power.bj {SetTest}

Statistical power of Berk and Jones test.

Description

Statistical power of Berk and Jones test.

Usage

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

Arguments

alpha

- type-I error rate.

n

- dimension parameter, i.e. the number of input statitics to construct B-J 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.

Value

Power of BJ test.

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).

3. Jager, Leah; Wellner, Jon A. "Goodness-of-fit tests via phi-divergences". Annals of Statistics 35 (2007).

4. Berk, R.H. & Jones, D.H. Z. "Goodness-of-fit test statistics that dominate the Kolmogorov statistics". Wahrscheinlichkeitstheorie verw Gebiete (1979) 47: 47.

See Also

stat.bj for the definition of the statistic.

Examples

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