liu {CompQuadForm} | R Documentation |
Distribution function (survival function in fact) of quadratic forms in normal variables using Liu et al.'s method.
liu(q, lambda, h = rep(1, length(lambda)), delta = rep(0, length(lambda)))
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 |
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.
Qq |
P[Q>q] |
Pierre Lafaye de Micheaux (lafaye@dms.umontreal.ca) and Pierre Duchesne (duchesne@dms.umontreal.ca)
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
# 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)