Le Cam's two-point lower bound on a finite sample space — morie_two_point

Two laws \(p, q\) on the same finite space with parameter values \(\theta_p, \theta_q\). For every estimator \(T\), \(E_p|T-\theta_p| + E_q|T-\theta_q| \ge |\theta_p-\theta_q|(1 - TV)\) (Research.P1LeCam.two_point), so the worse of the two risks is at least \(|\theta_p-\theta_q|(1-TV)/2\) (minimax), using \(\sum\min(p,q) = 1 - TV\) (sum_min). For the dark figure: two recording mechanisms whose recorded-crime laws are close in total variation but whose true rates differ by \(\delta\) leave every estimator a worst-case error of at least \(\delta(1-TV)/2\).

Usage

morie_two_point_bound(p, q, theta_p, theta_q, estimator = NULL)

Arguments

p, q: Probability vectors on the same finite space.
theta_p, theta_q: The parameter under each law.
estimator: Optional vector of the estimator's value at each point.

Value

A list with tv, sum_min, delta, bound (\(\delta(1-TV)/2\)), and when estimator is given risk_p, risk_q, minimax_risk and satisfied; plus theorems.

Examples

# recorded counts 0..4 under two reporting rates for the same true rate 2 vs 3 offences
p <- dbinom(0:4, 4, 0.5); q <- dbinom(0:4, 4, 0.6)
b <- morie_two_point_bound(p, q, theta_p = 2, theta_q = 3, estimator = 0:4 * 1.25)
c(tv = b$tv, bound = b$bound, minimax_risk = b$minimax_risk, satisfied = b$satisfied)
#>           tv        bound minimax_risk    satisfied 
#>      0.16270      0.41865      1.12500      1.00000