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.
In this paper, we study the self-normalized Cramér-type moderate deviations for centered independent random variables X1,X2,... with 0 <E | Xi | 3< ∞. The main results refine Theorems 1.1 and 1.2 of ...Wang Q. Wang, J. Theoret. Probab. 24 (2011) 307–329, the Berry−Esseen bound (2.11) and Corollaries 2.2 and 2.3 of Jing, et al. B.Y. Jing, Q.M. Shao and Q. Wang, Ann. Probab. 31 (2003) 2167–2215 under stronger moment conditions.
The generative adversarial network (GAN) is an important model developed for
high-dimensional distribution learning in recent years. However, there is a
pressing need for a comprehensive method to ...understand its error convergence
rate. In this research, we focus on studying the error convergence rate of the
GAN model that is based on a class of functions encompassing the discriminator
and generator neural networks. These functions are VC type with bounded
envelope function under our assumptions, enabling the application of the
Talagrand inequality. By employing the Talagrand inequality and Borel-Cantelli
lemma, we establish a tight convergence rate for the error of GAN. This method
can also be applied on existing error estimations of GAN and yields improved
convergence rates. In particular, the error defined with the neural network
distance is a special case error in our definition.
Large and moderate deviation probabilities play an important role in many applied areas, such as insurance and risk analysis. This paper studies the exact moderate, and large deviation asymptotics in ...non-logarithmic form for linear processes with independent innovations. The linear processes we analyze are general and they include the long memory case. We give an asymptotic representation for the probability of the tail of the normalized sums and specify the zones in which it can be approximated either by a standard normal distribution or by the marginal distribution of the innovation process. The results are then applied to regression estimates, moving averages, fractionally integrated processes, linear processes with regularly varying exponents, and functions of linear processes. We also consider the computation of value at risk and expected shortfall, fundamental quantities in risk theory and finance.
We consider regression estimation with modified ReLU neural networks in which network weight matrices are first modified by a function \(\alpha\) before being multiplied by input vectors. We give an ...example of continuous, piecewise linear function \(\alpha\) for which the empirical risk minimizers over the classes of modified ReLU networks with \(l_1\) and squared \(l_2\) penalties attain, up to a logarithmic factor, the minimax rate of prediction of unknown \(\beta\)-smooth function.
It is shown that the Hall, Hu and Marron Hall, P., Hu, T., and Marron J.S. (1995), 'Improved Variable Window Kernel Estimates of Probability Densities', Annals of Statistics, 23, 1-10 modification ...of Abramson's Abramson, I. (1982), 'On Bandwidth Variation in Kernel Estimates - A Square-root Law', Annals of Statistics, 10, 1217-1223 variable bandwidth kernel density estimator satisfies the optimal asymptotic properties for estimating densities with four uniformly continuous derivatives, uniformly on bounded sets where the preliminary estimator of the density is bounded away from zero.
The United States Department of Agriculture’s National Agricultural Statistics Service (NASS) conducts the June Agricultural Survey (JAS) annually. Substantial misclassification occurs during the ...prescreening process and from field-estimating farm status for nonresponse and inaccessible records, resulting in a biased estimate of the number of US farms from the JAS. Here, the Annual Land Utilization Survey (ALUS) is proposed as a follow-on survey to the JAS to adjust the estimates of the number of US farms and other important variables. A three-phase survey design-based estimator is developed for the JAS-ALUS with nonresponse adjustment for the second phase (ALUS). A design-unbiased estimator of the variance is provided in explicit form.
We establish the asymptotic theory of least absolute deviation estimators for AR(1) processes with autoregressive parameter satisfying \(n(\rho_n-1)\to\gamma\) for some fixed \(\gamma\) as ...\(n\to\infty\), which is parallel to the results of ordinary least squares estimators developed by Andrews and Guggenberger (2008) in the case \(\gamma=0\) or Chan and Wei (1987) and Phillips (1987) in the case \(\gamma\ne 0\). Simulation experiments are conducted to confirm the theoretical results and to demonstrate the robustness of the least absolute deviation estimation.
Let \(\{X_n: n\in \N\}\) be a linear process with density function \(f(x)\in L^2(\R)\). We study wavelet density estimation of \(f(x)\). Under some regular conditions on the characteristic function ...of innovations, we achieve, based on the number of nonzero coefficients in the linear process, the minimax optimal convergence rate of the integrated mean squared error of density estimation. Considered wavelets have compact support and are twice continuously differentiable. The number of vanishing moments of mother wavelet is proportional to the number of nonzero coefficients in the linear process and to the rate of decay of characteristic function of innovations. Theoretical results are illustrated by simulation studies with innovations following Gaussian, Cauchy and chi-squared distributions.
In this paper we study the ideal variable bandwidth kernel density estimator introduced by McKay (1993) and Jones, McKay and Hu (1994) and the plug-in practical version of the variable bandwidth ...kernel estimator with two sequences of bandwidths as in Giné and Sang (2013). Based on the bias and variance analysis of the ideal and true variable bandwidth kernel density estimators, we study the central limit theorems for each of them.
PDF
