# Statistical model

**modelprobabilistic modelstatistical modelingstatistical modelsstatistical modellingmodelsstatisticalnestedprobability modelmodeling**

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).wikipedia

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### Mathematical model

**mathematical modelingmodelmathematical models**

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).

Mathematical models can take many forms, including dynamical systems, statistical models, differential equations, or game theoretic models.

### Statistical hypothesis testing

**hypothesis testingstatistical teststatistical tests**

All statistical hypothesis tests and all statistical estimators are derived via statistical models.

A statistical hypothesis, sometimes called confirmatory data analysis, is a hypothesis that is testable on the basis of observing a process that is modeled via a set of random variables.

### Statistical inference

**inferential statisticsinferenceinferences**

More generally, statistical models are part of the foundation of statistical inference.

Given a hypothesis about a population, for which we wish to draw inferences, statistical inference consists of (first) selecting a statistical model of the process that generates the data and (second) deducing propositions from the model.

### Estimator

**estimatorsestimateestimates**

All statistical hypothesis tests and all statistical estimators are derived via statistical models.

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.

### All models are wrong

**driven by the relevant variables and assumptions**

Indeed, as Burnham & Anderson state, "A model is a simplification or approximation of reality and hence will not reflect all of reality" —whence the saying "all models are wrong".

It is usually considered to be applicable to not only statistical models, but to scientific models generally.

### Identifiability

**identifiableModel identificationnonidentifiable**

A parameterization that meets the requirement is said to be identifiable.

In statistics, identifiability is a property which a model must satisfy in order for precise inference to be possible.

### Statistical assumption

**assumptionsStatistical assumptionsdistributional assumption**

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).

Both approaches rely on some statistical model to represent the data-generating process.

### Statistical parameter

**parametersparameterparametrization**

The set \Theta defines the parameters of the model.

It can be regarded as a numerical characteristic of a statistical population or a statistical model.

### Independent and identically distributed random variables

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

For instance, we might assume that the ε i distributions are i.i.d. Gaussian, with zero mean.

In practical applications of statistical modeling, however, the assumption may or may not be realistic.

### Parametric model

**parametricregular parametric modelparameters**

The model is said to be parametric if \Theta has a finite dimension.

In statistics, a parametric model or parametric family or finite-dimensional model is a particular class of statistical models.

### Coefficient of determination

**R-squaredR'' 2 R 2**

Common criteria for comparing models include the following: R 2, Bayes factor, and the likelihood-ratio test together with its generalization relative likelihood.

It is a statistic used in the context of statistical models whose main purpose is either the prediction of future outcomes or the testing of hypotheses, on the basis of other related information.

### Semiparametric model

**semiparametricsemi-parametricsemi-parametric model**

A statistical model is semiparametric if it has both finite-dimensional and infinite-dimensional parameters.

In statistics, a semiparametric model is a statistical model that has parametric and nonparametric components.

### Likelihood function

**likelihoodlikelihood ratiolog-likelihood**

Common criteria for comparing models include the following: R 2, Bayes factor, and the likelihood-ratio test together with its generalization relative likelihood.

In statistics, the likelihood function (often simply called the likelihood) is formed from the joint probability of a sample of data given a set of model parameter values; it is viewed and used as a function of the parameters given the data sample.

### Likelihood-ratio test

**likelihood ratio testlikelihood ratiolikelihood-ratio**

Common criteria for comparing models include the following: R 2, Bayes factor, and the likelihood-ratio test together with its generalization relative likelihood.

In statistics, the likelihood-ratio test assesses the goodness of fit of two competing statistical models based on the ratio of their likelihoods, specifically one found by maximization over the entire parameter space and another found after imposing some constraint.

### Bayes factor

**Bayesian model comparisonBayes factorsBayesian model selection**

The models under consideration are statistical models.

### Design of experiments

**experimental designdesignExperimental techniques**

Charles S. Peirce also contributed the first English-language publication on an optimal design for regression models in 1876.

### Statistical model specification

**Model specificationmisspecifiedmisspecification**

In statistics, model specification is part of the process of building a statistical model: specification consists of selecting an appropriate functional form for the model and choosing which variables to include.

### Statistical model validation

**model validationModel checkingtested and validated**

In statistics, model validation is the task of confirming that the outputs of a statistical model are acceptable with respect to the real data-generating process.

### Predictive modelling

**predictive modelingpredictive modelpredictive models**

Nearly any statistical model can be used for prediction purposes.

### Statistical theory

**statisticalstatistical theoriesmathematical statistics**

### Scientific modelling

**modelmodelingmodels**

Considerations that may influence the structure of a model might be the modeller's preference for a reduced ontology, preferences regarding statistical models versus deterministic models, discrete versus continuous time, etc. In any case, users of a model need to understand the assumptions made that are pertinent to its validity for a given use.

### Abductive reasoning

**abductionabductiveinference to the best explanation**

### Deterministic system

**deterministicdeterministic processdeterministic model**

What distinguishes a statistical model from other mathematical models is that a statistical model is non-deterministic.

### Sample (statistics)

**samplesamplesstatistical sample**

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).

### Statistical population

**populationsubpopulationsubpopulations**