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

L-moments of the 3-Parameter Student t Distribution

Description

This function estimates the first six L-moments of the 3-parameter Student t distribution given the parameters (\xi, \alpha, \nu) from parst3. The L-moments in terms of the parameters are

\lambda_1 = \xi\mbox{,}

\lambda_2 = 2^{6-4\nu}\pi\alpha\nu^{1/2}\,\Gamma(2\nu-2)/[\Gamma(\frac{1}{2}\nu)]^4\mbox{\, and}

\tau_4 = \frac{15}{2} \frac{\Gamma(\nu)}{\Gamma(\frac{1}{2})\Gamma(\nu - \frac{1}{2})} \int_0^1 \! \frac{(1-x)^{\nu - 3/2}[I_x(\frac{1}{2},\frac{1}{2}\nu)]^2}{\sqrt{x}}\; \mathrm{d} x - \frac{3}{2}\mbox{,}

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

The functional relation \tau_4(\nu) was solved numerically and a polynomial approximation made. The polynomial in turn with a root-solver is used to solve \nu(\tau_4) in parst3. The other two parameters are readily solved for when \nu is available. The polynomial based on \log{\tau_4} and \log{\nu} has nine coefficients with a residual standard error (in natural logarithm units of \tau_4) of 0.0001565 for 3250 degrees of freedom and an adjusted R-squared of 1. A polynomial approximation is used to estimate the \tau_6 as a function of \tau_4; the polynomial was based on the theoLmoms estimating \tau_4 and \tau_6. The \tau_6 polynomial has nine coefficients with a residual standard error units of \tau_6 of 1.791e-06 for 3593 degrees of freedom and an adjusted R-squared of 1.

Usage

lmomst3(para, bypoly=TRUE)

Arguments

para	The parameters of the distribution.
bypoly	A logical as to whether a polynomial approximation of \tau_4 as a function of \nu will be used. The default is TRUE because this polynomial is used to reverse the estimate for \nu as a function of \tau_4. A polynomial of \tau_6(\tau_4) is always used.

Value

An R list is returned.

lambdas	Vector of the L-moments. First element is \lambda_1, second element is \lambda_2, and so on.
ratios	Vector of the L-moment ratios. Second element is \tau, third element is \tau_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.

See Also

parst3, cdfst3, pdfst3, quast3

Examples

lmomst3(vec2par(c(1124,12.123,10), type="st3"))

lmomco documentation built on Aug. 30, 2023, 5:10 p.m.