AS204 | R Documentation |
Distribution of a positive linear combination of χ^2 random variables.
AS204( c, lambda, mult = rep(1, length(lambda)), delta = rep(0, length(lambda)), maxit = 1e+05, eps = 1e-14, mode = 1 )
c |
value point at which distribution is to be evaluated. |
lambda |
the weights λ_j. |
mult |
the multiplicities m_j. |
delta |
the non-centrality parameters δ^2_j. |
maxit |
the maximum number of terms K (see Details). |
eps |
the desired level of accuracy. |
mode |
if " |
Algorithm AS 204 evaluates the expression
P [X < c] = P [ ∑_{j=1}^n λ_j χ^2(m_j, δ^2_j) < c ]
where λ_j and c are positive constants and χ^2(m_j, δ^2_j) represents an independent χ^2 random variable with m_j degrees of freedom and non-centrality parameter δ^2_j. This can be approximated by the truncated series
∑_{k=0}^{K-1} a_k P [χ^2(m+2k) < c/β]
where m = ∑_{j=1}^n m_j and β is an arbitrary constant (as given by argument "mode").
The C++
implementation of
algorithm AS 204 used here is identical
to the one employed by the
farebrother
method
in the CompQuadForm
package,
with minor modifications.
The function returns the
probability P[X > c] = 1 - P[X < c]
if the AS 204 fault indicator
is 0 (see Note below), and NULL
if
the fault indicator is 4, 5 or 9,
as the corresponding faults can be
corrected by increasing "eps
".
Other faults raise an error.
The algorithm AS 204 defines the following fault indicators: -j) one or more of the constraints λ_j > 0, m_j > 0 and δ^2_j ≥ 0 is not satisfied. 1) non-fatal underflow of a_0. 2) one or more of the constraints n > 0, c > 0, maxit > 0 and eps > 0 is not satisfied. 3) the current estimate of the probability is < -1. 4) the required accuracy could not be obtained in maxit iterations. 5) the value returned by the procedure does not satisfy 0 ≤ P [X < c] ≤ 1. 6) the density of the linear form is negative. 9) faults 4 and 5. 10) faults 4 and 6. 0) otherwise.
Diego Garrido-Martín
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, Vol. 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), 332-339
farebrother
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