 x Estimator Encyclopedia
In statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

, an estimator is a rule for calculating an estimate of a given quantity based on observed data: thus the rule and its result (the estimate) are distinguished.

There are point and interval estimators. The point estimators yield single-valued results, although this includes the possibility of single vector-valued results and results that can be expressed as a single function. This is in contrast to an interval estimator, where the result would be a range of plausible values (or vectors or functions).

Statistical theory
Statistical theory
The theory of statistics provides a basis for the whole range of techniques, in both study design and data analysis, that are used within applications of statistics. The theory covers approaches to statistical-decision problems and to statistical inference, and the actions and deductions that...

is concerned with the properties of estimators; that is, with defining properties that can be used to compare different estimators (different rules for creating estimates) for the same quantity, based on the same data. Such properties can be used to determine the best rules to use under given circumstances. However, in robust statistics
Robust statistics
Robust statistics provides an alternative approach to classical statistical methods. The motivation is to produce estimators that are not unduly affected by small departures from model assumptions.- Introduction :...

, statistical theory goes on to consider the balance between having good properties, if tightly defined assumptions hold, and having less good properties that hold under wider conditions.

## Background

An "estimator" or "point estimate" is a statistic
Statistic
A statistic is a single measure of some attribute of a sample . It is calculated by applying a function to the values of the items comprising the sample which are known together as a set of data.More formally, statistical theory defines a statistic as a function of a sample where the function...

(that is, a function of the data) that is used to infer the value of an unknown parameter
Parameter
Parameter from Ancient Greek παρά also “para” meaning “beside, subsidiary” and μέτρον also “metron” meaning “measure”, can be interpreted in mathematics, logic, linguistics, environmental science and other disciplines....

in a statistical model
Statistical model
A statistical model is a formalization of relationships between variables in the form of mathematical equations. A statistical model describes how one or more random variables are related to one or more random variables. The model is statistical as the variables are not deterministically but...

. The parameter being estimated is sometimes called the estimand. It can be either finite-dimensional (in parametric
Parametric model
In statistics, a parametric model or parametric family or finite-dimensional model is a family of distributions that can be described using a finite number of parameters...

and semi-parametric models), or infinite-dimensional (semi-nonparametric and non-parametric models). If the parameter is denoted θ then the estimator is typically written by adding a "hat
Circumflex
The circumflex is a diacritic used in the written forms of many languages, and is also commonly used in various romanization and transcription schemes. It received its English name from Latin circumflexus —a translation of the Greek περισπωμένη...

" over the symbol: $\scriptstyle\hat\theta$. Being a function of the data, the estimator is itself a random variable; a particular realization of this random variable is called the "estimate". Sometimes the words "estimator" and "estimate" are used interchangeably.

The definition places virtually no restrictions on which functions of the data can be called the "estimators". The attractiveness of different estimators can be judged by looking at their properties, such as unbiasedness, mean square error, consistency
Consistent estimator
In statistics, a sequence of estimators for parameter θ0 is said to be consistent if this sequence converges in probability to θ0...

, asymptotic distribution
Asymptotic distribution
In mathematics and statistics, an asymptotic distribution is a hypothetical distribution that is in a sense the "limiting" distribution of a sequence of distributions...

, etc.. The construction and comparison of estimators are the subjects of the estimation theory
Estimation theory
Estimation theory is a branch of statistics and signal processing that deals with estimating the values of parameters based on measured/empirical data that has a random component. The parameters describe an underlying physical setting in such a way that their value affects the distribution of the...

. In the context of decision theory
Decision theory
Decision theory in economics, psychology, philosophy, mathematics, and statistics is concerned with identifying the values, uncertainties and other issues relevant in a given decision, its rationality, and the resulting optimal decision...

, an estimator is a type of decision rule
Decision rule
In decision theory, a decision rule is a function which maps an observation to an appropriate action. Decision rules play an important role in the theory of statistics and economics, and are closely related to the concept of a strategy in game theory....

