DISCRETE GENERALIZED RAYLEIGH DISTRIBUTION

Transcription

1 Pak. J. Statst. 06 Vol. 3(), -0 DISCRETE GENERALIZED RAYLEIGH DISTRIBUTION M.H. Alamatsaz, S. Dey, T. Dey 3 and S. Shams Harand 4 Department of Statstcs, Unversty of Isfahan, Isfahan, Iran Naghshejahan Insttute of Hgher Educaton, Baharestan, Isfahan, Iran. Emal: alamatho@sc.u.ac.r Department of Statstcs, St. Anthony s College, Shllong, Meghalaya, Inda. Emal: sanku_deyk003@yahoo.co.n 3 Department of Quanttatve Health Scences, Cleveland Clnc, Cleveland, Oho, USA. Emal: tanujt.dey@gmal.com 4 Department of Statstcs, Qom Unversty, Qom, Iran Emal: salmeh.shams03@yahoo.com Correspondng author ABSTRACT In ths paper, a new two-parameter dscrete dstrbuton s ntroduced whch s n fact an analogue of the contnuous generalzed Raylegh dstrbuton, called dscrete generalzed Raylegh dstrbuton. The one parameter dscrete Raylegh dstrbuton s obtaned as a partcular case. Several mportant dstrbutonal and relablty propertes such as survval and hazard rate functons, moments, order statstcs, unmodalty and nfnte dvsblty of the proposed dstrbuton are examned. A related characterzaton of the dstrbuton s also provded. Parameter estmaton, usng dfferent methods, namely methods of moments, maxmum lkelhood and proporton, s dscussed. Performance of the dfferent estmaton methods are compared by means of a Monte Carlo smulaton. Fnally, two real data sets are analyzed to nvestgate the sutablty of the proposed dstrbuton n modelng count data. KEYWORDS Maxmum lkelhood estmaton, Method of moments, Method of Proporton, Observed Fsher nformaton matrx, Unmodalty, Infntely dvsblty..introduction In scentfc research, we frequently come across varables that are dscrete n nature. In lfe testng or relablty experments, t s often dffcult to quantfy the lfe length of a devce on a contnuous scale. For example, n measurng relablty of an on/off-swtchng devce, the lfetme of the swtch s a dscrete random varable. Smlarly, n survval analyss, one may be nterested n recordng the number of days that a patent has survved snce therapy, or the number of days taken from remsson to relapse. In all such cases the lfetmes are not measured on contnuous scale but are smply counted and hence are dscrete random varables. 06 Pakstan Journal of Statstcs

2 Dscrete Generalzed Raylegh Dstrbuton Not many of the known dscrete dstrbutons can provde accurate models for both tmes and counts. For example, Posson dstrbuton s used to model counts but not tmes. Bnomal and negatve bnomal dstrbutons are not consdered to be popular models for relablty, falure tmes, counts, etc. Ths s partly because they are not defned over the set of all non-negatve ntegers. Besdes, bnomal and negatve bnomal dstrbutons can be approxmated well by Posson dstrbuton under sutable condtons. Further, the applcablty of conventonal dscrete dstrbutons lke geometrc, Posson, etc. has lmted use as models for relablty, falure tmes, counts, etc. Ths has led to the development of some new dscrete dstrbutons based on popular contnuous models for relablty, falure tmes, etc. Among these new dstrbutons, the dscrete Webull dstrbuton s the most popular. Besdes Webull, another newly developed dstrbuton s the dscrete gamma dstrbuton whch has receved sgnfcant attenton n applcatons. It was frst used by Yang (994, Equatons (8)-()) n the area of molecular bology and evoluton. Snce then, many researchers n molecular bology and evoluton have used the dscrete gamma dstrbuton. More recently, Kemp (008) studed the dscrete half-normal dstrbuton. Krshna and Pundr (009) constructed dscrete analogues of the contnuous Burr and Pareto dstrbutons. Aghababae Jaz et al. (00) developed a dscrete analogue of the contnuous nverse Webull dstrbuton. Nekoukhou et al. (03) ntroduced the dscrete generalzed exponental dstrbuton. Furthermore, Gómez-Dénz and Caldern-Ojeda (0), Al-Hunt and Al-Dayan (0), Hossen and Ahmad (04), Anwar and Ahmad (04) are related references. For any contnuous dstrbuton on [0, ) wth probablty densty functon (pdf) f, one can construct a dscrete counterpart supported on the set of ntegers N0 {0,,,...}, whose probablty mass functon (pmf) s of the form p P( X x) S( x) S( x ), x 0,,,..., () x where S s the survval functon of f. The generalzed Raylegh (GR) dstrbuton s one of the well-known lfetme dstrbutons wth pdf ( x) ( x) f ( x) xe e, x 0 and survval functon ( x) S( x) e, x 0 Here α and λ are the shape and scale parameters, respectvely. The two-parameter GR dstrbuton s denoted by GR(α, λ). It s observed by Raqab and Kundu (003) that for the pdf of a GR dstrbuton s a decreasng functon and t s a rght skewed unmodal functon for. It s also observed that the hazard functon of a GR dstrbuton can be ether bathtub type or an ncreasng functon, dependng on the value of α.

