Estimating Parameters of A Hyper-Poisson Distribution in SAS

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.

HP