# lmomst3: L-moments of the 3-Parameter Student t Distribution In lmomco: L-Moments, Censored L-Moments, Trimmed L-Moments, L-Comoments, and Many Distributions

## Description

This function estimates the first six L-moments of the 3-parameter Student t distribution given the parameters (ξ, α, ν) from parst3. The L-moments in terms of the parameters are

λ_1 = ξ\mbox{,}

λ_2 = 2^{6-4ν}παν^{1/2}\,Γ(2ν-2)/[Γ(\frac{1}{2}ν)]^4\mbox{\, and}

τ_4 = \frac{15}{2} \frac{Γ(ν)}{Γ(\frac{1}{2})Γ(ν - \frac{1}{2})} \int_0^1 \! \frac{(1-x)^{ν - 3/2}[I_x(\frac{1}{2},\frac{1}{2}ν)]^2}{√{x}}\; \mathrm{d} x - \frac{3}{2}\mbox{,}

where I_x(\frac{1}{2}, \frac{1}{2}ν) is the cumulative distribution function of the Beta distribution. The distribution is symmetrical so that τ_r = 0 for odd values of r: r ≥ 3.

The functional relation τ_4(ν) was solved numerically and a polynomial approximation made. The polynomial in turn with a root-solver is used to solve ν(τ_4) in parst3. The other two parameters are readily solved for when ν is available. The polynomial based on \log{τ_4} and \log{ν} has nine coefficients with a residual standard error (in natural logarithm units of τ_4) of 0.0001565 for 3250 degrees of freedom and an adjusted R-squared of 1. A polynomial approximation is used to estimate the τ_6 as a function of τ_4; the polynomial was based on the theoLmoms estimating τ_4 and τ_6. The τ_6 polynomial has nine coefficients with a residual standard error units of τ_6 of 1.791e-06 for 3593 degrees of freedom and an adjusted R-squared of 1.

## Usage

 1 lmomst3(para, bypoly=TRUE) 

## Arguments

 para The parameters of the distribution. bypoly A logical as to whether a polynomial approximation of τ_4 as a function of ν will be used. The default is TRUE because this polynomial is used to reverse the estimate for ν as a function of τ_4. A polynomial of τ_6(τ_4) is always used.

## Value

An R list is returned.

 lambdas Vector of the L-moments. First element is λ_1, second element is λ_2, and so on. ratios Vector of the L-moment ratios. Second element is τ, third element is τ_3 and so on. trim Level of symmetrical trimming used in the computation, which is 0. leftrim Level of left-tail trimming used in the computation, which is NULL. rightrim Level of right-tail trimming used in the computation, which is NULL. source An attribute identifying the computational source of the L-moments: “lmomst3”.

## Author(s)

W.H. Asquith with A.R. Biessen

## References

Asquith, W.H., 2011, Distributional analysis with L-moment statistics using the R environment for statistical computing: Createspace Independent Publishing Platform, ISBN 978–146350841–8.

parst3, cdfst3, pdfst3, quast3
 1 lmomst3(vec2par(c(1124,12.123,10), type="st3"))