3 Alamatsaz, Dey, Dey and Harand 3 In ths paper, we shall propose a dscrete verson of the generalzed Raylegh dstrbuton (DGR) usng relaton (). Note that the two-parameter generalzed Raylegh dstrbuton s a dstnct member of the class of exponentated Webull dstrbutons, orgnally proposed by Mudholkar and Srvastava (993) and Mudholkar et al. (995). The rest of the paper s organzed as follows. Secton ntroduces the DGR dstrbuton and dscusses several of ts characterstcs. We shall settle two mportant structural propertes of the dstrbuton,.e., ts unmodalty and nfnte dvsblty. Several mathematcal propertes of the dstrbuton, specfcally quantle functon, hazard rate functon, order statstcs, moments and assocaton to other dstrbutons are nvestgated. Usng the method of moments, method of maxmum lkelhood and the method of proporton, estmator of parameters of the dstrbuton are derved n Secton 3. In Secton 4, performance of the dfferent estmaton methods are compared va a smulaton study. Two real lfe data sets are also analyzed showng applcablty of the new dstrbuton n ths secton. The paper ends wth a bref concluson n Secton 5.. DISCRETE GENERALIZED RAYLEIGH DISTRIBUTION The DGR dstrbuton can be defned as a non-negatve nteger valued dstrbuton wth pmf ( x) x p p x 0,,,... px P( X x) 0 ow. e j j j x j j(x) ( ) p p x 0,,,... 0 ow. where 0 p, 0. Note that when α s an nteger number, the summaton n (4) stops at α (see, e.g., Nekoukhou et al. (03)). If α =, we have the dscrete Raylegh dstrbuton as ntroduced by Roy (004). Let [ ] be the greatest nteger value of. If s a random varable from generalzed Raylegh dstrbuton (), then [ ] s dstrbuted as (4). The cumulatve dstrbuton functon (cdf) of a random varable followng a DGR(α, p) dstrbuton s gven by ( x) F( x) p, x N. 0 Furthermore, the quantle functon of the DGR(α, p) dstrbuton, say Q(u), s obtaned from Qu ( ) p u, where 0u. By solvng the equaton, we have

4 4 Dscrete Generalzed Raylegh Dstrbuton / u / log Qu ( ). log( p) Fgure llustrates the behavor of the pmf of DGR(α, p) for several values of p and α. The unmodalty property of the DGR(α, p) dstrbuton s consstent wth that of the contnuous generalzed Raylegh dstrbuton (Raqab and Kundu (006)). Proposton. DGR(α, p) dstrbuton s unmodal for all values of α and p. In partcular, the mass probablty functon s non-ncreasng when. Another mportant structural property of a dstrbuton s ts nfnte dvsblty. We refer to the monograph of Steutel and Van Harn (004) for a good and complete ntroducton of the ssue at hand. We frst recall the followng nterestng result from the above mentoned monograph (page 56). Lemma. If p, k, s nfntely dvsble, then we have k Z pk e Proposton. A DGR(α, p) dstrbuton s not nfntely dvsble n general., for all k N. Proof: By Lemma., we show that p / e for some values of k N, α and p. For ths, k we take, p 0.8 and k. Then, we see that p As seen n Fgure, n fact DGR(α, p) dstrbuton s not nfntely dvsble for α = and all p.. Hazard Rate Functon The survval and hazard rate functons of the DGR(α, p) dstrbuton are gven by e and respectvely. [ x] S( x;, p) P( X x) p, x 0 ( x) x p p ( x) p h( x;, p), x N, 0 (6) (7)

5 Alamatsaz, Dey, Dey and Harand 5 Fgure : Illustraton of the Probablty Mass Functon of DGR(α, p) for Dfferent Values of p and α.

