The output corresponds to the theoretical portion of the Table 3, 4,
5, 6 in the reference paper. We also have another set of functions
(available upon request) that put observed and theoretical statistics
together for making tables that resemble those in the reference.
Let \(P_n\) be the probability of a consumer buying the product
category \(n\) times. Then \(P_n\) has a Negative Binomial
Distribution (NBD). Let \(p(r_j|n)\) be the probability of making
\(r_j\) purchases of brand \(j\), gien that \(n\) purchases
of the category having been make (\(r_j\leq n\)). Then \(p(r_j|n)\)
has a Beta-Binomial distribution.
The theoretical brand penetration \(b\) is then
$$b = 1 - \sum_{n=0} P_n p(0|n)$$
The theoretical brand buying rate \(w\) is
$$w = \frac{\sum_{n=1} \{ P_n \sum_{r=1}^n r p(r|n) \}}{b} $$
and the category buying rate per brand buyer \(w_P\) is
$$w_P = \frac{\sum_{n=1} \{ n P_n [ 1 - p(0|n)] \}}{b} $$
The brand purchase frequency distribution is
$$ f(r) = \sum_{n \geq r} P_n p(r|n) $$
The brand penetration given a specific category purchase frequency range
\(R=\{i_1, i_2, i_3, \ldots\}\) is
$$1 - \frac{\sum_{n \in R} P(n) p(0|n)}{\sum_{n \in R} P(n)}$$
The brand buying rate given a specific category purchase frequency range
\(R=\{i_1, i_2, i_3, \ldots\}\) is
$$\frac{\sum_{n \in R} P(n) [\sum_{r=1}^n r p(r|n)]}{\sum_{n \in R}
P(n) [1 - p(0|n)] }$$
To calculate the brand duplication measure, we first get the penetration \(b_{(j+k)}\)
of the "composite" brand of two brands \(j\) and \(k\) as:
$$ b_{(j+k)} = 1 - \sum_n P_n p_k(0|n) p_j(0|n)$$
Then the theoretical proportion \(b_{jk}\) of the population buying both brands at
least once is:
$$b_{jk} = b_j + b_k - b_{(j+k)}$$
and the brand duplication \(b_{j/k}\) (where brand \(k\) is the focal
brand) is:
$$b_{j/k} = b_{jk} / b_k$$