QF: compute_MellinQF_ratio – R documentation

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compute_MellinQF_ratio

Mellin Transform of the Independent Positive QFs Ratio

Description

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

Usage

compute_MellinQF_ratio(
  lambdas_num,
  lambdas_den,
  etas_num = rep(0, length(lambdas_num)),
  etas_den = rep(0, length(lambdas_den)),
  eps = 1e-06,
  rho = 1 - 1e-04,
  maxit_comp = 1e+05,
  eps_quant = 1e-06,
  maxit_quant = 10000,
  lambdas_tol = NULL
)

Arguments

`lambdas_num`	vector of positive weights for the numerator.
`lambdas_den`	vector of positive weights for the denominator.
`etas_num`	vector of non-centrality parameters for the numerator. Default all zeros (central chi square).
`etas_den`	vector of non-centrality parameters for the denominator. 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 for both the numerator and the denominator. 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 Mellin transform of the ratio of two independent quadratic forms having positive weights lambdas_num and lambdas_den and non-centrality parameters etas_num and etas_den is computed 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_ratio that contains information on the Mellin transform of the ratio of two linear combinations of positively weighted chi-square random variables. This information can be used in order to evaluate the density, cumulative distribution and quantile functions of the desired quadratic form.

An object of the class MellinQF_ratio 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 numerator weight (lambda_min).
the inputs rho, lambdas_num, lambdas_den, etas_num, etas_den, 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.

Examples

library(QF)
# Definition of the QFs
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)
print(Mellin_ratio)

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