Discrete alpha-skew-laplace distribution

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1 Statistics & Oerations Research Transactions SORT 39 () January-June 25, 7-84 ISSN: eissn: Statistics & Oerations Research c Institut d Estadstica de Catalunya Transactions sort@idescat.cat Discrete alha-skew-lalace distribution S. Shams Harandi and M. H. Alamatsaz Abstract Classical discrete distributions rarely suort modelling data on the set of whole integers. In this aer, we shall introduce a fleible discrete distribution on this set, which can, in addition, cover bimodal as well as unimodal data sets. The roosed distribution can also be fitted to ositive and negative skewed data. The distribution is indeed a discrete counterart of the continuous alha-skew-lalace distribution recently introduced in the literature. The roosed distribution can also be viewed as a weighted version of the discrete Lalace distribution. Several distributional roerties of this class such as cumulative distribution function, moment generating function, moments, modality, infinite divisibility and its truncation are studied. A simulation study is also erformed. Finally, a real data set is used to show alicability of the new model comaring to several rival models, such as the discrete normal and Skellam distributions. MSC: 6E, 62E Keywords: Discrete Lalace distribution, discretization, maimum likelihood estimation, uni-bimodality, weighted distribution.. Introduction The traditional discrete distributions (geometric, Poisson, etc.) have limited alicability in modelling certain real situations such as data on the set of integersz={,, 2,...} or bimodal data sets. Thus, several researchers have attemted to develo new classes of discrete distributions to cover such situations. Recall that any continuous distribution on R with robability density function (df) f admits a discrete counterart suorted on the set of integersz={,, 2,...} whose robability mass function (mf) is defined as P(X = )= f(), Z. () y= f(y) Corresonding author: alamatho@sci.ui.ac.ir (and mh -alamatsaz@yahoo.com) Deartment of Statistics, University of Isfahan, Isfahan, Iran. salimeh.shams23@yahoo.com Received: December 23 Acceted: November 24

2 72 Discrete alha-skew-lalace distribution For instance, Roy (23) introduced a discrete version of normal distribution to cover discrete data on the whole set of integersz = {,, 2,...} and, similarly, Inusah and Kozubowski (26) considered a discrete analogue of Lalace (DL) distribution. Kozubowski and Inusah (26) roosed a discrete version of the skew Lalace (skewdl) distribution as a generalization of discrete Lalace distribution which is useful for unimodal data sets. Also, Barbiero (24) and Jayakumar and Jacob (22) introduced other discrete distributions based on skew Lalace and wraed skew Lalace distributions on the integers, resectively. The aim of this aer is to roose a more fleible distribution onzwhich can cover unimodal as well as bimodal data. The new discrete distribution can also fit both ositively and negatively skewed data. In fact, using (), we rovide a discrete version of the alha-skew-lalace distribution which was recently introduced by Shams Harandi and Alamatsaz (23). The robability density function of the alha-skew-lalace distribution is f(;α,µ,σ)= 4σ(+α 2 ) [+( α σ ( µ))2 ]e µ σ, R, (2) whereα R is the skewness arameter andµ R andσ > are its location and scale arameters, resectively. The discrete version of (2) which is considered here can be fitted to unimodal as well as bimodal data sets having ositive as well as negative skewness. The rest of the article is organized as follows. Section 2 introduces the discrete alhaskew-lalace (DASL(, γ)) distribution and discusses some of its imortant features and roerties. In Section 3, we shall rovide some distributional roerties such as moment generating function and moments. Maimum likelihood estimations of arameters involved will be discussed in section 4. Section 5 describes a simulation study. In Section 6, we shall consider some interesting modification of DASL distribution. In Section 7, we attemt to fit the roosed model and its secial cases to a real data set and comare it with several rival models such as the discrete normal, DL, skewdl and Skellam distributions. 2. The family of discrete alha-skew-lalace distributions In this section, we resent the mf of our new class of discrete distributions onzby discretizing alha-skew-lalace distribution (2). We let µ = and use relation () to obtain (;,α)= C(,α)[+(+α log ) 2 ], Z, (3) where C(,α)= 2 + [+α2 (log ) 2 ], < =e ( ) 2 σ < andα R.

