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 AΣA\Sigma

h

respective orders of multiplicity of the λ\lambdas

delta

non-centrality parameters (should be positive)

epsabs

absolute accuracy requested

epsrel

relative accuracy requested

limit

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

Details

Let X=(X1,,Xn)\boldsymbol{X}=(X_1,\ldots,X_n)' be a column random vector which follows a multidimensional normal law with mean vector 0\boldsymbol{0} and non-singular covariance matrix Σ\boldsymbol{\Sigma}. Let μ=(μ1,,μn)\boldsymbol{\mu}=(\mu_1,\ldots,\mu_n)' be a constant vector, and consider the quadratic form

Q=(x+μ)A(x+μ)=r=1mλrχhr;δr2.Q=(\boldsymbol{x}+\boldsymbol{\mu})'\boldsymbol{A}(\boldsymbol{x}+\boldsymbol{\mu})=\sum_{r=1}^m\lambda_r\chi^2_{h_r;\delta_r}.

The function imhof computes P[Q>q]P[Q>q].

The λr\lambda_r's are the distinct non-zero characteristic roots of AΣA\Sigma, the hrh_r's their respective orders of multiplicity, the δr\delta_r's are certain linear combinations of μ1,,μn\mu_1,\ldots,\mu_n and the χhr;δr2\chi^2_{h_r;\delta_r} are independent χ2\chi^2-variables with hrh_r degrees of freedom and non-centrality parameter δr\delta_r. The variable χh,δ2\chi^2_{h,\delta} is defined here by the relation χh,δ2=(X1+δ)2+i=2hXi2\chi^2_{h,\delta}=(X_1 + \delta)^2+\sum_{i=2}^hX_i^2, where X1,,XhX_1,\ldots,X_h are independent unit normal deviates.

Value

Qq

P[Q>q]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 1.4.3 Index]