pQF_depratio: Cumulative Distribution Function of the Dependent QFs Ratio

Description

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

Usage

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

Value

The values of the CDF at quantiles q.

Arguments

q: quantile.
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.

Details

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.