Probability for Statisticians

By G. Shorack. 2000

259 citations recorded by the source corpus.

Subfields: Finance, Management Science and Operations Research, Statistics and Probability

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Citations in the corpus

On the empirical estimation of integral probability metrics (2012, methodology): “In statistics, IPMs have been used in nonparametric two-sample testing, including the Kolmogorov-Smirnov test [7, 39, 40] and the kernel test [20, 21]; as well as in independence testing [20, Section 7.”
Hilbert Space Embeddings and Metrics on Probability Measures (2009, background): “(a) F = Cb(M), the space of bounded continuous functions on (M,ρ), where ρ is a metric (Shorack, 2000, Chapter 19, Definition 1.1).”
On integral probability metrics, φ-divergences and binary classification (2009, methodology): “hat the classical central limit theorem and the Berry-Esse´en theorem in Rd use the Kolmogorov distance. The Kolmogorov distance also appears in hypothesis testing as the Kolmogorov-Smirnov statistic [21]. 4) Maximum mean discrepancy: γF is called the maximum mean discrepancy (MMD) [26], [27] when F”
Quantile tomography: using quantiles with multivariate data (2008, background): “The concept of the quantile function of a univariate probability distribution is known well beyond any need of exposition; a casual reference like Shorack (2000) or the encyclopedic entry of Eubank (1986) results in the following.”
Genealogical particle analysis of rare events (2005, methodology): “Returning to the Gaussian situation, and using Mill’s inequalities 1 t + 1/t ≤ P(N (0, 1) ≥ t) √ 2πe t2 2 ≤ 1 t which are valid for any t > 0, and any reduced Gaussian random variable N (0, 1) (see for instance (6) on page 237 in [25]), we find that σ n (a) 2 ≤ e−a2/(2n) Pn(a) − P 2 n(a) ≤ P 2 n(a) ”
Normal approximation for nonlinear statistics using a concentration inequality approach (2007, background): “The example also provides a counterexample to a result of Shorack [22] and of Bolthausen and Götze [5].”
$L$-functions, processes, and statistics in measuring economic inequality and actuarial risks (2009, background): “, Serfling [84], Shorack [87], Shao [86].”
A Martingale Representation for Matching Estimators (2012, unknown): “(3) In view of this result, to apply a martingale central limit theorem to DN , it is sufficient to check the Lindeberg’s condition, 2N∑ k=1 E [ ξ 2N,k1{|ξN,k |≥ε} ] → 0, for all ε > 0 [see Billingsley (1995), Hall and Heyde (1980), and Shorack (2000) for alternative conditions].”
Maximum likelihood estimation of a spatial autoregressive Tobit model (2015, background): “It is sufficient to show that supi,n E||qi,n(θ)||2+δ ∞ (see Exercise 5.4, page 54, Shorack, 2000).”
Algebraic factor analysis: tetrads, pentads and beyond (2005, methodology): “Hence, by the delta method (Shorack, 2000, p.279), √ N f(S) = √ N [f(S)− f(Ψ)] −→d N (0, vf ) with asymptotic variance vf = lim N→∞ N VarΨ[f(S)] > 0.”
Support points (2016, background): “Since (nk)k=1 was arbitrarily chosen, it follows by the proof of Corollary 1 in Chapter 9 of [62] that Xn d − → X, which is as desired.”
Extremiles: A New Perspective on Asymmetric Least Squares (2018, methodology): “Thanks to this special closed-form expression for ξ̂Lτ , we can rely, for example, on Serfling (1980) or Shorack (2000) for proving its consistency, asymptotic normality, and deriving the Berry– Esséen rate of uniform convergence O(n−1/2).”