AS204 | R Documentation |

Distribution of a positive linear combination
of `\chi^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 [ \sum_{j=1}^n \lambda_j \chi^2(m_j, \delta^2_j) < c ]
```

where `\lambda_j`

and `c`

are positive constants and
`\chi^2(m_j, \delta^2_j)`

represents an independent `\chi^2`

random variable with `m_j`

degrees of freedom and non-centrality
parameter `\delta^2_j`

.
This can be approximated by the truncated series

```
\sum_{k=0}^{K-1} a_k P [\chi^2(m+2k) < c/\beta]
```

where `m = \sum_{j=1}^n m_j`

and
`\beta`

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 `\lambda_j > 0`

,
`m_j > 0`

and `\delta^2_j \ge 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 \le P [X < c] \le 1`

.
**6)** the density of the linear form is negative.
**9)** faults 4 and 5.
**10)** faults 4 and 6.
**0)** otherwise.

Diego Garrido-MartÃn

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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