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for a bridge of

Local likelihood method and asmyptotical theory. for a bridge of. parametric and nonparametric inference. Shinto Eguchi (ISM, Tokyo). This talk is based on a joint work with TaeYoon Kim, Keimyung University. a covariate vector X = ( X ,…., X ). d. 1. a response variable Y .

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for a bridge of

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  1. Local likelihood method and asmyptotical theory for a bridge of parametric and nonparametric inference Shinto Eguchi (ISM, Tokyo) This talk is based on a joint work with TaeYoon Kim,Keimyung University

  2. a covariate vector X = (X ,….,X ) d 1 a response variable Y true regression function the model function Tibshirani & Hastie(1987) Local fitting technique Y scatterplor smoother X

  3. Fan et al. (1998) , Loader (1999)

  4. Given , whichi h is the optimal ? Given there exists an st A relation of m (x) & m(x,b ) Eguchi & Copas (1998) White (1982) nonparametric ? parametric Model complexity Estimation flexibility

  5. near parametric near nonparametric Under Under

  6. Simulation 25 20 15 10 5 -6 -4 -2 2 4 6 8 -5

  7. 60 50 40 30 0.5 1.5 2 2.5 3 Generalization error GE(h) h h = 1.0 opt

  8. h opt 20 h = 0.05 15 10 5 -7.5 -5 -2.5 2.5 5 7.5 -5 -10 -15

  9. 20 15 10 5 -8 -6 -4 -2 2 4 6 x -5 4 3 2 1 -8 -6 -4 -2 2 4 6 x Parametric understanding from local likelihood the behaviour locally fitted to the parametrric model

  10. 20 15 10 5 -8 -6 -4 -2 2 4 6 -5 true regession function vs. local estimate

  11. EDF ( Gaussian, Bernoulli, Cox, Poisson regression, ….) the conditional distribution of Y given X = x GLM the deviance function

  12. with Gaussian Bernoulli with local estimator local trend

  13. The average deviance loss function Hence the expected loss or the risk function is where E dentotes the expectation wrt data. data The risk function of local estimate true function m(x) local trend m (x) h

  14. Theorem 1. where m is the local trend. h

  15. Gauss-Markov model

  16. THEOREM 2.

  17. Under our assumption , satisfies for a fixed h:

  18. (A1) with and that THEOREM 4. Assume The optimal h is Cor.

  19. (A1) with and that Theorem 5. Assume Then

  20. (A1) with and that Theorem 6. Assume The optimal h is Cor.

  21. Model complexity Estimation flexibility Given , whichi h is the optimal ? The optimal h is Conjecture h = O(1) if a= 0. opt

  22. The cross validation for Under the assumption A(a),

  23. References Fan, J., Farmen, M. and Gibels, I. (1998). Local Likelihood estimation and inference. J. R. Statist. Soc. B 60, 591-608. Eguchi, S and Copas, J. B. (1998). A class of local likelihood methods and near-parametric asymptotics. J. R. Statist. Soc. B, 60, 709-724. Lindsey, J. K. (1996). Parametric Statistical Inference. Oxford Unversity Press, oxford. Loader, C. (1999). Local Regression and Likelihood. Springer, New York . McCullagh, P. and Nelder, J. A. (1989). Generalized Linear Models. Capman and Hall, London. Silverman, B. W. (1986). Density estimation for statistics and data analysis. Chapman nad Hall, London. Tibishirani, R. and Hastie, T. (1987). Local likelihood estimation. J. Amer. Statist.Soc. 82, 559-567. White, H. (1982). Maximum lkielihood estimation of misspecified models. Econometrica50, 1-25

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