CompQuadForm (version 1.4.3)

davies: Davies method

Description

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

Usage

davies(q, lambda, h = rep(1, length(lambda)), delta = rep(0,
         length(lambda)), sigma = 0, lim = 10000, acc = 0.0001)

Arguments

q

value point at which distribution function is to be evaluated

lambda

the weights \(\lambda_1, \lambda_2, ..., \lambda_n\), i.e. distinct non-zero characteristic roots of \(A\Sigma\)

h

respective orders of multiplicity \(n_j\) of the \(\lambda\)s

delta

non-centrality parameters \(\delta_j^2\) (should be positive)

sigma

coefficient \(\sigma\) 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.

Value

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: round-off error possibly significant, 3: invalid parameters, 4: unable to locate integration parameters

Qq

\(P[Q>q]\)

Details

Computes \(P[Q>q]\) where \(Q = \sum_{j=1}^r\lambda_jX_j+\sigma X_0\) where \(X_j\) are independent random variables having a non-central \(chi^2\) distribution with \(n_j\) degrees of freedom and non-centrality parameter \(delta_j^2\) for \(j=1,...,r\) and \(X_0\) having a standard Gaussian distribution.

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 Davies R.B., Algorithm AS 155: The Distribution of a Linear Combination of chi-2 Random Variables, Journal of the Royal Statistical Society. Series C (Applied Statistics), 29(3), p. 323-333, (1980)

Examples

# 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)