CompQuadForm (version 1.4.3)

imhof: 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 \(\lambda\)s

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

Value

Qq

\(P[Q>q]\)

abserr

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

Details

Let \(\boldsymbol{X}=(X_1,\ldots,X_n)'\) be a column random vector which follows a multidimensional normal law with mean vector \(\boldsymbol{0}\) and non-singular covariance matrix \(\boldsymbol{\Sigma}\). Let \(\boldsymbol{\mu}=(\mu_1,\ldots,\mu_n)'\) be a constant vector, and consider the quadratic form $$Q=(\boldsymbol{x}+\boldsymbol{\mu})'\boldsymbol{A}(\boldsymbol{x}+\boldsymbol{\mu})=\sum_{r=1}^m\lambda_r\chi^2_{h_r;\delta_r}.$$ The function imhof computes \(P[Q>q]\). The \(\lambda_r\)'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 \(\mu_1,\ldots,\mu_n\) 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 \(\chi^2_{h,\delta}=(X_1 + \delta)^2+\sum_{i=2}^hX_i^2\), where \(X_1,\ldots,X_h\) are independent unit normal deviates.

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)