The formulas used for computing sample means and other sample statistics are examples of estimators. The particular value calculated from sample observations using an estimator is called an estimate. An estimate is a fixed number pertaining to a given sample and has no sampling distribution whereas an estimator has a sampling distribution. The sample mean used as an estimate of the population, is called a point estimate of the population mean.
The three desirable properties of an estimator are unbiasedness, efficiency, and consistency.
An unbiased estimator is one whose expected value (the mean of its sampling distribution) equals the parameter it is intended to estimate. The sample variance calculated using a divisor of n-1 is an unbiased estimator of the population variance. If the sample variance is calculated using a divisor n, then it would be biased estimator because its expected value would be smaller than the population variance.
There can be many unbiased estimators. To choose among the unbiased estimators, we look at the efficiency.
An unbiased estimator is efficient if no other unbiased estimator of the same parameter has a sampling distribution with smaller variance. The variance of the sampling distribution of the sample mean is smaller for larger sample sizes.
An estimator is consistent if its accuracy (probability of estimates close to the value of the population parameter) increases with an increase in the sample size. For a consistent estimator, the sampling distribution becomes concentrated on the value of the parameter it is intended to estimate as the sample size approaches infinity. Because when the sample size (n) approaches infinity, the standard error (σ2/n) goes to zero.
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