distibution is close to its mean, ie ID 1 It should be concen with the distibutions skewness fo testing the fit of sample obsevations that aise fom Poisson distibution In addition, cae must be taken when testing against beta-binomial altenatives since Poisson ( ) appoximation to the betabinomial BB ( a,, ( ) a ) is accuate when and ( 1) a+ takes on small values [1] ACKNOLEDGEMENTS The authos would like to thank edito and efeee fo thei helpful valuable comments and constuctive suggestions Also, sincee thanks ae extended to Chiang Mai Univesity and the National Institute of Development Administation fo financially suppot REFERENCES [1] Fishe RA, Thonton HG and Mackenzie WA,The accuacy of the plating method of estimating the density of bacteial populations, Ann Appl Biol, 19; 9: 5-59 DOI /j tb0596x [] Gat J and Pettigew H, On the conditional moments of the k-statistics fo the Poisson distibution, Biometika, 1970; 57: DOI 10109/ biomet/57661 [] Bohning, D, A note on a test fo Poisson ovedispesion, Biometika, 1994; 81: DOI 10109/biomet/81418 [4] Gupta AK, Mói TF and Székely GZ, Testing fo Poissonity-nomality vs othe infinite divisibility, Stat Pobab Lett, 1994; 19: DOI / (94)90111-

9 78 Chiang Mai J Sci 015; 4() [5] Kalis D and Xekalaki E, A simulation compaison of seveal pocedues fo testing the Poisson assumption,the Statistician, 000; 49: 55-8 DOI / [6] Kochelakota S and Kochelakota K, Goodness of fit fo discete distibutions, Commun Statist Theo Meth, 1986; 15: DOI / [7] Gutle,N and Henze, N, Recent and classical goodness-of-fit tests fo the Poisson distibution, J Stat Plann Infeence, 000; 90: 07-5 DOI /S (00) [8] Bown LD and Zhao LH, A Test fo the Poisson distibution, Sankhya Se A, 00; 64: [9] Lehmann EL and Casella G,Theoy of Point Estimation, nd Edn, Spinge-Velag, New Yok, 1998 [10] R Development Coe Team, R: A Language and Envionment fo Statistical Computing, R Foundation fo Statistical Computing, Vienna, Austia, 009 [11] Sasom N, Finite-dimesional simple Poisson modules, Chiang Mai J Sci, 01; 9: DOI1010/A: [1] Baksh MF, Bohning D and Ledsuwansi R, An extension of an ove-dispesion test fo count data, Comput Stat Data Anal, 011; 55: DOI /j csda [1] Teeapabolan K, Poisson apoximation to the beta binomial distibution, Int Math Foum, 009; 5: