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

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

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)

Results

R version 3.3.1 (2016-06-21) -- "Bug in Your Hair"
Copyright (C) 2016 The R Foundation for Statistical Computing
Platform: x86_64-pc-linux-gnu (64-bit)

R is free software and comes with ABSOLUTELY NO WARRANTY.
You are welcome to redistribute it under certain conditions.
Type 'license()' or 'licence()' for distribution details.

R is a collaborative project with many contributors.
Type 'contributors()' for more information and
'citation()' on how to cite R or R packages in publications.

Type 'demo()' for some demos, 'help()' for on-line help, or
'help.start()' for an HTML browser interface to help.
Type 'q()' to quit R.

> library(CompQuadForm)
> png(filename="/home/ddbj/snapshot/RGM3/R_CC/result/CompQuadForm/imhof.Rd_%03d_medium.png", width=480, height=480)
> ### Name: imhof
> ### Title: Imhof method.
> ### Aliases: imhof
> ### Keywords: distribution htest
> 
> ### ** Examples
> 
> # Some results from Table 1, p.424, Imhof (1961)
> 
> # Q1 with x = 2
> round(imhof(2,c(0.6,0.3,0.1))$Qq,4)
[1] 0.124
> 
> # Q2 with x = 6
> round(imhof(6,c(0.6,0.3,0.1),c(2,2,2))$Qq,4)
[1] 0.0161
> 
> # Q6 with x = 15
> round(imhof(15,c(0.7,0.3),c(1,1),c(6,2))$Qq,4)
[1] 0.0223
> 
> 
> 
> 
> 
> 
> dev.off()
null device 
          1 
>