DMAR (version 1.0.0)

moments_nc_chisq: Moments of the Noncentral Chi Square Distribution

Description

Returns the mean, variance, standard deviation, skewness, and excess kurtosis of a noncentral chi square distribution with df degrees of freedom and noncentrality parameter ncp. The noncentral chi square is the distribution of a sum of squared independent normals with nonzero means (\(\sum (Z_i + \mu_i)^2\), with \(\lambda = \sum \mu_i^2\)); it is the building block of the noncentral F (whose numerator is a noncentral chi square) and the reference distribution for likelihood ratio and Wald statistics under the alternative. Unlike the noncentral t and F, every moment exists, so none of the returned values is ever NA.

Usage

moments_nc_chisq(df, ncp = 0)

Value

A data.frame (class dmar_tbl) in

term / value layout with the mean, variance,

sd, skewness, and excess_kurtosis, followed by the

df and ncp that produced them.

Arguments

df

Degrees of freedom, a single positive number (need not be a whole number).

ncp

Noncentrality parameter \(\lambda\), a single non-negative number. Defaults to 0, the central chi square.

Author

Ken Kelley [email protected]

Details

The cumulants of the noncentral chi square are \(\kappa_n = 2^{n-1}(n-1)!\,(\nu + n\lambda)\) for \(n \ge 1\), from which the moments follow directly: the mean is \(\kappa_1 = \nu + \lambda\), the variance is \(\kappa_2 = 2(\nu + 2\lambda)\), the skewness is \(\kappa_3 / \kappa_2^{3/2} = \sqrt{8}\,(\nu + 3\lambda)/(\nu + 2\lambda)^{3/2}\), and the excess kurtosis is \(\kappa_4 / \kappa_2^{2} = 12(\nu + 4\lambda)/(\nu + 2\lambda)^{2}\). At \(\lambda = 0\) these reduce to the central chi square values: mean \(\nu\), variance \(2\nu\), skewness \(\sqrt{8/\nu}\), and excess kurtosis \(12/\nu\).

References

Johnson, N. L., Kotz, S., & Balakrishnan, N. (1995). Continuous univariate distributions (Vol. 2, 2nd ed., Chapter 29). Wiley.

See Also

moments_nc_F (whose numerator is a noncentral chi square) and moments_nc_t for the other noncentral moments; ci_nc_chisq for the noncentral chi square confidence limits; dchisq for the density.

Other noncentral distribution moments: moments_nc_F(), moments_nc_t()

Examples

# A noncentral chi square with 5 df and noncentrality 3.
moments_nc_chisq(df = 5, ncp = 3)

# ncp = 0 is the central chi square: mean df, variance 2 * df.
moments_nc_chisq(df = 5)

# Every moment exists for any positive df, so nothing is ever NA.
anyNA(moments_nc_chisq(df = 1, ncp = 10)$value)