LRT05_Sphericity: p-value of the log-transformed LRT statistic and/or its null...

In gajdosandrej/CharFunToolR: Numerical Computation Cumulative Distribution Function and Probability Density Function from Characteristic Function

p-value of the log-transformed LRT statistic and/or its null distribution CF/PDF/CDF

Description

LRT05_Sphericity(W, n, p, options) computes p-value of the log-transformed LRT statistic W = -log(\Lambda), for testing the null hypothesis of equality of q (q > 1) p-dimensional normal populations, and/or its null distribution CF/PDF/CDF.

Usage

LRT05_Sphericity(W, n, p, options)

Arguments

W	observed value of the minus log-transformed LRT statistic W = -log(\Lambda). If empty, the algorithm evaluates the CF/PDF/CDF and the quantiles of the null distribution of W.
n	sample size, n > min(p+q-1).
p	common dimension of the vectors X_k, k = 1,...q.
options	option structure, for more details see cf2DistGP. Moreover, x set vector of values where PDF/CDF is evaluated, prob set vector of probabilities for the quantiles, coef set arbitrary multiplicator of the argument t of the characteristic function. If empty, default value is -n/2 (standard value for minus log-transform of LRT). Possible alternative is e.g. coef = -1, leading to W = -(2/n)*log(LRT).

Details

In particular, let X ~ N_p(\mu,\Sigma), then, the null hypothesis is given as H0: \Sigma = \sigma^2 I_p (\sigma^2 unspecified). Here, the LRT test statistic is given by \Lambda = (det(S) / trace((1/p)*S)^p)^n/2, where S is MLE of Sigma based on sample size n from X ~ N_p(\mu,\Sigma). Under null hypothesis, distribution of the test statistic \Lambda is \Lambda ~ prod_{j=2}^{p} (B_j)^{n/2}, with B_j ~ Beta((n-j)/2,(j-1)/p + (j-1)/2), where j = (2,...,p). Here we assume that n > p. Hence, the exact characteristic function of the null distribution of minus log-transformed LRT statistic \Lambda, say W = -log(\Lambda) is given by cf = function(t) {cf_LogRV_Beta(-(n/2)*t, (n-j)/2, (j-1)/p + (j-1)/2)}, where j = (2,..., p)'.

Value

p-value of the log-transformed LRT statistic, W = -log(\Lambda) and/or its null distribution CF/PDF/CDF.

Note

Ver.: 16-Sep-2018 21:11:16 (consistent with Matlab CharFunTool v1.3.0, 20-Jan-2018 12:43:15).

References

[1] ANDERSON, Theodore Wilbur. An Introduction to Multivariate Statistical Analysis. New York: Wiley, 3rd Ed., 2003.

[2] MARQUES, Filipe J.; COELHO, Carlos A.; ARNOLD, Barry C. A general near-exact distribution theory for the most common likelihood ratio test statistics used in Multivariate Analysis. Test, 2011, 20.1:180-203.

[3] WITKOVSKY, Viktor. Exact distribution of selected multivariate test criteria by numerical inversion of their characteristic functions.
arXiv preprint arXiv:1801.02248, 2018.

See Also

Other Likelihood Ratio Test: LRT01_Independence(), LRT02_EqualityMeans(), LRT03_EqualityCovariances(), LRT04_EqualityPopulations()

Examples

## EXAMPLE
# LRT for testing hypothesis on sphericity of covariance matrix
# Null distribution of the minus log-transformed LRT statistic
n <- 30                    # total sample size
p <- 8                     # dimension of X ~ N_p(mu,Sigma)
# W <- vector()            # observed value of W = -log(Lambda)
options <- list()
# options$coef = -1
options$prob <- c(0.9, 0.95, 0.99)
output <- LRT05_Sphericity(n = n, p = p, options = options)
str(output)

gajdosandrej/CharFunToolR documentation built on June 3, 2024, 7:46 p.m.