Dispersion of place counts against the Poisson null and its mixtures

For any finite mixture of Poisson counts (places with unequal intensities) the variance is at least the mean, the excess being the variance of the intensity, with equality exactly when the intensity is constant on the support (Research.P7.Mixture.variance_eq, mixture_var_ge_mean, mixture_var_eq_mean_iff); and the zero share is at least \(e^{-\mu}\) (mixture_zero_ge_exp_neg_mean). So a dispersion index above one and a zero share above the Poisson null are implied by any heterogeneity of exposure and cannot by themselves separate a "criminology of place" from unequal exposure. This function reports the sample dispersion index, the implied intensity variance under the mixture reading, and the zero-share gap.

Usage

morie_concentration_dispersion(x)

Arguments

x

Non-negative counts per place; at least one positive.

Value

A list with mean_count, variance (sample, denominator n - 1), dispersion_index (variance over mean), implied_intensity_variance (variance minus mean, floored at 0), implied_intensity_sd, zero_share, null_zero_share (\(e^{-\mu}\), a lower bound for every mixture with the same mean), zero_share_gap and theorems.

Examples

set.seed(1)
x <- rpois(2000, rgamma(2000, shape = 2, rate = 5))   # heterogeneous places
unlist(morie_concentration_dispersion(x)[
  c("dispersion_index", "implied_intensity_sd", "zero_share_gap")])
#>     dispersion_index implied_intensity_sd       zero_share_gap 
#>           1.16114814           0.25610016           0.02385581