imhof {CompQuadForm} | R Documentation |
Distribution function (survival function in fact) of quadratic forms in normal variables using Imhof's method.
imhof(q, lambda, h = rep(1, length(lambda)), delta = rep(0, length(lambda)), epsabs = 10^(-6), epsrel = 10^(-6), limit = 10000)
q |
value point at which the survival function is to be evaluated |
lambda |
distinct non-zero characteristic roots of A.Sigma |
h |
respective orders of multiplicity of the lambdas |
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 |
Let \strong{X}=(X_1,...,X_n)' be a column random vector which follows a multidimensional normal law with mean vector \strong{0} and non-singular covariance matrix \strong{Sigma}. Let \strong{mu}=(mu_1,...,mu_n)' be a constant vector, and consider the quadratic form
Q = (\strong{x}+\strong{mu})'\strong{A}(\strong{x}+\strong{mu}) = sum_{r=1}^m lambda_r chi^2_{h_r;δ_r}.
The function imhof
computes P[Q>q].
The λ_r's are the distinct non-zero characteristic roots of A.Sigma, the h_r's their respective orders of multiplicity, the delta_r's are certain linear combinations of mu_1,...,mu_n and the chi^2_{h_r;delta_r} are independent chi^2-variables with h_r degrees of freedom and non-centrality parameter delta_r. The variable chi^2_{h;delta} is defined here by the relation chi^2_{h,delta}=(X_1 + delta)^2+ sum_{i=2}^h X_i^2, where X_1,...,X_h are independent unit normal deviates.
Qq |
P[Q>q] |
abserr |
estimate of the modulus of the absolute error, which should equal or exceed abs(i - result) |
Pierre Lafaye de Micheaux (lafaye@dms.umontreal.ca) and Pierre Duchesne (duchesne@dms.umontreal.ca)
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
# 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)