partri: Estimate the Parameters of the Asymmetric Triangular Distribution

Description

This function estimates the parameters of the Asymmetric Triangular distribution given the L-moments of the data in an L-moment object such as that returned by lmoms. The relations between distribution parameters and L-moments are seen under lmomtri.

The estimtion by the partri function is built around simultaneous numerical optimization of an objective function defined as $$\epsilon = \biggl(\frac{\lambda_1 - \hat\lambda_1}{\hat\lambda_1}\biggr)^2 + \biggl(\frac{\lambda_2 - \hat\lambda_2}{\hat\lambda_2}\biggr)^2 + \biggl(\frac{\tau_3 - \hat\tau_3}{1}\biggr)^2$$ for estimation of the three parameters ($\nu$, minimum; $\omega$, mode; and $\psi$, maximum) from the sample L-moments ($\hat\lambda_1$, $\hat\lambda_2$, $\hat\tau_3$). The divisions shown in the objective function are used for scale removal to help make each L-moment order somewhat similar in its relative contribution to the solution. The coefficient of L-variation is not used because the distribution implementation by the lmomco package supports entire real number line and the loss of definition of $\tau_2$ at $x = 0$, in particular, causes untidiness in coding.

The function is designed to support both left- or right-hand right triangular shapes because of (1) paracheck argument availability in lmomtri, (2) the sorting of the numerical estimates if the mode is no compatable with either of the limits, and (3) the snapping of $\nu = \omega \equiv (\nu^\star + \omega^\star)/2$ when $\hat\tau_3 > 0.142857$ or $\psi = \omega \equiv (\psi^\star + \omega^\star)/2$ when $\hat\tau_3 < 0.142857$ where the $\star$ versions are the optimized values if the $\tau_3$ is very near to its numerical bounds.

Usage

partri(lmom, checklmom=TRUE, ...)

Value

An R

list is returned.

type: The type of distribution: tri.
para: The parameters of the distribution.
obj.val: The value of the objective function, which is the error of the optimization.
source: The source of the parameters: “partri”.

Arguments

lmom: An L-moment object created by lmoms or vec2lmom.
checklmom: Should the lmom be checked for validity using the are.lmom.valid function. Normally this should be left as the default and it is very unlikely that the L-moments will not be viable (particularly in the $\tau_4$ and $\tau_3$ inequality). However, for some circumstances or large simulation exercises then one might want to bypass this check.
...: Other arguments to pass.

Author

W.H. Asquith

Examples

Run this code

lmr <- lmomtri(vec2par(c(10,90,100), type="tri"))
partri(lmr)

partri(lmomtri(vec2par(c(-11, 67,67), type="tri")))$para
partri(lmomtri(vec2par(c(-11,-11,67), type="tri")))$para