Weights {MVT}R Documentation

Distribution of the weights from a multivariate t-distribution

Description

Density, distribution function and quantile function for the weights distribution arising from the multivariate t-distribution with dimension dim and shape parameter eta.

Usage

dweights(x, eta = .25, dim, log = FALSE, scaled = TRUE)
pweights(q, eta = .25, dim, lower.tail = TRUE, log.p = FALSE, scaled = TRUE)
qweights(p, eta = .25, dim, lower.tail = TRUE, log.p = FALSE, scaled = TRUE)

Arguments

x, q

vector of quantiles.

p

vector of probabilities.

eta

shape parameter of the multivariate t-distribution, must be in the interval [0,1/2). Default value is eta = 0.25.

dim

dimension of the multivariate t-distribution.

log, log.p

logical; if TRUE, probabilities p are given as \log(p).

lower.tail

logical; if TRUE (default), probabilities are P(X ≤ x), otherwise, P(X > x).

scaled

logical; if TRUE, the weights are scaled to belong the interval (0,1).

Details

The weights' distribution with parameters eta and dim = p has density

f(x) = m^(1-(1/η + p)/2)/B(1/(2η),p/2)x^(1/(2η)-1)(m-x)^(p/2-1)

for 0 ≤ η < 1/2, p > 0 and 0 < x < m, where m = (1 + pη)/(1 - 2η).

The mean is E(X) = 1/(1-2η) and the variance is

2pη^2/((1+(p+2)η)(1-2η)^2)

The scaled version of the weights distribution has a Beta distribution with parameters 1/(2η) and p/2.

Value

dweights gives the density, pweights the distribution function, and qweights the quantile function.

Invalid arguments will result in return value NaN, with a warning.

References

Abramowitz, M., and Stegun, I.A. (1972). Handbook of Mathematical Functions. Dover, New York. Chapter 6: Gamma and related functions.

Johnson, N.L., Kotz, S., and Balakrishnan, N. (1995). Continuous Univariate Distributions, volume 2, 2nd Ed. Wiley, New York. Chapter 25: Beta distributions.

Osorio, F., and Galea, M. (2015). Statistical inference in multivariate analysis using the t-distribution. Unpublished manuscript.

See Also

Distributions for other standard distributions.

beta for the Beta function.

Examples

data(companies)
fit <- studentFit(companies, family = Student(eta = .25))

# compute the 5% quantile from the estimated distribution of the weights
p <- fit$dims[2]
eta <- fit$eta
wts <- fit$weights
cutoff <- qweights(.05, eta = eta, dim = p, scaled = FALSE)

# identify observations with 'small' weights
n <- fit$dims[1]
which <- seq_len(n)[wts < cutoff]
which

[Package MVT version 0.3 Index]