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\centerline{\bf STANFORD UNIVERSITY}
\centerline{\bf DEPARTMENT OF STATISTICS}
\centerline{\big DEPARTMENT SEMINAR}
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\centerline{4:15 p.m., Tuesday, July 23, 2002}
\centerline{Sequoia Hall Room 200}
\centerline{(Cookies at 3:45 in 1st Floor Lounge)}

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\centerline{\sl Carlos Rivero Rodriguez}
\centerline{\sl Department of Statistics and Operations Research I }
\centerline{\sl Faculty of Mathematics University Complutense of Madrid}

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\centerline{\bf Iterative procedures in linear and non-linear models with grouped data. }
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In this seminar we introduce an iterative estimation procedures based on imputation techniques which are valid to fit linear models with grouped data. The distribution of errors may be general and also the dependent data stem from different sources and may be either non-grouped or grouped with different classification criteria. At each step the procedure requires the imputation of the exact values of the grouped data and it runs by means of a process that is similar to the EM algorithm when the errors are normal. The expectation step has been substituted by a step within a wide class of possibilities (including the conditional modes imputations), so avoiding eventual awkward integration with general errors. Also, we have substituted the maximisation step by an Ordinary Least Square step which only coincides with it when the errors are normally distributed. Notwithstanding the former modifications, we have proved that the iterative estimation algorithm converges to a point which is unique and non dependent on the starting values. Even though this point does not equal the maximum likelihood estimate, we have proved that enjoys good stochastic asymptotic properties. 

In the second part of the talk the initial procedure gives rise to two simplified versions tending to the Stochastic Approximation techniques. In the first we avoid the nested iteration, which implicitly appears in the initial procedure and also in the EM. In the second version we approach the Stochastic Approximations, where we need to select an adequate real step size. In spite of the simplifications the stochastic asymptotic properties of these estimators are similar to those ones of the maximum likelihood estimate. 

Finally, we show how these procedures could be generalized to non-linear models by means of the Stochastic Approximations techniques. As an example we propose to estimate a Radial Basic Function (RBF), which is a popular neural network supervised model.

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