Optimality

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8 E. L. Lehmann [5] Lehmann, E. L. (1986). Testing Statistical Hypotheses, 2nd Edition. Springer- Verlag, New York. [6] Menéndez, J. A., Rueda, C. and Salvador, B. (1992). Dominance of likelihood ratio tests under cone constraints. Amer. Statist. 20, 2087, 2099. [7] Perlman, M. D. and Wu, L. (1999). The Emperor’s new tests (with discussion). Statist. Sci. 14, 355–381. [8] Royall, R. (1997). Statistical Evidence. Chapman and Hall, Boca Raton.
IMS Lecture Notes–Monograph Series 2nd Lehmann Symposium – <strong>Optimality</strong> Vol. 49 (2006) 9–15 c○ Institute of Mathematical Statistics, 2006 DOI: 10.1214/074921706000000365 Student’s t-test for scale mixture errors Gábor J. Székely 1 Bowling Green State University, Hungarian Academy of Sciences Abstract: Generalized t-tests are constructed under weaker than normal conditions. In the first part of this paper we assume only the symmetry (around zero) of the error distribution (i). In the second part we assume that the error distribution is a Gaussian scale mixture (ii). The optimal (smallest) critical values can be computed from generalizations of Student’s cumulative distribution function (cdf), tn(x). The cdf’s of the generalized t-test statistics are (x), resp. As the sample size n → ∞ we get denoted by (i) tS n (x) and (ii) tGn the counterparts of the standard normal cdf Φ(x): (i) ΦS (x) := limn→∞ tS n (x), and (ii) ΦG (x) := limn→∞ tG n (x). Explicit formulae are given for the underlying new cdf’s. For example ΦG (x) = Φ(x) iff |x| ≥ √ 3. Thus the classical 95% confidence interval for the unknown expected value of Gaussian distributions covers the center of symmetry with at least 95% probability for Gaussian scale mixture distributions. On the other hand, the 90% quantile of ΦG is 4 √ 3/5 = 1.385 · · · > Φ−1 (0.9) = 1.282 . . . . 1. Introduction An inspiring recent paper by Lehmann [9] summarizes Student’s contributions to small sample theory in the period 1908–1933. Lehmann quoted Student [10]: “The question of applicability of normal theory to non-normal material is, however, of considerable importance and merits attention both from the mathematician and from those of us whose province it is to apply the results of his labours to practical work.” In this paper we consider two important classes of distributions. The first class is the class of all symmetric distributions. The second class consists of scale mixtures of normal distributions which contains all symmetric stable distributions, Laplace, logistic, exponential power, Student’s t, etc. For scale mixtures of normal distributions see Kelker [8], Efron and Olshen [5], Gneiting [7], Benjamini [1]. Gaussian scale mixtures are important in finance, bioinformatics and in many other areas of applications where the errors are heavy tailed. First, let X1, X2, . . . , Xn be independent (not necessarily identically distributed) observations, and let µ be an unknown parameter with Xi = µ + ξi, i = 1,2, . . . , n, where the random errors ξi, 1≤i≤nare independent, and symmetrically distributed around zero. Suppose that ξi = siηi, i = 1, 2, . . . , n, where si, ηi i = 1, 2, . . . , n are independent pairs of random variables, and the random scale, si≥ 0, is also independent of ηi. We also assume the ηi variables are 1 Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, OH 43403-0221 and Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, Hungary, e-mail: gabors@bgnet.bgsu.edu AMS 2000 subject classifications: primary 62F03; secondary 62F04. Keywords and phrases: generalized t-tests, symmetric errors, Gaussian scale mixture errors. 9
Page 1 and 2: Institute of Mathematical Statistic
Page 3 and 4: Institute of Mathematical Statistic
Page 5 and 6: iv Contents Copulas and Decoupling
Page 7 and 8: vi Finally, thanks go the Statistic
Page 9 and 10: SCIENTIFIC PROGRAM The Second Erich
Page 11 and 12: x Recent Advances in Longitudinal D
Page 13 and 14: The Second Lehmann Symposium—Opti
Page 15 and 16: xiv Richard Davis Colorado State Un
Page 17 and 18: xvi Matthias Matheas Rice Universit
Page 19 and 20: xviii Hui Zhao University of Texas
Page 21 and 22: IMS Lecture Notes-Monograph Series
Page 23 and 24: Proof. The maximum LR test rejects
Page 25 and 26: 4. Location-scale families On likel
Page 27: 6. Conclusions On likelihood ratio
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Page 41 and 42: power 0.02 0.03 0.04 0.05 0.06 0.07
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Page 47 and 48: Optimality in multiple testing 27 M
Page 49 and 50: 6. Summary Optimality in multiple t
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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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IMS Lecture Notes-Monograph Series
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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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1 . . . 1 i . . . . . . R j . . . O
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Optimal sampling strategies 269 as
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Optimal sampling strategies 271 γ1
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Optimal sampling strategies 273 Def
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Optimal sampling strategies 275 Usi
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Optimal sampling strategies 277 Def
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optimal 1 2 3 4 leaf sets 5 6 (leaf
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6. Related work Optimal sampling st
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Optimal sampling strategies 283 Cla
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Optimal sampling strategies 285 Sin
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Optimal sampling strategies 287 µ
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Optimal sampling strategies 289 on
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Linear prediction after model selec
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y the P× 1 vector (3.1) ηn(p) = L
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Appendix B: Proofs for Section 5 Li
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Local asymptotic minimax risk/asymm
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Then (3.5) Local asymptotic minimax
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Moment-density estimation 323 may n
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3. The bias and MSE of ˆf ∗ α M
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Moment-density estimation 327 Corol
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Moment-density estimation 329 Suppo
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6. Simulations Moment-density estim
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Moment-density estimation 333 [7] M
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for positive α, and for α = 0 as
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Asymptotics of the MDPDE 337 Lemma
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Asymptotics of the MDPDE 339 asympt
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