Existence and Uniqueness of the Maximum Likelihood Estimators of the Topp-Leone Distribution Parameters
Ahmad A. Zghoul, Mahmoud Mustafa Smadi · Mathematics and Statistics · 2025
The Topp-Leone (TL) distribution is a bounded probability model with flexible hazard rate which can assume increasing, decreasing, or bathtub-shaped forms. This flexibility makes the TL distribution useful in reliability analysis and lifetime data modeling. Despite its applicability and attractive theoretical features, maximum likelihood estimation (MLE) of its parameters must be handled with care because the support of the distribution depends on the scale parameter. This dependency violates classical regularity conditions that ensure the existence and uniqueness of the MLEs. Thus, before applying MLE, we must verify that the estimators exist and are unique. This article investigates these properties for the TL distribution parameters. We show that the MLEs do not exist for a single observation but prove their existence and uniqueness when the sample contains at least two distinct values. After establishing the existence and uniqueness of the MLEs, we assess their performance through simulations. The simulation studies confirm the consistency of the estimators, where their bias and mean squared error (MSE) both decline as the sample size increases. However, the shape parameter estimator shows increasing bias for larger true parameter values. In addition to simulation studies, we fit the TL distribution to a real-world dataset—aircraft window glass strength—and confirm model adequacy using the Anderson-Darling goodness-of-fit test.