3 S. Shams Harandi and M. H. Alamatsaz 73 Sinceσ=( log ), to simlify, we letγ= α. Then, we have σ (;,γ)= C(,γ)[+( γ) 2 ], Z, (,),γ R, (4) where C(,γ)= 2 + [+γ2 ]. We denote this distribution by X DASL(,γ). ( ) 2 Remark Recall that for a distribution with df (mf) f, we can construct a new distribution with df (mf) g(;θ,θ 2 )= w(;θ,θ 2 ) E Θ [w(;θ,θ 2 )] f(;θ ), where Θ and Θ 2 can be two vectors of arameters and w is called a weighted distribution of f. It is worth noting that mf (4) can also be viewed as the weighted version of the discrete Lalace distribution of Inusah and Kozubowski (26). To see this, it is sufficient to consider the weight function w(;θ,θ 2 )=(+( γ) 2 ) with Θ =, Θ 2 =γ and f(; )= +. Some secial cases of this new class of discrete distributions are revealed below:. Ifα= in (3), or equivalentlyγ= in (4), we obtain the discrete Lalace (DL) distribution. 2. Ifα in (3), or equivalentlyγ in (4), we have (;,γ) ( )3 2(+ ) 2, Z which is a symmetric and bimodal discrete distribution. 3. If X DASL(,γ), then X DASL(, γ). 4. By considering the continuous version of the alha-skew-lalace distribution of Shams and Alamatsaz (23), we can conclude that DASL(, γ) is unimodal for log <γ< log and bimodal forγ log orγ log, resectively. Equivalently, if we consider the mf in (3), then the distribution is unimodal for <α< and bimodal forα orα, resectively. Figure below illustrates several lots of DASL(, γ) distribution for selected values of the arameters andγwhich confirms our result on modality of the distribution. We note that forγ<, all lots are symmetric.

4 74 Discrete alha-skew-lalace distribution () () () (,γ)=(.,5) 5 5 (,γ)=(.5,5) (,γ)=(.9,5) () 5 5 () () (,γ)=(.,) (,γ)=(.5,) (,γ)=(.9,) () 5 5 () () (,γ)=(.,.5) 5 5 (,γ)=(.5,.5) (,γ)=(.9,.5) 5 5 Figure : Illustrations of mf of DASL(,γ) for different values of andγ. The cumulative distribution function (cdf) of the random variable X DASL(, γ) is given by F(;,γ)=P(X ) = C(,γ) [] [ 2 +γ2 2 ([]+) 2 (2[] 2 +2[] )+[] 2 ( ) 3 2γ ( )[] ( ) 2 ], < 2(+) + 2γ2 (+) []+ ( ) 3 [γ 2 ( ) 2 ([]+) 2 +2( )([]+)+(+) ( ) 3 2γ ( )([]+)+ ( ) ],. 3. Moments The moment generating function of a random variable X DASL(,γ) is given by [ 2 M X (t)=e(e tx )= C(,γ) e t + 2 e t + 2γ e t (e t ) 2 e t 2γ (e t ) 2 +γ2 et (+e t ) (e t ) 3 +γ 2 et (e t ] + ) ( e t ) 3, t >log.

5 S. Shams Harandi and M. H. Alamatsaz 75 Relacing t in M X (t) by it, i=, we can easily obtain the characteristic function of DASL(,γ). To find the moments, using the combinatorial identity = n = n =! S(n, ) ( ) +, (see, e.g., formula (7.46),. 337, of Graham et al., 989), where S(n,)=! k=( ) k( ) ( k) n k is the Stirling number of the second kind, we obtain the n-th moment of X DASL(,γ) for n as µ n = E(X n )= C(,γ) n =! [ ( ( ) + 2S(n,)+γ S(n+2,)) 2 ( +( ) n) 2γS(n+,) (+( ) n+)] (5) { +C(,γ) n+ (n+)! ( ) n+2 [γ 2 (+( ) n )(S(n+2,n+) + n+2 } S(n+2,n+2)) 2γ(+( )n+ )S(n+,n+)]. We can easily observe that for even n, and for odd n, { n µ n = 2C(,γ) =! [ ( ) + 2S(n,)+γ 2 S(n+2,) [ + n+ (n+)! ( ) n+2 γ 2 (S(n+2,n+)+ n+2 ]} S(n+2,n+2)). { n µ n = 4γC(,γ) =! S(n+,) ( ) + } + n+ (n+)! S(n+,n+) ( ) n+2 ]

