survey (version 3.1)

pchisqsum: 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.

Usage

pchisqsum(x, df, a, lower.tail = TRUE, method = c("satterthwaite", "integration","saddlepoint"))

Arguments

x

Observed values

df

Vector of degrees of freedom

a

Vector of coefficients

lower.tail

lower or upper tail?

method

See Details below

Value

• Vector of cumulative probabilities

Details

The "satterthwaite" method uses Satterthwaite's approximation, and this is also used as a fallback for the other methods.

"integration" inverts the characteristic function numerically. This is relatively slow, and not reliable for p-values below about 1e-5 in the upper tail, but is highly accurate for moderate p-values.

"saddlepoint" uses a saddlepoint approximation when x>1.05*sum(a) and the Satterthwaite approximation for smaller x. This is fast and is accurate in the upper tail, where accuracy is important.

References

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

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

x <- 5*rnorm(1001)^2+rnorm(1001)^2
x.thin<-sort(x[1+(0:100)*10])
p.invert<-pchisqsum(x.thin,df=c(1,1),a=c(5,1),method="int" ,lower=FALSE)
p.satt<-pchisqsum(x.thin,df=c(1,1),a=c(5,1),method="satt",lower=FALSE)
p.sadd<-pchisqsum(x.thin,df=c(1,1),a=c(5,1),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")