By Michael R. Kosorok
ISBN-10: 0387749772
ISBN-13: 9780387749778
ISBN-10: 0387749780
ISBN-13: 9780387749785
This ebook presents a self-contained, linear, and unified advent to empirical tactics and semiparametric inference. those robust study suggestions are strangely worthwhile for constructing equipment of statistical inference for complicated types and in realizing the homes of such equipment. The special viewers contains statisticians, biostatisticians, and different researchers with a historical past in mathematical statistics who've an curiosity in studying approximately and doing study in empirical techniques and semiparametric inference yet who want to have a pleasant and slow creation to the world. The e-book can be utilized both as a examine reference or as a textbook. the extent of the e-book is appropriate for a moment 12 months graduate direction in information or biostatistics, supplied the scholars have had a 12 months of graduate point mathematical records and a semester of probability.The publication includes 3 components. the 1st half is a concise review of the entire major ideas lined within the e-book with at least technicalities. the second one and 3rd elements conceal the 2 respective major themes of empirical approaches and semiparametric inference extensive. The connections among those issues is additionally verified and emphasised during the textual content. every one half has a last bankruptcy with a number of case reviews that use concrete examples to demonstrate the thoughts constructed to this point. The final elements additionally each one comprise a bankruptcy which covers the wanted mathematical preliminaries. each one major suggestion is brought with a non-technical motivation, and examples are given all through to demonstrate vital options. Homework difficulties also are integrated on the finish of every bankruptcy tohelp the reader achieve extra insights.
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Extra resources for Introduction to Empirical Processes and Semiparametric Inference
Sample text
This is a bounded Donsker class since F = {f1 f2 : f1 ∈ F1 , f2 ∈ F2 }, where Fj = {1{yj ≤ t}, t ∈ R} is a bounded Donsker class for j = 1, 2. We thus obtain consistency in probability of the bootstrap. 7, since F is bounded by 1. 4 The Functional Delta Method √ X for Suppose Xn is a sequence of random variables with n(Xn − θ) some θ ∈ Rp , and the function φ : Rp → Rq√has a derivative φ′ (θ) at θ. The standard delta method now tells us that n(φ(Xn ) − φ(θ)) φ′ (θ)X. d. data involve maps from empirical processes to spaces of functions, and hence cannot be handled by the standard delta method.
Estimation under the model P is more taxing than estimation under any parametric submodel P0 = {Pθ : θ ∈ Θ0 } ⊂ P, where Θ0 is finite dimensional. Thus the information for estimation under model P is worse than the information under any parametric submodel P0 . If the information for the regular estimator Tn is equal to the minimum of the information over all efficient estimators for all parametric submodels P0 , then Tn is semiparametric efficient. For semiparametric models, this minimizer is the best possible, since the only models with more information are parametric models.
Another interesting application—to “large p, small n” asymptotics for 32 2. 5 of Part II, although we will not address the theoretical derivation of the KMT construction. d. data are the U-processes. The mth order empirical U-process measure Un,m is defined, for a measurable function f : X m → R and a sample of obervations X1 , . . , Xn on X , as n m −1 f (Xi1 , . . ,im )∈In,m where In,m is the set of all m-tuples of integers (i1 , . . , im ) satisfying 1 ≤ i1 < · · · < im ≤ n. , Un,1 = Pn .
Introduction to Empirical Processes and Semiparametric Inference by Michael R. Kosorok
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