, and its performance may be evaluated through the use of loss function
Loss function
In statistics and decision theory a loss function is a function that maps an event onto a real number intuitively representing some "cost" associated with the event. Typically it is used for parameter estimation, and the event in question is some function of the difference between estimated and...

s.

When the word "estimator" is used without a qualifier, it usually refers to point estimation. The estimate in this case is a single point in the parameter space. Other types of estimators also exist: interval estimators, where the estimates are subsets of the parameter space.

The problem of density estimation
Density estimation
In probability and statistics,density estimation is the construction of an estimate, based on observed data, of an unobservable underlying probability density function...

arises in two applications. Firstly, in estimating the probability density function
Probability density function
In probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...

s of random variables and secondly in estimating the spectral density function
Spectral density
In statistical signal processing and physics, the spectral density, power spectral density , or energy spectral density , is a positive real function of a frequency variable associated with a stationary stochastic process, or a deterministic function of time, which has dimensions of power per hertz...

of a time series
Time series
In statistics, signal processing, econometrics and mathematical finance, a time series is a sequence of data points, measured typically at successive times spaced at uniform time intervals. Examples of time series are the daily closing value of the Dow Jones index or the annual flow volume of the...

. In these problems the estimates are functions that can be thought of as point estimates in an infinite dimensional space, and there are corresponding interval estimation problems.

## Definition

Suppose there is a fixed parameter that needs to be estimated. Then an "estimator" is a function that maps the sample space to a set of sample estimates. An estimator of is usually denoted by the symbol . It is often convenient to express the theory using the algebra of random variables
Algebra of random variables
In the algebraic axiomatization of probability theory, the primary concept is not that of probability of an event, but rather that of a random variable. Probability distributions are determined by assigning an expectation to each random variable...

: thus if X is used to denote a random variable
Random variable
In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...

corresponding to the observed data, the estimator (itself treated as a random variable) is symbolised as a function of that random variable, . The estimate for a particular observed dataset (i.e. for X=x) is then , which is a fixed value. Often an abbreviated notation is used in which is interpreted directly as a random variable, but this can cause confusion.

## Quantified properties

The following definitions and attributes apply:

Error
For a given sample , the "error
Errors and residuals in statistics
In statistics and optimization, statistical errors and residuals are two closely related and easily confused measures of the deviation of a sample from its "theoretical value"...

" of the estimator is defined as where is the parameter being estimated. Note that the error, e, depends not only on the estimator (the estimation formula or procedure), but on the sample.
Mean squared error
The mean squared error
Mean squared error
In statistics, the mean squared error of an estimator is one of many ways to quantify the difference between values implied by a kernel density estimator and the true values of the quantity being estimated. MSE is a risk function, corresponding to the expected value of the squared error loss or...

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 (samples). Then high MSE means the average distance of the arrows from the bull's-eye is high, and low MSE means the average distance from the bull's-eye 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).
Sampling deviation
For a given sample , the sampling deviation of the estimator is defined as where is the expected value
Expected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...

of the estimator. Note that the sampling deviation, d, depends not only on the estimator, but on the sample.
Variance
The variance
Variance
In probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...

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.
Bias
The bias
Bias of an estimator
In statistics, bias of an estimator is the difference between this estimator's expected value and the true value of the parameter being estimated. An estimator or decision rule with zero bias is called unbiased. Otherwise the estimator is said to be biased.In ordinary English, the term bias is...

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
Accuracy and precision
In the fields of science, engineering, industry and statistics, the accuracy of a measurement system is the degree of closeness of measurements of a quantity to that quantity's actual value. The precision of a measurement system, also called reproducibility or repeatability, is the degree to which...

.
Unbiased
The estimator is an unbiased estimator of if and only if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements.... . 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 just a little bias is permitted, then an estimator can be found with lower MSE and/or fewer outlier sample estimates.

