The Gompertz distribution is widely applied in describing human mortality, establishing actuarial tables, and various other fields. Historically, it was originally introduced by Benjamin Gompertz (1825) in connection with human mortality. This study aims to derive and analyze the mathematical and statistical properties of the Gompertz distribution, providing explicit expressions for parameter estimation from both frequentist and Bayesian perspectives. We then apply these estimation methodologies to analyze COVID-19 data in Cameroon. We investigate and compare numerous frequentist approaches for parameter estimation, including maximum likelihood, method of moments, pseudo-moments, modified moments, L-moments, percentile-based, least squares (including weighted), maximum product of spacings, minimum spacing absolute distance, minimum spacing absolute-log distance, Cramér-von-Mises, and Anderson-Darling (including right-tail) estimators. Their performance is evaluated using extensive numerical simulations, and their coverage probabilities are also assessed. Our results indicate that among the frequentist estimators, modified moments and moments estimators generally perform better than their counterparts. For Bayesian estimators, those based on the Mean Squared Error Loss Function (MSELF) and Kullback-Leibler Loss Function (KLF) demonstrate superior performance. The maximum product of spacings estimators also exhibit competitive performance.