pQF_depratio: Cumulative Distribution Function of the Dependent QFs Ratio
In QF: Density, Cumulative and Quantile Functions of Quadratic Forms

pQF_depratio

R Documentation

Cumulative Distribution Function of the Dependent QFs Ratio

The function computes the CDF of the ratio of two dependent and possibly indefinite quadratic forms.

pQF_depratio(
  q = NULL,
  lambdas = NULL,
  A = NULL,
  B = NULL,
  eps = 1e-06,
  maxit_comp = 1e+05,
  lambdas_tol = NULL
)

`q`	vector of quantiles.
`lambdas`	vector of eigenvalues of the matrix (A-qB).
`A`	matrix of the numerator QF. If not specified but `B` is passed, it is assumed to be the identity.
`B`	matrix of the numerator QF. If not specified but `A` is passed, it is assumed to be the identity.
`eps`	requested absolute error.
`maxit_comp`	maximum number of iterations.
`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.

The distribution function of the following ratio of dependent quadratic forms is computed:

P\left(\frac{Y^TAY }{Y^TBY}<q\right),

where Y\sim N(0, I).

The transformation to the following indefinite quadratic form is exploited:

P\left(Y^T(A-qB)Y <0\right).

The following inputs can be provided:

vector lambdas that contains the eigenvalues of the matrix (A-qB). Input q is ignored.
matrix A and/or matrix B: in these cases q is required to be not null and an eventual missing specification of one matrix make it equal to the identity.