Description Usage Arguments Details Value Author(s) References Examples
Distribution function (survival function in fact) of quadratic forms in normal variables using Davies's method.
1 2 
q 
value point at which distribution function is to be evaluated 
lambda 
the weights λ_1, λ_2, ..., λ_n, i.e. distinct nonzero characteristic roots of A.Sigma 
h 
respective orders of multiplicity n_j of the lambdas 
delta 
noncentrality parameters δ_j^2 (should be positive) 
sigma 
coefficient σ of the standard Gaussian 
lim 
maximum number of integration terms. Realistic values for 'lim' range from 1,000 if the procedure is to be called repeatedly up to 50,000 if it is to be called only occasionally 
acc 
error bound. Suitable values for 'acc' range from 0.001 to 0.00005 which should be adequate for most statistical purposes. 
Computes P[Q>q] where Q = sum_{j=1}^r lambda_j X_j+ sigma X_0 where X_j are independent random variables having a noncentral chi^2 distribution with n_j degrees of freedom and noncentrality parameter delta_j^2 for j=1,...,r and X_0 having a standard Gaussian distribution.
trace 
vector, indicating performance of procedure, with the following components: 1: absolute value sum, 2: total number of integration terms, 3: number of integrations, 4: integration interval in main integration, 5: truncation point in initial integration, 6: standard deviation of convergence factor term, 7: number of cycles to locate integration parameters 
ifault 
fault indicator: 0: no error, 1: requested accuracy could not be obtained, 2: roundoff error possibly significant, 3: invalid parameters, 4: unable to locate integration parameters 
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
Davies R.B., Algorithm AS 155: The Distribution of a Linear Combination of chi2 Random Variables, Journal of the Royal Statistical Society. Series C (Applied Statistics), 29(3), p. 323333, (1980)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37  # Some results from Table 3, p.327, Davies (1980)
round(1  davies(1, c(6, 3, 1), c(1, 1, 1))$Qq, 4)
round(1  davies(7, c(6, 3, 1), c(1, 1, 1))$Qq, 4)
round(1  davies(20, c(6, 3, 1), c(1, 1, 1))$Qq, 4)
round(1  davies(2, c(6, 3, 1), c(2, 2, 2))$Qq, 4)
round(1  davies(20, c(6, 3, 1), c(2, 2, 2))$Qq, 4)
round(1  davies(60, c(6, 3, 1), c(2, 2, 2))$Qq, 4)
round(1  davies(10, c(6, 3, 1), c(6, 4, 2))$Qq, 4)
round(1  davies(50, c(6, 3, 1), c(6, 4, 2))$Qq, 4)
round(1  davies(120, c(6, 3, 1), c(6, 4, 2))$Qq, 4)
round(1  davies(20, c(7, 3), c(6, 2), c(6, 2))$Qq, 4)
round(1  davies(100, c(7, 3), c(6, 2), c(6, 2))$Qq, 4)
round(1  davies(200, c(7, 3), c(6, 2), c(6, 2))$Qq, 4)
round(1  davies(10, c(7, 3), c(1, 1), c(6, 2))$Qq, 4)
round(1  davies(60, c(7, 3), c(1, 1), c(6, 2))$Qq, 4)
round(1  davies(150, c(7, 3), c(1, 1), c(6, 2))$Qq, 4)
round(1  davies(70, c(7, 3, 7, 3), c(6, 2, 1, 1), c(6, 2, 6, 2))$Qq, 4)
round(1  davies(160, c(7, 3, 7, 3), c(6, 2, 1, 1), c(6, 2, 6, 2))$Qq, 4)
round(1  davies(260, c(7, 3, 7, 3), c(6, 2, 1, 1), c(6, 2, 6, 2))$Qq, 4)
round(1  davies(40, c(7, 3, 7, 3), c(6, 2, 1, 1), c(6, 2, 6,
2))$Qq, 4)
round(1  davies(40, c(7, 3, 7, 3), c(6, 2, 1, 1), c(6, 2, 6, 2))$Qq,
4)
round(1  davies(140, c(7, 3, 7, 3), c(6, 2, 1, 1), c(6, 2, 6,
2))$Qq, 4)
# You should sometimes play with the 'lim' parameter:
davies(0.00001,lambda=0.2)
imhof(0.00001,lambda=0.2)$Qq
davies(0.00001,lambda=0.2, lim=20000)

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