imhof {CompQuadForm}R Documentation

Imhof method.

Description

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

Usage

imhof(q, lambda, h = rep(1, length(lambda)), delta = rep(0, length(lambda)), epsabs = 10^(-6), epsrel = 10^(-6), limit = 10000)

Arguments

q

value point at which the survival function is to be evaluated

lambda

distinct non-zero characteristic roots of A.Sigma

h

respective orders of multiplicity of the lambdas

delta

non-centrality parameters

epsabs

absolute accuracy requested

epsrel

relative accuracy requested

limit

limit determines the maximum number of subintervals in the partition of the given integration interval

Details

Let \strong{X}=(X_1,...,X_n)' be a column random vector which follows a multidimensional normal law with mean vector \strong{0} and non-singular covariance matrix \strong{Sigma}. Let \strong{mu}=(mu_1,...,mu_n)' be a constant vector, and consider the quadratic form

Q = (\strong{x}+\strong{mu})'\strong{A}(\strong{x}+\strong{mu}) = sum_{r=1}^m lambda_r chi^2_{h_r;δ_r}.

The function imhof computes P[Q>q].

The λ_r's are the distinct non-zero characteristic roots of A.Sigma, the h_r's their respective orders of multiplicity, the delta_r's are certain linear combinations of mu_1,...,mu_n and the chi^2_{h_r;delta_r} are independent chi^2-variables with h_r degrees of freedom and non-centrality parameter delta_r. The variable chi^2_{h;delta} is defined here by the relation chi^2_{h,delta}=(X_1 + delta)^2+ sum_{i=1}^h X_i^2, where X_1,...,X_n are independent unit normal deviates.

Value

Qq P[Q>q]
abserr

estimate of the modulus of the absolute error, which should equal or exceed abs(i-result)

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

J. P. Imhof, Computing the Distribution of Quadratic Forms in Normal Variables, Biometrika, Volume 48, Issue 3/4 (Dec., 1961), 419-426

Examples

# Some results from Table 1, p.424, Imhof (1961)

# Q1 with x = 2
round(imhof(2,c(0.6,0.3,0.1))$Qq,4)

# Q2 with x = 6
round(imhof(6,c(0.6,0.3,0.1),c(2,2,2))$Qq,4)

# Q6 with x = 15
round(imhof(15,c(0.7,0.3),c(1,1),c(6,2))$Qq,4)


[Package CompQuadForm version 1.3 Index]