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In
statistics Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, indust ...
, an estimator is a rule for calculating an estimate of a given quantity based on observed data: thus the rule (the estimator), the quantity of interest (the estimand) and its result (the estimate) are distinguished. For example, the
sample mean The sample mean (or "empirical mean") and the sample covariance are statistics computed from a sample of data on one or more random variables. The sample mean is the average value (or mean value) of a sample of numbers taken from a larger popu ...
is a commonly used estimator of the
population mean In statistics, a population is a set of similar items or events which is of interest for some question or experiment. A statistical population can be a group of existing objects (e.g. the set of all stars within the Milky Way galaxy) or a hypothe ...
. There are point and interval estimators. The point estimators yield single-valued results. This is in contrast to an interval estimator, where the result would be a range of plausible values. "Single value" does not necessarily mean "single number", but includes vector valued or function valued estimators. ''
Estimation theory Estimation theory is a branch of statistics 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 valu ...
'' 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 are statistics with good performance for data drawn from a wide range of probability distributions, especially for distributions that are not normal. Robust statistical methods have been developed for many common problems, su ...
, 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 A statistic (singular) or sample statistic is any quantity computed from values in a sample which is considered for a statistical purpose. Statistical purposes include estimating a population parameter, describing a sample, or evaluating a hypo ...
(that is, a function of the data) that is used to infer the value of an unknown
parameter A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
in a
statistical model A statistical model is a mathematical model that embodies a set of statistical assumptions concerning the generation of sample data (and similar data from a larger population). A statistical model represents, often in considerably idealized form ...
. A common way of phrasing it is "the estimator is the method selected to obtain an estimate of an unknown parameter". The parameter being estimated is sometimes called the '' estimand''. It can be either finite-dimensional (in parametric and semi-parametric models), or infinite-dimensional ( semi-parametric and non-parametric models). If the parameter is denoted \theta then the estimator is traditionally written by adding a
circumflex The circumflex () is a diacritic in the Latin and Greek scripts that is also used in the written forms of many languages and in various romanization and transcription schemes. It received its English name from la, circumflexus "bent around" ...
over the symbol: \widehat. Being a function of the data, the estimator is itself a
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
; 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 In statistics, the mean squared error (MSE) or mean squared deviation (MSD) of an estimator (of a procedure for estimating an unobserved quantity) measures the average of the squares of the errors—that is, the average squared difference between ...
,
consistency In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent ...
,
asymptotic distribution In mathematics and statistics, an asymptotic distribution is a probability distribution that is in a sense the "limiting" distribution of a sequence of distributions. One of the main uses of the idea of an asymptotic distribution is in providing ...
, etc. The construction and comparison of estimators are the subjects of the
estimation theory Estimation theory is a branch of statistics 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 valu ...
. In the context of
decision theory Decision theory (or the theory of choice; not to be confused with choice theory) is a branch of applied probability theory concerned with the theory of making decisions based on assigning probabilities to various factors and assigning numerical ...
, an estimator is a type of
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 the ...
, and its performance may be evaluated through the use of
loss function In mathematical optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one or more variables onto a real number intuitively representing some "cos ...
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 The parameter space is the space of possible parameter values that define a particular mathematical model, often a subset of finite-dimensional Euclidean space. Often the parameters are inputs of a function, in which case the technical term for ...
. There also exists another type of estimator: interval estimators, where the estimates are subsets of the parameter space. The problem of density estimation arises in two applications. Firstly, in estimating the
probability density function In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) ca ...
s of random variables and secondly in estimating the spectral density function of a
time series In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Ex ...
. 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 a fixed ''parameter'' \theta needs to be estimated. Then an "estimator" is a function that maps the sample space to a set of ''sample estimates''. An estimator of \theta is usually denoted by the symbol \widehat. It is often convenient to express the theory using the algebra of random variables: thus if ''X'' is used to denote a
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
corresponding to the observed data, the estimator (itself treated as a random variable) is symbolised as a function of that random variable, \widehat(X). The estimate for a particular observed data value x (i.e. for X=x) is then \widehat(x), which is a fixed value. Often an abbreviated notation is used in which \widehat is interpreted directly as a
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
, but this can cause confusion.


