1. M ODEL IS W RONG ?! S. Eguchi, ISM & GUAS

2. What is MODEL? No Model is True ! Feature of interests can reflect on Model Patterns of interests can incorporate into Model Observations can only be made to finite precision ● ● ● Cf. J K Lindsay “ Parametric Statistical Inference ”

3. Asymptotics on correct model Large sample asymptotics Asymptotic consistency, normality Asymptotic efficiency (Higher-order asymptotics) Non-parametric asymptotics

4. Outline ● Near-Model Bridge para and non-parametrics Non-efficiency under Near model ● ●

5. Near model parametric non-parametric near-parametric

6. Tubular Neighborhood M g

7. Density estimation Estimate g(y) Kernel estimate

8. Local Likelihood The main body Localization versions (Eguchi, Copas, 1998)

9. Local likelihood density estimate Maximum Local Likelihood Estimator The density estimator normalizing const )

10. h y

11. Global vs Local likelihood Global (h =  ) Local (h = 3.65) opt

12. Regression function Estimate  (x) = E(Y|x) GLM Cf.Eguchi,Kim,Park (2002)

13. Bridge of nonpara / parametric

14. Discriminant Analysis Input vectorlabel Logistic model Almost logistic model

15. A class of loss functions For a given data Estimate the score

16. Logistic loss

17. Error rate Medical screening where

18. Empirical loss For a training data score

19. Estimating function IRLS where Logistic

20. Asymptotic efficiency Cramer-Rao type （ logistic loss) ．

21. Risk under correct model Under the correct model Expected D-loss Let

22. Risk under near model where Let

23. λ-family Target risk λ-family score

24. λ (Proof ) opt (Eguchi, Copas, 2002)

25. Some analysis False positive rate 0.435% 0.423% λ

26. Conclusions ● Near-Model Bridge para and non-parametrics Non-efficiency under Near model ● ● α-neighborhood

27. Future project??