Topics in Statistical Theory
نویسنده
چکیده
1 Deviation bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1 Markov and generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Class of sub-Gaussian random variables . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.1 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.2 Chernoff and Hoeffding-type bounds . . . . . . . . . . . . . . . . . . . . . . . 5 1.2.3 Bernstein-type bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Class of sub-exponential random variables . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3.1 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3.2 Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4 Maximal inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.4.1 Maximum over a finite set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.4.2 Maximum over a polytope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.4.3 Maximum over the Euclidean ball . . . . . . . . . . . . . . . . . . . . . . . . . 11 2 Estimation and detection in high-dimension . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1 Constrained estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1.1 General bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.1.3 Convex sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2 Hypothesis testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2.1 Statistics for testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3 Information-theoretic lower bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.1 Bounds for hypothesis testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.1.1 Total variation distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.1.2 Divergence between distributions . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.1.3 Kullback-Leibler divergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.2 Bounds for estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.2.1 Fano’s lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.2.2 Packing techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.2.3 Lower bounds for estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.3 Techniques for detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
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