imhof {CompQuadForm} | R Documentation |
Imhof method.
Description
Distribution function (survival function in fact) of quadratic forms in normal variables using Imhof's method.
Usage
imhof(q, lambda, h = rep(1, length(lambda)),
delta = rep(0, length(lambda)),
epsabs = 10^(-6), epsrel = 10^(-6), limit = 10000)
Arguments
q |
value point at which the survival function is to be evaluated |
lambda |
distinct non-zero characteristic roots of |
h |
respective orders of multiplicity of the |
delta |
non-centrality parameters (should be positive) |
epsabs |
absolute accuracy requested |
epsrel |
relative accuracy requested |
limit |
determines the maximum number of subintervals in the partition of the given integration interval |
Details
Let be a column random vector which follows a multidimensional normal law with mean vector
and non-singular covariance matrix
.
Let
be a constant vector, and consider the quadratic form
The function imhof
computes .
The 's are the distinct non-zero characteristic roots of
, the
's their respective orders of
multiplicity, the
's are certain linear combinations
of
and the
are independent
-variables with
degrees of freedom and
non-centrality parameter
. The variable
is defined here by the
relation
, where
are
independent unit normal deviates.
Value
Qq |
|
abserr |
estimate of the modulus of the absolute error, which should equal or exceed abs(i - result) |
Author(s)
Pierre Lafaye de Micheaux (lafaye@dms.umontreal.ca) and Pierre Duchesne (duchesne@dms.umontreal.ca)
References
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, Volume 54, (2010), 858-862
J. P. Imhof, Computing the Distribution of Quadratic Forms in Normal Variables, Biometrika, Volume 48, Issue 3/4 (Dec., 1961), 419-426
Examples
# Some results from Table 1, p.424, Imhof (1961)
# Q1 with x = 2
round(imhof(2, c(0.6, 0.3, 0.1))$Qq, 4)
# Q2 with x = 6
round(imhof(6, c(0.6, 0.3, 0.1), c(2, 2, 2))$Qq, 4)
# Q6 with x = 15
round(imhof(15, c(0.7, 0.3), c(1, 1), c(6, 2))$Qq, 4)