R: Ruben/Farebrother method

farebrother {CompQuadForm} | R Documentation

## Ruben/Farebrother method

### Description

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

### Usage

    farebrother(q, lambda, h = rep(1, length(lambda)),
                delta = rep(0, length(lambda)), maxit = 100000,
                eps = 10^(-10), mode = 1)

### Arguments

`q` | value point at which distribution function is to be evaluated
`lambda` | the weights λ_1, λ_2, ..., λ_n, i.e. the distinct non-zero characteristic roots of A.Sigma
`h` | vector of the respective orders of multiplicity m_i of the lambdas
`delta` | the non-centrality parameters δ_i (should be positive)
`maxit` | the maximum number of term K in equation below
`eps` | the desired level of accuracy
`mode` | if 'mode' > 0 then β=mode*λ_{min} otherwise β=β_B=2/(1/λ_{min}+1/λ_{max})

### Details

Computes P[Q>q] where Q=sum_{j=1}^n lambda_j chi^2(m_j,delta_j^2). P[Q<q] is approximated by ∑_k=0^{K-1} a_k P[χ^2(m+2k)<q/β] where m=∑_{j=1}^n m_j and β is an arbitrary constant (as given by argument mode).

### Value

`dnsty` | the density of the linear form
`ifault` | the fault indicator. -i: one or more of the constraints λ_i>0

, m_i>0 and δ_i^2≥q0 is not satisfied. 1: non-fatal underflow of a_0. 2: one or more of the constraints n>0, q>0, maxit>0 and eps>0 is not satisfied. 3: the current estimate of the probability is greater than 2. 4: the required accuracy could not be obtained in 'maxit' iterations. 5: the value returned by the procedure does not satisfy 0≤q RUBEN≤q 1. 6: 'dnsty' is negative. 9: faults 4 and 5. 10: faults 4 and 6. 0: otherwise.

`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

Farebrother R.W., Algorithm AS 204: The distribution of a Positive Linear Combination of chi-squared random variables, Journal of the Royal Statistical Society, Series C (applied Statistics), Vol. 33, No. 3 (1984), p. 332-339

### Examples

    # Some results from Table 3, p.327, Davies (1980)

     1 - farebrother(1, c(6, 3, 1), c(1, 1, 1), c(0, 0, 0))$Qq

[Package CompQuadForm version 1.4.3 Index]