Measurements are drawn independently from a Gaussian with unknown mean and known…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Probability, statistics and stochastic methods · Probability, statistics and stochastic methods

Error, Chance and the Machine, Ch Estimators and the information in data

Problem

N = 16

measurements

x_1,\dots,x_{16}

are drawn independently from a Gaussian with unknown mean

\mu

and known standard deviation

\sigma = 2

. Someone argues: "The Fisher information per measurement is

I(\mu) = 1/\sigma^2 = 0.25

, so the Cramér-Rao bound on the variance of any estimator of

\mu

\sigma^2/N = 4/16 = 0.25

. Now consider the estimator

\hat\mu' = 0.9\,\bar x

, where

\bar x

is the sample mean. Its variance is

0.9^2\times 0.25 = 0.2025

, which is less than

0.25

. So

\hat\mu'

beats the Cramér-Rao bound." ((a)) Find the error in this argument. ((b)) Compute the bias

b(\mu)

\hat\mu'

and the correctly bias-adjusted Cramér-Rao bound,

(1+b'(\mu))^2/(NI)

, and show that

\hat\mu'

does not in fact beat it. ((c)) Compute the mean squared error of

\hat\mu'

as a function of

\mu

and state the value of

|\mu|

above which it exceeds the mean squared error of the unbiased sample mean.

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