# Distribution of quadratic forms

### Description

The distribution of a quadratic form in p standard Normal variables is a linear combination of p chi-squared distributions with 1df. When there is uncertainty about the variance, a reasonable model for the distribution is a linear combination of F distributions with the same denominator.

### Usage

1 2 3 4 |

### Arguments

`x` |
Observed values |

`df` |
Vector of degrees of freedom |

`a` |
Vector of coefficients |

`ddf` |
Denominator degrees of freedom |

`lower.tail` |
lower or upper tail? |

`method` |
See Details below |

`...` |
arguments to |

### Details

The `"satterthwaite"`

method uses Satterthwaite's
approximation, and this is also used as a fallback for the other
methods. The accuracy is usually good, but is more variable depending
on `a`

than the other methods and is anticonservative in the
extreme tail. The Satterthwaite approximation requires all `a>0`

.

`"integration"`

requires the `CompQuadForm`

package. For
`pchisqsum`

it uses Farebrother's algorithm if all
`a>0`

. For `pFsum`

or when some `a<0`

it inverts the
characteristic function using the algorithm of Davies (1980). If
the `CompQuadForm`

package is not present, a warning is given
and the saddlepoint approximation is used. These algorithms are not
accurate for very large `x`

or when some `a`

are close to
zero: a warning is given if the relative error bound is more than
10% of the result.

`"saddlepoint"`

uses Kuonen's saddlepoint approximation. This
is accurate even very far out in the upper tail or with some
`a=0`

and does not require any additional packages. It is
implemented in pure R and so is slower than the `"integration"`

method.

The distribution in `pFsum`

is standardised so that a likelihood
ratio test can use the same `x`

value as in `pchisqsum`

.
That is, the linear combination of chi-squareds is multiplied by
`ddf`

and then divided by an independent chi-squared with
`ddf`

degrees of freedom.

### Value

Vector of cumulative probabilities

### References

Davies RB (1973). "Numerical inversion of a characteristic function" Biometrika 60:415-7

Davies RB (1980) "Algorithm AS 155: The Distribution of a Linear Combination of chi-squared Random Variables" Applied Statistics,Vol. 29, No. 3 (1980), pp. 323-333

P. Duchesne, P. Lafaye de Micheaux (2010) "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. (1984) "Algorithm AS 204: The distribution of a Positive Linear Combination of chi-squared random variables". Applied Statistics Vol. 33, No. 3 (1984), p. 332-339

Kuonen D (1999) Saddlepoint Approximations for Distributions of Quadratic Forms in Normal Variables. Biometrika, Vol. 86, No. 4 (Dec., 1999), pp. 929-935

### See Also

`pchisq`

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 | ```
x <- 2.7*rnorm(1001)^2+rnorm(1001)^2+0.3*rnorm(1001)^2
x.thin<-sort(x)[1+(0:50)*20]
p.invert<-pchisqsum(x.thin,df=c(1,1,1),a=c(2.7,1,.3),method="int" ,lower=FALSE)
p.satt<-pchisqsum(x.thin,df=c(1,1,1),a=c(2.7,1,.3),method="satt",lower=FALSE)
p.sadd<-pchisqsum(x.thin,df=c(1,1,1),a=c(2.7,1,.3),method="sad",lower=FALSE)
plot(p.invert, p.satt,type="l",log="xy")
abline(0,1,lty=2,col="purple")
plot(p.invert, p.sadd,type="l",log="xy")
abline(0,1,lty=2,col="purple")
pchisqsum(20, df=c(1,1,1),a=c(2.7,1,.3), lower.tail=FALSE,method="sad")
pFsum(20, df=c(1,1,1),a=c(2.7,1,.3), ddf=49,lower.tail=FALSE,method="sad")
pFsum(20, df=c(1,1,1),a=c(2.7,1,.3), ddf=1000,lower.tail=FALSE,method="sad")
``` |