Katz Family of Distributions and Processes

Transcription

1 CHAPTER 7 Katz Family of Distributions and Processes 7. Introduction The Poisson distribution and the Negative binomial distribution are the most widely used discrete probability distributions for the analysis of count data. Bardwell and Crow (964) derived a two-parameter family of univariate hyper-poisson distribution covering Poisson and the left truncated Poisson distributions as particular cases. The Katz families of distributions cover a wide spectrum including binomial, negative binomial, and Poisson distributions. A major motivation of Katz s (965) work was the problem of discriminating among binomial, negative binomial, and Poisson distributions when a given set of data is known to come from one or other of them. It has been used as a basis of developing more general families of distributions such as distributions defined by a discrete analogue to the Pearson system of continuous distributions studied in Ord (967a,b,972). Gurland and Tripathi (975) generalized the Katz s family to three and four parameter families of distributions, which covers the hyper-poisson distribution as a special case, (see Tripathi 25

2 and Gurland (977,979)). Fang (2003a) developed Generalized method of moments (GMM) tests for the Katz family of distributions. Katz family often serves as embryonic forms of data generating distributions and provide an ideal benchmark in many applications such as econometric modeling, industrial quality control, risk and insurance, and sampling theory among others. A more general extension of the Katz family is the Kemp families of distribution. The family of generalized hypergeometric probability distributions by Kemp (968) developed a wide variety of existing discrete distributions. More specifically, Katz suggested the use of a test based on the first two sample moments as a discriminating test statistic for testing the null hypothesis of equi-dispersion (the Poisson distribution) against the alternative hypothesis of under-or overdispersions (the binomial or negative binomial distribution). One of the important questions in statistical analysis of count data is how to formulate an adequate probability model to describe observed variation of counts. The Poisson family of discrete distributions is used as a benchmark for statistical analysis of count data. Integer valued time series models are commonly used for modelling time series data in the form of counts. Discrete time series models with binomial, Poisson, negative binomial and geometric distributions are discussed by McKenzie (986, 2003), Al-Osh and Alzaid (987), Alzaid and Al-Osh (990) etc. Some recent works are Ristić et al. (2009, 202) and Nastić et al. (202). A first order autoregressive model with count (or integer-valued) data is usually developed through the binomial thinning operator. Let X be an Z + -valued random variable and γ (0, ), then the thinning operator is defined as γ X Y i Y i where Y i is an i.i.d. Bernoulli random variable with P (Y i ) P (Y i 0) γ, that 26

3 is independent of X. With this operator, the INAR() model has the general structure X n γ X n + ɛ n, γ (0, ) and n (7..) where {ɛ n } will be referred to as innovation sequence, throughout the rest of the Chapter. Thus discrete time series modelling is based on binomial thinning which utilizes the concept of discrete self-decomposability introduced by Steutel and van Harn (979). If φ(s) E(s X ) is the probability generating function (pgf) of X, then under stationarity, (7..) can be rewritten as φ(s) φ( γ + γs)φ ɛ (s), (7..2) for γ (0, ) and with φ ɛ (s) is a proper pgf (for more details on self-decomposability, see Steutel and van Harn, 2004, p.246). If η(s) φ( s) is the apgf, we can rewrite (7..2) as η(s) η(γs)η ɛ (s) (7..3) where η ɛ (s) is a properly defined apgf. The above equation is similar to the condition of self decomposability and hence random variables satisfying (7..3) are known as discrete self-decomposable. In this chapter, some basic concepts of Katz family of distributions and its properties are portrayed. We introduce the first order integer-valued autoregressive processes with Katz family of distributions. The geometric Katz family of distributions is developed and its properties are studied. The INAR() processes with geometric Katz family marginals is discussed. Then, the corresponding higher order autoregressive processes are described. Applications of the Katz family to real data set is provided. 27

4 7.2 Katz Family of Distribution Let p 0, p, p 2,... denote a probability distribution on the nonnegative integers. The condition p(x + ) p(x) α + βx ; x 0,,... ; (7.2.) + x where α > 0, β < (β does not yield a valid distribution) and p(x) is the probability distribution of X, characterizes the Katz family of distributions (KF(α, β)) (Johnson et al. (992)). Some properties of Katz family of distributions are given in Table. Several extensions of Katzs recursion have been proposed to enlarge the hierarchy of models available for modelling and analyzing count data. Katz (965) has also suggested another criterion for identifying the discrete distributions. That is, µ 2 µ µ β β ; β < (7.2.2) where µ and µ 2 are the mean and variance respectively and β is a constant. The ratio (7.2.2) takes the value zero for the Poisson distribution, positive for the negative binomial and negative for the binomial distribution. This criterion is helpful in deciding which of the three distributions to use, for fitting the sample data. Recently, Sudheesh and Tibiletti (202) discuss moment properties of this family in detail. The pgf for the Katz family is φ(s) ( ) α/β βs. (7.2.3) β Distribution Parameters Fisher Dispersion Index Skewness Kurtosis (Ratio of Variance to Mean) Binomial α > 0, β < 0 < 0 iff β 2 Poisson α > 0, β 0 > 0 > 3 Negative Binomial α > 0, 0 < β < > > 0 > 3 3 iff β 3 3 or β

