Kevin Ford
We give sharp, uniform estimates for the probability that the empirical distribution function for n uniform-[0,1] random variables stays to one side of a given line.
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Crossing the Kolmogorov-Smirnov Boundary: Exact Tails, Sharp Bounds, and Broken Pivots
本文通过有限样本穿越账本重写 Kolmogorov-Smirnov 统计量的精确分布，给出单样本和双样本的精确尾部计算、指数界解释，并指出在复合零假设下参数拟合会破坏路径从而失去分布自由性质。
Non-degenerate U-empirical Kolmogorov-Smirnov tests are studied and their large deviation asymptotics under the null-hypothesis is described. Several examples of such statistics used for testing goodness-of-fit and symmetry are considered. It is shown how to calculate their local Bahadur efficiency.
We apply recent bounds of the author (math.PR/0609224) for generalized Smirnov statistics to the distribution of integers whose prime factors satisfy certain systems of inequalities.
Motivated by the weak limit of the Kolmogorov-Smirnov test statistics, in this contribution, we concern the asymptotics of \begin{align*} \mathbb{P}\left\{\sup_{\boldsymbol{x}\in [0,1]^n}\left(W(\boldsymbol{x})\Big| W(\boldsymbol{1})=w\right)>u\right\}, \ w\in\mathbb{R}, \end{align*} for large $u$ where $W(\boldsymbol{x})$ is the multivariate Brownian sheet based on a distribution function $F$. The results related to general $F$ are investigated and some important examples are also showed.
Consider $n$ iid random variables, where $ξ_1, \ldots, ξ_n$ are $n$ realisations of a random variable $ξ$ and $ζ_1, \ldots, ζ_n$ are $n$ realisations of a random variable $ζ$. The distribution of each realisation of $ξ$, that is the distribution of \emph{one} $ξ_i$, depends on the value of the corresponding $ζ_i$, that is the probability $P\left(ξ_i\leq x\right)=F(x,ζ_i)$. We develop a statistical test to see if the $ξ_1, \ldots, ξ_n$ are distributed according to the distribution function $F(x,ζ_i)$. We call this new statistical test the condition Kolmogorov-Smirnov test.
Gene Set Enrichment Analysis (GSEA) is a basic tool for genomic data treatment. From a statistical point of view, the centering of its test statistic does not allow the derivation of asymptotic results. A test statistic with a different centering is proposed. Under the null hypothesis, the convergence in distribution of the new test statistic is proved, using the theory of empirical processes. The limiting distribution can be computed by Monte-Carlo simulation. The test defined in this way has been called Weighted Kolmogorov Smirnov (WKS) test. The fact that the evaluation of the asymptotic di…
In this paper, we analyze a multivariate counterpart of the generalized weighted Kolmogorov-Smirnov statistic, which is the supremum of weighted locally stationary chi-square process over non-compact interval. The boundedness and the exact tail asymptotic behavior of the statistics are derived. We illustrate our findings by two examples where the statistic is defined by Brownian bridge and fractional Brownian motion respectively.
This paper considers a new bootstrap procedure to estimate the distribution of high-dimensional $\ell_p$-statistics, i.e. the $\ell_p$-norms of the sum of $n$ independent $d$-dimensional random vectors with $d \gg n$ and $p \in [1, \infty]$. We provide a non-asymptotic characterization of the sampling distribution of $\ell_p$-statistics based on Gaussian approximation and show that the bootstrap procedure is consistent in the Kolmogorov-Smirnov distance under mild conditions on the covariance structure of the data. As an application of the general theory we propose a bootstrap hypothesis test …
Exploring the Statistical Properties of Outputs from a Process Inspired by Geometrical Interpretation of Newton's Method
本文研究特定情形下牛顿法算法输出的统计特性，证明其概率密度服从柯西分布，提出生成均匀分布的新方法，经统计检验验证结果，并推导连续输出距离分布的变换求解方法。
Functionals of Brownian bridge arise as limiting distributions in nonparametric statistics. In this paper we will give a derivation of distributions of extrema of the Brownian bridge based on excursion theory for Brownian motion. Only the Poisson character of the excursion process will be used. Particular cases of calculations include the distributions of the Kolmogorov-Smirnov statistic, the Kuiper statistic, and the ratio of the maximum positive ordinate to the minumum negative ordinate.
