LRT01_Independence: 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

LRT01_Independence(W, n, p, options) computes p-value of the log-transformed LRT statistic, W = -log(\Lambda), for testing the null hypothesis of independence (under normality assumptions) of m groups of variables (m > 1), and/or its null distribution CF/PDF/CDF.

Usage

LRT01_Independence(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 > q_k).
p	vector p = (p_1,...,p_k) of dimensions of X_k, k = 1,...m.
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_k ~ N_{p_k}(\mu_k,\Sigma_k) are p_k dimensional random vectors, for k = 1,...,m. Let us denote X = (X_1,...,X_m) and assume X ~ N_p(mu,\Sigma). Then, the null hypothesis is given as H0: \Sigma = diag(\Sigma_1,...,\Sigma_m), i.e. the off-diagonal blocks of Sigma are blocks of zeros. Here, the LRT test statistic is given by \Lambda = det(S) / prod(det(S_k)), where S is MLE of Sigma, and S_k are MLEs of \Sigma_k, for k = 1,...,m, based on n samples from the compound vector X = (X_1,...,X_m).

Under null hypothesis, distribution of the test statistic \Lambda is \Lambda ~ prod_{k=1}^{m-1} prod_{j=1}^{p_k} (B_{jk})^{n/2}, with B_{jk} ~ Beta((n-q_k-j)/2,q_k/2), where q_k = p_{k+1} + ... + p_m. Here we assume that n > q_k, for all k = 1,...,m-1.

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-q-j)/2, q/2)}, where q = (q_1,...,q_m) with q_k = p_{k+1} + ... + p_m, and j = (j_1,...,j_m) with j_k = 1:p_k.

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 21:08:45 (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: LRT02_EqualityMeans(), LRT03_EqualityCovariances(), LRT04_EqualityPopulations(), LRT05_Sphericity()

Examples

## EXAMPLE
# LRT for testing hypothesis about independence
# Null distribution of the minus log-transformed LRT statistic
n <- 30                  # sample size
p <- c(3,4,5,6,7)        # dimensions of X_k,k = 1,...,m where m = 5
# W <- vector()          # observed value of W = -log(Lambda)
options <- list()
# options$coef <- -1
options$prob <- c(0.9,0.95,0.99)
output <- LRT01_Independence(n = n, p = p, options = options)
str(output)

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