5 Table. Properties of the Katz family of distributions. The factorial moment generating function is ( ) α/β β βt φ( + t). β From Table, it is clear that the Poisson distribution can be used for modeling data with equi-dispersion (variance mean ) whereas binomial distribution is suitable for modeling under-dispersed (variance< mean) data only. The negative binomial distribution is suitable only for over-dispersed (variance > mean). But Katz distribution offers greater scope since it can be used for all the three types of data. Theorem The KF(α, β) distribution is discrete self-decomposable. Proof: Consider the pgf of KF(α, β), ] α β βs φ X (s) β ] α β β βs β β + γβ γβs φ( γ + γs)φ γ (s) ] α ( )] α β β β γ + ( γ) βs where φ γ (s) is the pgf of α -fold convolution of a negative binomial random variable. It β may be verified that irrespective of whether k is integer-valued or not, the random variable can be generated using a compound Poisson representation. For details see Jose and Thomas (20). Hence the KF(α, β) is discrete self-decomposable. 29

6 7.3 Integer Valued Autoregressive Process with Katz Family of Distributions Now we shall construct a first order integer-valued time series model with KF marginals. The first order integer-valued autoregressive process with Katz family of distributions (KFAR()) is constituted by {X n, n } where X n satisfies the equation, X n γ X n + ɛ n, γ (0, ) and n. where {ɛ n } is a sequence of independently and identically distributed (iid) random variables such that X n is stationary Markovian with KF marginal distributions. Now from (7.2.3), we have φ X (s) ] α β β βs + β β ( s) + ( s) ] α β ] α(+) where β. In terms of apgf, the above equation becomes, β η(s) + s ] α(+) (7.3.) 30

7 Under stationary equilibrium, from (7..3) we have η ɛ (s) η X(s) η X (γs) ] α(+) + γs + s γ + ( γ) + s ] α(+). (7.3.2) we can regard the innovations {ɛ n } as the α(+) -fold convolutions of random variables T n s such that 0, with probability γ T n X n, with probability γ where X n s are iid Katz family of random variables. Remark If X 0 is distributed arbitrarily, then also the process is asymptotically stationary Markovian with KF marginal distribution. Proof: From (7..) we have, X n γ X n + ɛ n n γ n X 0 + γ k ɛ n k. k0 Writing in terms of apgf, n η Xn (s) η X0 (γ n s) η ɛn (γ k s) k0 n ] + γ η X0 (γ n k+ s α(+) s) + γ k s + s k0 ] α(+) as n. 3

8 Hence it follows that even if X 0 is arbitrarily distributed, the process is asymptotically stationary Markovian with KF marginals. We therefore have the following theorem. Theorem The first order autoregressive process X n γ X n +ɛ n, γ (0, ), is strictly stationary Markovian with marginal distribution given by (7.2.3) if and only if the innovations {ɛ n } are distributed iid as the α(+) -fold convolution of random variables {T n } where 0, with probability γ T n X n, with probability γ where {X n } are iid Katz family of random variables provided X 0 d KF ( α, β ) + and independent of ɛ n. 7.4 Geometric Katz Family of Distributions Definition A random variable X on Z + is said to follow geometric Katz family of distributions if it has the pgf φ(s) ( ); s, 0 < β. (7.4.) + α ln βs β β Theorem Geometric Katz family of distributions is the limit distribution of geometric sum of random variables following KF(α, α βn ). Proof: From we have, ( ) α/β βs + β ( ) ] α n βs βn β is the pgf of a probability distribution since Katz family of distribution is infinitely divisible. 32

