Mellin Transform of a Positive QF
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.
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 )
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 |
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. |
The quadratic form having positive weights lambdas and non-centrality parameters etas is considered:
Q=∑_{i=1}^r λ_iχ^2_{1,η_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.
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.
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.
The function print.MellinQF can be used to summarize the basic information on the Mellin transform.
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)
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