Statistical Inference for Geometric Process with the Power Lindley Distribution

Area of Science:

• Statistics
• Probability Theory
• Reliability Engineering

Background:

• The geometric process (GP) models inter-arrival times with monotonic trends, serving as an alternative to non-homogeneous Poisson processes.
• Parameter estimation is crucial for applying GP effectively in various fields.
• The Power Lindley distribution is introduced as the distribution for the first occurrence time in the GP.

Purpose of the Study:

• To investigate parameter estimation for the geometric process (GP) when the first occurrence time follows a Power Lindley distribution.
• To compare the performance of different estimation methods: maximum likelihood, modified moments, modified L-moments, and modified least-squares.
• To evaluate the bias, mean squared error (MSE), and mean of these estimators across various sample sizes.

Main Methods:

• Maximum Likelihood Estimation (MLE)
• Modified Moments Estimation
• Modified L-Moments Estimation
• Modified Least-Squares Estimation
• Monte Carlo simulations to assess estimator performance
• Analysis of bias, mean squared error (MSE), and mean values

Main Results:

• The study obtained estimators for the GP parameters (a, α, λ) using four distinct methods.
• Performance of estimators was evaluated via Monte Carlo simulations for small, moderate, and large sample sizes.
• The paper includes two real-data examples to demonstrate the practical application of the proposed methods.

Conclusions:

• The research provides a comprehensive comparison of parameter estimation techniques for the geometric process with a Power Lindley first occurrence time distribution.
• The findings offer valuable guidance for selecting appropriate estimation methods based on sample size and desired statistical properties.
• The inclusion of real-data examples enhances the practical applicability of the study's results in fields utilizing geometric process modeling.