Two count laws with the same mean and variance: what a K-function cannot tell apart

Contagion and heterogeneity can give the same first and second moments (Bartlett 1964), which is all a dispersion index or a K-function sees. The witness laws on {0, 1, 2, 3} are \(p = (1/2, 0, 1/2, 0)\) and \(q = (1/3, 1/2, 0, 1/6)\): mean 1 and variance 1 in both, dispersion index one, yet \(P(0)\) of 1/2 against 1/3 (Research.P10.Moments.second_moments_do_not_identify).

Usage

morie_contagion_moment_witness(statistic = NULL)

Arguments

statistic

Optional function of (mean, variance) to evaluate on both laws.

Value

A list with the two laws, their moments, dispersion (both 1), zero_probability (1/2 and 1/3), and, with statistic, its two values.

Examples

morie_contagion_moment_witness(function(m, v) v / m)$statistic
#> p q 
#> 1 1