, 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:
- Select a random sample from the population of interest.
- Calculate the point estimate of the parameter.
- Calculate a measure of its variability, often a confidence interval.
- Associate with this estimate a measure of variability.
There are two types of estimators: point estimators and interval estimators.
Suppose we have a fixed parameter 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 is usually denoted by the symbol . A sample design can be thought of as an ordered pair where is a set of samples (or outcomes), and is the probability density function. The probability density function maps the set to the closed interval [0,1], and has the property that the sum (or integral) of the values of , over all in , is equal to 1. For any given subset of , the sum or integral of over all in is .
For all the properties below, the value , 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 in the notation. Hence, the estimate for a given sample is denoted as .
We have the following definitions and attributes.
- For a given sample , the error of the estimator is defined as , where is the estimate for sample , and 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 is defined as the expected value (probability-weighted average, over all samples) of the squared errors; that is, . 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 , the sampling deviation of the estimator is defined as , where is the estimate for sample , and 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 is simply the expected value of the squared sampling deviations; that is, . 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 is defined as . 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 . 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.
- is an unbiased estimator of if and only if . 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:
- 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.
A consistent sequence of estimators is a sequence of estimators that converge in probability
to the quantity being estimated as the index (usually the sample size
) grows without bound.
Mathematically, a sequence of estimators is a consistent estimator for parameter if and only if, for all , no matter how small, we have
The consistency defined above may be called weak consistency. The sequence is strongly consistent, if it converges almost surely to the true value.
An estimator that converges to a multiple of a parameter can be made into a consistent estimator by multiplying the estimator by a scale factor, namely the true value divided by the asymptotic value of the estimator. This occurs frequently in estimation of scale parameters by measures of statistical dispersion.
An asymptotically normal estimator is a consistent estimator whose distribution around the true parameter approaches a normal distribution with standard deviation shrinking in proportion to as the sample size grows. Using to denote convergence in distribution, is asymptotically normal if
for some , which is called the asymptotic variance of the estimator.
The central limit theorem implies asymptotic normality of the sample mean as an estimator of the true mean.
More generally, maximum likelihood estimators are asymptotically normal under fairly weak regularity conditions — see the asymptotics section of the maximum likelihood article.
Two naturally desirable properties of estimators are for them to be unbiased and have minimal mean squared error (MSE). These cannot in general both be satisfied simultaneously: a biased estimator may have lower mean squared error (MSE) than any unbiased estimator: despite having bias, the estimator variance may be sufficiently smaller than that of any unbiased estimator, and it may be preferable to use, despite the bias; see estimator bias.
Among unbiased estimators, there often exists one with the lowest variance, called the MVUE. In some cases an unbiased efficient estimator exists, which, in addition to having the lowest variance among unbiased estimators, satisfies the Cramér-Rao bound, which is an absolute lower bound on variance for statistics of a variable.
Concerning such "best unbiased estimators", see also Cramér-Rao bound, Gauss-Markov theorem, Lehmann-Scheffé theorem, Rao-Blackwell theorem.
See: Robust estimator, Robust statistics
Sometimes, estimators should satisfy further restrictions (restricted estimators) - eg, one might require an estimated probability
to be between zero and one, or an estimated variance to be nonnegative. Sometimes this conflicts with the requirement of unbiasedness,
see the example in estimator bias concerning the estimation of the exponent of minus twice lambda based on a sample of size one from the Poisson distribution with mean lambda.