6 6 Dscrete Generalzed Raylegh Dstrbuton Fgure : Illustraton of the Probablty Mass Functon p of DGR(α, p) as a Functon of p and α aganst the Unform Plane e. We also observe that the reversed hazard rate functon of ths dstrbuton s a nondecreasng functon, because ( ) x p x x x f( x) p p p hx ( x), xn Fx ( ) ( ) ( x) p x p and ( x) p s a non-ncreasng functon of x, we have the requred result.. Order Statstcs of the DGR dstrbuton F x;, p be the cumulatve dstrbuton functon of the th order statstc for a Let random sample X, X,..., X n from DGR(α, p). Snce n n k nk F ( x;, p) F( x; p, ) F( x; p, ), kk usng the bnomal expanson for ( ;, ) n Fxp k, we obtan 0

7 Alamatsaz, Dey, Dey and Harand 7 n nk n n k j k j F ( x;, p) ( ) F( x; p, ), k j0 k j n nk n n k ( k j) j ([ x] ) ( ) p, k j0 k j n nk n n k j ( ) F x; p, ( k j). k j0 k j Note that F x;, p s a lnear combnaton of a fnte number of DGR ( k j), p dstrbutons. Ths helps one to obtan certan nterestng propertes of the order statstcs, such as the moment generatng functon and the moments, from those of the correspondng DGR dstrbuton. Suppose that X, X,..., Xn s a random sample from DGR(α, p) dstrbuton. Let X: n, X: n,..., Xn: n denote the correspondng order statstcs. Then, the pmf of X, s n : gven by n! F( x) n P Xn : x u ( u) du, F( x) ( )!( n )! n! n j j/( j) F( x) j ( ) u du, F( x) ( )!( n )! n j0 p p n! n j ( j) ( j) j/( j) ( x ) x ( ), ( )!( n )! j0 n n! n j j ( ) F x; p, ( j) F x ; p, ( j). ( )!( n )! j0 n Proposton.3 Let X s be ndependent random varables from DGR, p dstrbuton,,,..., n, respectvely. Then, V max X, X,..., Xn follows a dstrbuton. Proof: We smply have as requred. [ ] [ ] n v v FV ( v) P( V v) p p n n DGR, p

8 8 Dscrete Generalzed Raylegh Dstrbuton Proposton.4 Let α be an nteger value. If X s,,,...,, are ndependent and dentcal random varables from a geometrc dstrbuton wth parameter p, then max X has DGR(α, p) dstrbuton. Proof: Pmax X z Pz max X z, Pz max X z, P max X z P max X z, P X z P X z. Snce Roy (004) proved that f be a random varable from geometrc dstrbuton wth parameter p, then has dscrete Raylegh dstrbuton, ths clearly completes the proof. Proposton.5 Let be a contnuous random varable and t be a postve constant. Then, dstrbuton. Proof: Let where t X t X s DGR f, and only f, X has a generalzed Raylegh (GR) ~ GR(α, λ) and S(x) be ts survval functon. We have P X x P x X / t x S( xt) S ( x ) t. Accordngly, t S t( x) P Xt x S ( x ) t e pt ( x) t p, t 0 t( x) ( t) e. Thus, X t follows a DGR, p t dstrbuton for every t 0. Conversely, let X ~ DGR, p t ( x ) wth survval functon S ( x) P X x p, x 0,,,...; t t t t

