What is the use of Cramer-Rao inequality?

What is the use of Cramer-Rao inequality?

The Cramér-Rao Inequality provides a lower bound for the variance of an unbiased estimator of a parameter. It allows us to conclude that an unbiased estimator is a minimum variance unbiased estimator for a parameter.

How do you calculate the Cramer-Rao lower bound?

= p(1 − p) m . Alternatively, we can compute the Cramer-Rao lower bound as follows: ∂2 ∂p2 log f(x;p) = ∂ ∂p ( ∂ ∂p log f(x;p)) = ∂ ∂p (x p − m − x 1 − p ) = −x p2 − (m − x) (1 − p)2 .

What is the Cramer-Rao lower bound of the variance of an unbiased estimator of theta?

The function 1/I(θ) is often referred to as the Cramér-Rao bound (CRB) on the variance of an unbiased estimator of θ. I(θ) = −Ep(x;θ) { ∂2 ∂θ2 logp(X;θ) } . and, by Corollary 1, X is a minimum variance unbiased (MVU) estimator of λ.

Does MLE achieve Cramer-Rao lower bound?

Maximum Likelihood Estimation Therefore, all ML estimators achieve the Cramér-Rao lower bound. In this sense then, ML estimators are optimal. No other consistent estimator can have a smaller variance.

What is unbiased estimator in statistics?

An unbiased estimator is an accurate statistic that’s used to approximate a population parameter. “Accurate” in this sense means that it’s neither an overestimate nor an underestimate. If an overestimate or underestimate does happen, the mean of the difference is called a “bias.”

What is efficient estimator in statistics?

A measure of efficiency is the ratio of the theoretically minimal variance to the actual variance of the estimator. This measure falls between 0 and 1. An estimator with efficiency 1.0 is said to be an “efficient estimator”. The efficiency of a given estimator depends on the population.

Are unbiased estimators unique?

A very important point about unbiasedness is that unbiased estimators are not unique. That is, there may exist more than one unbiased estimator for a parameter. It is also to be noted that unbiased estimator does not always exists.

What is the difference between minimum variance unbiased estimator and minimum variance bound estimator?

One is a bound on the variance of an estimator, and one is an unbiased estimator with minimum variance. If we’re speaking about unbiased estimators in particular, if the UMVUE (or MVUE, the estimator) exists, it satisfies the bound.

What is asymptotic variance?

Though there are many definitions, asymptotic variance can be defined as the variance, or how far the set of numbers is spread out, of the limit distribution of the estimator.

What is a Rao-Blackwell estimator?

The Rao–Blackwell theorem states that if g(X) is any kind of estimator of a parameter θ, then the conditional expectation of g(X) given T(X), where T is a sufficient statistic, is typically a better estimator of θ, and is never worse. The transformed estimator is called the Rao–Blackwell estimator.

What are the generalizations of the Cramér–Rao inequality?

There are different generalizations of the Cramér–Rao inequality to the case of a vector parameter, or to that of estimating a function of the parameter. Refinements of the lower bound in (2) play an important role in such cases. The inequality (1) was independently obtained by M. Fréchet, C.R. Rao and H. Cramér.

What is the Cramér Rao bound in statistics?

In estimation theory and statistics, the Cramér–Rao bound (CRB) expresses a lower bound on the variance of unbiased estimators of a deterministic (fixed, though unknown) parameter, stating that the variance of any such estimator is at least as high as the inverse of the Fisher information.

What is the Cram´ER-Rao inequality of a random sample?

The answer is given by the following result. The Cram´er-Rao Inequality Let X= (X1,X2,…,Xn) be a random sample from a distribution with p.m/d.f. f(x|θ), where θ is a scalar parame- ter. Under certain regularity conditions on f(x|θ), for any unbiasedestimator φˆ(X) of φ(θ) Var.

What is risk indicator inequality?

An inequality in mathematical statistics that establishes a lower bound for the risk corresponding to a quadratic loss function in the problem of estimating an unknown parameter.