Certified Bounds for the Largest DP Population Weight Tail

Computes distribution-free bounds on \(P(W_{max}>threshold)\). If \(q_t=P(W_{SB}>threshold)\), then $$q_t \le P(W_{max}>threshold) \le \min(q_t/threshold,1).$$ The identity remains valid after mixing alpha over a Gamma hyperprior.

Usage

wmax_tail_bounds(threshold, alpha = NULL, a = NULL, b = NULL)

Arguments

threshold

One finite probability in the closed unit interval.

alpha

Optional positive scalar fixed DP concentration.

a, b

Optional positive scalar Gamma shape and rate. Supply either alpha or both a and b, never both specifications.

Value

A wmax_tail_bounds object. The fields lower_bound and upper_bound are certified bounds, not point estimates. size_biased_tail is \(P(W_{SB}>threshold)\).

See also

prob_wmax_exceeds, prob_wsb_exceeds

Other weights_w1: cdf_w1(), density_w1(), mean_w1(), prob_w1_exceeds(), prob_wmax_exceeds(), prob_wsb_exceeds(), quantile_w1(), summary_w1(), var_w1()

Examples

wmax_tail_bounds(0.4, alpha = 1)
#> Certified bounds for P(W_max > threshold)
#>   Conditioning: fixed alpha = 1
#>   Threshold: 0.4
#>   Rounded P(W_SB > threshold): 0.6
#>   Certified bounds: [0.6, 1]
wmax_tail_bounds(0.5, a = 2, b = 1)
#> Certified bounds for P(W_max > threshold)
#>   Conditioning: Gamma(shape = 2, rate = 1)
#>   Threshold: 0.5
#>   Rounded P(W_SB > threshold): 0.348827
#>   Certified bounds: [0.348827, 0.697655]