R: Liu's method

liu {CompQuadForm} | R Documentation

## Liu's method

### Description

Distribution function (survival function in fact) of quadratic forms in normal variables using Liu et al.'s method.

### Usage

    liu(q, lambda, h = rep(1, length(lambda)),
        delta = rep(0, length(lambda)))

### Arguments

`q` | value point at which the survival function is to be evaluated
`lambda` | distinct non-zero characteristic roots of A.Sigma, i.e. the λ_i's
`h` | respective orders of multiplicity h_i's of the lambda's
`delta` | non-centrality parameters δ_i's (should be positive)

### Details

New chi-square approximation to the distribution of non-negative definite quadratic forms in non-central normal variables.

Computes P[Q>q] where Q=sum_{j=1}^n lambda_j chi^2(h_j,delta_j).

This method does not work as good as the Imhof's method. Thus Imhof's method should be recommended.

### Value

`Qq` | P[Q>q]

### 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

H. Liu, Y. Tang, H.H. Zhang, A new chi-square approximation to the distribution of non-negative definite quadratic forms in non-central normal variables, Computational Statistics and Data Analysis, Volume 53, (2009), 853-856

### Examples

    # Some results from Liu et al. (2009)
    # Q1 from Liu et al.
    round(liu(2, c(0.5, 0.4, 0.1), c(1, 2, 1), c(1, 0.6, 0.8)), 6)
    round(liu(6, c(0.5, 0.4, 0.1), c(1, 2, 1), c(1, 0.6, 0.8)), 6)
    round(liu(8, c(0.5, 0.4, 0.1), c(1, 2, 1), c(1, 0.6, 0.8)), 6)

    # Q2 from Liu et al.
    round(liu(1, c(0.7, 0.3), c(1, 1), c(6, 2)), 6)
    round(liu(6, c(0.7, 0.3), c(1, 1), c(6, 2)), 6)
    round(liu(15, c(0.7, 0.3), c(1, 1), c(6, 2)), 6)

    # Q3 from Liu et al.
    round(liu(2, c(0.995, 0.005), c(1, 2), c(1, 1)), 6)
    round(liu(8, c(0.995, 0.005), c(1, 2), c(1, 1)), 6)
    round(liu(12, c(0.995, 0.005), c(1, 2), c(1, 1)), 6)

    # Q4 from Liu et al.
    round(liu(3.5, c(0.35, 0.15, 0.35, 0.15), c(1, 1, 6, 2), c(6, 2, 6, 2)),
    6)
    round(liu(8, c(0.35, 0.15, 0.35, 0.15), c(1, 1, 6, 2), c(6, 2, 6, 2)), 6)
    round(liu(13, c(0.35, 0.15, 0.35, 0.15), c(1, 1, 6, 2), c(6, 2, 6, 2)), 6)

[Package CompQuadForm version 1.4.3 Index]