6 76 Discrete alha-skew-lalace distribution In articular, we have and thus E(X)= 2γ ( ) 2[+γ2 ( ) 2], [ E(X ) ] )= ( ) 2 +γ2(2 ( ) 4 [+γ 2 ( ) 2], E(X 3 )= 2γ ( ) 4 [+γ 2 ( ) 2], E(X 4 )= [ ( ) 4 2( 2 + +)+ γ2 ( ) ] ( ) 2 [+γ 2 ( ) 2] [ Var(X) = ( ) 2 2+γ ( ) 2 4γ 2 ] ( ) 2 +γ 2 [+γ 2 ( ) 2]. Skewness and kurtosis of our distribution can be evaluated easily. But since their formulas are too long, they are omitted and we only show their behaviour by their grahs in Figure 2. skew 5 γ= 2 γ=2 γ=.5 kur γ= 2 γ=.5 γ= skew =. =.5 = γ kur =. =.5 = γ Figure 2: Illustrations of the skewness and kurtosis as functions of andγ.

7 4. Maimum likelihood estimation S. Shams Harandi and M. H. Alamatsaz 77 To aly maimum likelihood for estimating andγ, assume that, 2,..., n are the observed values of a random samle of size n from a DASL(,γ) distribution. The loglikelihood function becomes l(,γ)= nlog2+nlog( ) nlog(+ ) nlog(+γ 2 ( ) 2) + n i= log(+( γ i ) 2 )+ n i= i log. Then, the likelihood equations for andγare given by and l(,γ) l(,γ) γ = n i= i 2n 2 + nγ2 ( ) 3 + γ 2 ( ) = (6) n 2nγ = ( ) 2 + γ 2 2 i= ( γ i ) i =. (7) +( γ i ) 2 The solutions of likelihood equations (7) and (8) rovide the maimum likelihood estimators (MLEs) of andγ, which can be obtained by a numerical method such as the Newton-Rahson tye rocedure. Since the MLEs of the unknown arameters (,γ) can not be obtained in closed forms, it is not easy to derive the eact distributions of MLEs. One can show that the DASL family satisfies the regularity conditions which are fulfilled for arameters in the interior of the arameter sace but not on the boundary (see, e.g., Ferguson, 996,. 2). Hence, by using the simlest large samle aroach, the MLE vector ( ˆ, ˆγ) is consistent and asymtotically normal, i.e., (, γ γ) N 2 (,I (, γ)), where I is the variance covariance matri of the unknown arameters (,γ) and the covariance matri I, as the Fisher information matri, can be obtained by I (,γ)= ( 2 ) l E 2 ( 2 ) l E γ ( 2 ) l E γ ( 2 ) l E γ 2 = [ Var( ) Cov(, γ) Cov(, γ) Var( γ) ],

