Abstract
This paper explores the complexities and limitations of Maximum Likelihood Estimation (MLE) when the uniqueness of the estimator cannot be guaranteed. While standard statistical theory often assumes that MLE provides a single, optimal point estimate, this research demonstrates scenarios where multiple values or intervals can maximize the likelihood function. Drawing on foundational works by Bickel, Doksum, Hogg, and Craig, the author provides a detailed analysis of distributions—such as the uniform distribution and the Cauchy distribution with a location parameter—where the MLE is inherently non-unique. The study highlights that even with a small sample size, such as n=2, the likelihood function can fail to produce a single peak, leading to estimation challenges. Furthermore, the paper discusses the structural reasons behind these occurrences, noting that while some densities naturally yield unique estimates for single observations, others remain problematic. By presenting these unconventional examples, the work serves to caution researchers against the blind application of MLE and encourages a more nuanced understanding of the likelihood surface, particularly in cases involving non-standard density functions and discrete or interval-based maximums.
DOI
10.33095/jeas.v16i60.1520
Subject Area
Statistical
First Page
194
Last Page
198
Recommended Citation
AlGharabi, S. I. (2010). Examples of Non-Unique Maximum Likelihood Estimates. Journal of Economics and Administrative Sciences, 16(60), 194-198. https://doi.org/10.33095/jeas.v16i60.1520
