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 in 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 Models
Famous quotes containing the words formal and/or definition:
“The conviction that the best way to prepare children for a harsh, rapidly changing world is to introduce formal instruction at an early age is wrong. There is simply no evidence to support it, and considerable evidence against it. Starting children early academically has not worked in the past and is not working now.”
—David Elkind (20th century)
“Although there is no universal agreement as to a definition of life, its biological manifestations are generally considered to be organization, metabolism, growth, irritability, adaptation, and reproduction.”
—The Columbia Encyclopedia, Fifth Edition, the first sentence of the article on life (based on wording in the First Edition, 1935)