Maximum likelihood estimation of parameters of truncated and censored gamma and exponential distributions using the expectation maximisation algorithm.

This study aimed to determine the maximum likelihood (ML) estimates for the Gamma and Exponential distributions under conditions of left-truncated and right-censored data, employing the Expectation-Maximisation (EM) algorithm. Additionally, it sought to compare the performance of the EM algorithm with two alternative estimation techniques, namely the Newton-Raphson (NR) algorithm and the Stochastic Expectation-Maximisation (SEM) algorithm. Simulated left-truncated and right-censored data were generated from the Exponential and Gamma distributions. Performance comparisons among the algorithms were conducted based on simulations involving varying degrees of censoring, truncation, and sample sizes, utilizing metrics such as mean square error (MSE), bias, Akaike Information Criterion (AIC), and Bayesian Information Criterion (BIC). The validation of AIC and BIC was assessed under both candidate models (Gamma and Exponential distributions) using the EM algorithm. All computations were performed using R-4.3 version. The EM algorithm consistently yielded estimates with low bias under moderate levels of censoring and truncation, indicating its accuracy in estimation. Comparison of the EM algorithm with the NR and SEM algorithms revealed similar estimates for larger sample
sizes. In terms of convergence speed, both the EM and NR algorithms converged faster than the SEM algorithm. Moreover, as the sample size increased, bias and MSE decreased for all three algorithms. Furthermore, the EM algorithm demonstrated superior performance in terms of MSE, bias,
AIC, BIC, and convergence speed. However, in scenarios with high levels of censoring and truncation, all three algorithms encountered difficulties in accurately estimating parameters due to data loss. Ultimately, the findings of this study are anticipated to advance statistical methodology and
enhance the accuracy of parameter estimation techniques for censored and truncated data scenarios.

Key words : Truncation, censoring, left-truncated, right-censored, maximum likelihood esitimation, expectation maximisation algorithm, stochastic expectation algorithm, NewtonRaphson algorithm, likelihood function, simulations, Gamma and Exponential distribution.