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IMS Lecture Notes–Monograph Series 2nd Lehmann Symposium – <strong>Optimality</strong> Vol. 49 (2006) 312–321 c○ Institute of Mathematical Statistics, 2006 DOI: 10.1214/074921706000000527 Local asymptotic minimax risk bounds in a locally asymptotically mixture of normal experiments under asymmetric loss Debasis Bhattacharya 1 and A. K. Basu 2 Visva-Bharati University and Calcutta University Abstract: Local asymptotic minimax risk bounds in a locally asymptotically mixture of normal family of distributions have been investigated under asymmetric loss functions and the asymptotic distribution of the optimal estimator that attains the bound has been obtained. 1. Introduction There are two broad issues in the asymptotic theory of inference: (i) the problem of finding the limiting distributions of various statistics to be used for the purpose of estimation, tests of hypotheses, construction of confidence regions etc., and (ii) problems associated with questions such as: how good are the estimation and testing procedures based on the statistics under consideration and how to define ‘optimality’, etc. Le Cam [12] observed that the satisfactory answers to the above questions involve the study of the asymptotic behavior of the likelihood ratios. Le Cam [12] introduced the concept of ‘Limit Experiment’, which states that if one is interested in studying asymptotic properties such as local asymptotic minimaxity and admissibility for a given sequence of experiments, it is enough to prove the result for the limit of the experiment. Then the corresponding limiting result for the sequence of experiments will follow. One of the many approaches which are used in asymptotic theory to judge the performance of an estimator is to measure the risk of estimation under an appropriate loss function. The idea of comparing estimators by comparing the associated risks was considered by Wald [19, 20]. Later this idea has been discussed by Hájek [8], Ibragimov and Has’minskii [9] and others. The concept of studying asymptotic efficiency based on large deviations has been recommended by Basu [4] and Bahadur [1, 2]. In the above context it is an interesting problem to obtain a lower bound for the risk in a wide class of competing estimators and then find an estimator which attains the bound. Le Cam [11] obtained several basic results concerning asymptotic properties of risk functions for LAN family of distributions. Jeganathan [10], Basawa and Scott [5], and Le Cam and Yang [13] have extended the results of Le Cam for Locally Asymptotically Mixture of Normal (LAMN) experiments. Basu and Bhattacharya [3] further extended the result for Locally Asymptotically Quadratic (LAQ) family of distributions. A symmetric loss structure (for example, 1Division of Statistics, Institute of Agriculture, Visva-Bharati University, Santiniketan, India, Pin 731236 2Department of Statistics, Calcutta University, 35 B. C. Road, Calcutta, India, Pin 700019 AMS 2000 subject classifications: primary 62C20, 62F12; secondary 62E20, 62C99. Keywords and phrases: locally asymptotically mixture of normal experiment, local asymptotic minimax risk bound, asymmetric loss function 312
Local asymptotic minimax risk/asymmetric loss 313 squared error loss) has been used to derive the results in the above mentioned references. But there are situations where the loss can be different for equal amounts of over-estimation and under-estimation, e. g., there exists a natural imbalance in the economic results of estimation errors of the same magnitude and of opposite signs. In such cases symmetric losses may not be appropriate. In this context Bhattacharya et al. [7], Levine and Bhattacharya [15], Rojo [16], Zellner [21] and Varian [18] may be referred to. In these works the authors have used an asymmetric loss, known as the LINEX loss function. Let ∆ = ˆ θ− θ, a�= 0 and b > 0. The LINEX loss is then defined as: (1.1) l(∆)=b[exp(a∆)−a∆−1]. Other types of asymmetric loss functions that can be found in the literature are as follows: � C1∆, for ∆≥0 l(∆) = −C2∆, for ∆ < 0, C1, C2 are constants, or l(∆) = � λw(θ)L(∆), for ∆≥0, (over-estimation) w(θ)L(∆), for ∆ < 0, (under-estimation), where ‘L’ is typically a symmetric loss function, λ is an additional loss (in percentage) due to over-estimation, and w(θ) is a weight function. The problem of finding the lower bound for the risk with asymmetric loss functions under the assumption of LAN was discussed by Lepskii [14] and Takagi [17]. In the present work we consider an asymmetric loss function and obtain the local asymptotic minimax risk bounds in a LAMN family of distributions. The paper is organized as follows: Section 2 introduces the preliminaries and the relevant assumptions required to develop the main result. Section 3 is dedicated to the derivation of the main result. Section 4 contains the concluding remarks and directions for future research. 2. Preliminaries Let X1, . . . , Xn be n random variables defined on the probability space (X,A, Pθ) and taking values in (S,S), where S is the Borel subset of a Euclidean space and S is the σ-field of Borel subsets of S. Let the parameter space be Θ, where Θ is an open subset of R 1 . It is assumed that the joint probability law of any finite set of such random variables has some known functional form except for the unknown parameter θ involved in the distribution. LetAn be the σ-field generated by X1, . . . , Xn and let Pθ,n be the restriction of Pθ toAn. Let θ0 be the true value of θ and let θn = θ0 + δnh (h∈R 1 ), where δn→ 0 as n→∞. The sequence δn may depend on θ but is independent of the observations. It is further assumed that, for each n≥1, the probability measures Pθo,n and Pθn,n are mutually absolutely continuous for all θ0 and θn. Then the sequence of likelihood ratios is defined as Ln(Xn;θ0, θn) = Ln(θ0, θn) = dPθn,n , dPθ0,n where Xn = (X1, . . . , Xn) and the corresponding log-likelihood ratios are defined as Λn(θ0, θn) = log Ln(θ0, θn) = log dPθn,n . dPθ0,n
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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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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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X1 Massive multiple hypotheses test
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Then π0α ∗ cal π0α ∗ cal +
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Frequentist statistics: theory of i
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together with the probabilistic ass
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Statistical models: problem of spec
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6.3. Part (ii) Put (6.1) (6.2) ˙Γ
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Bayesian transformation hazard mode
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Hazard function 0.0 0.5 1.0 1.5 2.0
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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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2.1. Maximin efficiency robust test
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