Formal Definition
A Statistical model, is a collection of probability distribution functions or probability density functions (collectively referred to as distributions for brevity). A parametric model is a collection of distributions, each of which is indexed by a unique finite-dimensional parameter:, where is a parameter and is the feasible region of parameters, which is a subset of d-dimensional Euclidean space. A statistical model may be used to describe the set of distributions from which one assumes that a particular data set is sampled. For example, if one assumes that data arise from a univariate Gaussian distribution, then one has assumed a Gaussian model: .
A non-parametric model is a set of probability distributions with infinite dimensional parameters, and might be written as . A semi-parametric model also has infinite dimensional parameters, but is not dense in the space of distributions. For example, a mixture of Gaussians with one Gaussian at each data point is dense is the space of distributions. Formally, if d is the dimension of the parameter, and n is the number of samples, if as and as, then the model is semi-parametric.
Read more about this topic: Statistical Model
Famous quotes containing the words formal and/or definition:
“Then the justice,
In fair round belly with good capon lined,
With eyes severe and beard of formal cut,
Full of wise saws and modern instances;
And so he plays his part.”
—William Shakespeare (15641616)
“Was man made stupid to see his own stupidity?
Is God by definition indifferent, beyond us all?
Is the eternal truth mans fighting soul
Wherein the Beast ravens in its own avidity?”
—Richard Eberhart (b. 1904)