Quantified properties

The following definitions and attributes are relevant.


Error

For a given sample x , the " error" of the estimator \widehat is defined as :e(x)=\widehat(x) - \theta, where \theta is the parameter being estimated. The error, ''e'', depends not only on the estimator (the estimation formula or procedure), but also on the sample.


Mean squared error

The
mean squared error In statistics, the mean squared error (MSE) or mean squared deviation (MSD) of an estimator (of a procedure for estimating an unobserved quantity) measures the average of the squares of the errors—that is, the average squared difference between ...
of \widehat is defined as the expected value (probability-weighted average, over all samples) of the squared errors; that is, :\operatorname(\widehat) = \operatorname \widehat(X) - \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 (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. However, if the MSE is relatively low then the arrows are likely more highly clustered (than highly dispersed) around the target.


Sampling deviation

For a given sample x , the ''sampling deviation'' of the estimator \widehat is defined as :d(x) =\widehat(x) - \operatorname( \widehat(X) ) =\widehat(x) - \operatorname( \widehat ), where \operatorname( \widehat(X) ) is the
expected value In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a ...
of the estimator. The sampling deviation, ''d'', depends not only on the estimator, but also on the sample.


Variance

The
variance In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbe ...
of \widehat is the expected value of the squared sampling deviations; that is, \operatorname(\widehat) = \operatorname \widehat_-_\operatorname
\widehat_-_\operatorname[\widehat_^2">widehat.html"_;"title="\widehat_-_\operatorname[\widehat">\widehat_-_\operatorname[\widehat_^2/math>._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._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,_even_if_all_arrows_grossly_miss_the_target,_if_they_nevertheless_all_hit_the_same_point,_the_variance_is_zero.


_Bias

The_bias_of_an_estimator.html" "title="widehat_^2.html" ;"title="widehat.html" ;"title="\widehat - \operatorname[\widehat">\widehat - \operatorname[\widehat ^2">widehat.html" ;"title="\widehat - \operatorname[\widehat">\widehat - \operatorname[\widehat ^2/math>. 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. 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, even if all arrows grossly miss the target, if they nevertheless all hit the same point, the variance is zero.


Bias

The bias of an estimator">bias Bias is a disproportionate weight ''in favor of'' or ''against'' an idea or thing, usually in a way that is closed-minded, prejudicial, or unfair. Biases can be innate or learned. People may develop biases for or against an individual, a group ...
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. The bias of \widehat is a function of the true value of \theta so saying that the bias of \widehat is b means that for every \theta the bias of \widehat is b. There are two kinds of estimators: biased estimators and unbiased estimators. Whether an estimator is biased or not can be identified by the relationship between \operatorname(\widehat) - \theta and 0: * If \operatorname(\widehat) - \theta\neq0, \widehat is biased. * If \operatorname(\widehat) - \theta=0, \widehat is unbiased. The bias is also 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. The estimator \widehat is an estimator bias, unbiased estimator of \theta if and only if B(\widehat) = 0. 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. That 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, an estimator's being biased does not preclude the error of an estimate from being zero in a particular instance. The ideal situation 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 mean squared error and/or fewer outlier sample estimates. An alternative to the version of "unbiased" above, is "median-unbiased", where the
median In statistics and probability theory, the median is the value separating the higher half from the lower half of a data sample, a population, or a probability distribution. For a data set, it may be thought of as "the middle" value. The basic f ...
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 of a distribution: see median-unbiased estimators. In a practical problem, \widehat can always have functional relationship with \theta. For example, A genetic theory states there is a type of leave, starchy green, occur with probability p_1=1/4\cdot(\theta + 2), with 0<\theta<1. For n leaves, the random variable N_1, the number of starchy green leaves, can be modeled with a Bin(n,p_1) distribution. The number can be used to express the following estimator for \theta: \widehat=4/n\cdot N_1-2. One can show that \widehat is an unbiased estimator for \theta: E widehatE /n\cdot N_1-2/math> =4/n\cdot E _12 =4/n\cdot np_1-2 =4\cdot p_1-2 =4\cdot1/4\cdot(\theta+2)-2 =\theta+2-2 =\theta.