We describe the statistics of repetition times of a string of symbols in a stochastic process. Denote by T(A) the time elapsed until the process spells the finite string A and by S(A) the number of consecutive repetitions of A. We prove that, if the length of the string grows unbondedly, (1) the distribution of T(A), when the process starts with A, is well aproximated by a certain mixture of the point measure at the origin and an exponential law, and (2) S(A) is approximately geometrically distributed. We provide sharp error terms for each of these approximations. The errors we obtain are poin…
In this article, we establish a near-optimal convergence rate for the CLT of linear eigenvalue statistics of Wigner matrices, in Kolmogorov-Smirnov distance. For all test functions $f\in C^5(\mathbb R)$, we show that the convergence rate is either $N^{-1/2+\varepsilon}$ or $N^{-1+\varepsilon}$, depending on the first Chebyshev coefficient of $f$ and the third moment of the diagonal matrix entries. The condition that distinguishes these two rates is necessary and sufficient. For a general class of test functions, we further identify matching lower bounds for the convergence rates. In addition, …
Sharp Anti-Concentration Inequalities for Extremum Statistics via Copulas
该研究推导了d个同分布随机变量最大值集中函数的精确界，提出凸对角截面copula类以得到更紧上界，建立了新的反集中不等式并改进了相关结果，可应用于高维统计推断。
Under the Kolmogorov--Smirnov metric, an upper bound on the rate of convergence to the Gaussian distribution is obtained for linear statistics of the matrix ensembles in the case of the Gaussian, Laguerre, and Jacobi weights. The main lemma gives an estimate for the characteristic functions of the linear statistics; this estimate is uniform over the growing interval. The proof of the lemma relies on the Riemann--Hilbert approach.
This paper addresses the following question: given a sample of i.i.d. random variables with finite variance, can one construct an estimator of the unknown mean that performs nearly as well as if the data were normally distributed? One of the most popular examples achieving this goal is the median of means estimator. However, it is inefficient in a sense that the constants in the resulting bounds are suboptimal. We show that a permutation-invariant modification of the median of means estimator admits deviation guarantees that are sharp up to $1+o(1)$ factor if the underlying distribution posses…
The purpose of this paper is to provide a sharp analysis on the asymptotic behavior of the Durbin-Watson statistic. We focus our attention on the first-order autoregressive process where the driven noise is also given by a first-order autoregressive process. We establish the almost sure convergence and the asymptotic normality for both the least squares estimator of the unknown parameter of the autoregressive process as well as for the serial correlation estimator associated to the driven noise. In addition, the almost sure rates of convergence of our estimates are also provided. It allows us …
The theory for multiplier empirical processes has been one of the central topics in the development of the classical theory of empirical processes, due to its wide applicability to various statistical problems. In this paper, we develop theory and tools for studying multiplier $U$-processes, a natural higher-order generalization of the multiplier empirical processes. To this end, we develop a multiplier inequality that quantifies the moduli of continuity of the multiplier $U$-process in terms of that of the (decoupled) symmetrized $U$-process. The new inequality finds a variety of applications…
Sharp total variation rates of convergence for fluctuations of linear statistics of $β$-ensembles
本文在一般势函数的单切口与非临界状态下，为$β$-系综线性统计量波动建立了全变差意义下速度为$1/n$的精确中心极限定理，并证明了多元CLT及密度的超收敛。
Sharp large deviation estimates for Gaussian extrema
本文研究了高斯分布极值的大偏差估计，通过精确渐进行为提供了比经典Gumbel极限更准确的尾部近似方法。
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