Estimate the left-tail parameters of the alternative p-value density

Under the two-group model \(f(p) = \pi_0 + (1-\pi_0) f_1(p)\) with the local approximation \(f_1(p) \approx C\alpha p^{\alpha-1}\) near zero, \((\alpha, C)\) solve the conditional moment equations $$J^{-1}\sum_j \log(p_j) I(p_j \le r) - \pi_0(r\log r - r) - C(1-\pi_0) r^{\alpha}(\log r - \alpha^{-1}) = 0,$$ $$J^{-1}\sum_j I(p_j \le r) = \pi_0 r + (1-\pi_0) C r^{\alpha},$$ which have a closed-form solution. The threshold \(r\) is the \(m\)-th smallest p-value with \(m = \lceil k\sqrt{J}\rceil\). The solutions are truncated to alpha_bounds and C_bounds.

Usage

heart_tail_estimate(
  p,
  pi0,
  k = 50,
  pi0_cap = 0.996,
  alpha_bounds = c(1e-05, 1 - 1e-05),
  C_bounds = c(0.1, 3)
)

Arguments

p

Vector of p-values.

pi0

Estimated null proportion of p.

k

Multiplier defining the tail size \(m = k\sqrt{J}\). k = 1 corresponds to the \(J^{-1/2}\) quantile.

pi0_cap

Upper cap applied to pi0 (avoids division by zero when the p-values look uniform).

alpha_bounds, C_bounds

Truncation ranges for \(\alpha\) and \(C\).

Value

Named numeric vector c(alpha = , C = , r = ).