Bias-Corrected Maximum Likelihood Estimation in Actuarial Science

In modeling the rate of return associated with financial instruments, common probability distributions include the lognormal, gamma, and Weibull distributions. Furthermore, the method of maximum likelihood is widely used to estimate the unknown parameters of distributions due to the highly desirable properties of maximum likelihood estimators (MLEs). These properties include asymptotic unbiasedness, consistency, and asymptotic normality. Many of these properties, specifically unbiasedness, may not be valid for small sample sizes. We consider the Cox and Snell / Cordeiro and Klein (CSCK) methodology for determining analytic MLE bias expressions in small samples. We provide a module using Mathematica 8.0 which can calculate the CSCK MLE bias for each parameter of a given distribution. We determine the CSCK MLE biases for the lognormal, two-parameter gamma, and twoparameter Weibull distributions. By subtracting the bias (evaluated at the MLEs) from the MLE, a bias-corrected MLE (BMLE) is obtained. We also provide two simulation analyses. The first simulation demonstrates that BMLEs have preferable empirical properties when compared to MLEs for the lognormal, two-parameter gamma, and two-parameter Weibull distributions. The second simulation shows that BMLEs are preferable to MLEs for the loss reserving of an illustrative 20-period equity-linked insurance contract for both the lognormal and two-parameter Weibull distributions, but not for the two-parameter gamma distribution. JEL Classification: C13, G22