What is an efficient estimator in statistics?

What is an efficient estimator in statistics?

An efficient estimator is an estimator that estimates the quantity of interest in some “best possible” manner. The notion of “best possible” relies upon the choice of a particular loss function — the function which quantifies the relative degree of undesirability of estimation errors of different magnitudes.

How do you show an estimator is efficient?

For an unbiased estimator, efficiency indicates how much its precision is lower than the theoretical limit of precision provided by the Cramer-Rao inequality. A measure of efficiency is the ratio of the theoretically minimal variance to the actual variance of the estimator.

Why sample mean is efficient estimator?

Example 1: The variance of the sample mean¯X is σ2/n, which decreases to zero as we increase the sample size n. Hence, the sample mean is a consistent estimator for µ.

What is the efficiency of a sample estimator?

The efficiency of an estimator is a measure of how ‘tight’ are it’s estimates around the true population value of the parameter that it is estimating, as compared to a perfectly efficient estimator. A perfectly efficient estimator is one whose variance is equal to the Cramér–Rao bound for that class of estimators.

What is the most efficient estimator?

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.

What does it mean for an estimator to be inefficient?

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.

Why mean is more efficient than median?

Either not much or nothing at all depending on which data point was changed. The sample median is more robust than the mean because it is more resilient to this kind of change. Asymptotic relative efficiency (ARE) is a way of measuring how much statistics bounce around as the amount of data increases.

What is an efficient estimator how is it different from a consistent estimator?

An estimate is unbiased if its expected value equals the true parameter value. This will be true for all sample sizes and is exact whereas consistency is asymptotic and only is approximately equal and not exact.

How are efficiency and sufficiency related in statistics?

EFFICIENCY: An estimator is said to be efficient if in the class of unbiased estimators it has minimum variance. Clearly, is the more efficient since it has the smaller variance. SUFFICIENCY: We say that an estimator is sufficient if it uses all the sample information. so the sequence is a consistent estimator for .

Which estimator is more efficient?

What is the difference between consistency and Unbiasedness?

Consistency of an estimator means that as the sample size gets large the estimate gets closer and closer to the true value of the parameter. Unbiasedness is a finite sample property that is not affected by increasing sample size. An estimate is unbiased if its expected value equals the true parameter value.