Description Usage Arguments Details Value Author(s) References Examples
Distribution function (survival function in fact) of quadratic forms in normal variables using Liu et al.'s method.
1 2 
q 
value point at which the survival function is to be evaluated 
lambda 
distinct nonzero characteristic roots of A.Sigma, i.e. the λ_i's 
h 
respective orders of multiplicity h_i's of the lambda's 
delta 
noncentrality parameters δ_i's (should be positive) 
New chisquare approximation to the distribution of nonnegative definite quadratic forms in noncentral 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 LiuTangZhang approximation and exact methods, Computational Statistics and Data Analysis, Volume 54, (2010), 858862
H. Liu, Y. Tang, H.H. Zhang, A new chisquare approximation to the distribution of nonnegative definite quadratic forms in noncentral normal variables, Computational Statistics and Data Analysis, Volume 53, (2009), 853856
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21  # 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)

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