9 Alamatsaz, Dey, Dey and Harand 9 from (8) we have where ( x ) t S ( x ) t P X ( x ) t p, x 0,,,..., t 0, pt ( t) e wth λ > 0. If true. Now, let F(t) be the dstrbuton functon of Y X and, thus, F t S t, t 0. / t 0, S x t S 0 and relaton (8) s trvally. We have ( x) F ( x ) t F t, t 0. As a result, we obtan / / ( x) F ( x ) t F t, t 0. Now, let / and G(y) F (y). Then, ( x) G ( x ) t G t, t 0. Hence, usng Result 4 of Roy (004), the dstrbuton functon G s an exponental dstrbuton functon, and consequently Y ~ F G s a generalzed exponental random varable. Therefore, we have x P( X x) P Y x P Y x e, whch s the cdf of the GR dstrbuton..3 Moments We have the frst and the second moments of the dstrbuton (3) as and E( X ) p x0 x x E X x p E( X ). x0 The above expressons are nfnte seres and cannot be wrtten n closed forms. From these equatons and a result of Jaz et al. (00), we have

10 0 Dscrete Generalzed Raylegh Dstrbuton x x p p dx p x 0. x x0 Thus, the mean of the contnuous generalzed Raylegh dstrbuton satsfes µ d µ c µ d, where µ d and µ c are the means of dscrete and contnuous Raylegh dstrbutons, re-spectvely..3. Index of Dsperson The ndex of dsperson (ID) s defned as varance dvded by the mean of a dstrbuton. If ID value s greater than one, the correspondng dstrbuton s overdspersed, and f t s less than one, the dstrbuton s underdspersed. Fgure 3 represents the ID plot for the DGR dstrbuton for dfferent values of p and α. It s observed that for the values of α greater than, the dstrbuton s always overdspersed but for the values of α less than, the dstrbuton may be over or underdspersed. The ID seems to ncrease wth the ncrease of α for fxed p. Fgure 3: Index of Dsperson Plot of DGR Dstrbuton for Dfferent Values of p and α. 3. PARAMETER ESTIMATION 3. Method of Proportons Khan et al. (989) proposed a method of proportons to estmate the parameters of the dscrete Webull dstrbuton, that Jaz et al. (00) mplemented to estmate the

11 Alamatsaz, Dey, Dey and Harand parameters of the dscrete nverse Webull dstrbuton. Here, we use the same technque to estmate the parameters of dscrete generalzed Raylegh dstrbuton. Let x, x,..., x n be n observatons from dstrbuton (4). Defne p I x x 0 0 ow. Then, Z I x 0 p n denotes the number of s n the sample. It s lkely that can be estmated wth Z / n. But, snce we have two parameters, we defne another ndcator functon as follows: x I( x ) 0 ow. Akn to the frst case, t s apparent that the estmator for p p 4 p can be obtaned by equatng t to / smultaneous equatons Thus, ( p) Z / n 4 p ( W Z) / n. n ˆ and pˆ ( Z / n) / W n I x n. Subsequently, we have W Z n p 4 ˆ log / log ˆ. Snce Z / n and W Z / n are unbased and consstent emprcal estmators of probabltes P X 0 and P X, the above estmators of the parameters are also consstent. 3. Method of Moments To apply the method of moments for estmatng the parameters p and α of DGR dstrbuton, we need to equate the populaton moments to the correspondng sample moments and subsequently solve the two equatons smultaneously. Snce the moments of the dscrete generalzed Raylegh dstrbuton cannot be obtaned n closed forms, the equatons can not be solved va ordnary technques. So, we resort to a method of pseudomoment by mnmzng S(, p) M E( X ) M E X n n wth respect to p and α, where M x and sample moments. M n x n are the frst and second

12 Dscrete Generalzed Raylegh Dstrbuton 3.3 Maxmum Lkelhood Estmaton One popular method of estmatng parameters of dstrbutons s the method of maxmum lkelhood. To apply ths method for estmatng p and α, assume that T x x, x,..., x n s a random sample of sze n from a DGR(α, p) dstrbuton. The log-lkelhood functon becomes and n log p. (0) x x p Hence, the normal equatons are p log p p log p n 0 () x x p p x x x x x x x x p p x p p x n 0. () p x x p p The solutons of lkelhood equatons () and () present the maxmum lkelhood estmators (MLEs) of α and p, whch can be obtaned va the numercal method of two dmensonal Newton-Raphson type procedure. Snce the MLE of the vector of unknown parameters α p cannot be derved n closed forms, t s therefore hard to derve the exact dstrbuton of the MLEs. Hence, we can not fnd the exact bounds for the parameters. However, usng large sample approxmaton, we see that the DGR famly satsfes the regularty condtons of the parameters n the nteror of the parameter space but not on the boundary (see, e.g., Ferguson, 996, pp. ). It s known that the asymptotc dstrbuton of the MLE ˆ s N I ˆ 0, ( ), (see Lawless (98) ), where the unknown parameters α p as follows: I ( ) s the nverse of the Fsher s nformaton matrx of I X E E p ( ). E E p p

