Similar to COM-Poisson, Double-Poisson, and Generalized Poisson distributions discussed in my previous post (https://statcompute.wordpress.com/2016/11/27/more-about-flexible-frequency-models/), the Hyper-Poisson distribution is another extension of the standard Poisson and is able to accommodate both under-dispersion and over-dispersion that are common in real-world problems. Given the complexity of parameterization and computation, the Hyper-Poisson is somewhat under-investigated. To the best of my knowledge, there is no off-shelf computing routine in SAS for the Hyper-Poisson distribution and only a R function available in http://www4.ujaen.es/~ajsaez/hp.fit.r written by A.J. Sáez-Castillo and A. Conde-Sánchez (2013).

The SAS code presented below is the starting point of my attempt on the Hyper-Poisson and its potential applications. The purpose is to replicate the calculation result shown in the Table 6 of “On the Hyper-Poisson Distribution and its Generalization with Applications” by Bayo H. Lawal (2017) (http://www.journalrepository.org/media/journals/BJMCS_6/2017/Mar/Lawal2132017BJMCS32184.pdf). As a result, the parameterization employed in my SAS code will closely follow Bayo H. Lawal (2017) instead of A.J. Sáez-Castillo and A. Conde-Sánchez (2013).

data d1; input y n @@; datalines; 0 121 1 85 2 19 3 1 4 0 5 0 6 1 ; run; data df; set d1; where n > 0; do i = 1 to n; output; end; run; proc nlmixed data = df; parms lambda = 1 beta = 1; theta = 1; do k = 1 to 100; theta = theta + gamma(beta) * (lambda ** k) / gamma(beta + k); end; prob = (gamma(beta) / gamma(beta + y)) * ((lambda ** y) / theta); ll = log(prob); model y ~ general(ll); run; /* Standard Parameter Estimate Error DF t Value Pr > |t| Alpha lambda 0.3752 0.1178 227 3.19 0.0016 0.05 beta 0.5552 0.2266 227 2.45 0.0150 0.05 */

As shown, the estimated Lambda = 0.3752 and the estimated Beta = 0.5552 are identical to what is presented in the paper. The next step is be to explore applications in the frequency modeling as well as its value in business cases.