compute_MellinQF {QF}

Mellin Transform of a Positive QF

Description

The function computes the Mellin transform of a positive definite quadratic form producing a MellinQF object. The output can be used to evaluate the density, cumulative and quantile functions of the target quadratic form.

Usage

compute_MellinQF(
  lambdas,
  etas = rep(0, length(lambdas)),
  eps = 1e-06,
  rho = 1 - 1e-04,
  maxit_comp = 1e+05,
  eps_quant = 1e-06,
  maxit_quant = 10000,
  lambdas_tol = NULL
)

Arguments

lambdas

vector of positive weights.

etas

vector of non-centrality parameters. Default all zeros (central chi square).

eps

required absolute error for density and cumulative functions.

rho

distribution total probability mass for which it is desired to keep the error eps.

maxit_comp

maximum number of iterations.

eps_quant

required numerical error for quantile computation.

maxit_quant

maximum number of iterations before stopping the quantile computation.

lambdas_tol

maximum value admitted for the weight skewness. When it is not NULL (default), elements of lambdas such that the ratio max(lambdas)/lambdas is greater than the specified value are removed.

Details

The quadratic form having positive weights lambdas and non-centrality parameters etas is considered:

Q=\sum_{i=1}^r \lambda_i\chi^2_{1,\eta_i}.

Its Mellin transform is computed by exploiting the density formulation by Ruben (1962). The numerical error is controlled in order to provide the requested precision (eps) for the interval of quantiles that contains the specified total probability rho.

The argument eps_quant controls the relative precision requested for the computation of quantiles that determine the range in which the error eps is guaranteed, whereas maxit_quant sets the maximum number of Newton-Raphson iterations of the algorithm.

Value

The function returns an object of the class MellinQF that contains information on the Mellin transform of a linear combination of positively weighted chi-square random variables. This information can be used in order to evaluate the density, cumulative distribution and quantile functions.

An object of the class MellinQF has the following components:

• range_q: the range of quantiles that contains the specified mass of probability rho in which it is possible to compute density and CDF preserving the error level eps.

• Mellin: a list containing the values of the Mellin transform (Mellin), the corresponding evaluation points (z), the integration step delta and the lowest weight (lambda_min).

• the inputs rho, lambdas, etas, eps needed for CDF, PDF and quantile function computation.

Source

Ruben, Harold. "Probability content of regions under spherical normal distributions, IV: The distribution of homogeneous and non-homogeneous quadratic functions of normal variables." The Annals of Mathematical Statistics 33.2 (1962): 542-570.

See Also

The function print.MellinQF can be used to summarize the basic information on the Mellin transform.

The object can be used in the function dQF to compute the density function of the QF, pQF for the CDF and qQF for the quantile function.

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)
print(Mellin)