cf_LogRV_WilksLambda: Characteristic function of a linear combination of...

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

Characteristic function of a linear combination of independent LOG-TRANSFORMED WILK's LAMBDA distributed random variables

Description

cf_LogRV_WilksLambda(t, p, n, q, coef, niid) evaluates characteristic function of a linear combination (resp. convolution) of independent LOG-TRANSFORMED WILK's LAMBDA distributed random variables.

Usage

cf_LogRV_WilksLambda(t, p, n, q, coef, niid)

Arguments

t	vector or array of real values, where the CF is evaluated.
p	vector of the dimension parameters p = (p_1,...,p_N). If empty, default value is p = (1,...,1).
n	vector of degrees of freedom of the Wishart matrices E_i, n = (n_1,...,n_N). If empty, default value is n = (1,...,1).
q	vector of degrees of freedom of the Wishart matrices H_i, q = (q_1,...,q_N). If empty, default value is q = (1,...,1).
coef	vector of the coefficients of the linear combination of the log-transformed random variables. If coef is scalar, it is assumed that all coefficients are equal. If empty, default value is coef = 1.
niid	scalar convolution coeficient niid, such that Z = Y + ... + Y is sum of niid random variables Y, where each Y = sum_{i=1}^N coef(i) * log(X_i) is independently and identically distributed random variable. If empty, default niid = 1.

Details

cf_LogRV_WilksLambda evaluates the characteristic function of a random variable Y = coef_1*W_1 +...+ coef_N*W_N, such that cf_Y(t) = cf_W_1(coef_1*t) *...* cf_W_N(coef_N*t), where cf_W_i(t) is CF of W_i = log(\Lambda_i), and each \Lambda_i has central WILK's LAMBDA distribution, \Lambda_i ~ Lambda(p_i,n_i,q_i), for i = 1,...,N. In particular \Lambda_i = det(E_i)/det(E_i + H_i), with E_i and H _i being independent random matrices with central Wishart distributions, E_i ~ Wp_i(n_i,\Sigma_i) and H_i ~ Wp_i(q_i,\Sigma_i) with n_i >= p_i and \Sigma_i > 0 (an unknown positive definite symmetric covariance matrix), for all i = 1,...,N.

Each particular Lambda_i statistic can be used to test specific null hypothesis (by measuring its effect expressed by the matrix H_i). In this case, the null hypothesis is rejected for small values of the observed statistic \Lambda_i, or large value of -log(\Lambda_i).

The central Wilks' distribution of \Lambda_i ~ Lambda(p_i,n_i,q_i), with Lambda_i in (0,1), is defined by \Lambda_i ~ Prod_{j=1}^p_i B_{i,j}, where B_{i,j} ~ Beta{(n_i+1-j)/2, q_i/2)}, i.e. B_{i,j} follow independent Beta distributions for all i = 1,...,N and j = 1,...,p_i.

Value

Characteristic function cf(t) of a linear combination of independent LOG-TRANSFORMED WILK's LAMBDA distributed random variables.

Note

Ver.: 04-Oct-2018 13:30:38 (consistent with Matlab CharFunTool v1.3.0, 19-Jul-2018 17:23:23).

References

[1] Witkovsky, V., 2018. Exact distribution of selected multivariate test criteria by numerical inversion of their characteristic functions. arXiv preprint arXiv:1801.02248.

See Also

For more details see WIKIPEDIA: https://en.wikipedia.org/wiki/Wilks

Other Continuous Probability Distribution: cfS_Arcsine(), cfS_Beta(), cfS_Gaussian(), cfS_Laplace(), cfS_Rectangular(), cfS_Student(), cfS_TSP(), cfS_Trapezoidal(), cfS_Triangular(), cfS_Wigner(), cfX_ChiSquare(), cfX_Exponential(), cfX_FisherSnedecor(), cfX_Gamma(), cfX_InverseGamma(), cfX_LogNormal(), cf_ArcsineSymmetric(), cf_BetaNC(), cf_BetaSymmetric(), cf_Beta(), cf_ChiSquare(), cf_Exponential(), cf_FisherSnedecorNC(), cf_FisherSnedecor(), cf_Gamma(), cf_InverseGamma(), cf_Laplace(), cf_LogRV_BetaNC(), cf_LogRV_Beta(), cf_LogRV_ChiSquareNC(), cf_LogRV_ChiSquare(), cf_LogRV_FisherSnedecorNC(), cf_LogRV_FisherSnedecor(), cf_LogRV_MeansRatioW(), cf_LogRV_MeansRatio(), cf_LogRV_WilksLambdaNC(), cf_Normal(), cf_RectangularSymmetric(), cf_Student(), cf_TSPSymmetric(), cf_TrapezoidalSymmetric(), cf_TriangularSymmetric(), cf_vonMises()

Examples

## EXAMPLE 1
# CF of log Wilks Lambda RV distribution Lambda(p,n,q)
p <- 5
n <- 10 # d.f. of within SS&P
q <- 3 # d.f. of between SS&P
t <- seq(-10, 10, length.out = 201)
plotReIm(function(t)
        cf_LogRV_WilksLambda(t, p, n, q),
        t,
        title = 'CF of log Wilks Lambda RV with Lambda(p,n,q), p=5, n=10, q=3')

## EXAMPLE 2
# CF of a weighted linear combination of minus log Wilks Lambda RVs
p <- c(5, 5, 5)
n <- c(10, 15, 20)
q <- c(3, 2, 1)
coef <- -c(10, 15, 20) / 45
t <- seq(-20, 20, length.out = 201)
plotReIm(function(t)
        cf_LogRV_WilksLambda(t, p, n, q, coef),
        t,
        title = 'CF of a weighted linear combination of -log Wilks Lambda RVs')

## EXAMPLE 3
# PDF/CDF of minus log Wilks Lambda RV (p=5, n=10, q=3) from its CF
p <- 5
n <- 10
q <- 3
coef <- -1
cf <- function(t) {cf_LogRV_WilksLambda(t, p, n, q, coef)}
x <- seq(0, 5, length.out = 100)
prob <- c(0.9, 0.95, 0.99)
options <- list()
options$xMin <- 0
result <- cf2DistGP(cf, x, prob, options)
str(result)

## EXAMPLE 4
# Compare the exact distribution with the Bartlett's approximation
# The Bartlett's approximation (see e.g. Wikipedia) is given by:
# ((p-q+1)/2 - n)*log(Lambda(p,n,q)) ~ chi^2_{q*p}
p <- 15
n <- 30
q <- 3
coef <- (p - q + 1) / 2 - n
cf <- function(t) {cf_LogRV_WilksLambda(t, p, n, q, coef)}
prob <- c(0.9, 0.95, 0.99)
options <- list()
options$xMin <- 0
result <- cf2DistGP(cf, prob = prob, options = options)
str(result)
x <- result$x
matplot(cbind(x, x), cbind(result$cdf, pchisq(x, q * p)),
        main = 'Exact CDF vs. the Bartlett approximation',
        xlab = expression(paste("transformed ", Lambda, "(p,n,q)")),
        ylab = 'CDF')
matplot(cbind(x, x), cbind(result$cdf, pchisq(x, q * p)),
        main = 'Exact CDF vs. the Bartlett approximation',
        xlab = expression(paste("transformed ", Lambda, "(p,n,q)")),
        ylab = 'CDF', lwd = 2, type = "l")
prob
result$qf
qchisq(prob, q * p)

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