Accurate Inference for the Mean of the Poisson-Exponential Distribution

Although the random sum distribution has been well-studied in probability theory, inference for the mean of such distribution is very limited in the literature. In this paper, two approaches are proposed to obtain inference for the mean of the Poisson-Exponential distribution. Both proposed approaches require the log-likelihood function of the Poisson-Exponential distribution, but the exact form of the log-likelihood function is not available. An approximate form of the log-likelihood function is then derived by the saddlepoint method. Inference for the mean of the Poisson-Exponential distribution can either be obtained from the modified signed likelihood root statistic or from the Bartlett corrected likelihood ratio statistic. The explicit form of the modified signed likelihood root statistic is derived in this paper, and a systematic method to numerically approximate the Bartlett correction factor, hence the Bartlett corrected likelihood ratio statistic is proposed. Simulation studies show that both methods are extremely accurate even when the sample size is small.