An alternative to the version of "unbiased" above, is "median-unbiased", where the median
Median
In probability theory and statistics, a median is described as the numerical value separating the higher half of a sample, a population, or a probability distribution, from the lower half. The median of a finite list of numbers can be found by arranging all the observations from lowest value to...

of the distribution of estimates agrees with the true value; thus, in the long run half the estimates will be too low and half too high. While this applies immediately only to scalar-valued estimators, it can be extended to any measure of central tendency
Central tendency
In statistics, the term central tendency relates to the way in which quantitative data is clustered around some value. A measure of central tendency is a way of specifying - central value...

of a distribution: see median-unbiased estimators.

Relationships
• The MSE, variance, and bias, are related: i.e. mean squared error = variance + square of bias. In particular, for an unbiased estimator, the variance equals the MSE.
• The standard deviation
Standard deviation
Standard deviation is a widely used measure of variability or diversity used in statistics and probability theory. It shows how much variation or "dispersion" there is from the average...

of an estimator of θ (the square root
Square root
In mathematics, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square is x...

of the variance), or an estimate of the standard deviation of an estimator of θ, is called the standard error
Standard error (statistics)
The standard error is the standard deviation of the sampling distribution of a statistic. The term may also be used to refer to an estimate of that standard deviation, derived from a particular sample used to compute the estimate....

of θ.

## Behavioural properties

Consistency

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
Sample size
Sample size determination is the act of choosing the number of observations to include in a statistical sample. The sample size is an important feature of any empirical study in which the goal is to make inferences about a population from a sample...

) grows without bound. In other words, increasing the sample size increases the probability of the estimator being close to the population parameter.

Mathematically, a sequence of estimators } is a consistent estimator for parameter
Parameter
Parameter from Ancient Greek παρά also “para” meaning “beside, subsidiary” and μέτρον also “metron” meaning “measure”, can be interpreted in mathematics, logic, linguistics, environmental science and other disciplines....

θ 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
Scale factor
A scale factor is a number which scales, or multiplies, some quantity. In the equation y=Cx, C is the scale factor for x. C is also the coefficient of x, and may be called the constant of proportionality of y to x...

, 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.

Asymptotic normality

An asymptotically normal estimator is a consistent estimator whose distribution around the true parameter θ approaches a normal distribution with standard deviation shrinking in proportion to as the sample size n grows. Using to denote convergence in distribution, tn is asymptotically normal if for some V, which is called the asymptotic variance of the estimator.

The central limit theorem
Central limit theorem
In probability theory, the central limit theorem states conditions under which the mean of a sufficiently large number of independent random variables, each with finite mean and variance, will be approximately normally distributed. The central limit theorem has a number of variants. In its common...

implies asymptotic normality of the sample mean as an estimator of the true mean.
More generally, maximum likelihood
Maximum likelihood
In statistics, maximum-likelihood estimation is a method of estimating the parameters of a statistical model. When applied to a data set and given a statistical model, maximum-likelihood estimation provides estimates for the model's parameters....

estimators are asymptotically normal under fairly weak regularity conditions — see the asymptotics section of the maximum likelihood article. However, not all estimators are asymptotically normal, the simplest examples being case where the true value of a parameter lies in the boundary of the allowable parameter region.

Efficiency
Two naturally desirable properties of estimators are for them to be unbiased and have minimal mean squared error
Mean squared error
In statistics, the mean squared error of an estimator is one of many ways to quantify the difference between values implied by a kernel density estimator and the true values of the quantity being estimated. MSE is a risk function, corresponding to the expected value of the squared error loss or...

(MSE). These cannot in general both be satisfied simultaneously: a biased estimator may have lower mean squared error
Mean squared error
In statistics, the mean squared error of an estimator is one of many ways to quantify the difference between values implied by a kernel density estimator and the true values of the quantity being estimated. MSE is a risk function, corresponding to the expected value of the squared error loss or...

(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 minimum variance unbiased estimator (MVUE). In some cases an unbiased efficient estimator
Efficiency (statistics)
In statistics, an efficient estimator is an estimator that estimates the quantity of interest in some “best possible” manner. The notion of “best possible” relies upon the choice of a particular loss function — the function which quantifies the relative degree of undesirability of estimation errors...

