Sample Size Requirements for Zero Order Models
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As indicated earlier the probability of purchasing Brand 1 is allowed to vary over the population of consumers according to the beta distribution with parameters a and 3 .
f(p;a, e) = p°'"^(l-p)^"VB(a, 3), 0
0, where B(a, B) = r(a)r(3)/r(a+3) is the beta function. The marginal distribu- tion of j purchases of Brand 1 on k trials is obtained by compounding the bi- nomial and beta distributions. The unconditional probability of j purchases of Brand 1 on k trials is given by P(j;k, a, 6)
...= ($) ^^^^T[!tIT^ ' 3 = o, U.. -X (9) This is the beta-binomial distribution whose mean and variance are given by and UAR Til - M(a+B+k) and VAR LjJ - {a+3)^(a+B+l ) • The expression for the unconditional probability of j successes on k trials in equation (9) can also be written in terms of the transformed parameters y and ({) by making the substitutions for a and B as follows: , = lilMi, and 6 =(1:m1(M) (10) Parameter Estimation Given the purchase frequency data (i. E. , N-'s) the likelihood function J can be written as where the term P.
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