CompQuadForm (version 1.4.3)

imhof: 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 $A\mathrm{\Sigma }$

h

respective orders of multiplicity of the $\lambda$s

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

Value

Qq

$P[Q>q]$

abserr

estimate of the modulus of the absolute error, which should equal or exceed abs(i - result)

Details

Let $\mathit{X}=(X_1,\dots ,X_n{)}^{\prime }$ be a column random vector which follows a multidimensional normal law with mean vector $\mathbf{0}$ and non-singular covariance matrix $\mathbf{\Sigma }$. Let $\mathit{\mu }=({\mu }_1,\dots ,{\mu }_n{)}^{\prime }$ be a constant vector, and consider the quadratic form $$Q=(\mathit{x}+\mathit{\mu }{)}^{\prime }\mathit{A}(\mathit{x}+\mathit{\mu })=\sum _{r=1}^m{\lambda }_r{\chi }_{h_r;{\delta }_r}^2.$$ The function imhof computes $P[Q>q]$. The ${\lambda }_r$'s are the distinct non-zero characteristic roots of $A\mathrm{\Sigma }$, the $h_r$'s their respective orders of multiplicity, the ${\delta }_r$'s are certain linear combinations of ${\mu }_1,\dots ,{\mu }_n$ and the ${\chi }_{h_r;{\delta }_r}^2$ are independent ${\chi }^2$-variables with $h_r$ degrees of freedom and non-centrality parameter ${\delta }_r$. The variable ${\chi }_{h,\delta }^2$ is defined here by the relation ${\chi }_{h,\delta }^2=(X_1+\delta {)}^2+\sum _{i=2}^h{X}_i^2$, where $X_1,\dots ,X_h$ are independent unit normal deviates.

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)