cf_LogRV_MeansRatioW: Characteristic function of a linear combination of N...

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

Characteristic function of a linear combination of N independent LOG-TRANSFORMED WEIGHTED MEANS-RATIO random variables

Description

cf_LogRV_MeansRatioW(t, n, alpha, weight, coef, niid) evaluates characteristic function of a linear combination (resp. convolution) of independent LOG-TRANSFORMED WEIGHTED MEANS-RATIO random variables (RVs) W_i = log(R_i), for i = 1,...,N, where each R_i = G_i/A_i is a ratio of the weighted geometric mean G_i and the (unweighted) arithmetic mean A_i of independent RVs X_{i,1},...,X_{i,n_i} with GAMMA distributions, with the shape parameters \alpha_{i,j} and the (common) rate parameter \beta_i for i = 1,...,N and j = 1,...,n_i.

That is, cf_LogRV_MeansRatioW 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(R_i), where R_i is the ratio statistic of the weighted geometric mean and the arithmetic mean of independent gamma distributed RVs, which distribution depends on the shape parameter \alpha_{i,j} and the vector of weights w_{i,j}, for i = 1,...,N and j = 1,...,n_i.

Here, for each fixed i = 1,...,N, the weighted geometric mean is defined by G_i = (X_{i,1}^w_{i,1} *...* X_{i,n_i}^w_{i,n_i}), where the weights w_{i,j}, j = 1,...,n_i, are such that w_{i,1} +...+ w_{i,n_i} = 1, and the arithmetic mean is defined by A_i = (X_{i,1} +...+ X_{i,n_i})/n_i. The random variables X_{i,j} are mutually independent with X_{i,j} ~ \Gamma(\alpha_{i,j},\beta_i) for all i = 1,...,N and j = 1,...,n_i, where \alpha_{i,j} are the shape parameters and \beta_i is the (common) rate parameter of the gamma distributions.

Note that the weighted ratio random variables R_i are scale invariant, so their distribution does not depend on the common rate (or scale) parameter \beta_i for each i = 1,...,N.

The distribution of the logarithm of the means ratio log(R_i) is defined by its characteristic function, see e.g. Chao and Glaser (JASA 1978), which is

cf_{log(R_i)}(t) = (n_i)^(1i*t) * ... \Gamma(sum(\alpha_{i,j}))/\Gamma(sum(\alpha_{i,j})+1i*t)) * ... Prod_{j=1}^n_i \Gamma(\alpha_{i,j}+1i*w_{i,j}*t)/\Gamma(\alpha_{i,j})

, for each i = 1,...,N.

Usage

cf_LogRV_MeansRatioW(t, n, alpha, weight, coef, niid)

Arguments

t	vector or array of real values, where the CF is evaluated.
n	vector of sample size parameters n = (n_1,...,n_N). If empty, default value is n = (1,...,1).
alpha	list of weights vectors with shape parameters alpha[[i]] = (\alpha_{i,1},...,alpha_{i,n_i}). If empty, default value is alpha[[i]] = c(1,...,1).
weight	list of weights vectors with weights[[i]] = (w_{i,1},...,w_{i,n_i}), for i = 1,...,N. If empty, default value is weights[[i]] = c(1/k_i,...,1/k_i).
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 iid 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 value is niid = 1.

Value

Characteristic function cf(t) of a linear combination of N independent LOG-TRANSFORMED WEIGHTED MEANS-RATIO random variables.

Note

Ver.: 05-Oct-2018 17:54:23 (consistent with Matlab CharFunTool v1.3.0, 17-Jun-2017 17:18:39).

References

Glaser, R. E. (1976a). The ratio of the geometric mean to the arithmetic mean for a random sample from a gamma distribution. Journal of the American Statistical Association, 71(354), 480-487.

Glaser, R. E. (1976b). Exact critical values for Bartlett's test for homogeneity of variances. Journal of the American Statistical Association, 71(354), 488-490.

