Show that the following random variables have the given...

Show that the following random variables have the given Cramer-Rao lower bound and determine whether the associated maximum likelihood estimator is efficient: (a) Binomial with parameters $n$ and unknown $p: p(1-p) / n^2$. (b) Gaussian with known variance $\sigma^2$ and unknown mean: $\sigma^2 / n$. (c) Gaussian with unknown variance: $2 \sigma^4 / n$. Consider two cases: mean known; mean unknown. Does the standard unbiased estimator for the variance achieve the Cramer-Rao lower bound? Note that $E\left[(X-\mu)^4\right]=3 \sigma^4$. (d) Gamma with parameters known $\alpha$ and unknown $\beta=1 / \lambda: \beta^2 / n \alpha$. (e) Poisson with parameter unknown $\alpha: \alpha / n$.

Show that the following random variables have the given Cramer-Rao lower bound and determine whether the associated maximum likelihood estimator is efficient:
(a) Binomial with parameters $n$ and unknown $p: p(1-p) / n^2$.
(b) Gaussian with known variance $\sigma^2$ and unknown mean: $\sigma^2 / n$.
(c) Gaussian with unknown variance: $2 \sigma^4 / n$. Consider two cases: mean known; mean unknown. Does the standard unbiased estimator for the variance achieve the Cramer-Rao lower bound? Note that $E\left[(X-\mu)^4\right]=3 \sigma^4$.
(d) Gamma with parameters known $\alpha$ and unknown $\beta=1 / \lambda: \beta^2 / n \alpha$.
(e) Poisson with parameter unknown $\alpha: \alpha / n$.

Key Concepts

Unbiased Estimator

An unbiased estimator is one for which the expected value of the estimator equals the true parameter value. Although unbiasedness is a desirable property, it does not guarantee minimum variance. Evaluating whether an unbiased estimator attains the Cramer-Rao lower bound is crucial in determining if it is the best among all unbiased estimators, particularly when comparing standard estimators like that for the variance in Gaussian distributions.

Maximum Likelihood Estimation

Maximum likelihood estimation is a method for estimating the parameters of a statistical model by maximizing the likelihood function, which measures how likely it is that given the observed data, the parameters take on certain values. Under regular conditions, maximum likelihood estimators are asymptotically unbiased, consistent, and efficient, meaning they achieve the Cramer-Rao lower bound in large samples.

Fisher Information

Fisher information quantifies the amount of information that an observable random variable carries about an unknown parameter upon which the probability depends. It is calculated from the expected value of the squared derivative (or score) of the log-likelihood function with respect to the parameter. This measure is central to the derivation of the Cramer-Rao lower bound and helps determine the best possible accuracy achievable for an unbiased estimator.

Efficiency of Estimators

An estimator is termed efficient if it achieves the Cramer-Rao lower bound, meaning its variance is as low as theoretically possible for unbiased estimators. Efficiency is an important property because it indicates that the estimator is utilizing all available information from the data in estimating the parameter, often making it the preferred estimator in practice.

Cramer-Rao Lower Bound

The Cramer-Rao lower bound is a fundamental result in estimation theory that provides a lower limit on the variance of any unbiased estimator of a parameter. It is derived from the Fisher information and acts as a benchmark to assess the efficiency of estimators; if an estimator’s variance reaches the bound, it is considered to be efficient.

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