The constant in the Hirano-Porter treatment-choice bound

Under local asymptotics the welfare regret of the rule that treats when \(\hat\tau > 0\), against a truth \(\tau = h/\sqrt n\) with \(\hat\tau \sim N(h/\sqrt n, V/n)\), is \(|h| \Phi(-|h|/\sqrt V)/\sqrt n\). The worst case over the local parameter therefore scales as \(c\sqrt{V/n}\) with \(c = \max_{t \ge 0} t\,\Phi(-t)\), whose stationary condition is \(\Phi(-t) = t\,\phi(t)\).

Usage

morie_minimax_regret_constant(tol = 1e-14)

Arguments

tol

Bisection tolerance.

Value

A list with t_star, constant, stationarity_residual.

Details

The constant is solved rather than quoted. The two-figure 0.17 that appears in the literature is not precise enough to check an attainment claim against simulation.

References

Hirano K, Porter JR (2009) Econometrica 77(5):1683-1701, doi:10.3982/ECTA6630.

Examples

morie_minimax_regret_constant()
#> $t_star
#> [1] 0.7517915
#> 
#> $constant
#> [1] 0.1699712
#> 
#> $stationarity_residual
#> [1] 1.276756e-15
#>