13 Alamatsaz, Dey, Dey and Harand 3 On the other hand, the Fsher s nformaton matrx can be computed usng the approxmaton I X ˆ, pˆ ˆ, pˆ ˆ p, ˆ, pˆ ˆ, pˆ p p θ where α and p are the MLEs of α and p, respectvely. and The second partal dervatves of the log-lkelhood functon (0) are gven below: ( x ) ( x ) x x p log p p log p p p n ( x ) x ( ) ( ) x x x x p log p p log p, ( x) x p p 4 ( ) p ( p ) ( x) ( x) ( x) x p ( p ) 4 ( x) ( x) x p p ( x) x p p p x x x x n x x p p x p p ( x ) x ( x ) (( x ) ) p p ( )( ) ( ) x x ( x ) ( x ) x p p x p p ( x) x p p

14 4 n p ( x ) x Dscrete Generalzed Raylegh Dstrbuton ( x ) p p log p x x x y p p log p p p ( x ) ( x ) ( x ) ( x) x p p x x ( x ) ( x ) x p p p log p ( x ) x p p ( x ) ( x ) ( x ) x p p log p ( x) x p p ( x ) ( x ) x x x p p p log p x x p p log x x x x p p p. x x p p 4. NUMERICAL EXPERIMENTS 4. Smulaton Study We perform a smulaton study to compare dfferent methods of estmaton for the parameters of DGR dstrbuton. We have consdered dfferent sample szes; n 0, 50, 80, dfferent values of p=0.5, 0.8 and dfferent values of α=,. The performance of the estmators are compared based on the bas and the mean squared error (MSE) of the estmators usng three dfferent methods. In Table, M OM stands for the method of moments technque, M OP represents the method of proportons technque, and fnally M LE corresponds to the method of maxmum lkelhood technque. All results are based on averagng over 0,000 replcatons. From Table, we see that the performance of the MLE method s better than the other two methods. Under the MLE method, the estmator of p s slghtly negatve based and so s the case under the MOM method. The performance of the MOP method s nferor

15 Alamatsaz, Dey, Dey and Harand 5 wth respect to the other methods; one reason beng that t uses only the nformaton of s and s from the samples and dscards all other nformaton. Ths mght be the reason behnd ncreased bases and MSEs. It s also observed that gven a fxed value of α, as p ncreases, the precson of the estmates also ncreases. 4. An Example We are usng two real lfe count data sets () to llustrate several estmaton procedures proposed n ths artcle; and () to nvestgate how well the proposed DGR model works n comparson to other exstng dstrbutons. The frst data set n Table represents the number of European red mtes on apple leaves (Chakraborty (00)). Chakraborty and Chakravarty (0), deem ths an over-dspersed data set. The second data set n Table 3 represents the number of outbreaks of strkes n UK coal mnng ndustres n four successve week perods durng (Rdout and Besbeas (004)). Chakraborty and Chakravarty (0) mentoned that ths s an underdspersed data set. Our goal s to llustrate how DGR can be useful n modelng two unque and dstnct data sets. In evaluatng DGR wth other dstrbutons, we took nto consderaton Dscrete Burr (Krshna and Pundr (009)), Dscrete generalzed exponental (Nekoukhoua et al (03)) and Dscrete Webull (Nakagawa and Osak (975), Khan et al (989)) dstrbutons. The comparson has been performed by usng goodness-of-ft test along wth two nformaton crtera: Akake Informaton Crteron (AIC) and Bayesan Informaton Crteron (BIC). These crtera are well known for mplementng model selecton. In the set of competng models, a model s selected as the best model that has the smallest AIC and BIC values. For the frst data set, the MLEs of the parameters of DGR dstrbuton are gven n Table. The MOM estmates are α = and p = 0.938, whle the MOP estmates are α =.033 and p = 0.5. The varance-covarance matrx of the MLEs s gven by

