QF {QF}

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")