Raw moments, limited moments and moment generating function for the
chi-squared (\(\chi^2\)) distribution with df
degrees
of freedom and optional non-centrality parameter ncp
.
mchisq(order, df, ncp = 0)
levchisq(limit, df, ncp = 0, order = 1)
mgfchisq(t, df, ncp = 0, log= FALSE)
order of the moment.
limit of the loss variable.
degrees of freedom (non-negative, but can be non-integer).
non-centrality parameter (non-negative).
numeric vector.
logical; if TRUE
, the cumulant generating function
is returned.
mchisq
gives the \(k\)th raw moment,
levchisq
gives the \(k\)th moment of the limited loss
variable, and
mgfchisq
gives the moment generating function in t
.
Invalid arguments will result in return value NaN
, with a warning.
The \(k\)th raw moment of the random variable \(X\) is \(E[X^k]\), the \(k\)th limited moment at some limit \(d\) is \(E[\min(X, d)]\) and the moment generating function is \(E[e^{tX}]\).
Only integer moments are supported for the non central Chi-square
distribution (ncp > 0
).
The limited expected value is supported for the centered Chi-square
distribution (ncp = 0
).
Klugman, S. A., Panjer, H. H. and Willmot, G. E. (2012), Loss Models, From Data to Decisions, Fourth Edition, Wiley.
Johnson, N. L. and Kotz, S. (1970), Continuous Univariate Distributions, Volume 1, Wiley.
# NOT RUN {
mchisq(2, 3, 4)
levchisq(10, 3, order = 2)
mgfchisq(0.25, 3, 2)
# }
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