9 Hence by lemma 3.2 of Pillai(990), φ n (s) + n ( ) ] α βs βn β is the pgf of a geometric sum of iid Katz random variables. Then by taking limit as n Hence the theorem follows. φ(s) lim φ n (s) n { ( βs + lim n n β + α ( )] βs β ln. β 7.5 Geometric KF Autoregressive Processes ) α βn }] In this section, we develop a first order new integer-valued autoregressive process with geometric Katz family of marginals. The geometric Katz family first order new autoregressive process(gkfar()) is constituted by {X n, n } where X n satisfies the equation, 0, with probability ρ X n ɛ n + X n, with probability ρ (7.5.) where 0 < ρ and {ɛ n } is iid random variables such that {X n } is stationary Markovian with Katz family of marginal distribution. Theorem Consider a stationary integer-valued autoregressive process {X n } with structure given by A necessary and sufficient condition that {X n } is stationary Markovian with geometric KF marginal distribution is that {ɛ n } is distributed a geometric KF provided X 0 is distributed as geometric KF (GKF). 33

10 Proof: Let us denote the pgf of X n by φ Xn (s) and that of ɛ n by φ ɛn (s), (7.5.) in terms of pgf becomes φ Xn (s) ρφ ɛn (s) + ( ρ)φ Xn (s)φ ɛn (s). Assuming stationarity, it becomes φ X (s) φ ɛ (s) ρ + ( ρ)φ X (s)]. That is, φ ɛ (s) φ X (s) ρ + ( ρ)φ X (s) ( + ρα ln β ). βs β Hence it follows that ɛ n d GKF ( ) β, ρα. β The converse part can be proved by the method of mathematical induction as follows. Now assume that X n d GKF (α, β). φ Xn (s) φ ɛn (s) ρ + ( ρ)φ Xn 2 (s) ] ( ρ + ( ρ) ( + ρα ln + α ln β β + α β ln ( ) βs β ). βs β βs β ) 34

11 7.6 Generalization to a k th Order Geometric Katz Family Processes Now we shall extend the results by considering higher order autoregressive processes in (7..) having the following structure, γ X n + ɛ n, with probability ρ X n γ 2 X n 2 + ɛ n, with probability ρ 2 γ k X n k + ɛ n, with probability ρ k (7.6.) where 0 < γ i, ρ i, i, 2,, k; Weiß(2008). k i ρ i. For details see Zhu & Joe (2006), In terms of pgf the above equation can be written as k ] φ Xn (s) φ ɛn (s) ρ i φ Xn i (γ i s). i Assuming stationarity it reduces to k ] φ X (s) φ ɛ (s) ρ i φ X (γ i s). i Hence φ ɛ (s) φ X (s) k i ρ iφ X (γ i s). For the GKF marginals, the innovation sequence of the process has pgf, φ ɛ (s) + αβ ln ( )] βs β k i ρ i + α β ln ( βγ i s β )]. (7.6.2) 35

12 Figure 7.: The empirical pdf and theoretical pdf of KF distribution. For the particular case of γ i γ, for i, 2,, p, (7.6.2) yields a similar pattern of apgf defined in (7.3). Hence with an error sequence {ɛ n } distributed as GKF random variables, the k th order GKF autoregressive processes can be properly defined. 7.7 Applications of Katz Family The Katz family of distribution is a more general family of distributions which can be applied in a wide variety of contexts. This family has been used for modeling count data on birth rates. Fang (2003b) investigate the relative merits of the 3-sigma c-chart compared to the X-chart for count data generated from a variety of distributions of the Katz family. Ghahfarokhi et al. (200) use Katz family of distributions for detecting and testing overdispersion in Poisson regression models. To illustrate the usefulness and flexibility of the KF(α, β) distribution, we apply it to a set of real data on the inter-arrival times of customers in a bank counter measured in 36

13 Figure 7.2: The empirical cdf and theoretical cdf of KF distribution. terms of number of months from January 994 to October 2003, which is available in The empirical probability density function (pdf ) and theoretical pdf are plotted in Figure 7.. The empirical pdf shows a decreasing trend in the probabilities. In Figure 7.2 the curved line segment denote the theoretical cumulative density function (cdf ) and dotted line represents empirical cdf. It can be seen from Figures 7. and 7.2 that a linear relationship is reasonable for the data set. From the data set we estimate the parameter values as ˆα 0.5 and ˆβ 0.5. For details see Fang (2003a,b). We use the Kolmogorov-Smirnov K.S.] test for testing H 0 : KF(0.5, 0.5) is a good fit for the given data. Since the computed value of the K.S.test statistic has the value and does not exceed the critical value at level % of , the KF assumption for inter-arrival times is justified. Using this we can obtain the probabilities associated with the stationary distribution of the INAR() model as well as predict the future values of the process. This will help in developing optimal service policies for ensuring customer sat- 37

