estimator n : an expert at calculation (or at operating calculating machines) [syn: calculator, reckoner, figurer, computer]
In statistics, an estimator is a function of the observable sample data that is used to estimate an unknown population parameter (which is called the estimand); an estimate is the result from the actual application of the function to a particular sample of data. Many different estimators are possible for any given parameter. Some criterion is used to choose between the estimators, although it is often the case that a criterion cannot be used to clearly pick one estimator over another. To estimate a parameter of interest (e.g., a population mean, a binomial proportion, a difference between two population means, or a ratio of two population standard deviation), the usual procedure is as follows:
Suppose we have a fixed parameter \theta \ that we wish to estimate. Then an estimator is a function that maps a sample design to a set of sample estimates. An estimator of \theta \ is usually denoted by the symbol \widehat. A sample design can be thought of as an ordered pair \ ( S , p ) where \ S is a set of samples (or outcomes), and \ p is the probability density function. The probability density function maps the set \ S to the closed interval [0,1], and has the property that the sum (or integral) of the values of \ p(s) , over all \ s in \ S , is equal to 1. For any given subset \ A of \ S , the sum or integral of \ p(s) over all \ s in \ A is the \ Prob(A) .
For all the properties below, the value \theta\ , the estimation formula, the set of samples, and the set probabilities of the collection of samples, can be considered fixed. Yet since some of the definitions vary by sample (yet for the same set of samples and probabilities), we must use \ s in the notation. Hence, the estimate for a given sample \ s is denoted as \widehat(s).
We have the following definitions and attributes.
- For a given sample s \ , the error of the estimator \widehat is defined as \widehat(s) - \theta, where \widehat(s) \ is the estimate for sample \ s , and \theta \ is the parameter being estimated. Note that the error depends not only on the estimator (the estimation formula or procedure), but on the sample.
- The mean squared error of \widehat is defined as the expected value (probability-weighted average, over all samples) of the squared errors; that is, \operatorname(\widehat) = \operatorname[(\widehat - \theta)^2]. It is used to indicate how far, on average, the collection of estimates are from the single parameter being estimated. Consider the following analogy. Suppose the parameter is the bull's-eye of a target, the estimator is the process of shooting arrows at the target, and the individual arrows are estimates. Then high MSE means the average distance of the arrows from the target is high, and low MSE means the average distance from the target is low. The arrows may or may not be clustered. For example, even if all arrows hit the same point, yet grossly miss the target, the MSE is still relatively large. Note, however, that if the MSE is relatively low, then the arrows are likely more highly clustered (than highly dispersed).
- For a given sample s \ , the sampling deviation of the estimator \widehat is defined as \widehat(s) - \operatorname( \widehat ) , where \widehat(s) \ is the estimate for sample \ s , and \operatorname( \widehat ) is the expected value of the estimator. Note that the sampling deviation depends not only on the estimator, but on the sample.
- The variance of \widehat is simply the expected value of the squared sampling deviations; that is, \operatorname(\widehat) = \operatorname[(\widehat - \operatorname(\widehat) )^2]. It is used to indicate how far, on average, the collection of estimates are from the expected value of the estimates. Note the difference between MSE and variance. If the parameter is the bull's-eye of a target, and the arrows are estimates, then a relatively high variance means the arrows are dispersed, and a relatively low variance means the arrows are clustered. Some things to note: even if the variance is low, the cluster of arrows may still be far off-target, and even if the variance is high, the diffuse collection of arrows may still be unbiased. Finally, note that even if all arrows grossly miss the target, if they nevertheless all hit the same point, the variance is zero.
- The bias of \widehat is defined as B(\widehat) = \operatorname(\widehat) - \theta. It is the distance between the average of the collection of estimates, and the single parameter being estimated. It also is the expected value of the error, since \operatorname(\widehat) - \theta = \operatorname(\widehat - \theta ) . If the parameter is the bull's-eye of a target, and the arrows are estimates, then a relatively high absolute value for the bias means the average position of the arrows is off-target, and a relatively low absolute bias means the average position of the arrows is on target. They may be dispersed, or may be clustered. The relationship between bias and variance is analogous to the relationship between accuracy and precision.
- \widehat is an unbiased estimator of \theta \ if and only if B(\widehat) = 0. Note that bias is a property of the estimator, not of the estimate. Often, people refer to a "biased estimate" or an "unbiased estimate," but they really are talking about an "estimate from a biased estimator," or an "estimate from an unbiased estimator." Also, people often confuse the "error" of a single estimate with the "bias" of an estimator. Just because the error for one estimate is large, does not mean the estimator is biased. In fact, even if all estimates have astronomical absolute values for their errors, if the expected value of the error is zero, the estimator is unbiased. Also, just because an estimator is biased, does not preclude the error of an estimate from being zero (we may have gotten lucky). The ideal situation, of course, is to have an unbiased estimator with low variance, and also try to limit the number of samples where the error is extreme (that is, have few outliers). Yet unbiasedness is not essential. Often, if we permit just a little bias, then we can find an estimator with lower MSE and/or fewer outlier sample estimates.
- The MSE, variance, and bias, are related: \operatorname(\widehat) = \operatorname(\widehat\theta) + (B(\widehat))^2,
- i.e. mean squared error = variance + square of bias.
The standard deviation of an estimator of θ (the square root of the variance), or an estimate of the standard deviation of an estimator of θ, is called the standard error of θ.
ConsistencyA consistent estimator is an estimator that converges in probability to the quantity being estimated as the sample size grows without bound.
An estimator t_n (where n is the sample size) is a consistent estimator for parameter \theta if and only if, for all \epsilon > 0, no matter how small, we have