STEIN'S IDENTITY AND THE CRAMER-RAO LOWER BOUND

Abstract

Stein’s identity is a way to characterize continuous probability distributions. It helps in verifying or identifying the underlying distribution of a given random variable. Initially, we take a look at Stein’s identity and how the equation varies with different distributions. The Cramér Rao Lower Bound (CRLB) provides us with the least possible variance of an estimator of a distribution. Its key use lies in the comparison of the variance of estimators in order to choose the one with ideal variance. Our goal in this paper is to observe how despite being discovered in a different time period and serving an entirely different purpose from that of the CRLB, Stein’s Identity can be framed as a restatement of the CRLB. We will look at how Stein’s Identity provides an equation that is analogous to the one that the general formula of the CRLB gives us. We will analyze an unbiased estimator for the normal and calculate its CRLB using the general formula. Then, we will apply Stein’s identity on the estimator and see how we obtain an analogous equation. We have benefited from following the developments in the interesting paper (Mukhopadhyay, 2021).