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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

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