The objective of this article is to compare numerical and statistical methods for estimating the maximum likelihood parameters in a given distribution when closed-form solutions are not available due to the shape of the density function. To address this, we utilize numerical and statistical methods such as the Newton-Raphson method, the Expectation-Maximization algorithm, and the Stochastic Expectation-Maximization algorithm. We aim to compare these methods in terms of bias mean, average of mean squared error, runtime of the program, and the maximum value of the log- likelihood function. Specifically, we investigate the application of these methods for estimating parameters in the Poisson-Exponential distribution under the joint type-II censoring scheme. By analyzing the performance of these methods, we contribute to the understanding of their effectiveness and limitations in this context.
