On comparison of estimators for proportional error nonlinear regression models in the limit of small measurement error

TL;DR

This paper compares maximum likelihood, quasi likelihood, weighted least squares, and data weighted least squares estimators for proportional error nonlinear regression models, focusing on their bias and variance in small measurement error scenarios.

Contribution

It derives bias and variance formulas for these estimators in the limit of small measurement errors, highlighting conditions where ML and QL estimators outperform others.

Findings

ML estimator has less bias than QL.

ML has smaller variance than QL when errors are normally distributed.

In small error limit, ML and QL estimators are asymptotically normal.

Abstract

In this paper, we compare maximum likelihood (ML), quasi likelihood (QL) and weighted least squares (WLS) estimators for proportional error nonlinear regression models. Literature on thermoluminescence sedimentary dating revealed another estimator similar to weighted least squares but observed responses used as weights. This estimator that we refer to as data weighted least squares (DWLS) is also included in the comparison. We show that on the order $\sigma, $ all four estimators behave similar to ordinary least squares estimators for standard linear regression models. On the order of $\sigma^2, $ the estimators have biases. Formulae that are valid in the limit of small measurement error are derived for the biases and the variances of the four estimators. The maximum likelihood estimator has less bias compared to the quasi likelihood estimator. Conditions are derived under which weighted least squares and maximum likelihood estimators have similar biases. On the order of $\sigma^ {2} $, all estimators have similar standard errors. On higher order of $\sigma$, maximum likelihood estimator has smaller variance compared to quasi likelihood estimator, provided that the random errors have the same first four moments as the normal distribution. The maximum likelihood and quasi-likelihood estimating equations are unbiased. In large samples, these two estimators are distributed as multivariate normal. The estimating equations for weighted least squares and data weighted least squares are biased. However, in the limit of $\sigma \to 0$ and $n \to \infty, $ if $ n^ {1/2} \sigma$ remains bounded, these two estimators are also distributed as multivariate normal. A simulation study justified the applicability of the derived formulae in the presence of measurement errors typical in sedimentary data. Results are illustrated with a data set from thermoluminescence sedimentary dating.