power.hc: Statistical power of Higher Criticism test.

Description

Statistical power of Higher Criticism test.

Usage

power.hc(
  alpha,
  n,
  beta,
  method = "gaussian-gaussian",
  eps = 0,
  mu = 0,
  df = 1,
  delta = 0
)

Value

Power of HC test.

Arguments

alpha: - type-I error rate.
n: - dimension parameter, i.e. the number of input statitics to construct Higher Criticism statistic.
beta: - search range parameter. Search range = (1, beta*n). Beta must be between 1/n and 1.
method: - different alternative hypothesis, including mixtures such as, "gaussian-gaussian", "gaussian-t", "t-t", "chisq-chisq", and "exp-chisq". By default, we use Gaussian mixture.
eps: - mixing parameter of the mixture.
mu: - mean of non standard Gaussian model.
df: - degree of freedom of t/Chisq distribution and exp distribution.
delta: - non-cenrality of t/Chisq distribution.

Details

We consider the following hypothesis test, $$H_0: X_i\sim F, H_a: X_i\sim G$$ Specifically, $F = F_0$ and $G = (1-\epsilon)F_0+\epsilon F_1$, where $\epsilon$ is the mixing parameter, $F_0$ and $F_1$ is speified by the "method" argument:

"gaussian-gaussian": $F_0$ is the standard normal CDF and $F = F_1$ is the CDF of normal distribution with $\mu$ defined by mu and $\sigma = 1$.

"gaussian-t": $F_0$ is the standard normal CDF and $F = F_1$ is the CDF of t distribution with degree of freedom defined by df.

"t-t": $F_0$ is the CDF of t distribution with degree of freedom defined by df and $F = F_1$ is the CDF of non-central t distribution with degree of freedom defined by df and non-centrality defined by delta. "chisq-chisq": $F_0$ is the CDF of Chisquare distribution with degree of freedom defined by df and $F = F_1$ is the CDF of non-central Chisquare distribution with degree of freedom defined by df and non-centrality defined by delta.

"exp-chisq": $F_0$ is the CDF of exponential distribution with parameter defined by df and $F = F_1$ is the CDF of non-central Chisqaure distribution with degree of freedom defined by df and non-centrality defined by delta.

References

1. Hong Zhang, Jiashun Jin and Zheyang Wu. "Distributions and Statistical Power of Optimal Signal-Detection Methods In Finite Cases", submitted.

2. Donoho, David; Jin, Jiashun. "Higher criticism for detecting sparse heterogeneous mixtures". Annals of Statistics 32 (2004).

Examples

Run this code

power.hc(0.05, n=10, beta=0.5, eps = 0.1, mu = 1.2)

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