14 isfaction. Such discrete probability models can also be used for applications in various domains as ecology, linguistics, information sciences, statistical physics, etc Concluding Remarks In this chapter, we proposed INAR() process with Katz family of distribution. Katz (965) has formulated one of the most prominent families of discrete distributions whose successive probabilities satisfy first-order recurrence relations. These distributions often serve as statistical models for data sets from various contexts and provide an ideal benchmark in many applications such as econometric modeling, industrial quality control, risk and insurance, and sampling theory, among others. References Al-Osh, M.A., Alzaid, A.A. (987). First order integer-valued autoregressive (INAR()) process. Journal of Time Series Analysis, 8, Alzaid, A.A., Al-Osh, M.A. (990). An integer-valued p th order autoregressive structure(inar(p)) process. Journal of Applied Probability, 27, Bardwell, G.E., Crow, E.L. (964). A two-parameter family of hyper-poisson distributions. Journal of the American Statistical Association, 59, Fang,Y. (2003a). GMM tests for the Katz family of distributions. Journal of Statistical Planning and Inference, 0, Fang,Y. (2003b). C-chart, X-chart, and the Katz Family of Distributions. Journal of Quality Technology, 35(), -5. Ghahfarokhi, M.A.B., Iravani, H., Sepehri, M.R. (200). Application of Katz family of distributions for detecting and testing overdispersion in Poisson models. Journal of Statistical Modeling and Analytics, (),

15 Gurland, J., Tripathi, R., (975). Estimation of parameters on some extensions of the Katz family of discrete distributions involving hypergeometric functions. In: Patil, G., Kotz, S., Ord, J. (Eds.), Statistical Distributions in Scientific Work, Vol. : Models and Structures. Reidel, Dordrecht, pp Johnson, N., Kotz, S., Kemp, A., (992). Univariate Discrete Distributions, 2nd Edition. Wiley, New York. Jose, K.K., Manu, M.T. (20). Generalized Laplacian Distributions and Autoregressive Processes. Communications in Statistics - Theory and Methods, 40, Katz, L. (965). Unified treatment of a broad class of discrete probability distributions. In Classical and Contagious Discrete Distributions, Pergamon Press, Oxford, Kemp, A.W. (968). A wide class of discrete distributions and their associated differential equations. Sankhya, Series A, 30, McKenzie, E. (986). Autoregressive-moving average processes with negative binomial and geometric marginal distributions. Advances in Applied Probability, 8, McKenzie, E. (2003). Discrete Variate time series. In: Shanbhag, D.N.; Rao, C.R. (Eds), Handbook of Statistics. Elsevier, Amsterdam, pp, 2, Nastić, A.S., Ristić, M.M., Bakouch, H.S. (202). A combined geometric INAR(p) model based on negative binomial thinning. Mathematical and Computer Modelling, 55, Ord, J. (967a). Graphical methods for a class of discrete distributions. Journal of the Royal Statistical Society: Series A, 30, Ord, J. (967b). On a system of discrete distributions. Biometrika, 54, Ord, J. (972). Families of Frequency Distributions. Griffin, London. 39

16 Pillai, R.N. (990). Harmonic mixtures and geometric infinite divisibility. Journal of Indian Statistical Association, 28, Ristić, M.M., Bakouch, H.S., Nastić, A.S. (2009). A new geometric first-order integervalued autoregressive (NGINAR()) process. Journal of Statistical Planning and Inference, 39, Ristić, M.M., Nastić, A.S., Bakouch, H.S. (202). Estimation in an integer-valued autoregressive process with negative binomial marginals (NBINAR()). Communications in Statistics - Theory and Methods, 4, Steutel, F.W., van Harn, K. (979). Discrete analogues of self- decomposability and stability. Annals of Probability, 7, Steutel, F.W., van Harn, K. (2004). Infinite Divisibility of Probability Distributions in the Real Line. New York: Dekker. Sudheesh, K.K., Tibiletti, L. (202). Moment identity for discrete random variable and its applications. Statistics: A Journal of Theoretical and Applied Statistics, 46(6), Tripathi, R., Gurland, J. (977). A general family of discrete distributions with hypergeometric probabilities. Journal of the Royal Statistical Society: Series B, 39, Tripathi, R., Gurland, J. (979). Some aspects of the Kemp families of distributions. Communications in Statistics - Theory and Methods, 8, Weiß, C.H. (2008). The combined INAR(p) models for time series of counts. Statistics and Probability Letters, 78(3), Zhu, R., Joe, H. (2006). Modelling count data time series with Markov processes based on binomial thinning. Journal of Time Series Analysis, 27(5),