Estimation of two ordered quantile residual life functions based on mixtures

Abstract

This paper investigates some properties of the mixture models from the quantile residual life perspective. It is shown that a mixture model is bounded by its components via the quantile residual life. We investigate how mixture models are ordered in terms of the quantile residual life function when their components are ordered. Besides, we prove that the limiting quantile residual life of a mixture is similar to that of the greatest component at infinity. Based on these results, it is possible to construct estimators of two quantile residual life functions subject to an order restriction. Such estimators are shown to be strongly uniformly consistent and asymptotically unbiased. We develop the weak convergence theory for these estimators. Simulations seem to indicate that both of the restricted estimators improve on the empirical (unrestricted) estimators in terms of the mean squared error, uniformly at all quantiles, and for a variety of distributions.

Keywords:

• Mixture model
• quantile residual life
• stochastic orders
• order restricted inference
• asymptotic theory

2010 Mathematics Subject Classifications:

• 62G05
• 62H30
• 60E15
• 62N05

Acknowledgments

We thank the anonymous referee for the helpful comments, and constructive remarks on this manuscript.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was partially supported by grant MTM2017-89422-P of the Spanish Ministry of Economy and Competitiveness (Ministerio de Economía y Competitividad (Spain)) and the European Union (European Regional Development Fund – ERDF).