Empirical Processes with Applications to Statistics

By G. Shorack, Jon A. Wellner. 2009

2,243 citations recorded by the source corpus.

Subfields: Management Science and Operations Research, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/56defcc4d278c24e8517a43ad6e739b760622f1d

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In Management Science and Operations Research

In Statistics and Probability

Citations in the corpus

Distribution free specification tests of conditional models (2008, methodology): “It follows from classical empirical process theory, see Shorack and Wellner (1986), that bn !dK in the space D½0; 1 2, where K is the standard Kiefer process, a centered biparameter Gaussian process on the unit square with covariance function E½Kðu1; v1Þ Kðu2; v2Þ ¼ ðu1 ^ u2Þðv1 ^ v2 v1 v2Þ.”
Proportion of non‐zero normal means: universal oracle equivalences and uniformly consistent estimators (2008, background): “1=n/ by the Dvoretzky–Kiefer–Wolfowitz theorem (Shorack and Wellner, 1986) and (b) "̂n="→1 in probability; combining these with expression (A.”
A new approach to strong embeddings (2007, methodology): “ the SRW couples a standardized empirical distribution function with a Brownian Bridge. For a discussion of the numerous applications of this result in theoretical statistics, see Shorack and Wellner [14], Chapter 12. A further signiﬁcant advance was made by Nussbaum [12], who used this result to es”
Maximum likelihood estimators based on the block maxima method (2017, methodology): “We use Shorack and Wellner [21] (inequality 1 on p.”
Optimal weighted nearest neighbour classifiers (2011, methodology): “2d = O(n−M), (5.24) for every M>0. Moreover, using the moment bound in (A.3), 1− pt = Z u:kuk>t f¯(x+u)du= O(t−ρ), (5.25) ast→ ∞, uniformlyforx∈ Sǫn. Therefore we canapply Bennett’s inequality (Shorack and Wellner, 1986, p.440) to show that there exist c 3,c 4 >0 such that for suﬃciently la”
Regression diagnostics meets forecast evaluation: conditional calibration, reliability diagrams, and coefficient of determination (2021, methodology): “Evidently, there are alternatives based on empirical process theory (Shorack and Wellner, 2009).”
Adaptive density estimation: a curse of support? ✩ (2009, neutral): “If nFjk ≤ n−γ−1, using properties of Binomial random variables (see page 482 of Shorack and Wellner (1986)), for n ≥ 2, P(Njk − nFjk ≥ (1− θ′)C ′log n) ≤ P(Njk > (1− θ′)C ′log n) ≤ P(Njk ≥ 2) ≤ (1− Fjk)C nF 2 jk(1− Fjk)n−2 1− 3−1(n+ 1)Fjk ≤ n2F 2 jk 2(1 − 2−1nFjk) ≤ (nFjk) and the result is true.”
A Martingale Representation for Matching Estimators (2012, positive): “Using Bennett’s bound for binomial tails (e.g., Shorack and Wellner 1986, p. 440), we obtain Pr ( B ( N1, f̄ /N k 1 ) > Nα1 ) = Pr ( B ( N1, f̄ /N k 1 ) − f̄ /Nk−11√ N1 > Nα1 − f̄ /Nk−11√ N1 ) ≤ exp { − f̄ /N k−1 1 1 − f̄ /Nk1 [ Nα+k−11 f̄ ( log ( Nα+k−11 f̄ ) − 1 ) + 1 ]} = exp { − 1 1 − f̄ /Nk1 […”
Uniform change point tests in high dimension (2015, background): “,1). It then follows from Theorem 4.4 that |P|X(T ⋄,bσ∗ d (t)≤zbα,L(t))−(1−α)|=Op(n−C), C >0, where T⋄,σb ∗ d (t) is a copy of T bσ∗ d (t), independent of X. Standard empirical process theory (cf. [47]) now implies that we are able to consistently estimate zbα,L(t) based on TM for large enough M ”
Fractional diffusion equations and processes with randomly varying time. (2011, methodology): “7) and in the first step we applied the wellknown result for the maximal distribution of the absolute value of Brownian motion [see Shorack and Wellner (1986), page 34].”