How do you know if an estimator is efficient?

Essentially, a more efficient estimator, experiment, or test needs fewer observations than a less efficient one to achieve a given performance. An efficient estimator is characterized by a small variance or mean square error, indicating that there is a small deviance between the estimated value and the “true” value.

How do you calculate relative efficiency?

We can compare the quality of two estimators by looking at the ratio of their MSE. If the two estimators are unbiased this is equivalent to the ratio of the variances which is defined as the relative efficiency. rndr = n + 1 n · n n + 1 θ.

What is an inefficient estimator?

inefficient estimator. A statistical estimator whose variance is greater than that of an efficient estimator. In other words, for an inefficient estimator equality in the Rao–Cramér inequality is not attained for at least one value of the parameter to be estimated.

What does it mean when an estimator is unbiased?

An unbiased estimator of a parameter is an estimator whose expected value is equal to the parameter. That is, if the estimator S is being used to estimate a parameter θ, then S is an unbiased estimator of θ if E(S)=θ. Remember that expectation can be thought of as a long-run average value of a random variable.

Can an efficient estimator be biased?

The fact that any efficient estimator is unbiased implies that the equality in (7.7) cannot be attained for any biased estimator. However, in all cases where an efficient estimator exists there exist biased estimators that are more accurate than the efficient one, possessing a smaller mean square error.

Which estimator is most efficient?

Efficiency: The most efficient estimator among a group of unbiased estimators is the one with the smallest variance. For example, both the sample mean and the sample median are unbiased estimators of the mean of a normally distributed variable. However, X has the smallest variance.

What is the efficiency of a sample estimator?

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.

What is the most efficient estimator?

Can a biased estimator be efficient?

An estimator in statistics is a way of guessing a parameter based on data. The estimator alternates between two ridiculous values, but in the long run these values average out to the true value. Exact in the limit, useless on the way there.

How to calculate the efficiency of an estimator?

Thus, if we have two estimators α 1 ^ and α 2 ^ with variances V a r ( α 1 ^) and V a r ( α 2 ^) respectively, and if V a r ( α 1 ^) < V a r ( α 2 ^), then α 1 ^ will be an efficient estimator. The ratio of the variances of two estimators denoted by e ( α 1 ^, α 2 ^) is known as the efficiency of α 1 ^ and α 2 ^ is defined as follows:

What does it mean by more efficient estimator cross validated?

It says in the above Wikipedia article that: If T 1 and T 2 are estimators for the parameter θ, then T 1 is said to dominate T 2 if: 1) If we don’t know θ , then how can we show one is smaller than the other in the above inequality.

Are there any finite sample efficient estimators?

However this criterion has some limitations: Finite-sample efficient estimators are extremely rare. In fact, it was proved that efficient estimation is possible only in an exponential family, and only for the natural parameters of that family. This notion of efficiency is sometimes restricted to the class of unbiased estimators.

Is it possible to calculate efficiency in an exponential family?

In fact, it was proved that efficient estimation is possible only in an exponential family, and only for the natural parameters of that family. This notion of efficiency is sometimes restricted to the class of unbiased estimators. (Often it isn’t.)