Cramer-Rao下界的扩充解释及其对帕累托分布参数倒数的估计

The Expanded Interpretation of Cramer-Rao Lower Bound and its Estimation of the Reciprocal of Pareto Distribution Parameters

Cramer-Rao下界是统计学中非常重要的下界,也是Fisher信息量的倒数。针对Cramer-Rao正则族成立的5个条件以及参数函数无偏估计的方差下限值,从微积分计算的角度对Cramer-Rao下界进行扩充解释,证明了帕累托分布参数倒数的最大似然估计能达到Cramer-Rao下界,并给出了该参数倒数的另一个无偏估计。结果表明只有在无偏估计与参数本身无关的情况下,该估计的方差下限值才是Cramer-Rao下界。

Cramer-Rao lower bound(CRLB)is a significant lower bound in statistics,which is the reciprocal of Fisher information.Pinpointing at the five conditions satisfied by Cramer-Rao regular family and the lower bound value for the variance of unbiased parameter s function,this paper provides the expanded interpretation of CRLB from the perspective of calculus calculation.It proves that the maximum likelihood estimation of the reciprocal of Pareto distribution parameter could attain CRLB.In addition,it offers another unbiased estimator for the reciprocal.The results show that the unbiased estimator s variance could attain CRLB,only if the estimator is irrelevant to the parameter itself.