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What is efficiency in statistical inference?

What is efficiency in statistical inference?

In the comparison of various statistical procedures, efficiency is a measure of quality of an estimator, of an experimental design, or of a hypothesis testing procedure.

How do you measure efficiency 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.

What is meant by relative efficiency?

1. for two tests (A and B) of the same hypothesis operating at the same significance level, the ratio of the number of cases needed by Test A to the number needed by Test B for each to have the same statistical power.

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.

What is the efficiency formula?

Efficiency is often measured as the ratio of useful output to total input, which can be expressed with the mathematical formula r=P/C, where P is the amount of useful output (“product”) produced per the amount C (“cost”) of resources consumed.

Is the estimator unbiased?

An unbiased estimator is an accurate statistic that’s used to approximate a population parameter. That’s just saying if the estimator (i.e. the sample mean) equals the parameter (i.e. the population mean), then it’s an unbiased estimator.

Which estimator is more 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.

How do you interpret 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. indicating that for n > 1, ˆθ2 has a lower variance.

Can a biased estimator be efficient?

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.

What is efficiency with example?

Efficiency is defined as the ability to produce something with a minimum amount of effort. An example of efficiency is a reduction in the number of workers needed to make a car. with a minimum of effort, expense, or waste; quality or fact of being efficient.

How do you describe efficiency?

Efficiency signifies a peak level of performance that uses the least amount of inputs to achieve the highest amount of output. Efficiency requires reducing the number of unnecessary resources used to produce a given output including personal time and energy.

How do you determine an unbiased estimator?

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.

Which is the best definition of efficiency in statistics?

Efficiency (statistics) In the comparison of various statistical procedures, efficiency is a measure of quality of an estimator, of an experimental design, or of a hypothesis testing procedure.

Why is efficiency of estimators important in statistics?

Efficiency in Statistics Efficiency in statistics is important because they allow one to compare the performance of various estimators. Although an unbiased estimator is usually favored over a biased one, a more efficient biased estimator can sometimes be more valuable than a less efficient unbiased estimator.

Is the minimum variance unbiased estimator always efficient?

Efficient estimators are always minimum variance unbiased estimators. However the converse is false: There exist point-estimation problems for which the minimum-variance mean-unbiased estimator is inefficient. Historically, finite-sample efficiency was an early optimality criterion.

How to calculate the efficiency of a normal distribution?

Consider the model of a normal distribution with unknown mean but known variance: { Pθ = N(θ, σ2) | θ ∈ R }. The data consists of n independent and identically distributed observations from this model: X = (x1, …, xn).