Compute moments and absolute moments of standardised-t and normal distributions

Compute moments and absolute moments of standardised-t, t and normal distributions.

Usage

stdnormmoment(k)
stdnormabsmoment(k)
stdtmoment(nu, k)
stdtabsmoment(nu, k)
tabsmoment(nu, k)

Arguments

k

numeric vector, moments to compute.

nu

a number, degrees of freedom.

Details

These functions compute moments of standardised-t and standard normal distibutions. These distributions have mean zero and variance 1. Standardised-t is often prefferred over Student-t for innovation distributions, since its variance doesn't depend on its parameter (degrees of freedom). The absolute moments of the usual t-distributions are provided, as well.

The names of the functions start with an abbreviated name of the distribution concerned: stdnorm (N(0,1)), stdt (standardised-t), t (Student-t).

The functions with names ending in absmoment() (stdnormabsmoment(), stdtabsmoment() and tabsmoment()) compute absolute moments, The rest (stdnormmoment() and stdtmoment()) compute ordinary moments.

The absolute moments are valid for (at least) k >= 0, not necessarily integer. The ordinary moments are currently intended only for integer moments and return NaN's for fractional ones, with warnings.

Note that the Student-t and standardised-t with \(\nu\) degrees of freedom have finite (absolute) moments only for \(k<\nu\). As a consequence, standardised-t is defined only for \(\nu>2\) (otherwise the variance is infinite).

stdtabsmoment returns Inf for any \(k \ge \nu\). stdtmoment returns Inf for even integer k's, such that \(k \ge \nu\). However, for odd integers it returns zero and for non-integer moments it returns NaN. Here is an example, where the first two k's are smaller than nu, while the others are not: stdtabsmoment(nu = 5, k = c(4, 4.5, 5, 5.5));stdtmoment(nu = 5, k = c(4, 4.5, 5, 5.5))

These functions are designed to work with scalar nu but this is not enforced.

Value

numeric vector of the same length as k.

References

WuertzEtAll2006mixAR

Author

Georgi N. Boshnakov

Examples

## some familiar positive integer moments
stdnormmoment(1:6)
#> [1]  0  1  0  3  0 15
## fractional moments of N(0,1) currently give NaN
stdnormmoment(seq(1, 6, by = 0.5))
#>  [1]   0 NaN   1 NaN   0 NaN   3 NaN   0 NaN  15
## abs moments don't need to be integer
curve(stdnormabsmoment, from = 0, to = 6, type = "l", col = "blue")


## standardised-t
stdtmoment(5, 1:6)
#> [1]   0   1   0   9   0 Inf
stdtabsmoment(5, 1:6)
#> [1] 0.7351052 1.0000000 2.2053156 9.0000000       Inf       Inf
stdtabsmoment(5, 1:6)
#> [1] 0.7351052 1.0000000 2.2053156 9.0000000       Inf       Inf

## Student-t
tabsmoment(5, 1:6)
#> [1]  0.9490167  1.6666667  4.7450836 25.0000000        Inf        Inf