liu {CompQuadForm}R Documentation

Liu's method

Description

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

Usage

liu(q, lambda, h = rep(1, length(lambda)), delta = rep(0, length(lambda)))

Arguments

q

value point at which the survival function is to be evaluated

lambda

distinct non-zero characteristic roots of A.Sigma, i.e. the λ_i's

h

respective orders of multiplicity h_i's of the lambda's

delta

non-centrality parameters δ_i's

Details

New chi-square approximation to the distribution of non-negative definite quadratic forms in non-central normal variables.

Computes P[Q>q] where Q=sum_{j=1}^n lambda_j chi^2(h_j,delta_j).

This method does not work as good as the Imhof's method. Thus Imhof's method should be recommended.

Value

Qq P[Q>q]

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

H. Liu, Y. Tang, H.H. Zhang, A new chi-square approximation to the distribution of non-negative definite quadratic forms in non-central normal variables, Computational Statistics and Data Analysis, Volume 53, (2009), 853-856

Examples

# Some results from Liu et al. (2009)
# Q1 from Liu et al.
round(liu(2,c(0.5,0.4,0.1),c(1,2,1),c(1,0.6,0.8)),6)
round(liu(6,c(0.5,0.4,0.1),c(1,2,1),c(1,0.6,0.8)),6)
round(liu(8,c(0.5,0.4,0.1),c(1,2,1),c(1,0.6,0.8)),6)

# Q2 from Liu et al.
round(liu(1,c(0.7,0.3),c(1,1),c(6,2)),6)
round(liu(6,c(0.7,0.3),c(1,1),c(6,2)),6)
round(liu(15,c(0.7,0.3),c(1,1),c(6,2)),6)

# Q3 from Liu et al.
round(liu(2,c(0.995,0.005),c(1,2),c(1,1)),6)
round(liu(8,c(0.995,0.005),c(1,2),c(1,1)),6)
round(liu(12,c(0.995,0.005),c(1,2),c(1,1)),6)

# Q4 from Liu et al.
round(liu(3.5,c(0.35,0.15,0.35,0.15),c(1,1,6,2),c(6,2,6,2)),6)
round(liu(8,c(0.35,0.15,0.35,0.15),c(1,1,6,2),c(6,2,6,2)),6)
round(liu(13,c(0.35,0.15,0.35,0.15),c(1,1,6,2),c(6,2,6,2)),6)


[Package CompQuadForm version 1.3 Index]