Weights {MVT} | R Documentation |
Density, distribution function and quantile function for the weights distribution arising from the multivariate t-distribution with dimension dim
and
shape parameter eta
.
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)
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 |
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 |
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.
dweights
gives the density, pweights
the distribution function,
and qweights
the quantile function.
Invalid arguments will result in return value NaN
, with a warning.
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 Unpublished manuscript.
Distributions for other standard distributions.
beta
for the Beta function.
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