Properties and parameter estimation of the partly-exponential distribution

The partly exponential distribution was introduced in the paper by Atikankul et al. in 2021. This distribution was conducted using a negatively exponential distribution, which is a special case of a negatively mixed distribution. In this work, we propose various theoretical properties such as the cumulative distribution function, the moment generating function, the first three moments, the characteristic function, the mode, the likelihood, and the moment equations for partly‑exponential distribution. This dissertation primarily seeks to propose parameter estimation techniques for the partly‑exponential distribution, notably Maximum Likelihood Estimation (MLE), the jackknife method based on MLE, and Bayesian estimation, using both numerical methods and the Metropolis‑Hastings Markov Chain Monte Carlo (MCMC) algorithm. The study examines point estimate and the construction of confidence intervals. The MLE method applies Wald and profile likelihood confidence intervals, whereas the Bayesian approach use credible intervals. Despite the analytical complexity of the partly‑exponential distribution’s probability density function, which precludes the derivation of closed‑form expressions for the MLEs, numerical methods provide a viable means of obtaining accurate parameter estimates within the context of simulation studies. The performance of point estimators for the parameters of the partly‑exponential distribution is evaluated by assessing their precision through mean squared error (MSE) and bias. Additionally, the performance of the confidence intervals is examined in terms of coverage probability (CP) and average length (AL). The objective of the study is to compare the result of point estimation and confidence interval methods across the four proposed estimate methods. For the simulation study, parameter settings were chosen across 60 distinct scenarios to thoroughly assess estimator performance. Specifically, values for the parameter δ were set at 0.1, 1, 10, and 50, with each δ level accompanied by three corresponding levels of τ (low, medium, and high), determined relative to the value of δ. Additionally, the parameter γγ was varied across five levels: 0, 0.25, 0.5, 0.75, and 1. The simulation incorporated sample sizes of 25, 50, 75, 100, and 1000. The simulation study indicates that for the point estimation, the parameter accuracy is heavily influenced by both the true parameter values and sample size. As the values of γ and δ increase, estimators tend to exhibit higher Mean Squared Error (MSE) and bias, especially in smaller samples. Among the methods compared, Bayesian approaches, particularly the Metropolis‑Hastings algorithm provide lower MSE and more stable bias across parameter configurations. The MLE and Jackknife methods show higher variability, especially under extreme parameter settings. Increasing sample size substantially reduces both MSE and bias, improving estimator reliability. When estimating multiple parameters simultaneously, MLE generally outperforms the Jackknife method in stability and efficiency, although the Jackknife occasionally yields better performance under specific conditions with high γ and large samples. Overall, robust estimation requires adequate sample size and careful method selection, with Bayesian methods proving especially reliable for challenging parameter settings. The comparison of 95% confidence and credible intervals reveals that interval accuracy and precision are highly sensitive to the choice of method, sample size, and parameter values. Wald confidence intervals often suffer from undercoverage and overly wide average lengths (AL) in small samples, particularly for extreme values of δ or τ. In contrast, Bayesian credible intervals (Numerical and Metropolis‑Hastings) consistently maintain high coverage probabilities with shorter ALs. Profile likelihood (PL) intervals demonstrate superior performance in large samples, often achieving nominal coverage with minimal interval width, but their instability in small samples limits their practical use. When estimating multiple parameters simultaneously, coverage for δ remains poor in many cases, while γ and τ are estimated more reliably, especially with large sample size. These findings highlight the importance of method selection and sufficient sample size in achieving accurate and efficient interval estimates for the partly‑exponential distribution. This study uses real datasets, including March precipitation (inches) in Minneapolis/St Paul and waiting times (in minutes) for 100 bank customers.We applied both Maximum Likelihood Estimation and the Bayesian methodology. The results indicated that the real data set is appropriate for estimation with a fixed γ=1, and the maximum likelihood is accurately estimated in these instances.