At a recent conference on Bayes, fiducial and frequentist inference, David Cox presented eight illustrative examples, chosen to highlight potential difficulties for the theory of inference. We discuss these examples in light of the efforts of the conference, and related meetings, to study the similarities and differences between the approaches to inference. Emphasis is placed on the goal of finding a distribution for an unknown parameter.
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.
SummaryFor an exponential model with scalar parameter, WelchP:1963 examined the role of Bayesian analysis in statistical inference, more specifically the use of the Jeffreys:1946 prior. They determined that Bayesian intervals and thus in effect Bayesian quantiles had second order confidence accuracy. We use a Taylor series expansion of the log-model to develop a second order version of the vector exponential model; this is developed as a contribution to theory in statistics at a time when algorithms are prominent, and it provides a basis for generalizing the Welch-Peers approach to the vector parameter context.
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