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The Distribution of a Positive Linear Combination of Chiqaure Random Variables

pwchisq.Rd

The cumulative distribution function for the distribution of a positive linear combination of \(\chi^2\) random variables with the weights (\(\lambda_1, \ldots, \lambda_K\)), degrees of freedom (\(\nu_1, \ldots, \nu_K\)), and non-centrality parameters (\(\delta_1, \ldots, \delta_K\)).

Usage

pwchisq(
  x,
  lambda = 1,
  nu = 1,
  delta = 0,
  mode = 1,
  maxit1 = 1e+05,
  eps = 10^(-10)
)

Arguments

x

numeric; value of x > 0 (\(P[X \le x]\)).

lambda

numeric vector; weights (\(\lambda_1, \ldots, \lambda_K\)).

nu

integer vector; degrees of freedom (\(\nu_1, \ldots, \nu_K\)).

delta

numeric vector; non-centrality parameters (\(\delta_1, \ldots, \delta_K\)).

mode

numeric; the mode of calculation (see Farabrother, 1984)

maxit1

integer; the maximum number of iteration.

eps

numeric; the desired level of accuracy.

Value

  • prob: the distribution function.

References

Farebrother, R. W. (1984). Algorithm AS 204: the distribution of a positive linear combination of \(\chi^2\) random variables. J R Stat Soc Ser C Appl Stat. 33(3): 332–339. https://rss.onlinelibrary.wiley.com/doi/10.2307/2347721.

Examples

# Table 1 of Farabrother (1984)
# Q6 (The taget values are 0.0061, 0.5913, and 0.9779)

pimeta::pwchisq( 20, lambda = c(7,3), nu = c(6,2), delta = c(6,2))
#> [1] 0.006117973
pimeta::pwchisq(100, lambda = c(7,3), nu = c(6,2), delta = c(6,2))
#> [1] 0.5913421
pimeta::pwchisq(200, lambda = c(7,3), nu = c(6,2), delta = c(6,2))
#> [1] 0.9779184
# [1] 0.006117973
# [1] 0.5913421
# [1] 0.9779184