Optimality

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310 H. Leeb G∗ n,θ (n) ∗ ,σ (n)(t|p)πn,θ (n) ,σ (n)(p) converges weakly to (B.2) � z∈R k z≤t−Aδ (p) � 1−∆σζ∞,p(δ (p) � p + ψp + b∞,pz, cpσξ∞,p) Φ∞,p(dz) × P� q=p+1 ∆σξ∞,q(δ (q) q + ψq, cqσξ∞,q). In case p = p∗ and p∗ >O, we again use Proposition 5.1 of Leeb [1] and Proposition 5.4 to obtain that the weak limit of G∗ n,θ (n) ∗ ,σ (n)(t|p∗)πn,θ (n) ,σ (n)(p∗) is of the form (B.2) with p∗ replacing p. Since|ψp∗| is infinite, the integrand in (B.2) reduces to one, i.e., the limit is given by Φ∞,p∗(t−Aδ (p∗) ) P� q=p∗+1 ∆σξ∞,q(δ (q) q + ψq, cqσξ∞,q). Finally, consider the case p = p∗ and p∗ = O. Arguing as above, we see that G∗ n,θ (n) ∗ ,σ (n)(t|O)πn,θ (n) ,σ (n)(O) converges weakly to Φ∞,O(t−Aδ (O) ) P� q=O+1 ∆σξ∞,q(δ (q) q + ψq, cqσξ∞,q). Because the individual model selection probabilities π∗ n,θ (n) ,σ (n)(p),O≤p≤P, sum up to one, the same is true for their large-sample limits. In particular, note that (5.2) is a convex combination of cdfs, and that all the weights in the convex combination are positive. From this, we obtain that G∗ n,θ (n) ,σ (n)(t) converges to the expression in (5.2) at each continuity point t of the limit expression, i.e., G∗ n,θ (n) ,σ (n)(t) converges weakly. (Note that a convex combination of cdfs on Rk is continuous at a point t if each individual cdf is continuous at t; the converse is also true, provided that all the weights in the convex combination are positive.) To establish that weak convergence can be strengthened to convergence in total variation under the conditions given in Proposition 5.1, it suffices to note, under these conditions, that G∗ n,θ (n) ,σ (n)(t|p), p∗ ≤ p ≤ P, converges not only weakly but also in total variation in view of Proposition 5.1 of Leeb [1]. References [1] Leeb, H., (2005). The distribution of a linear predictor after model selection: conditional finite-sample distributions and asymptotic approximations. J. Statist. Plann. Inference 134, 64–89. [2] Leeb, H. and Pötscher, B. M., Can one estimate the conditional distribution of post-model-selection estimators? Ann. Statist., to appear. [3] Leeb, H. and Pötscher, B. M., (2003). The finite-sample distribution of post-model-selection estimators, and uniform versus non-uniform approximations. Econometric Theory 19, 100–142. [4] Leeb, H. and Pötscher, B. M., (2005). Can one estimate the unconditional distribution of post-model-selection estimators? Manuscript. [5] Leeb, H. and Pötscher, B. M., (2005). Model selection and inference: Facts and fiction. Econometric Theory, 21, 21–59.
Linear prediction after model selection 311 [6] Pötscher, B. M., (1991). Effects of model selection on inference. Econometric Theory 7, 163–185. [7] Rao, C. R., (1973). Linear Statistical Inference and Its Applications, 2nd edition. John Wiley & Sons, New York.
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Institute of Mathematical Statistic
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Institute of Mathematical Statistic
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iv Contents Copulas and Decoupling
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vi Finally, thanks go the Statistic
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SCIENTIFIC PROGRAM The Second Erich
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x Recent Advances in Longitudinal D
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The Second Lehmann Symposium—Opti
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xiv Richard Davis Colorado State Un
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xvi Matthias Matheas Rice Universit
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xviii Hui Zhao University of Texas
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IMS Lecture Notes-Monograph Series
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Proof. The maximum LR test rejects
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4. Location-scale families On likel
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6. Conclusions On likelihood ratio
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Student’s t-test for scale mixtur
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Optimality in multiple testing 17 w
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Optimality in multiple testing 19 I
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power 0.02 0.03 0.04 0.05 0.06 0.07
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Optimality in multiple testing 23 i
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Optimality in multiple testing 25 (
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Optimality in multiple testing 27 M
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6. Summary Optimality in multiple t
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Optimality in multiple testing 31 [
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On the false discovery proportion 3
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≤ N� � N� P m=1 On the fals
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Since j≤ s, it follows that On th
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0.025 0.02 0.015 0.01 0.005 On the
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Massive multiple hypotheses testing
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P value EDF qdf 0.0 0.2 0.4 0.6 0.8
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Proof. See Appendix. Massive multip
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X1 Massive multiple hypotheses test
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0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4
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Then π0α ∗ cal π0α ∗ cal +
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Frequentist statistics: theory of i
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2.3. Failure and confirmation Frequ
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3.1. Types of null hypothesis Frequ
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together with the probabilistic ass
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Statistical models: problem of spec
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Modeling inequality, spread in mult
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Semiparametric transformation model
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2.1. The model Semiparametric trans
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6.3. Part (ii) Put (6.1) (6.2) ˙Γ
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8. Proof of Proposition 2.3 Semipar
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Bayesian transformation hazard mode
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Cumulative Hazard 0.0 0.2 0.4 0.6 0
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Hazard function 0.0 0.5 1.0 1.5 2.0
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Copulas, information, dependence an
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Regression trees 211 right model in
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Abs(effects) 0.00 0.05 0.10 0.15 0.
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Regression trees 215 of the tree. T
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Regression trees 217 GUIDE methods
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Regression trees 219 and E appear a
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Regression trees 221 Table 3 Result
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Solder =thick 2.5 Regression trees
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Observed values 0 10 20 30 40 50 60
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Regression trees 227 effects and in
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On Competing risk and degradation p
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Restricted estimation in competing
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9. Conclusion 0.0 0.1 0.2 0.3 0.4 0
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2.1. Maximin efficiency robust test
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Robust tests for genetic associatio
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