Asymptotic Properties of a Family of Minimum Quantile Distance Estimators

Abstract

A family of estimators based upon M specified quantiles is defined. These procedures take as the estimate the vector that minimizes a quadratic distance measure between M sample quantiles and a parametric family of quantile functions. Under regularity conditions these estimators are consistent, asymptotically normal, and robust. For a specific quadratic form the estimator is optimal among a class of asymptotically normal estimators, and it approaches full efficiency as M approaches infinity. The asymptotic relative efficiency is computed for various sets of quantiles and various parameter values of the three-parameter lognormal distribution. The small-sample properties and robustness of the optimal M-quantile estimator for the three-parameter lognormal distribution are investigated in Monte Carlo studies.

Key Words:

• Quantile functions
• Minimum distance
• Three-parameter lognormal distribution