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On the choice of a model to fit data from an exponential family. (English) Zbl 0657.62037
Let $$X_ 1$$, $$X_ 2$$,... be i.i.d. observations from an exponential family with densities $$f(X,\psi)=\exp (X\psi -b(\psi))$$, $$\psi\in \Theta$$, with respect to a finite measure on $$R^ k$$, where $$\Theta$$ is the natural parameter space. A finite number of competing models $$m_ j\cap \Theta$$, where $$m_ j$$ is a $$C^{\infty}$$, $$k_ j$$-dimensional manifold in $$R^ k$$, is considered. The correct choice between models is the model of lowest dimension which contains the true value of the parameter. The results are as follows:
1) Using the Schwarz criterion $\gamma (n,j)=n\sup_{\psi \in m_ j\cap \Theta}(n^{-1}\sum^{n}_{i=1}X_ i\psi -b(\psi))-2^{-1}k_ j\log n,$ the probability of correct choice between two models is asymptotically one;
2) Let S(n,j) be the log of the posterior probability of the jth model. The asymptotic expansion $$S(n,j)=T(n,j)+O_ p(n^{-1/2})$$ is given;
3) The choice based on $$\gamma$$ (n,j) and the choice based on T(n,j) coincide asymptotically.
An application to choice of degree in polynomial regression is discussed.
Reviewer: R.Zielinski

##### MSC:
 62F99 Parametric inference 62F12 Asymptotic properties of parametric estimators 62H99 Multivariate analysis 62J99 Linear inference, regression
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