Any intensity heterogeneity forces overdispersion

A count whose intensity is drawn from a finite mixture (weights w, intensities lam) has \(Var\,N = E\,N + Var\,\Lambda\) (Research.P7.Cox.total_variance), so the variance is at least the mean (variance_ge_mean) with equality iff the intensity is constant on the support (equality_iff_constant). A Gini or zero share above the Poisson null is therefore implied by any random intensity and is not by itself evidence of contagion.

Usage

morie_cox_overdispersion(w, lam)

Arguments

w

Mixture weights (normalised to sum to one).

lam

Non-negative intensities, one per weight.

Value

A list: mean, variance, intensity_var, dispersion (variance over mean) and constant_intensity.

References

Grimmett G, Stirzaker D (2020). Probability and Random Processes, 4th ed., Problem 6.15.22. Oxford.

Examples

morie_cox_overdispersion(w = c(0.7, 0.3), lam = c(1, 5))
#> $mean
#> [1] 2.2
#> 
#> $variance
#> [1] 5.56
#> 
#> $intensity_var
#> [1] 3.36
#> 
#> $dispersion
#> [1] 2.527273
#> 
#> $constant_intensity
#> [1] FALSE
#> 
#> attr(,"theorem")
#> [1] "Research.P7.Cox.total_variance"