Estimator

In statistics, 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. In statistics, 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. 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). Estimation 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. 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. 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 θ {displaystyle heta } then the estimator is traditionally written by adding a circumflex over the symbol: θ ^ {displaystyle {widehat { heta }}} . 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, 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 functions. 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. 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 functions of random variables and secondly in estimating the 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. Suppose there is a fixed parameter θ {displaystyle heta } 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 θ {displaystyle heta } is usually denoted by the symbol θ ^ {displaystyle {widehat { heta }}} . 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, θ ^ ( X ) {displaystyle {widehat { heta }}(X)} . The estimate for a particular observed data value x {displaystyle x} (i.e. for X = x {displaystyle X=x} ) is then θ ^ ( x ) {displaystyle {widehat { heta }}(x)} , which is a fixed value. Often an abbreviated notation is used in which θ ^ {displaystyle {widehat { heta }}} is interpreted directly as a random variable, but this can cause confusion. The following definitions and attributes are relevant.