8 78 Discrete alha-skew-lalace distribution whose elements are evaluated by using the following eressions: and 2 l i= i 2 = n 2 2 l γ = 2nγ 2 [( ) 2 +γ 2 ] 2 4n ( 2 ) 2 2( nγ2 )2 (2 )+γ 2 ( ) [( ) 3 + γ 2 ( )] 2, 2 l γ 2 = 2n ( ) 2 +γ 2 [( )2 γ 2 n ( ) 2 +γ 2 ] 2 i= 2 i ( γ i ) 2 [+( γ i ) 2 ] 2. To find eectations of the above eressions, we need to comute E X and E{X 2 ( γx) 2 }. The Fisher s information matri can be comuted using the aroimation [+( γx) 2 ] 2 I( ˆ, ˆγ)= 2 l 2 ˆ, ˆγ 2 l γ ˆ, ˆγ 2 l γ ˆ, ˆγ 2 l γ 2 ˆ, ˆγ as the observed Fisher s information matri. The normal aroimation can then be used to construct confidence intervals for and q to test hyothesis of the kind H : = and H :γ=γ, resectively, as and ( ˆ z α/2 I( ˆ), ˆ+z α/2 I( ˆ)) ( ˆγ z α/2 I( ˆγ), ˆγ+z α/2 I( ˆγ)). where I( ˆ) and I( ˆγ) refer to the roots of diagonal elements of the inverse Fisher s information matri., 5. Simulation Here, we assess the erformance of the maimum-likelihood estimate given by Equations (7) and (8) with resect to the samle size n. The simulation study assessment is based on the inversion method with iterations.

9 S. Shams Harandi and M. H. Alamatsaz 79 γ Table : MLEs of andγin DASL(,γ) for different values of n n ˆ ˆγ Var( ˆ ˆ) Var( ˆ ˆγ) ˆ ˆγ Var( ˆ ˆ) 2 ˆ Var( ˆγ) γ.5 n ˆ ˆγ Var( ˆ ˆ) Var( ˆ ˆγ) ˆ ˆγ Var( ˆ ˆ) ˆ Var( ˆγ) γ.5 n ˆ ˆγ Var( ˆ ˆ) Var( ˆ ˆγ) ˆ ˆγ Var( ˆ ˆ) ˆ Var( ˆγ) γ.5 n ˆ ˆγ Var( ˆ ˆ) Var( ˆ ˆγ) ˆ ˆγ Var( ˆ ˆ) ˆ Var( ˆγ) These results are resented in Table accomanied by their estimated variances ( Var), ˆ for different values of n. Table shows how the MLEs and estimated variances of arameters vary with resect to n. The difference between real and estimated values of the arameters are not too large and, thus, the method works well. 6. Some secial cases In this section, we consider the distribution of the random variable X DASL(,γ) truncated at zero. This distribution is an imortant case, because it is a weighted version of geometric distribution and may be useful in fitting count or time data sets. Let Y = X X, then the mf of Y is given by Y (y;,γ)= C (,γ)(+( γy) 2 ) y, y=,,2,..., where C (,γ)= 2 2γ ( ) 2 +γ 2 (+) ( ) 3, < < andγ R. This distribution is called weighted geometric distribution and is denoted by Y WGD(,γ).

10 8 Discrete alha-skew-lalace distribution This can be used as a discrete lifetime distribution which contains geometric distribution by settingγ=. The survival and failure rate functions of this random variable are given by and R Y (y;,γ)=p(y > y)= C (,γ) y+ [2+2γy y H Y (y;,γ)=( ) +γ 2 y2 2 +( 2y 2 2y+)+(y+) 2 ( ) 2 ], y=,,... +( γy) 2 2+2γ y y +γ 2 y2 2 +( 2y 2 2y+)+(y+) 2 ( ) 2, y=,,..., resectively. The behaviour of the failure rate function of X WGD(,γ) is described in Figure 3. As we can see the failure rate function of WGD distribution can be increasing or U-shaed. Further, we note that ifγ =, WGD distribution will reduce to the geometric distribution with constant failure rate function which deends only on. Another imortant structural roerty of a distribution, both in theory and alication, is its infinite divisibility. We refer, for eamle, to the monograh of Steutel and Van Harn (24) for a good and comlete introduction of the subject. Since most of the well-known distributions ossess this roerty, one has to be concerned with the infinite divisibility or non-infinite divisibility roerty of any distribution newly introduced. Here, we note that WGD(,γ) distribution is not infinitely divisible. To see this, we first recall the following interesting result from the above-mentioned monograh (age 56). Lemma If k, k Z + is infinitely divisible, then we have k /e, for all k N. Now, we can show that Y (y;,γ) > /e for some values of y N, andγ. For instance, take y=, =. andγ=. Then, we see that Y (;.,).5525> /e Thus, a W GD(, γ) distribution is not infinitely divisible in general. In the case γ =, however, we have the geometric distribution, with robability of success, which is obviously infinitely divisible. It is also worth noting that, we can describe the distribution of the random variable Z = X as a new distribution on the set of non-negative integers as follows: {, z= (z;,γ)=p( X =z)=2c(,γ) {2+γ 2 z 2 } z, z=,2,... where and γ are given as before. This distribution is called a generalized geometric distribution and denoted by Z GGD(,γ). It is worth mentioning that if Z GGD(,γ),

