Simulating the Bootstrap Rejection Probability

A single bootstrap test is to be based on a statistic τ in asymptotic P value form. Rejection by an asymptotic test at level α is thus the event τ < α. Rejection by the bootstrap test is the event τ < Q(α, μ∗), where μ∗ is the bootstrap DGP, and Q(α, μ∗) is the (random) α--quantile of the distribution of the statistic τ as generated by μ∗. We define two random variables that are deterministic functions of the two random elements, τ and μ∗, needed for computing the bootstrap P value R(τ, μ∗), where R is the inverse function of Q, in the sense that, for all α ∈ [0, 1], and for all DGPs μ in the model we consider, α = R(Q(α, μ), μ) = Q(R(α, μ), μ). (1) The first of these random variables is distributed as U(0, 1) under μ; it is