Utilizing regression properties of order statistics, we characterize a family of distributions introduced by Akhundov, Balakrishnan, and Nevzorov (2004), that includes the t-distribution with two degrees of freedom as one of its members. Then we extend this characterization result to t-distribution with more than two degrees of freedom.
Two-sample $U$-statistics are widely used in a broad range of applications, including those in the fields of biostatistics and econometrics. In this paper, we establish sharp Cramér-type moderate deviation theorems for Studentized two-sample $U$-statistics in a general framework, including the two-sample $t$-statistic and Studentized Mann-Whitney test statistic as prototypical examples. In particular, a refined moderate deviation theorem with second-order accuracy is established for the two-sample $t$-statistic. These results extend the applicability of the existing statistical methodologies f…
Consider the multiple linear regression model $y_{i} = \boldsymbol{x}'_{i} \boldsymbolβ + ε_{i}$, where $ε_i$'s are independent and identically distributed random variables, $\mathbf{x}_i$'s are known design vectors and $\boldsymbolβ$ is the $p \times 1$ vector of parameters. An effective way of approximating the distribution of the M-estimator $\boldsymbol{\barβ}_n$, after proper centering and scaling, is the Perturbation Bootstrap Method. In this current work, second order results of this non-naive bootstrap method have been investigated. Second order correctness is important for reducing th…
Improved Scaling Laws via Weak-to-Strong Generalization in Random Feature Ridge Regression
Institute of Science and Technology Austria (ISTA)（奥地利科学与技术研究所）；Department of Applied and Computational Mathematics, Yale University（耶鲁大学应用与计算数学系）；Department of Statistics and Data Science, Yale University（耶鲁大学统计与数据科学系）
We propose a class of tests for linear regression on concomitants (induced order statistics). These tests are based on sequential sums of regression residuals. We self-center and self-normalize these sums. The resulting process is called an empirical bridge. We prove weak convergence of the empirical bridge in uniform metrics to a centered Gaussian process. The proposed tests are of chi-square type.
Consistency of the jackknife-after-bootstrap variance estimator for the bootstrap quantiles of a studentized statistic
埃夫隆[J.罗伊。中央集权主义者Soc.Ser.B54（1992）83- 111]提出了一种计算效率高的方法，称为自举后折刀法，用于估计独立数据的自举估计量的方差。对于相依数据，Lahiri最近提出了一种自举后折刀法[Econometric Theory 18（2002）79- 98]。本文证明了对于相依和独立数据，Bootstrap分位数方差的Jackknife-after-Bootstrap估计是相合的。模拟研究的结果也被提出。
Function-on-function linear regression is important for understanding the relationship between the response and the predictor that are both functions. In this article, we propose a reproducing kernel Hilbert space approach to function-on-function linear regressionvia the penalised least square, regularized by the thin-plate spline smoothness penalty. The minimax optimal convergence rate of our estimator of the coefficient function is studied. We derive the Bahadur representation, which allows us to propose statistical inference methods using bootstrap and the convergence of Banach-valued rando…
This paper presents a selective survey of recent developments in statistical inference and multiple testing for high-dimensional regression models, including linear and logistic regression. We examine the construction of confidence intervals and hypothesis tests for various low-dimensional objectives such as regression coefficients and linear and quadratic functionals. The key technique is to generate debiased and desparsified estimators for the targeted low-dimensional objectives and estimate their uncertainty. In addition to covering the motivations for and intuitions behind these statistica…
This paper presents a Bayesian approach to multiple-output quantile regression. The unconditional model is proven to be consistent and asymptotically correct frequentist confidence intervals can be obtained. The prior for the unconditional model can be elicited as the ex-ante knowledge of the distance of the tau-Tukey depth contour to the Tukey median, the first prior of its kind. A proposal for conditional regression is also presented. The model is applied to the Tennessee Project Steps to Achieving Resilience (STAR) experiment and it finds a joint increase in tau-quantile subpopulations for …
It is clear that conventional statistical inference protocols need to be revised to deal correctly with the high-dimensional data that are now common. Most recent studies aimed at achieving this revision rely on powerful approximation techniques, that call for rigorous results against which they can be tested. In this context, the simplest case of high-dimensional linear regression has acquired significant new relevance and attention. In this paper we use the statistical physics perspective on inference to derive a number of new exact results for linear regression in the high-dimensional regim…
Sharpe ratio is widely used in asset management to compare and benchmark funds and asset managers. It computes the ratio of the excess return over the strategy standard deviation. However, the elements to compute the Sharpe ratio, namely, the expected returns and the volatilities are unknown numbers and need to be estimated statistically. This means that the Sharpe ratio used by funds is subject to be error prone because of statistical estimation error. Lo (2002), Mertens (2002) derive explicit expressions for the statistical distribution of the Sharpe ratio using standard asymptotic theory un…
In this paper, we consider a weighted local linear estimator based on the inverse selection probability for nonparametric regression with missing covariates at random. The asymptotic distribution of the maximal deviation between the estimator and the true regression function is derived and an asymptotically accurate simultaneous confidence band is constructed. The estimator for the regression function is shown to be oracally efficient in the sense that it is uniformly indistinguishable from that when the selection probabilities are known. Finite sample performance is examined via simulation st…
Confidence intervals provide a way to determine plausible values for a population parameter. They are omnipresent in research articles involving statistical analyses. Appropriately, a key statistical literacy learning objective is the ability to interpret and understand confidence intervals in a wide range of settings. As instructors, we devote a considerable amount of time and effort to ensure that students master this topic in introductory courses and beyond. Yet, studies continue to find that confidence intervals are commonly misinterpreted and that even experts have trouble calibrating the…
Neural Networks as Linear Regression: An Introduction for Statisticians
Department of Mathematics Statistics and Computer Science（数学统计与计算机科学系）；Macalester College（麦克莱斯特学院）；Department of Biostatistics（生物统计学系）；University of Michigan（密歇根大学）
Imputation is a popular technique for handling missing data. We consider a nonparametric approach to imputation using the kernel ridge regression technique and propose consistent variance estimation. The proposed variance estimator is based on a linearization approach which employs the entropy method to estimate the density ratio. The root-n consistency of the imputation estimator is established when a Sobolev space is utilized in the kernel ridge regression imputation, which enables us to develop the proposed variance estimator. Synthetic data experiments are presented to confirm our theory.
