L-Moments and Maximum Likelihood Estimation for the Complementary Beta Distribution with Applications on Temperature Extremes

Josmar Mazucheli,Andr´E Felipe Berdusco Menezes

doi:10.6339/jds.201904_17(2).0009

Abstract

Although the two-parameter Beta distribution is the standard distribution for analyzing data in the unit interval, there are in the literature some useful and interesting alternatives which are often under-used. An example is the two parameter complementary Beta distribution, introduced by Jones (2002) and, to the best of our knowledge, used only by Iacobellis (2008) as a probabilistic model for the estimation of T year flow duration curves. In his paper the parameters of complementary Beta distribution were successfully estimated, perhaps due to its simplicity, by means of the L-moments method. The objective of this paper is to compare, using Monte Carlo simulations, the bias and mean-squared error, of the estimators obtained by the methods of L-moments and maximum likelihood. The simulation study showed that the maximum likelihood method has bias and mean -squared error lower than L-moments. It is also revealed that the parameters estimated by the maximum likelihood are negatively biased, while by the L-moments method the parameters are positively biased. Data on relative indices from annual temperature extremes (percentage of cool nights, percentage of warm nights, percentage of cool days and percentage of warm days) in Uruguay are used for illustrative purposes.

Highlights

It is well known that Beta distribution in its standard form is the most important probability distribution for data analysis with support on the unit interval
A probability distribution related to Beta the distribution that has not received much attention in the literature was introduced by Jones (2002) and it is obtained by switching the roles of c.d.f. and quantile function of the Beta distribution
The L-moments, whose theory was unified by Hosking (1990), are linear combinations of order statistics and have lower sample variances and are more robust against outliers when compared to the conventional moments

Summary

Introduction

It is well known that Beta distribution in its standard form is the most important probability distribution for data analysis with support on the unit interval. These behaviors are quite similar to the Beta distribution with parameters 1/α and 1/β. In this paper our main goal is to compare, mainly, by Monte Carlo simulation and real applications, the performance of estimators obtained by the methods of L -moments and maximum likelihood. In the same direction, Erişoǧlu and Erişoǧlu (2014) compared the L-moments estimation with maximum likelihood method to estimate the parameters related to mixture of Weibull distributions.

Method of L-moments

Method of maximum likelihood

Simulation results

Applications

Conclusions