On the asymptotic properties of the likelihood estimates and some inferential issues related to hidden truncated Pareto (type II) model

Abstract

The usefulness of a hidden truncated Pareto (type II) model along with its’ inference under both the classical and Bayesian paradigm have been discussed in the literature in great details. In the multivariate set-up, some discussions are made that are primarily based on constructing a multivariate hidden truncated Pareto (type II) models with — single variable truncation or more than one variable truncation. However, in all such previous discussions regarding bivariate hidden truncated Pareto models, in the classical estimation set-up, large bias and standard error values for the truncation parameter(s) as well as for the other parameters have been observed, and no discussion was made to address this issue. In this article, we try to address this issue of large bias values by considering constrained optimization via linear/non-linear transformation of the parameters following the strategy as proposed (the reference is given in Section 3), in efficiently implementing Newton-Raphson optimization algorithm in R. This plays a major motivation for the present paper. We also derive the observed Fisher Information Matrix. For illustrative purposes, we provide a simulation study to address this issue. A real-life data set is also re-analyzed to study the utility of such two-sided hidden truncation Pareto (type II) models.

Keywords:

• Bivariate Pareto (type II) distribution
• hidden truncation
• two-sided truncation
• method of maximum likelihood
• asymptotic distribution
• constrained optimization
• likelihood ratio test
• observed fisher information matrix

Acknowledgements

The author would like to thank the two anonymous referees for their valuable and constructive suggestions on the previous submitted version of the manuscript which has helped them to improve the manuscript significantly.