Chức năng thông tin nghịch đảo là thấp hơn Cramer-Rao ràng buộc đối với phương sai của một ước tính. Nó không nhận được tốt hơn so với điều này cho mẫu lớn, mặc dù đối với các mẫu nhỏ, phương pháp ước lượng khác có thể thực hiện tốt hơn so với khả năng tối đa cho cả hai thiên vị | Properties ofMaximum Likelihood Estimators 51 The inverse information function is the Cramer-Rao lower bound for the variance of an estimator. It doesn t get better than this for large samples although for small samples other estimation methods may perform better than maximum likelihood for both bias and variance. The asymptotic variance I d 1 is often used as an approximate variance of an estimator even where the sample size is not large. A problem in practice is that in general I 0 is a function of the unknown parameter 0. To put an approximate value on the variance of Ỗ we use the estimator Ồ in place of 0. Another problem arises if the likelihood function is very complicated because then the information matrix is difficult to find analytically. In these cases we can use numerical methods. Asymptotic Normal Distribution Estimates are asymptotically normal multivariate normal if 0 is a vector with mean equal to the parameter s being estimated and variance matrix I ớ _1 where I 0 is the information function defined above. For large samples this can be used to set confidence intervals for the parameters. MLE of g 0 The Delta Method The maximum likelihood estimate of a function of 0 say g 0 is simply g 0 . The value of this can be seen with the lognormal model for example. Given parameters Ị1 and T2 the mean and variance of the associated normal distribution the mean of the lognormal distribution is g M a e ơ2 2 If we use maximum likelihood to determine parameter estimates and T the maximum likelihood estimate of the mean is g A â e -2 2 The asymptotic variance of the MLE g 0 is V d Xd where ịdg e dg e dg e dhi dd2 dOs J and s nn 1 1 Investment Guarantees INTRODUCTION The objective of life insurance is to provide financial security to policyholders and their families. Traditionally this security has been provided by means of a lump sum payable contingent on the death or survival of the insured life. The sum insured would be fixed and guaranteed. The policyholder .