11 S. Shams Harandi and M. H. Alamatsaz 8.4 (,γ)=(.5,.5) (,γ)=(.5,5) 5 (,γ)=(.,.5) H(y) H(y) H(y) y y y (,γ)=(.,5).5 (,γ)=(.9,.5).5 (,γ)=(.9,5) H(y) 6 4 H(y).3.2 H(y) y y y Figure 3: Illustrations of the failure rate function of X WGD(,γ) for some selected values of andγ. then we have {, z=. Ifγ =, Z +, whose truncation at zero is a geometric 2 z, z=,2,... distribution with arameter and suort on{,2,...}. 2. GGD(,γ) = GGD(, γ). 3. Ifγ, then (z;,γ) ( )3 (+) z2 z, z=,2, Alication and comarison In this section, we attemt to eamine alication and advantage of DASL(, γ) and WGD(,γ) distributions comaring to several rival models using some real data sets. Eamle. The following data set is obtained based on a recent local research carried out on the etent of success of Iranian universities in transferring technology to industry and their effective factors. Out of 5 questionaries distributed, were returned. The data below show the difference between the desired and the eisting state values on each samle; a ositive number shows the etent of ositive imrovement and a negative number shows the etent of negative imrovement.. The data is art of an unublished research by A. Rafiei, an MSc student.

12 82 Discrete alha-skew-lalace distribution,, 3, 3,,, 2, 4, 5, 4, 5, 6,, 7, 4,, 9, 3, 5, 4,,,, 4, 5, 2, 4, 5, 3, 3, 2, 3, 4, 4, 5,, 5, 3, 4, 3, 4, 6, 6, 7,,,, 3,,, 5, 2, 2, 5,, 2, 2, 6, 2, 8, 7, 7, 2, 2, 9, 6,, 5, 5, 3, 3,, 4,, 8, 5, 2,, 2,, 8, 3,, 2, 4, 2,,, 8,, 2, 4, 6,,, 7, 8,, 9,, 9, 6,,, 7, 7,, 4, 7, 9, 28. Thus, it is logical to comare our distribution with some similar distributions such as SkewDL and ADSLalace distributions. SkewDL distribution was introduced by Kozubowski and Inusah (26) and has the following mf (;,q)= { ( )( q) q, =..., 2, q, =,,... ADSLalace distribution of Barbiero (24) has mf (;,q)= { log()[q (+) ( q)], =..., 2, log(q) log(q)[ ( )], =,,... Furthermore, we shall also consider the Skellam (Skellam, 946) with mf (;µ,µ 2 )=e µ µ 2 (µ /µ 2 )I (2 µ µ 2 ), Z. where I (2 µ µ 2 ) is modified Bessel function of the first kind, and discrete normal (Roy, 23) distribution with mf: (;µ,σ)=φ(+,µ,σ) Φ(,µ,σ), Z. where Φ(.,µ,σ) is the cdf of normal distribution with mean µ and variance σ 2, resectively. The results of comarison are illustrated in Table 2. We have also obtained maimum likelihood estimates and their estimation of standard errors for the arameters involved. We note that under regularity conditions, the standard error of the arameter estimators can be asymtotically comuted by root square of the diagonal elements of the inverted Fisher s matri. The Kolomogrov-Smirnov (K-S) statistic and Akaike information criterion as AIC= 2logL+2k, where k, the number of arameters in the model, n, the samle size, and L, the maimized value of the likelihood function for the estimated model, are used to comare the estimated models. Since DASL(, γ) distribution is an etension of DL distribution, in our iterative algorithm of Newton-Rahson, we have usedγ = and the MLE of arameters of DL distribution as initial values to find the MLEs of the arameters. As one can see from Table 2, our model is referable comaring to other models. Also, Figure 4 shows distribution lots of the data and the models in question.

