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| | My name's Velva Holland but everybody calls me Velva. I'm from Great Britain. I'm studying at the high school (3rd year) and I play the Euphonium for 8 years. Usually I choose songs from the famous films :D. <br>I have two brothers. I like Sewing, watching TV (How I Met Your Mother) and Parkour.<br><br>Have a look at my blog post; [http://Www.Hzgsp.com/plus/guestbook.php FIFA Coin Generator] |
| In [[statistics]], 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.
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| There are [[point estimator|point]] and [[interval estimator]]s. The [[point estimator]]s 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).
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| [[Statistical theory]] 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]], 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.
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| ==Background==
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| An "estimator" or "[[point estimate]]" is a [[statistic]] (that is, a function of the data) that is used to infer the value of an unknown [[parameter]] in a [[statistical model]]. The parameter being estimated is sometimes called the ''estimand''.{{Citation needed|date=September 2010}} It can be either finite-dimensional (in [[parametric model|parametric]] and [[semi-parametric model]]s), or infinite-dimensional ([[semi-nonparametric model|semi-nonparametric]] and [[non-parametric model]]s).{{Citation needed|date=September 2010}} If the parameter is denoted ''θ'' then the estimator is typically written by adding a [[circumflex]] over the symbol: <math style="vertical-align:0">\scriptstyle\hat\theta</math>. 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.
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| 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]], [[Consistent estimator|consistency]], [[asymptotic distribution]], etc.. The construction and comparison of estimators are the subjects of the [[estimation theory]]. In the context of [[decision theory]], an estimator is a type of [[decision rule]], and its performance may be evaluated through the use of [[loss function]]s.
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| 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 estimator]]s, where the estimates are subsets of the parameter space.
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| The problem of [[density estimation]] arises in two applications. Firstly, in estimating the [[probability density function]]s of random variables and secondly in estimating the [[Spectral density|spectral density function]] of a [[time series]]. 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.
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| ==Definition==
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| Suppose there is a fixed ''parameter'' <math> \theta \ </math> 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 <math> \theta \ </math> is usually denoted by the symbol <math>\widehat{\theta}</math>. It is often convenient to express the theory using the [[algebra of random variables]]: thus if ''X'' is used to denote a [[random variable]] corresponding to the observed data, the estimator (itself treated as a random variable) is symbolised as a function of that random variable, <math>\widehat{\theta}(X)</math>. The estimate for a particular observed dataset (i.e. for ''X''=''x'') is then <math>\widehat{\theta}(x)</math>, which is a fixed value. Often an abbreviated notation is used in which <math>\widehat{\theta}</math> is interpreted directly as a random variable, but this can cause confusion.
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| ==Quantified properties==
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| The following definitions and attributes apply:
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| ;Error
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| For a given sample <math> x \ </math>, the "[[Errors and residuals in statistics|error]]" of the estimator <math>\widehat{\theta}</math> is defined as
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| :<math>e(x)=\widehat{\theta}(x) - \theta,</math>
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| where <math>\theta \ </math> 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.
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| ;Mean squared error
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| The ''[[mean squared error]]'' of <math>\widehat{\theta}</math> is defined as the expected value (probability-weighted average, over all samples) of the squared errors; that is,
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| :<math>\operatorname{MSE}(\widehat{\theta}) = \operatorname{E}[(\widehat{\theta}(X) - \theta)^2].</math>
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| 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).
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| ;Sampling deviation
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| For a given sample <math> x \ </math>, the ''[[sampling deviation]]'' of the estimator <math>\widehat{\theta}</math> is defined as
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| :<math>d(x) =\widehat{\theta}(x) - \operatorname{E}( \widehat{\theta}(X) ) =\widehat{\theta}(x) - \operatorname{E}( \widehat{\theta} ),</math>
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| where <math> \operatorname{E}( \widehat{\theta}(X) ) </math> is the [[expected value]] of the estimator. Note that the sampling deviation, ''d'', depends not only on the estimator, but on the sample.
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| ;Variance
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| The ''[[variance]]'' of <math>\widehat{\theta}</math> is simply the expected value of the squared sampling deviations; that is, <math>\operatorname{var}(\widehat{\theta}) = \operatorname{E}[(\widehat{\theta} - \operatorname{E}(\widehat{\theta}) )^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. 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.
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| ;Bias
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| The ''[[bias of an estimator|bias]]'' of <math>\widehat{\theta}</math> is defined as <math>B(\widehat{\theta}) = \operatorname{E}(\widehat{\theta}) - \theta</math>. 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 <math> \operatorname{E}(\widehat{\theta}) - \theta = \operatorname{E}(\widehat{\theta} - \theta ) </math>. 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]].
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| ;Unbiased
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| The estimator <math>\widehat{\theta}</math> is an ''[[estimator bias|unbiased estimator]]'' of <math> \theta \ </math> [[if and only if]] <math>B(\widehat{\theta}) = 0</math>. 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.
