Estimation of Parameters of the Truncated Normal Distribution with Unknown Bounds

TL;DR

This paper introduces a new iterative algorithm within the expectation-solution framework to estimate both the bounds and parameters of a truncated normal distribution with unknown truncation limits, supported by theoretical convergence and asymptotic analysis.

Contribution

It develops a novel estimation method for unknown bounds and parameters of truncated normal distributions using an expectation-solution approach and empirical process theory.

Findings

Algorithm converges under certain conditions.

Estimators are asymptotically normal.

Method outperforms existing approaches with known bounds.

Abstract

Estimators of parameters of truncated distributions, namely the truncated normal distribution, have been widely studied for a known truncation region. There is also literature for estimating the unknown bounds for known parent distributions. In this work, we develop a novel algorithm under the expectation-solution (ES) framework, which is an iterative method of solving nonlinear estimating equations, to estimate both the bounds and the location and scale parameters of the parent normal distribution utilizing the theory of best linear unbiased estimates from location-scale families of distribution and unbiased minimum variance estimation of truncation regions. The conditions for the algorithm to converge to the solution of the estimating equations for a fixed sample size are discussed, and the asymptotic properties of the estimators are characterized using results on M- and Z-estimation from empirical process theory. The proposed method is then compared to methods utilizing the known truncation bounds via Monte Carlo simulation.