Relationships among the quantities

*The mean squared error, variance, and bias, are related: \operatorname(\widehat) = \operatorname(\widehat\theta) + (B(\widehat))^2, i.e. mean squared error = variance + square of bias. In particular, for an unbiased estimator, the variance equals the mean squared error. *The
standard deviation In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, whil ...
of an estimator \widehat of \theta (the
square root In mathematics, a square root of a number is a number such that ; in other words, a number whose '' square'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots of 16, because . ...
of the variance), or an estimate of the standard deviation of an estimator \widehat of \theta, is called the '' standard error'' of \widehat. *The bias-variance tradeoff will be used in model complexity, over-fitting and under-fitting. It is mainly used in the field of supervised learning and predictive modeling to diagnose the performance of algorithms.


Behavioral 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 determination is the act of choosing the number of observations or replicates 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 populati ...
) 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 A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
''θ'' if and only if, for all , no matter how small, we have : \lim_\Pr\left\=1 . 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.


Asymptotic normality

An asymptotically normal estimator is a consistent estimator whose distribution around the true parameter ''θ'' approaches a
normal distribution In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu ...
with standard deviation shrinking in proportion to 1/\sqrt as the sample size ''n'' grows. Using \xrightarrow to denote
convergence in distribution In probability theory, there exist several different notions of convergence of random variables. The convergence of sequences of random variables to some limit random variable is an important concept in probability theory, and its applications t ...
, ''tn'' is asymptotically normal if :\sqrt(t_n - \theta) \xrightarrow N(0,V), for some ''V''. In this formulation ''V/n'' can be called the ''asymptotic variance'' of the estimator. However, some authors also call ''V'' the ''asymptotic variance''. Note that convergence will not necessarily have occurred for any finite "n", therefore this value is only an approximation to the true variance of the estimator, while in the limit the asymptotic variance (V/n) is simply zero. To be more specific, the distribution of the estimator ''tn'' converges weakly to a
dirac delta function In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the enti ...
centered at \theta. The
central limit theorem In probability theory, the central limit theorem (CLT) establishes that, in many situations, when independent random variables are summed up, their properly normalized sum tends toward a normal distribution even if the original variables themsel ...
implies asymptotic normality of the
sample mean The sample mean (or "empirical mean") and the sample covariance are statistics computed from a sample of data on one or more random variables. The sample mean is the average value (or mean value) of a sample of numbers taken from a larger popu ...
\bar X as an estimator of the true mean. More generally,
maximum likelihood In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed stat ...
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 are found when the true value of a parameter lies on the boundary of the allowable parameter region.


