# Invariant estimator

**equivariantequivariant estimationInvarianceinvariantPitman 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.wikipedia

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### Equivariant map

**equivariantintertwining operatorintertwiner**

The term equivariant estimator is used in formal mathematical contexts that include a precise description of the relation of the way the estimator changes in response to changes to the dataset and parameterisation: this corresponds to the use of "equivariance" in more general mathematics.

In statistical inference, equivariance under statistical transformations of data is an important property of various estimation methods; see invariant estimator for details.

### Estimator

**estimatorsestimateestimates**

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.

Invariant estimator

### Location parameter

**locationlocation modelshift parameter**

Shift invariance: Notionally, estimates of a location parameter should be invariant to simple shifts of the data values. If all data values are increased by a given amount, the estimate should change by the same amount. When considering estimation using a weighted average, this invariance requirement immediately implies that the weights should sum to one. While the same result is often derived from a requirement for unbiasedness, the use of "invariance" does not require that a mean value exists and makes no use of any probability distribution at all.

Invariant estimator

### Loss function

**objective functioncost functionrisk function**

The problem is to estimate \theta given x. The estimate, denoted by a, is a function of the measurements and belongs to a set A. The quality of the result is defined by a loss function which determines a risk function.

Invariance: Choose the optimal decision rule which satisfies an invariance requirement.

### Statistics

**statisticalstatistical analysisstatistician**

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.

### Statistical inference

**inferenceinferential statisticsinferences**

In statistical inference, there are several approaches to estimation theory that can be used to decide immediately what estimators should be used according to those approaches.

### Estimation theory

**parameter estimationestimationestimated**

In statistical inference, there are several approaches to estimation theory that can be used to decide immediately what estimators should be used according to those approaches.

### Bayesian inference

**BayesianBayesian analysisBayesian methods**

For example, ideas from Bayesian inference would lead directly to Bayesian estimators.

### Bayes estimator

**BayesianBayesian estimationBayes**

For example, ideas from Bayesian inference would lead directly to Bayesian estimators.

### Statistical model

**modelprobabilistic modelstatistical modeling**

However, the usefulness of these theories depends on having a fully prescribed statistical model and may also depend on having a relevant loss function to determine the estimator.

### Robust statistics

**robustbreakdown pointrobustness**

In addition to these cases where general theory does not prescribe an estimator, the concept of invariance of an estimator can be applied when seeking estimators of alternative forms, either for the sake of simplicity of application of the estimator or so that the estimator is robust.

### Bias of an estimator

**unbiasedunbiased estimatorbias**

For example, a requirement of invariance may be incompatible with the requirement that the estimator be mean-unbiased; on the other hand, the criterion of median-unbiasedness is defined in terms of the estimator's sampling distribution and so is invariant under many transformations.

### Median

**averagesample medianmedian-unbiased estimator**

For example, a requirement of invariance may be incompatible with the requirement that the estimator be mean-unbiased; on the other hand, the criterion of median-unbiasedness is defined in terms of the estimator's sampling distribution and so is invariant under many transformations.

### Independent and identically distributed random variables

**independent and identically distributedi.i.d.iid**

Permutation invariance: Where a set of data values can be represented by a statistical model that they are outcomes from independent and identically distributed random variables, it is reasonable to impose the requirement that any estimator of any property of the common distribution should be permutation-invariant: specifically that the estimator, considered as a function of the set of data-values, should not change if items of data are swapped within the dataset.

### Weighted arithmetic mean

**averageaverage ratingweighted average**

Shift invariance: Notionally, estimates of a location parameter should be invariant to simple shifts of the data values. If all data values are increased by a given amount, the estimate should change by the same amount. When considering estimation using a weighted average, this invariance requirement immediately implies that the weights should sum to one. While the same result is often derived from a requirement for unbiasedness, the use of "invariance" does not require that a mean value exists and makes no use of any probability distribution at all.

### Scale invariance

**scale invariantscale-invariantscaling**

Scale invariance: Note that this topic about the invariance of the estimator scale parameter not to be confused with the more general scale invariance about the behavior of systems under aggregate properties (in physics).

### Maximum likelihood estimation

**maximum likelihoodmaximum likelihood estimatormaximum likelihood estimate**

Parameter-transformation invariance: Here, the transformation applies to the parameters alone. The concept here is that essentially the same inference should be made from data and a model involving a parameter θ as would be made from the same data if the model used a parameter φ, where φ is a one-to-one transformation of θ, φ=h. According to this type of invariance, results from transformation-invariant estimators should also be related by φ=h. Maximum likelihood estimators have this property when the transformation is monotonic. Though the asymptotic properties of the estimator might be invariant, the small sample properties can be different, and a specific distribution needs to be derived.

### Random variable

**random variablesrandom variationrandom**

Permutation invariance: Where a set of data values can be represented by a statistical model that they are outcomes from independent and identically distributed random variables, it is reasonable to impose the requirement that any estimator of any property of the common distribution should be permutation-invariant: specifically that the estimator, considered as a function of the set of data-values, should not change if items of data are swapped within the dataset.

### Multivariate random variable

**random vectorvectormultivariate**

The measurements x are modelled as a vector random variable having a probability density function f(x|\theta) which depends on a parameter vector \theta.

### Probability density function

**probability densitydensity functiondensity**

The measurements x are modelled as a vector random variable having a probability density function f(x|\theta) which depends on a parameter vector \theta.

### Statistical classification

**classificationclassifierclassifiers**

In statistical classification, the rule which assigns a class to a new data-item can be considered to be a special type of estimator.

### Prior knowledge for pattern recognition

**prior knowledge**

A number of invariance-type considerations can be brought to bear in formulating prior knowledge for pattern recognition.

### Equivalence class

**equivalence classesquotient setquotient**

Datasets x_1 and x_2 in X are equivalent if x_1=g(x_2) for some g\in G. All the equivalent points form an equivalence class.

### Group action

**actionorbitacts**

A group of transformations of X, to be denoted by G, is a set of (measurable) 1:1 and onto transformations of X into itself, which satisfies the following conditions:

### Multivariate normal distribution

**multivariate normalbivariate normal distributionjointly normally distributed**

a multivariate normal distribution with independent, unit-variance components) then