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Evaluates, at the statistic values t, the global-null CDF \(F_{00}(t) = \int_0^1 G_t(q)dq\), its empirical counterpart \(\hat Q(t) = J^{-1}\sum_j G_t(p_{de,j})\), the reconstructed \(\hat F_{01}(t) = \{\hat Q(t) - \hat\pi_{00}F_{00}(t)\}/(\hat\pi_{01}+\hat\pi_{11})\) (truncated to [0, 1] and made nondecreasing in \(t\) by a running maximum; set to 0 if \(\hat\pi_{01}+\hat\pi_{11} = 0\)), the null numerator \(\hat N(t) = \hat\pi_{00}F_{00}(t) + \hat\pi_{01}\hat F_{01}(t)\) and the composite null CDF \(\hat F_0(t) = \hat N(t) / (\hat\pi_{00} + \hat\pi_{01})\), where \(G_t(q) = \min\{1, (t/(\kappa_1 q^{1-\alpha_2} + \kappa_2))^{1/(1-\alpha_1)}\}\).

Usage

heart_null_cdf(log_t, p_de, par, exact = FALSE, n_exact = 5000L, n_grid = 500L)

Arguments

log_t

Values of \(\log t\).

p_de

All decorrelated p-values (used for \(\hat Q\)).

par

Parameter list as in heart_statistic(), also containing pi11 (or pi10, from which \(\pi_{11} = 1 - \pi_{00} - \pi_{01} - \pi_{10}\)).

exact

If FALSE, \(F_{00}\) is computed exactly at the n_exact smallest values and interpolated (log-log) on an n_grid grid elsewhere.

n_exact, n_grid

See exact.

Value

A data.frame with columns log_t, F00, Q, F01, N, F0, sorted by log_t.