A new credibility estimation of process variance with applications

In classical credibility theory, estimation is typically limited to the hypothetical mean, restricting its use for premium principles that depend on higher-order moments. To address this, we develop a credibility-based framework for estimating the process variance under both known and unknown hypothetical means and apply these estimators to a broad class of variance-related premium principles, including the expected value, variance, standard deviation, and modified-variance principles. The estimators are derived via constrained linear projection techniques, minimizing the mean squared error between the estimator and the true process variance. Explicit formulas are obtained that are optimal among affine transformations of the data. The proposed estimators exhibit desirable statistical properties, including conditional unbiasedness, consistency, mean squared error convergence, and asymptotic normality. Numerical studies demonstrate their favorable convergence behavior, and an empirical analysis with real insurance data highlights their practical relevance. This framework extends Bühlmann’s classical credibility theory to second-moment estimation while remaining computationally tractable and requiring only mild moment conditions, without specifying the population or prior distributions.