Abstract
The shape parameter of a symmetric probability distribution is often more difficult to estimate accurately than the location and scale parameters. In this paper, we suggest an intuitive but innovative matching quantile estimation method for this parameter. The proposed shape parameter estimate is obtained by setting its value to a level such that the central 1/���� portion of the distribution will just cover all n observations, while the location and scale parameters are estimated using existing methods such as maximum likelihood (ML). This hybrid estimator is proved to be consistent and is illustrated by two distributions, namely Student-t and Exponential Power. Simulation studies show that the hybrid method provides reasonably accurate estimates. In the presence of extreme observations, this method provides thicker tails than the full ML method and protect inference on the location and scale parameters. This feature offered by the hybrid method is also demonstrated in the empirical study using two real data sets.
Highlights
The paralogistic distribution is a sub-model of the generalized beta family which was introduced by McDonald (1984)
The shape parameter of a symmetric probability distribution is often more difficult to estimate than the location and scale parameters
This paper proposes a simple way to estimate the shape parameter using a matching quantiles (MQ) method applied to the most extreme observation and the location and scale parameters using the maximum likelihood (ML) method under a hybrid approach
Summary
The paralogistic distribution is a sub-model of the generalized beta family which was introduced by McDonald (1984). In a full ML approach, the shape parameter is of- ten more difficult to estimate accurately than the location and scale parameters because the NewtonRaphson (NR) or Fisher scoring procedures may sometimes fail due to non-differentiable loglikelihoods, arisen when the probability density functions (PDFs) have sharp peaks. This occurs for distributions such as exponential power (EP) when the shape parameter falls inside a certain range. Hill (1975) proposes measuring the tail thickness by approximating the tail cumulative distribution function (CDF) with a power function using extreme order statistics This semi-parametric approach does not assume any global form of a distribution.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
