statistical model

Let 𝐗=(X1,,Xn) be a random vector with a given realization 𝐗(ω)=(x1,,xn), where ω is the outcome (of an observation or an experiment) in the sample space Ω. A statistical model 𝒫 based on X is a set of probability distribution functions of X:


If it is known in advance that this family of distributionsDlmfPlanetmath comes from a set of continuous distributions, the statistical model 𝒫 can be equivalently defined as a set of probability density functions:


As an example, a coin is tossed n times and the results are observed. The probability of landing a head during one toss is p. Assume that each toss is independent of one another. If 𝐗=(X1,,Xn) is defined to be the vector of the n ordered outcomes, then a statistical model based on X can be a family of Bernoulli distributionsMathworldPlanetmath


where Xi(ω)=xi and xi=1 if ω is the outcome that the ith toss lands a head and xi=0 if ω is the outcome that the ith toss lands a tail.

Next, suppose X is the number of tosses where a head is observed, then a statistical model based on X can be a family binomial distributions:


where X(ω)=x, where ω is the outcome that x heads (out of n tosses) are observed.

A statistical model is usually parameterized by a functionMathworldPlanetmath, called a parameterization

Θ𝒫 given by θFθ so that 𝒫={FθθΘ},

where Θ is called a parameter space. Θ is usually a subset of n. However, it can also be a function space.

In the first part of the above example, the statistical model is parameterized by


If the parameterization is a one-to-one function, it is called an identifiable parameterization and θ is called a parameterMathworldPlanetmath. The p in the above example is a parameter.

Title statistical model
Canonical name StatisticalModel
Date of creation 2013-03-22 14:33:18
Last modified on 2013-03-22 14:33:18
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 10
Author CWoo (3771)
Entry type Definition
Classification msc 62A01
Defines identifiable parameterization
Defines parameter space