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A Symmetrization-Desymmetrization Procedure for Uniformly Good Approximation of Expectations Involving Arbitrary Sums of Generalized U-Statistics

Klass, Michael J. ; Nowicki, Krzysztof

The Annals of probability, 2000-10, Vol.28 (4), p.1884-1907 [Periódico revisado por pares]

Hayward, CA: Institute of Mathematical Statistics

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  • Título:
    A Symmetrization-Desymmetrization Procedure for Uniformly Good Approximation of Expectations Involving Arbitrary Sums of Generalized U-Statistics
  • Autor: Klass, Michael J. ; Nowicki, Krzysztof
  • Assuntos: 60E15 ; 60F25 ; 60G50 ; Approximation ; Distribution theory ; Exact sciences and technology ; expectations of functions of second-order sums ; Generalized U-statistics ; Limit theorems ; Matematik ; Mathematical functions ; Mathematics ; Natural Sciences ; Naturvetenskap ; Probability and statistics ; Probability Theory and Statistics ; Probability theory and stochastic processes ; Random variables ; Sannolikhetsteori och statistik ; Sciences and techniques of general use ; Stochastic processes ; symmetrization –desymmetrization
  • É parte de: The Annals of probability, 2000-10, Vol.28 (4), p.1884-1907
  • Descrição: Let Φ be a symmetric function, nondecreasing on [0, ∞) and satisfying Δ2growth condition, (X1, Y1), (X2, Y2), ..., (Xn, Yn) be independent random vectors such that (for each 1 ≤ i ≤ n) either Yi= Xior Yiis independent of all the other variates, and the marginal distributions of {Xi} and {Yj} are otherwise arbitrary. Let {fij(x, y)}1≤ i, j ≤ nbe any array of real valued measurable functions. We present a method of obtaining the order of magnitude of$E\Phi\Bigg(\underset{1\geq i, j\geq n}\sum f_{ij}(X_i, Y_j)\Bigg).$The proof employs a double symmetrization, introducing independent copies {X̃i, Ỹj} of {Xi, Yj}, and moving from summands of the form fij(Xi, Yj) to what we call f(s) ij(Xi, Yj, X̃i, Ỹj). Substitution of fixed constants x̃iand ỹjfor X̃iand Ỹj) results in f(s) ij(Xi, Yj, x̃i, ỹj), which equals fij(Xi, Yj) adjusted by a sum of quantities of first order separately in Xiand Yj. Introducing further explicit first-order adjustments, call them g1ij(Xi, x̃, ỹ) and g2ij(Yj, x̃, ỹ), it is proved that$E\Phi\Bigg( \underset{1\leq i, j \leq n}{\sum} \Big(f^{(s)}_{ij} (X_i, Y_j \tilde x_i, \tilde y_j) - g_{1ij} (X_i \tilde x, \tilde y) - g_{2ij} (Y_j, \tilde x, \tilde y)\Big)\Bigg)$ $\leq_\alpha E\Phi\Bigg(\sqrt {\underset{1\geq i, j\geq n}\sum f^{(s)}_{ij}(X_i, Y_j, \tilde{x}_i, \tilde{y}_j)\Big)^2\Bigg)}\approx_\alpha\Phi f^{(s)},X,Y,\tilde{x},\tilde{y})$where the latter is an explicitly computable quantity. For any x̃0and ỹ0which come within a factor of two of minimizing Φ f(s),X,Y,x̃,ỹ) it is shown that$E\Phi\Bigg(\underset{1\leq i, j\leq n}\sum f_{ij}(X_i, Y_j)\Bigg)$ $ \approx_\alpha\mathrm{max}\Bigg\{\Phi f^{(s)},X,Y,\tilde{x}^0,\tilde{y}^0), E\Phi\Bigg(\underset{1\leq i, j\leq n}\sum \Big(f_{ij}(X_i,\tilde{y}^0_j) +f_{ij}(\tilde{x}^0_i, Y_j)$- fij(x̃0 i,ỹ0 j) + g1ij(Xi, (x̃0 i,ỹ0 j) + g2ij(Yj, (x̃0 i, ỹ0 j)))}, which is computable (approximable) in terms of the underlying random variables. These results extend to the expectation of D of a sum of functions of k-components.
  • Editor: Hayward, CA: Institute of Mathematical Statistics
  • Idioma: Inglês
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