Model Selection via Information Criteria
نویسنده
چکیده
This is a survey of the information criterion approach to model selection problems. New results about context tree estimation and the estimation of the basic neighborhood of Markov random fields are also mentioned. 1. The model selection problem Let a stochastic process {Xt, t ∈ T } be given, where each Xt is a random variable with values a ∈ A, and T is an index set. The joint distribution of the random variables Xt, t ∈ T will be referred to as the distribution of the process and will be denoted by Q. A model of the process determines a hypothetical distribution of the process or a collection of hypothetical distributions. Typically, a model is determined by a structure parameter k with values in some set K, and by a parameter vector θk ∈ Θk ⊂ Rk ; this model is denoted by Mθk . Given the feasible models of the process, they can be arranged into model classes according to the structure parameter: Mk = { Mθk , θk ∈ Θk ⊂ Rk } . Statistical inference about the process is drawn based on a realization { xt, t ∈ T } of the process observed in the range Rn ⊂ T , where Rn extends with n. Thus the n’th sample is x(n) = { xt, t ∈ Rn }. Some typical examples for processes and their models are listed below. In the case of density function estimation, T = N and the random variablesXt, t ∈ N are independent and identically distributed (i.i.d.) with density function fθk . The n’th sample is { xi, i = 1, . . . , n }. Mathematics subject classification number: 62M99, 60J10, 62M40.
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