The classical credibility premium provides a simple and efficient method for predicting future damages and losses. However, when dealing with a nonhomogeneous population, this widely used technique has been challenged by the Regression Tree Credibility (RTC) model and the Logistic Regression Credibility (LRC) model. This article introduces the Mixture Credibility Formula (MCF), which represents a convex combination of the classical credibility premiums of several homogeneous subpopulations derived from the original population. We also compare the performance of the MCF method with the RTC and LRC methods. Our analysis demonstrates that the MCF method consistently outperforms these approaches in terms of the quadratic loss function, highlighting its effectiveness in refining insurance premium calculations and enhancing risk assessment strategies.

Second order Bayes estimators, being the main tool in second order optimal statistical theory, provide a natural basis for a new approach to the problem of the prediction of functions of expectation functional for members of an exponential dispersion family. A general formula, providing such prediction up to the term of the order 1/n, is suggested and the application to the problem of the prediction of the tail of distributions is demonstrated. The results are illustrated with normal and gamma claim sizes. The numerical experiment demonstrates the high effectiveness of the approach even for small sample sizes.