imhof: Imhof method.

In CompQuadForm: Distribution Function of Quadratic Forms in Normal Variables

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 (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 \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=2}^h X_i^2, where X_1,...,X_h 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 ([e-mail]) and Pierre Duchesne ([e-mail])

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)

CompQuadForm documentation built on May 1, 2019, 7:57 p.m.