1 Double the confidence region S. Eguchi, ISM & GUAS This talk is a part of co-work with J. Copas, University of Warwick

2 Tubular Neighborhood M

3 Sensitivity method

4 Confidence region For the empirical distribution from The conventional confidence region is where satisfies

5 Strong model for incomplete observation If Z has Z f (z,  Let Y = h(Z) be many-to-one mapping. then Y has Cf. EM algorithm

6 Mis-specification General misspecification model

7 Actual incomplete observation where

8 Ideal Model and Our Situation Ideal model Our situation Semi-parametric

9 Confidence region (weak model)

10 MCAR

11 MAR

12 Random design m groups comparison

13 Two MLEs Unobserved MLE Observed MLE

14 Ideals of two MLEs

15 Expectation of Scores

16 Covariance of Scores

17 Biases of two MLEs

18 Asymptotic variance

19 Joint (S,T )

20 Conditional S | T The conditional distribution If T were observed, the C. R. would be

21 Acceptability T has the asymptotic distribution We assume

22 Envelope region

23 The worst case

24

25

26 Confidence region For the empirical distribution from The conventional confidence region is Where satisfies

27 Bound of Envelope region where

28  1 2

29   1 2

30  1 2

31 Discussion Double the confidence interval The worst case occurs when