Chao, M. T., & Glaser, R. E. (1978). The exact distribution of Bartlett's test statistic for homogeneity of variances with unequal sample sizes. Journal of the American Statistical Association, 73(362), 422-426.

See Also

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

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_MeansRatio(), cf_LogRV_WilksLambdaNC(), cf_LogRV_WilksLambda(), cf_Normal(), cf_RectangularSymmetric(), cf_Student(), cf_TSPSymmetric(), cf_TrapezoidalSymmetric(), cf_TriangularSymmetric(), cf_vonMises()

Examples

## EXAMPLE 1
# CF of log MeansRatio RV with n = 5 and alpha = 7/2
n <- 5
alpha <- 7/2
t <- seq(-100, 100, length.out = 201)
plotReIm(function(t) cf_LogRV_MeansRatioW(t, n, alpha), t,
         title = 'CF of log MeansRatio RV with n = 5 and alpha = 7/2')

## EXAMPLE 2
# CF of a weighted linear combination of minus log MeansRatio RVs
rm(list=ls())
n <- c(5, 7, 10)
alpha <- list(c(7/2), c(10/2), c(3/2))
weight = list()
coef <- -1/3
t <- seq(-100, 100, length.out = 201)
plotReIm(function(t) cf_LogRV_MeansRatioW(t, n, alpha, weight, coef), t,
         title = 'CF of a weighted linear combination of minus log MeansRatio RVs')

## EXAMPLE 3
# PDF/CDF of minus log MeansRatio RV, n = 5 and alpha = 7/2, from its CF
rm(list=ls())
n <- 5
alpha <- 7/2
coef <- -1
cf <- function(t) cf_LogRV_MeansRatioW(t, n, alpha, coef = coef)
x <- seq(0, 0.6, 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
# PDF/CDF of minus log of the (weighted) means ratio RV, from its CF
rm(list=ls())
n <- 5
alpha <- list(c(3, 5, 7, 10, 3) / 2)
weight <- list(alpha[[1]] / sum(alpha[[1]]))
coef <- -1
cf <- function(t) cf_LogRV_MeansRatioW(t, n, alpha, weight, coef)
x <- seq(-0.15, 1.15, length.out = 200)
prob <- c(0.9, 0.95, 0.99)
options <- list()
options$N <- 2^12
result <- cf2DistGP(cf, prob = prob, options = options)
str(result)

## EXAMPLE 5
# Compare the exact distribution with the Bartlett's approximation
rm(list=ls())
k <- 15 # k normal populations with unequal sample sizes
df <- c(1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3) # degrees of freedom
DF <- sum(df)
alpha <- list(df / 2)
weight <- list(alpha[[1]] / sum(alpha[[1]]))
C <- DF * log(k * prod(weight[[1]] ^ weight[[1]]))
C_B <- 1 + 1 / (3 * (k - 1)) * (sum(1 / df) - 1 / DF)
shift <- C / C_B
coef <- -DF / C_B
cf_R <- function(t) cf_LogRV_MeansRatioW(t, k, alpha, weight, coef)
cf <- function(t) exp(1i * t * shift) * cf_R(t)
prob <- c(0.9, 0.95, 0.99)
options <- list()
options$xMin <- 0
options$N <- 2^12
result <- cf2DistGP(cf, prob = prob, options = options)
str(result)
x <- result$x
matplot(cbind(x, x), cbind(result$cdf, pchisq(x, k - 1)),
        xlab = 'corrected test statistic', ylab = 'CDF',
        main = 'Exact CDF vs. the Bartlett approximation')
matplot(cbind(x, x), cbind(result$cdf, pchisq(x, k - 1)),
        xlab = 'corrected test statistic', ylab = 'CDF',
        main = 'Exact CDF vs. the Bartlett approximation',
        type = "l", lwd = 2)
print(prob)
print(result$qf)
print(qchisq(prob, k - 1))

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