Distribution of quadratic forms
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.
pchisqsum(x, df, a, lower.tail = TRUE,
method = c("satterthwaite", "integration","saddlepoint"))
pFsum(x, df, a, ddf=Inf,lower.tail = TRUE,
method = c("saddlepoint","integration","satterthwaite"), ...)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 |
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
right tail (eg for upper tail probabilities less than 10^-5).
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).
These algorithms are highly accurate for the lower tail probability, but they obtain the upper tail probability by subtraction from 1 and so fail completely when the upper tail probability is comparable to machine epsilon or smaller.
If the CompQuadForm package is not present, a warning is given
and the saddlepoint approximation is used.
"saddlepoint" uses Kuonen's saddlepoint approximation. This
is moderately 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.
Vector of cumulative probabilities
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
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")
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