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.
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.
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_ncf (whose numerator is a noncentral chi
square) and moments_nct for the other noncentral moments;
conf_limits_nc_chisq for the noncentral chi square
confidence limits; dchisq for the density.
Other noncentral distribution moments:
moments_ncf(),
moments_nct()
Author
Ken Kelley kkelley@nd.edu
Examples
# A noncentral chi square with 5 df and noncentrality 3.
moments_nc_chisq(df = 5, ncp = 3)
#> term value
#> mean 8
#> variance 22
#> sd 4.69
#> skewness 1.09
#> excess_kurtosis 1.69
#> df 5
#> ncp 3
# ncp = 0 is the central chi square: mean df, variance 2 * df.
moments_nc_chisq(df = 5)
#> term value
#> mean 5
#> variance 10
#> sd 3.16
#> skewness 1.26
#> excess_kurtosis 2.4
#> df 5
#> ncp 0
# Every moment exists for any positive df, so nothing is ever NA.
anyNA(moments_nc_chisq(df = 1, ncp = 10)$value)
#> [1] FALSE