QF: Positive Definite Quadratic Forms Distribution

In QF: Density, Cumulative and Quantile Functions of Quadratic Forms

Positive Definite Quadratic Forms Distribution

Description

Density function, distribution function, quantile function and random generator for positive definite QFs.

Usage

dQF(x, obj)

pQF(q, obj)

qQF(p, obj, eps_quant = 1e-06, maxit_quant = 10000)

rQF(n, lambdas, etas = rep(0, length(lambdas)))

Arguments

x, q	vector of quantiles.
obj	MellinQF object produced by the compute_MellinQF function.
p	vector of probabilities.
eps_quant	relative error for quantiles.
maxit_quant	maximum number of Newton-Raphson iterations allowed to compute quantiles.
n	number of observations.
lambdas	vector of positive weights.
etas	vector of non-centrality parameters. Default all zeros.

Details

The quadratic form CDF and PDF are evaluated by numerical inversion of the Mellin transform. The absolute error specified in compute_MellinQF is guaranteed for values of q and x inside the range_q. If the quantile is outside range_q, computations are carried out, but a warning is sent.

The function uses the Newton-Raphson algorithm to compute the QF quantiles related to probabilities p.

Value

dQF provides the values of the density function at a quantile x.

pQF provides the cumulative distribution function at a quantile q.

qQF provides the quantile corresponding to a probability level p.

rQF provides a sample of n independent realizations from the QF.

See Also

See compute_MellinQF for details on the Mellin computation.

Examples

library(QF)
# Definition of the QF
lambdas_QF <- c(rep(7, 6),rep(3, 2))
etas_QF <- c(rep(6, 6), rep(2, 2))
# Computation Mellin transform
eps <- 1e-7
rho <- 0.999
Mellin <- compute_MellinQF(lambdas_QF, etas_QF, eps = eps, rho = rho)
xs <- seq(Mellin$range_q[1], Mellin$range_q[2], l = 100)
# PDF
ds <- dQF(xs, Mellin)
plot(xs, ds, type="l")
# CDF
ps <- pQF(xs, Mellin)
plot(xs, ps, type="l")
# Quantile
qs <- qQF(ps, Mellin)
plot(ps, qs, type="l")
#Comparison computed quantiles vs real quantiles
plot((qs - xs) / xs, type = "l")

QF documentation built on April 3, 2025, 9:23 p.m.