13 S. Shams Harandi and M. H. Alamatsaz 83 Table 2: Comaring criterions for the rival distributions. Model Parameter estimaties k K S log L AIC DL =.8496, S.E( ) = ADSLalace =.8787, S.E( ) = q =.753, S.E( q) =.394 SkewDL =.8732, S.E( ) = q =.765, S.E( q) =.368 dnormal µ = 4.248, S.E( µ) = σ=6.9769, S.E( σ)=.67 DASL =.7258, S.E( ) = γ =.32, S.E( γ) =.785 Skellam µ = , S.E( µ )= µ 2 = , S.E( µ 2 )=.387 Emirical DASL Emirical Skellam F().5 F() Emirical dnormal Emirical ADSLalace F().5 F() Emirical SkewDL Emirical DL F().5 F() Figure 4: Plots of emirical distribution functions for the data set and the fitted distributions.

14 84 Discrete alha-skew-lalace distribution In addition, we can use the likelihood ratio (LR) test statistic to confirm our claim. To do this, we consider the following test of hyotheses H :γ=(dl()) v.s H :γ (DASL(,γ)). Observed value of the likelihood ratio (LR) test statistic is while its tabulated value equalsχ 2 = Thus the null hyothesis is rejected. On the other hand, Figure 4 shows our different fitted distribution functions and the emirical distribution of the data set. From these lots, we can see that our distribution function better fits the data set. Acknowledgments The authors would like to sincerely thank two anonymous reviewers for their careful reading of the aer and their valuable oints and comments which led to considerable imrovements. The first author is also grateful to the Graduate Office of the University of Isfahan for their suort. References Barbiero, A. (24). An alternative discrete skew Lalace distribution. Statistical Methodology, 6, Ferguson, T. S. (996). A Course in Large Samle Theory. Chaman and Hall, London. Graham, R. L., Knuth, D. E. and Patashnik, O. (989). Concrete Mathematics. Addison-Wesley Publishing Comany, Reading, MA. Gradshteyn, I. S. and Ryzhik, I. M. (27). Table of Integrals, Series, and Products, 7 th ed. Academic Press, San Diego, CA. Inusah, S. and Kozubowski, T. J. (26). A discrete analogue of the Lalace distribution. Journal of Statistical Planning and Inference, 36, 9 2. Jayakumar, K. and Jacob, S. (22). Wraed skew Lalace distribution on integers: A new robability model for circular data. Oen Journal of Statistics, 2, 6 4. Kozubowski, T. J. and Inusah, S. (26). A skew Lalace distribution on integers. Annals of the Institute of Statistical Mathematics, 58, Lawless, J. F. (23). Statistical Models and Methods for Lifetime Data, 2 nd ed. John Wiley and Sons, New York. Molina, J. A. D and Arteaga, A. R. (27). On the infinite divisibility of some skewed symmetric distributions, Statistics and Probability Letters, 77, Roy, D. (23). The discrete normal distribution. Communications in Statistics: Theory and Methods, 32, Shams Harandi, S. and Alamatsaz, M. H. (23). Alha-Skew-Lalace distribution. Statistics and Probability Letters, 83, Skellam, J. G. (946). The frequency distribution of the difference between two Poisson variates belonging to different oulations. Journal of the Royal Statistical Society, Series A, 9, 296. Steutel, F. W and Van Harn, K. (24). Infinite Divisibility of Probability Distributions on the Real Line. Marcel Dekker, Inc, New York. Weisberg, S. (25). Alied Linear Regression, 3 rd edition. John Wiley and Sons, New York.