Efficiencies are often defined using the variance or mean square error as the measure of desirability. For experimental designs, efficiency relates to the ability of a design to achieve the objective of the study with minimal expenditure of resources such as time and money. In simple cases, the relative efficiency of designs can be expressed as the ratio of the sample sizes required to achieve a given objective. The efficiency of an unbiased estimator T is defined as.
Thus e T is the minimum possible variance for an unbiased estimator divided by its actual variance. If an unbiased estimator of a parameter attains for all values of the parameter, then the estimator is called efficient.
An efficient estimator is also the minimum variance unbiased estimator MVUE. For some estimators , they can attain efficiency asymptotically and are thus called asymptotically efficient estimators.
Consider a sample of size drawn from a normal distribution of mean and unit variance , i. The sample mean , , of the sample , defined as. Now consider the sample median.
This is an unbiased and consistent estimator for. For large the sample median is approximately normally distributed with mean and variance i. Note that this is the asymptotic efficiency — that is, the efficiency in the limit as sample size tends to infinity. The sample mean is thus more efficient than the sample median in this example. However, there may be measures by which the median performs better. For example, the median is far more robust to outliers , so that if the Gaussian model is questionable or approximate, there may advantages to using the median see Robust statistics.
If and are estimators for the parameter , then is said to dominate if:. Formally, dominates if. Although is in general a function of , in many cases the dependence drops out; if this is so, being greater than one would indicate that is preferable, whatever the true value of. Pitman efficiency [ 6 ] and Bahadur efficiency or Hodges—Lehmann efficiency [ 7 ] [ 8 ] relate to the comparison of the performance of Statistical hypothesis testing procedures.
There are various inefficient estimations of statistical parameters which can be calculated with far fewer mathematical operations than efficient estimates.
An estimator with efficiency 1. The efficiency of a given estimator depends on the population. For example, for a normally distributed population, the sample mean is an efficient estimator of the population mean. But the sample mean is usually not an efficient estimator of the mean of a non-normal population. Active 1 year, 8 months ago. Viewed 7k times. Could someone give an easy but very concrete example?
Improve this question. Add a comment. Active Oldest Votes. The relative efficiency of two unbiased estimators is the ratio of their precisions the bound cancelling out When you're dealing with biased estimators, relative efficiency is defined in terms of the ratio of MSE.
Could someone give an easy but very concrete example. Improve this answer. But I am just wondering could you explain in layman term what exactly it means by the number 0.
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