REPORTING UNCERTAINITY BY SPLINE FUNCTION APPROXIMATION OF LOG-LIKELIHOOD

by Sezer, Ahmet

Reporting uncertainty is one of the most important tasks in any statistical paradigm. Likelihood functions from independent studies can be easily combined, and the combined likelihood function serves as a meaningful indication of the support the observed data give to the various parameter values. This fact has led us to suggest using the likelihood function as a summary of post-data uncertainty concerning the parameter. However, a serious difficulty arises because likelihood functions may not be expressible in a compact, easily-understood mathematical form suitable for communication or publication. To overcome this difficulty, we propose to approximate log-likelihood functions by using piecewise polynomials governed by a minimal number of parameters. Our goal is to find the function of the parameter(s) that approximates the log-likelihood function with the minimum integrated (square) error over the parameter space. We achieve several things by approximating the log-likelihood; first, we significantly reduce the numerical difficulty associated with finding the maximum likelihood estimator. Second, in order to be able to combine the likelihoods that come from independent studies, it is important that the approximation of the log- likelihood should depend only upon a few parameters so that the results can be communicated compactly. By the simulation studies we compared natural cubic spline approximation with the conventional modified likelihood methods in terms of coverage probability and interval length of highest density region obtained from the likelihood and the mean squared error of the maximum likelihood estimator.