The best mean square error that the classical kernel density estimator achieves if the kernel is non-negative and f has only two continuous derivatives, is of the order of special characters omitted. ...If negative kernels are allowed, then this rate can be improved depending on the smoothness of f and the order of the kernel. Abramson and others modified the classical kernel estimator, assumed non-negative, by allowing the bandwidth hn to depend on the data. The last and best result in the literature is Hall, Hu and Marron who show that under suitable assumptions on a non-negative kernel K and the density f, |fˆn( t) − f(t)| = O P(special characters omitted) for fixed t. The main result of this thesis states that special characters omitted |fˆn(t) − f(t)| = OP((special characters omitted)4/9) where Dn and fˆn(t) are purely data driven and Dn can be taken as close as desired to the set { t : f(t) > 0}. This rate is best possible for estimating a density in the sup norm. The data driven fˆn(t) and Dn have 'ideal' counterparts that depend on f, and for the ideal estimator, slightly sharper results are proven.
Renz (Ann. Probab. 1996) has established a rate of convergence $1/\sqrt{n}$
in the central limit theorem for martingales with some restrictive conditions.
In the present paper a modification of the ...methods, developed by Bolthausen
(Ann. Probab. 1982) and Grama and Haeusler (Stochastic Process. Appl. 2000), is
applied for obtaining the same convergence rate for a class of more general
martingales. An application to linear processes is discussed.
We study a rank based univariate two-sample distribution-free test. The test
statistic is the difference between the average of between-group rank distances
and the average of within-group rank ...distances. This test statistic is closely
related to the two-sample Cram\'er-von Mises criterion. They are different
empirical versions of a same quantity for testing the equality of two
population distributions. Although they may be different for finite samples,
they share the same expected value, variance and asymptotic properties. The
advantage of the new rank based test over the classical one is its ease to
generalize to the multivariate case. Rather than using the empirical process
approach, we provide a different easier proof, bringing in a different
perspective and insight. In particular, we apply the H\'ajek projection and
orthogonal decomposition technique in deriving the asymptotics of the proposed
rank based statistic. A numerical study compares power performance of the rank
formulation test with other commonly-used nonparametric tests and
recommendations on those tests are provided. Lastly, we propose a multivariate
extension of the test based on the spatial rank.
Deep learning (DL) has gained much attention and become increasingly popular in modern data science. Computer scientists led the way in developing deep learning techniques, so the ideas and ...perspectives can seem alien to statisticians. Nonetheless, it is important that statisticians become involved -- many of our students need this expertise for their careers. In this paper, developed as part of a program on DL held at the Statistical and Applied Mathematical Sciences Institute, we address this culture gap and provide tips on how to teach deep learning to statistics graduate students. After some background, we list ways in which DL and statistical perspectives differ, provide a recommended syllabus that evolved from teaching two iterations of a DL graduate course, offer examples of suggested homework assignments, give an annotated list of teaching resources, and discuss DL in the context of two research areas.
In this paper, we study the self-normalized Cram\a'{e}r-type moderate deviations for centered independent random variables \(X_1, X_2,...\) with \(0<E |X_i|^3 <\infty\). The main results refine ...Theorems 1.1 and 1.2 of Wang (2011), the Berry-Esseen bound (2.11) and Corollaries 2.2 and 2.3 of Jing, Shao and Wang (2003) under stronger moment conditions.
We propose a sparse coefficient estimation and automated model selection procedure for autoregressive (AR) processes with heavy-tailed innovations based on penalized conditional maximum likelihood. ...Under mild moment conditions on the innovation processes, the penalized conditional maximum likelihood estimator (PCMLE) satisfies a strong consistency, \(O_P(N^{-1/2})\) consistency, and the oracle properties, where N is the sample size. We have the freedom in choosing penalty functions based on the weak conditions on them. Two penalty functions, least absolute shrinkage and selection operator (LASSO) and smoothly clipped average deviation (SCAD), are compared. The proposed method provides a distribution-based penalized inference to AR models, which is especially useful when the other estimation methods fail or under perform for AR processes with heavy-tailed innovations (see \cite{Resnick}). A simulation study confirms our theoretical results. At the end, we apply our method to a historical price data of the US Industrial Production Index for consumer goods, and obtain very promising results.
We propose a new covariance matrix called Gini covariance matrix (GCM), which is a natural generalization of univariate Gini mean difference (GMD) to the multivariate case. The extension is based on ...the covariance representation of GMD by applying the multivariate spatial rank function. We study properties of GCM, especially in the elliptical distribution family. In order to gain the affine equivariance property for GCM, we utilize the transformation-retransformation (TR) technique and obtain an affine equivariant version GCM that turns out to be a symmetrized M-functional. The influence function of those two GCM's are obtained and their estimation has been presented. Asymptotic results of estimators have been established. A closely related scatter Kotz functional and its estimator are also explored. Finally, asymptotical efficiency and finite sample efficiency of the TR version GCM are compared with those of sample covariance matrix, Tyler-M estimator and other scatter estimators under different distributions.
Standard Gini covariance and Gini correlation play important roles in measuring the dependence of random variables with heavy tails. However, the asymmetry brings a substantial difficulty in ...interpretation. In this paper, we propose a symmetric Gini-type covariance and a symmetric Gini correlation (\(\rho_g\)) based on the joint rank function. The proposed correlation \(\rho_g\) is more robust than the Pearson correlation but less robust than the Kendall's \(\tau\) correlation. We establish the relationship between \(\rho_g\) and the linear correlation \(\rho\) for a class of random vectors in the family of elliptical distributions, which allows us to estimate \(\rho\) based on estimation of \(\rho_g\). The asymptotic normality of the resulting estimators of \(\rho\) are studied through two approaches: one from influence function and the other from U-statistics and the delta method. We compare asymptotic efficiencies of linear correlation estimators based on the symmetric Gini, regular Gini, Pearson and Kendall's \(\tau\) under various distributions. In addition to reasonably balancing between robustness and efficiency, the proposed measure \(\rho_g\) demonstrates superior finite sample performance, which makes it attractive in applications.
In this paper we investigate the kernel estimator of the density for a stationary reversible Markov chain. The proofs are based on a new central limit theorem for a triangular array of reversible ...Markov chains obtained under conditions imposed to covariances, which has interest in itself.