Efficiency

The efficiency of an estimator is used to estimate the quantity of interest in a "minimum error" manner. In reality, there is not an explicit best estimator; there can only be a better estimator. The good or not of the efficiency of an estimator is based on the choice of a particular
loss function In mathematical optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one or more variables onto a real number intuitively representing some "cos ...
, and it is reflected by two naturally desirable properties of estimators: to be unbiased \operatorname(\widehat) - \theta=0 and have minimal
mean squared error In statistics, the mean squared error (MSE) or mean squared deviation (MSD) of an estimator (of a procedure for estimating an unobserved quantity) measures the average of the squares of the errors—that is, the average squared difference between ...
(MSE) \operatorname \widehat - \theta )^2/math>. These cannot in general both be satisfied simultaneously: an unbiased estimator may have a lower mean squared error than any biased estimator (see estimator bias). A function relates the mean squared error with the estimator bias. \operatorname \widehat - \theta )^2(\operatorname(\widehat) - \theta)^2+\operatorname(\theta)\ The first term represents the mean squared error; the second term represents the square of the estimator bias; and the third term represents the variance of the sample. The quality of the estimator can be identified from the comparison between the variance, the square of the estimator bias, or the MSE. The variance of the good estimator (good efficiency) would be smaller than the variance of the bad estimator (bad efficiency). The square of a estimator bias with a good estimator would be smaller than the estimator bias with a bad estimator. The MSE of a good estimator would be smaller than the MSE of the bad estimator. Suppose there are two estimator, \theta_1 is the good estimator and \theta_2 is the bad estimator. The above relationship can be expressed by the following formulas. \operatorname(\theta_1)<\operatorname(\theta_2) , \operatorname(_1) - \theta, <, \operatorname() - \theta, \operatorname(\theta_1)<\operatorname(\theta_2) Besides using formula to identify the efficiency of the estimator, it can also be identified through the graph. If an estimator is efficient, in the frequency vs. value graph, there will be a curve with high frequency at the center and low frequency on the two sides. For example: If an estimator is not efficient, the frequency vs. value graph, there will be a relatively more gentle curve. To put it simply, the good estimator has a narrow curve, while the bad estimator has a large curve. Plotting these two curves on one graph with a shared y-axis, the difference becomes more obvious. 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 exists, which, in addition to having the lowest variance among unbiased estimators, satisfies the
Cramér–Rao bound In estimation theory and statistics, the Cramér–Rao bound (CRB) expresses a lower bound on the variance of unbiased estimators of a deterministic (fixed, though unknown) parameter, the variance of any such estimator is at least as high as the ...
, which is an absolute lower bound on variance for statistics of a variable. Concerning such "best unbiased estimators", see also
Cramér–Rao bound In estimation theory and statistics, the Cramér–Rao bound (CRB) expresses a lower bound on the variance of unbiased estimators of a deterministic (fixed, though unknown) parameter, the variance of any such estimator is at least as high as the ...
, Gauss–Markov theorem, Lehmann–Scheffé 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-squ ...
.


Robustness


See also

* Best linear unbiased estimator (BLUE) * Invariant estimator * Kalman filter *
Markov chain Monte Carlo In statistics, Markov chain Monte Carlo (MCMC) methods comprise a class of algorithms for sampling from a probability distribution. By constructing a Markov chain that has the desired distribution as its equilibrium distribution, one can obtain ...
(MCMC) * Maximum a posteriori (MAP) * Method of moments, generalized method of moments *
Minimum mean squared error In statistics and signal processing, a minimum mean square error (MMSE) estimator is an estimation method which minimizes the mean square error (MSE), which is a common measure of estimator quality, of the fitted values of a dependent variable. I ...
(MMSE) *
Particle filter Particle filters, or sequential Monte Carlo methods, are a set of Monte Carlo algorithms used to solve filtering problems arising in signal processing and Bayesian statistical inference. The filtering problem consists of estimating the inte ...
*
Pitman closeness criterion In statistical theory, the Pitman closeness criterion, named after E. J. G. Pitman, is a way of comparing two candidate estimators for the same parameter. Under this criterion, estimator A is preferred to estimator B if the probability that estima ...
*
Sensitivity and specificity ''Sensitivity'' and ''specificity'' mathematically describe the accuracy of a test which reports the presence or absence of a condition. Individuals for which the condition is satisfied are considered "positive" and those for which it is not are ...
* Shrinkage estimator *
Signal Processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing '' signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, ...
* Testimator *
Wiener filter In signal processing, the Wiener filter is a filter used to produce an estimate of a desired or target random process by linear time-invariant ( LTI) filtering of an observed noisy process, assuming known stationary signal and noise spectra, and ...
*
Well-behaved statistic Although the term well-behaved statistic often seems to be used in the scientific literature in somewhat the same way as is well-behaved in mathematics (that is, to mean "non-pathological") it can also be assigned precise mathematical meaning, and ...


References


Further reading

* . * . * * * {{Citation , last = Shao , first = Jun , title = Mathematical Statistics , publisher =
Springer Springer or springers may refer to: Publishers * Springer Science+Business Media, aka Springer International Publishing, a worldwide publishing group founded in 1842 in Germany formerly known as Springer-Verlag. ** Springer Nature, a multinationa ...
, year = 1998 , isbn = 0-387-98674-X


External links


Fundamentals on Estimation Theory