# Estimator

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Editor-In-Chief: C. Michael Gibson, M.S., M.D. [1]

In statistics, an estimator is a function of the observable sample data that is used to estimate an unknown population parameter; an estimate is the result from the actual application of the function to a particular set 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:

1. Select a random sample from the population of interest.
2. Calculate the point estimate of the parameter.
3. Calculate a measure of its variability, often a confidence interval.
4. Associate with this estimate a measure of variability.

There are two types of estimators: point estimators and interval estimators.

## Point estimators

Suppose ${\displaystyle {\widehat {\theta }}}$ is an estimator of a parameter ${\displaystyle \theta \ }$. That is, ${\displaystyle {\widehat {\theta }}}$ is a function that maps each sample ${\displaystyle s\ }$ to its sample estimate ${\displaystyle {\widehat {\theta }}(s)\ }$.

1. For a given sample ${\displaystyle s\ }$, the error of the estimator ${\displaystyle {\widehat {\theta }}}$ is defined as ${\displaystyle {\widehat {\theta }}(s)-\theta }$, where ${\displaystyle {\widehat {\theta }}(s)\ }$ is the estimate for sample ${\displaystyle \ s}$, and ${\displaystyle \theta \ }$ is the parameter being estimated. Note that the error depends not only on the estimator (the estimation formula or procedure), but on the sample.
2. The mean squared error of ${\displaystyle {\widehat {\theta }}}$ is defined as the expected value (probability-weighted average, over all samples) of the squared errors; that is, ${\displaystyle \operatorname {MSE} ({\widehat {\theta }})=\operatorname {E} [({\widehat {\theta }}-\theta )^{2}]}$. 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).
3. For a given sample ${\displaystyle s\ }$, the sampling deviation of the estimator ${\displaystyle {\widehat {\theta }}}$ is defined as ${\displaystyle {\widehat {\theta }}(s)-\operatorname {E} ({\widehat {\theta }})}$, where ${\displaystyle {\widehat {\theta }}(s)\ }$ is the estimate for sample ${\displaystyle \ s}$, and ${\displaystyle \operatorname {E} ({\widehat {\theta }})}$ is the expected value of the estimator. Note that the sampling deviation depends not only on the estimator, but on the sample.
4. The variance of ${\displaystyle {\widehat {\theta }}}$ is simply the expected value of the squared sampling deviations; that is, ${\displaystyle \operatorname {var} ({\widehat {\theta }})=\operatorname {E} [({\widehat {\theta }}-\operatorname {E} ({\widehat {\theta }}))^{2}]}$. 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.
5. The bias of ${\displaystyle {\widehat {\theta }}}$ is defined as ${\displaystyle B({\widehat {\theta }})=\operatorname {E} ({\widehat {\theta }})-\theta }$. 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 ${\displaystyle \operatorname {E} ({\widehat {\theta }})-\theta =\operatorname {E} ({\widehat {\theta }}-\theta )}$. 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.
6. ${\displaystyle {\widehat {\theta }}}$ is an unbiased estimator of ${\displaystyle \theta \ }$ if and only if ${\displaystyle B({\widehat {\theta }})=0}$, for all values of θ in the parameter space or, equivalently, if and only if ${\displaystyle \operatorname {E} _{\theta }({\widehat {\theta }})}$ remains equal to θ regardless of the value of θ. 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.
7. The MSE, variance, and bias, are related: ${\displaystyle \operatorname {MSE} ({\widehat {\theta }})=\operatorname {var} ({\widehat {\theta }})+(B({\widehat {\theta }}))^{2},}$
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.

## Consistency

A consistent estimator is an estimator that converges in probability to the quantity being estimated as the sample size grows without bound.

An estimator ${\displaystyle t_{n}}$ (where n is the sample size) is a consistent estimator for parameter ${\displaystyle \theta }$ if and only if, for all ${\displaystyle \epsilon >0}$, no matter how small, we have

${\displaystyle \lim _{n\to \infty }\Pr \left\{\left|t_{n}-\theta \right|<\epsilon \right\}=1.}$

It is called strongly consistent, if it converges almost surely to the true value.

## Asymptotic normality

An asymptotically normal estimator is a consistent estimator whose distribution around the true parameter ${\displaystyle \theta }$ approaches a normal distribution with standard deviation shrinking in proportion to ${\displaystyle 1/{\sqrt {n}}}$ as the sample size ${\displaystyle n}$ grows. Using ${\displaystyle {\xrightarrow {D}}}$ to denote convergence in distribution, ${\displaystyle t_{n}}$ is asymptotically normal if

${\displaystyle {\sqrt {n}}(t_{n}-\theta ){\xrightarrow {D}}N(0,V),}$

for some ${\displaystyle V}$, which is called the asymptotic variance of the estimator.

Central limit theorem implies asymptotic normality of the sample mean ${\displaystyle {\bar {x}}}$ as an estimator of the true mean.

## Efficiency

The quality of an estimator is generally judged by its mean squared error.

However, occasionally one chooses the unbiased estimator with the lowest variance. Efficient estimators are those that have the lowest possible variance among all unbiased estimators. In some cases, a biased estimator may have a uniformly smaller mean squared error than does any unbiased estimator, so one should not make too much of this concept. For that and other reasons, it is sometimes preferable not to limit oneself to unbiased estimators; see estimator bias. Concerning such "best unbiased estimators", see also Cramér-Rao bound, Gauss-Markov theorem, Lehmann-Scheffé theorem, Rao-Blackwell theorem.

## Other properties

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.