A matching prior for the product of normal means based on the modified profile likelihood

In this paper, we develop a matching prior for the product of means in several normal distributions with unrestricted means and unknown variances. For this problem, properly assigning priors for the product of normal means has been issued because of the presence of nuisance parameters. Matching priors, which are priors matching the posterior probabilities of certain regions with their frequentist coverage probabilities, are commonly used but difficult to derive in this problem. We developed the first order probability matching priors for this problem; however, the developed matching priors are unproper. Thus, we apply an alternative method and derive a matching prior based on a modification of the profile likelihood. Simulation studies show that the derived matching prior performs better than the uniform prior and Jeffreys' prior in meeting the target coverage probabilities, and meets well the target coverage probabilities even for the small sample sizes. In addition, to evaluate the validity of the proposed matching prior, Bayesian credible interval for the product of normal means using the matching prior is compared to Bayesian credible intervals using the uniform prior and Jeffrey's prior, and the confidence interval using the method of Yfantis and Flatman.