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
Gauss–Markov theorem
In statistics, the Gauss–Markov theorem, named after Carl Friedrich Gauss and Andrey Markov, states that in a linear regression model in which the errors have expectation zero and are uncorrelated and have equal variances, the best linear unbiased estimator of the coefficients is given by the...

, Lehmann–Scheffé theorem
Lehmann–Scheffé theorem
In statistics, the Lehmann–Scheffé theorem is prominent in mathematical statistics, tying together the ideas of completeness, sufficiency, uniqueness, and best unbiased estimation...

, Rao–Blackwell theorem
Rao–Blackwell theorem
In statistics, the Rao–Blackwell theorem, sometimes referred to as the Rao–Blackwell–Kolmogorov theorem, is a result which characterizes the transformation of an arbitrarily crude estimator into an estimator that is optimal by the mean-squared-error criterion or any of a variety of similar...

.

Robustness
See: Robust estimator, Robust statistics
Robust statistics
Robust statistics provides an alternative approach to classical statistical methods. The motivation is to produce estimators that are not unduly affected by small departures from model assumptions.- Introduction :...

• Best linear unbiased estimator (BLUE)
• Invariant estimator
Invariant estimator
In statistics, the concept of being an invariant estimator is a criterion that can be used to compare the properties of different estimators for the same quantity. It is a way of formalising the idea that an estimator should have certain intuitively appealing qualities...

• Kalman filter
Kalman filter
In statistics, the Kalman filter is a mathematical method named after Rudolf E. Kálmán. Its purpose is to use measurements observed over time, containing noise and other inaccuracies, and produce values that tend to be closer to the true values of the measurements and their associated calculated...

• Markov chain Monte Carlo
Markov chain Monte Carlo
Markov chain Monte Carlo methods are a class of algorithms for sampling from probability distributions based on constructing a Markov chain that has the desired distribution as its equilibrium distribution. The state of the chain after a large number of steps is then used as a sample of the...

(MCMC)
• Maximum a posteriori
Maximum a posteriori
In Bayesian statistics, a maximum a posteriori probability estimate is a mode of the posterior distribution. The MAP can be used to obtain a point estimate of an unobserved quantity on the basis of empirical data...

(MAP)
• Method of moments, generalized method of moments
Generalized method of moments
In econometrics, generalized method of moments is a generic method for estimating parameters in statistical models. Usually it is applied in the context of semiparametric models, where the parameter of interest is finite-dimensional, whereas the full shape of the distribution function of the data...

• Minimum mean squared error (MMSE)
• Particle filter
Particle filter
In statistics, particle filters, also known as Sequential Monte Carlo methods , are sophisticated model estimation techniques based on simulation...

• Shrinkage estimator
Shrinkage estimator
In statistics, a shrinkage estimator is an estimator that, either explicitly or implicitly, incorporates the effects of shrinkage. In loose terms this means that a naïve or raw estimate is improved by combining it with other information. The term relates to the notion that the improved estimate is...

• Signal Processing
Signal processing
Signal processing is an area of systems engineering, electrical engineering and applied mathematics that deals with operations on or analysis of signals, in either discrete or continuous time...

• Testimator
Testimator
A testimator is an estimator whose value depends on the result of a test for statistical significance. In the simplest case the value of the final estimator is that of the basic estimator if the test result is significant, and otherwise the value is zero...

• Wiener filter
Wiener filter
In signal processing, the Wiener filter is a filter proposed by Norbert Wiener during the 1940s and published in 1949. Its purpose is to reduce the amount of noise present in a signal by comparison with an estimation of the desired noiseless signal. The discrete-time equivalent of Wiener's work was...

• Well-behaved statistic
Well-behaved statistic
A well-behaved statistic is a term sometimes used in the theory of statistics to describe part of a procedure. This usage is broadly similar to the use of well-behaved in more general mathematics...

• Sensitivity and specificity
Sensitivity and specificity
Sensitivity and specificity are statistical measures of the performance of a binary classification test, also known in statistics as classification function. Sensitivity measures the proportion of actual positives which are correctly identified as such Sensitivity and specificity are statistical... 