Distribution function (survival function in fact) of quadratic forms in normal variables using Farebrother's algorithm.
farebrother(q, lambda, h = rep(1, length(lambda)),
delta = rep(0, length(lambda)), maxit = 100000,
eps = 10^(-10), mode = 1)
the density of the linear form
the fault indicator. -i: one or more of the constraints \(\lambda_i>0\)
, \(m_i>0\) and \(\delta_i^2\geq0\) 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
\(0\leq RUBEN\leq 1\). 6: 'dnsty' is negative. 9: faults 4 and 5. 10: faults 4 and 6. 0: otherwise.
\(P[Q>q]\)
value point at which distribution function is to be evaluated
the weights \(\lambda_1, \lambda_2, ..., \lambda_n\), i.e. the distinct non-zero characteristic roots of \(A\Sigma\)
vector of the respective orders of multiplicity \(m_i\) of the \(\lambda\)s
the non-centrality parameters \(\delta_i\) (should be positive)
the maximum number of term K in equation below
the desired level of accuracy
if 'mode' > 0 then \(\beta=mode*\lambda_{min}\) otherwise \(\beta=\beta_B=2/(1/\lambda_{min}+1/\lambda_{max})\)
Pierre Lafaye de Micheaux (lafaye@dms.umontreal.ca) and Pierre Duchesne (duchesne@dms.umontreal.ca)
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 \(\sum_k=0^{K-1} a_k P[\chi^2(m+2k)<q/\beta]\) where \(m=\sum_{j=1}^n m_j\) and \(\beta\) is an arbitrary constant (as given by argument mode).
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
# 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
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