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 |

`mult` |
the multiplicities |

`delta` |
the non-centrality parameters |

`maxit` |
the maximum number of terms |

`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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