16 6 Dscrete Generalzed Raylegh Dstrbuton p Table The Average Bas and the Mean Squared Error (wthn Parenthess) of the Estmates for Three Dfferent Methods 0.5, 0.8, 0.5, 0.8, Sample Sze MLE MOM MOP ˆp ˆ ˆp ˆ ˆp ˆ (0.086) (0.3879) (0.0509) (0.3956) (0.0389) (0.490) (0.0068) (0.90) (0.033) (0.3645) (0.060) (0.03) (0.0043) (0.0630) (0.0066) (0.968) (0.08) (0.954) (0.0047) (0.783) (0.0057) (0.457) (0.03) (0.895) (0.004) (0.0490) (0.007) (0.076) (0.096) (0.645) (0.0008) (0.096) (0.0004) (0.0405) (0.0) (0.64) (0.004) (0.5489) (0.055) (0.58) (0.9) (0.06) (0.0038) (0.3007) (0.0054) (0.538) (0.896) (0.0) (0.00) (0.43) (0.0035) (0.3088) (0.65) (0.0664) (0.00) (0.7455) (0.0037) (0.3679) (0.0399) (0.06) (0.0006) (0.559) (0.0009) (0.0756) (0.03) (0.08) (0.0003) (0.097) (0.0005) (0.0478) (0.056) (0.0589)

17 Alamatsaz, Dey, Dey and Harand 7 Table Dstrbuton of Number of European Red Mtes on Apple Leaves and Goodness of ft Tests In the Table, DGR s abbrevated for DGR(α, p); DB s abbrevated for Dscrete Burr(, α); DGE s abbrevated for Dscrete Generalzed Exponental Type-II(α, p); and DW s abbrevated for Dscrete Webull(β, p). Ths provdes a 95% confdence nterval for α as (0.35, ) and for p as (0.8944, ). As per the comparson wth other dstrbutons, Table reveals that DGR gves the best ft compared to others wth respect to goodness-of-ft test statstc as well as AIC and BIC. It s noteworthy mentonng that bnomal, Posson and negatve bnomal dstrbutons are often used to model count data; Chakraborty and Chakravarty (0) used dscrete gamma dstrbuton to model ths data set and compared t wth negatve bnomal and generalzed Posson dstrbutons. They showed that dscrete gamma performed better than the other two dstrbutons. Comparng the result provded n Table 6 of Chakraborty and Chakravarty (0), DGR performs even better than dscrete gamma dstrbuton.

18 8 Dscrete Generalzed Raylegh Dstrbuton Table 3 Dstrbuton of Number of Outbreak of Strkes and Goodness of ft Tests In the Table, DGR s abbrevated for DGR(α, p); DB s abbrevated for Dscrete Burr(, α); DGE s abbrevated for Dscrete Generalzed Exponental Type-II(α, p); and DW s abbrevated for Dscrete Webull(β, p). For the second data set, the MLEs are reported n Table 3. The MOM estmates are ˆ and ˆp 0.77, whereas the MOP estmates are ˆ.579 and ˆp The varance-covarance matrx of the MLEs s Ths provdes a 95% confdence nterval for α as (0.673,.096) and for p as (0.6569, ). As per the comparson wth other dstrbutons, Table 3 reveals that f we choose goodness-of-ft test as the tool for comparson, the generalzed exponental dstrbuton gves better ft compared to others; and the DGR performance s qute compettve. If we choose AIC or BIC to be the tool of comparson, then DGR gves better ft compared to the others. Chakraborty and Chakravarty (0) also used dscrete gamma dstrbuton to model ths data set and compared t wth bnomal and Posson dstrbutons. They showed that dscrete gamma performed better than the other two dstrbutons. Comparng the result provded n Table 7 of Chakraborty and Chakravarty (0), DGR performs better than dscrete gamma dstrbuton. 5. CONCLUSION In ths paper, we proposed a dscretzaton verson of the generalzed Raylegh dstrbuton and dscussed several mportant dstrbutonal and structural propertes of the newly defned dstrbuton. Three dfferent methods have been conferred to estmate the

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