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| An alternative to the version of "unbiased" above, is "median-unbiased", where the [[median]] 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 [[Bias of an estimator#Median-unbiased estimators, and bias with respect to other loss functions|median-unbiased estimators]].
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| ;Relationships
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| *The MSE, variance, and bias, are related: <math>\operatorname{MSE}(\widehat{\theta}) = \operatorname{var}(\widehat\theta) + (B(\widehat{\theta}))^2,</math> i.e. mean squared error = variance + square of bias. In particular, for an unbiased estimator, the variance equals the MSE.
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| *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 (statistics)|standard error]]'' of θ.
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| ==Behavioural properties==
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| ;Consistency
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| {{Main|Consistent estimator}}
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| A consistent sequence of estimators is a sequence of estimators that [[convergence in probability|converge in probability]] to the quantity being estimated as the index (usually the [[sample size]]) grows without bound. In other words, increasing the sample size increases the probability of the estimator being close to the population parameter.
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| Mathematically, a sequence of estimators {{nowrap|{''t<sub>n</sub>''; ''n'' ≥ 0}}} is a consistent estimator for [[parameter]] ''θ'' if and only if, for all {{nowrap|''ϵ'' > 0}}, no matter how small, we have
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| :<math>
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| \lim_{n\to\infty}\Pr\left\{
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| \left|
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| t_n-\theta\right|<\epsilon
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| \right\}=1.
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| </math> | |
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| The consistency defined above may be called weak consistency. The sequence is ''strongly consistent'', if it [[Almost sure convergence|converges almost surely]] to the true value.
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| 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 [[Scale parameter#Estimation|estimation of scale parameters]] by [[Statistical dispersion#Measures of statistical dispersion|measures of statistical dispersion]].
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| ; Asymptotic normality
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| {{Main|Asymptotic normality}}
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| An [[asymptotic distribution#Asymptotic normality|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 <math>1/\sqrt{n}</math> as the sample size ''n'' grows. Using <math>\xrightarrow{D}</math> to denote [[Convergence of random variables#Convergence in distribution|convergence in distribution]], ''t<sub>n</sub>'' is [[Asymptotic normality|asymptotically normal]] if
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| :<math>\sqrt{n}(t_n - \theta) \xrightarrow{D} N(0,V),</math>
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| for some ''V''. Where ''V/n'' is called the ''asymptotic variance'' of the estimator.
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| The [[central limit theorem]] implies asymptotic normality of the [[sample mean]] <math>\bar x</math> as an estimator of the true mean.
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| More generally, [[maximum likelihood]] estimators are asymptotically normal under fairly weak regularity conditions — see the [[maximum likelihood#Asymptotics|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.
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| ; Efficiency
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| {{Main|Efficiency (statistics)}}
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| 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; see [[estimator bias]].
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| Among unbiased estimators, there often exists one with the lowest variance, called the minimum variance unbiased estimator ([[MVUE]]). In some cases an unbiased [[efficiency (statistics)|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.
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| Concerning such "best unbiased estimators", see also [[Cramér–Rao bound]], [[Gauss–Markov theorem]], [[Lehmann–Scheffé theorem]], [[Rao–Blackwell theorem]].
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| ; Robustness
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| See: [[Robust estimator]], [[Robust statistics]]
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| ==See also==
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| {{Refbegin|3}}
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| * [[Best linear unbiased estimator]] (BLUE)
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| * [[Invariant estimator]]
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| * [[Kalman filter]]
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| * [[Markov chain Monte Carlo]] (MCMC)
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| * [[Maximum a posteriori]] (MAP)
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| * [[Method of moments (statistics)|Method of moments]], [[generalized method of moments]]
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| * [[Minimum mean squared error]] (MMSE)
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| * [[Particle filter]]
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| * [[Pitman closeness criterion]]
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| * [[Shrinkage estimator]]
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| * [[Signal Processing]]
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| * [[Testimator]]
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| * [[Wiener filter]]
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| * [[Well-behaved statistic]]
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| * [[Sensitivity and specificity]]
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| {{Refend}}
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| ==References==
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| * {{Cite book
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| | last = Lehmann
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| | first = E. L.
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| | coauthors = Casella, G.
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| | title = Theory of Point Estimation
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| | edition = 2nd
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| | year = 1998
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| | publisher = Springer
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| | isbn = 0-387-98502-6
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| | pages = }}
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| * {{Citation
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| | last = Shao
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| | first = Jun
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| | title = Mathematical Statistics
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| | place = New York
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| | publisher = Springer
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| | year = 1998
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| | isbn = 0-387-98674-X }}
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| *{{SpringerEOM|title=Statistical Estimator|id=s/s087360|first=L.N.|last=Bol'shev}}
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| ==External links==
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| *[http://lmi.bwh.harvard.edu/papers/pdfs/2004/martin-fernandezCOURSE04b.pdf Fundamentals of Estimation Theory]
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| * India-Institute of Quantity Surveyors (IQSS)
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| [[Category:Estimation theory]]
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| [[Category:Statistical inference]]
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| [[ar:طرق التقدير]]
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