QF: QF_ratio – R documentation

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QF_ratio

Ratio of Positive Definite Quadratic Forms Distribution

Description

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

Usage

dQF_ratio(x, obj)

pQF_ratio(q, obj)

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

rQF_ratio(
  n,
  lambdas_num,
  lambdas_den,
  etas_num = rep(0, length(lambdas_num)),
  etas_den = rep(0, length(lambdas_den))
)

Arguments

`x, q`	vector of quantiles.
`obj`	`MellinQF_ratio` object produced by the `compute_MellinQF_ratio` 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_num`	vector of positive weights of the QF at the numerator.
`lambdas_den`	vector of positive weights of the QF at the denominator
`etas_num`	vector of non-centrality parameters of the QF at the numerator. Default all zeros.
`etas_den`	vector of non-centrality parameters of the QF at the denominator Default all zeros.

Details

The CDF and PDF of the ratio of positive QFs are evaluated by numerical inversion of the Mellin transform. The absolute error specified in compute_MellinQF_ratio is guaranteed for values of q and x inside 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 ratio of QFs quantiles related to probabilities p.

Value

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

pQF_ratio provides the cumulative distribution function at a quantile q.

qQF_ratio provides the quantile corresponding to a probability level p.

rQF_ratio provides a sample of n independent realizations the QFs ratio.

Examples

lambdas_QF_num <- c(rep(7, 6),rep(3, 2))
etas_QF_num <- c(rep(6, 6), rep(2, 2))
lambdas_QF_den <- c(0.6, 0.3, 0.1)
# Computation Mellin transform
eps <- 1e-7
rho <- 0.999
Mellin_ratio <- compute_MellinQF_ratio(lambdas_QF_num, lambdas_QF_den,
                                       etas_QF_num, eps = eps, rho = rho)
xs <- seq(Mellin_ratio$range_q[1], Mellin_ratio$range_q[2], l = 100)
# PDF
ds <- dQF_ratio(xs, Mellin_ratio)
plot(xs, ds, type="l")
# CDF
ps <- pQF_ratio(xs, Mellin_ratio)
plot(xs, ps, type="l")
# Quantile
qs <- qQF_ratio(ps, Mellin_ratio)
plot(ps, qs, type="l")
#Comparison computed quantiles vs real quantiles
plot((qs - xs) / xs, type = "l")

QF

Density, Cumulative and Quantile Functions of Quadratic Forms

v0.0.4

GPL-3

Authors

Aldo Gardini [aut, cre], Fedele Greco [aut], Carlo Trivisano [aut]

Initial release

2021-02-17