On the Joint Asymptotic Normality of the Method of Moments Estimators of the Two-Parameter Exponential Distribution

Two-parameter exponential distribution has been widely used in research involving lifetime data, survival analysis, medical research, and reliability assessment. Since variables following this distribution are continuous, the population mean is often regarded as a key parameter of interest. In this paper, confidence intervals for the mean of a two-parameter exponential population are developed based on the joint asymptotic normality of the method of moments estimators, derived using the Delta method. The proposed asymptotic confidence intervals are compared with Wald-type confidence intervals in terms of coverage probability and average interval width. The performance of the proposed intervals is evaluated using Monte Carlo simulation, where particular attention is given to their coverage probability and mean interval width. The performance is further illustrated through a real-data application using daily PM2.5 concentration data from Bangkok, Thailand. Simulation, and the theoretical results indicate that the performance of estimators in the two-parameter exponential distribution improves as the sample size increases. Small samples lead to biased and unstable estimates, with wider confidence intervals and lower coverage probabilities, whereas moderate and large samples yield more accurate and stable results. For large samples, bias becomes negligible, variances approach the corresponding theoretical values, and coverage probabilities and interval widths approach the nominal level, confirming consistency and asymptotic efficiency.