farebrother {CompQuadForm}R Documentation

Ruben/Farebrother method

Description

Distribution function (survival function in fact) of quadratic forms in normal variables using Farebrother's algorithm.

Usage

farebrother(q,lambda,h = rep(1, length(lambda)),delta = rep(0, length(lambda)),maxit=100000,eps=10^(-10),mode=1)

Arguments

q

value point at which distribution function is to be evaluated

lambda

the weights λ_1, λ_2, ..., λ_n, i.e. the distinct non-zero characteristic roots of A.Sigma

h

vector of the respective orders of multiplicity m_i of the lambdas

delta

the non-centrality parameters delta_i

maxit

the maximum number of term K in equation below

eps

the desired level of accuracy

mode

if mode>0 then β=mode*λ_{min} otherwise β=β_B=2/(1/λ_{min}+1/λ_{max})

Details

Computes P[Q>q] where Q=sum_{j=1}^n lambda_j chi^2(m_j,delta_j^2). P[Q<q] is approximated by ∑_k=0^{K-1} a_k P[χ^2(m+2k)<q/β] where m=∑_{j=1}^n m_j and β is an arbitrary constant (as given by argument mode).

Value

Qq P[Q>q]

Author(s)

Pierre Lafaye de Micheaux (lafaye@dms.umontreal.ca) and Pierre Duchesne (duchesne@dms.umontreal.ca)

References

P. Duchesne, P. Lafaye de Micheaux, Computing the distribution of quadratic forms: Further comparisons between the Liu-Tang-Zhang approximation and exact methods, Computational Statistics and Data Analysis, Volume 54, (2010), 858-862

Farebrother R.W., Algorithm AS 204: The distribution of a Positive Linear Combination of chi-squared random variables, Journal of the Royal Statistical Society, Series C (applied Statistics), Vol. 33, No. 3 (1984), p. 332-339

Examples

# Some results from Table 3, p.327, Davies (1980)

 farebrother(1,c(6,3,1),c(1,1,1),c(0,0,0))


[Package CompQuadForm version 1.3 Index]