Dependent Lindeberg central limit theorem for the fidis of empirical processes of cluster functionals

Drees H. and Rootzén H. [Limit theorems for empirical processes of cluster functionals (EPCF). Ann Stat. 2010;38(4):2145–2186] have proven central limit theorems (CLTs) for EPCF built from β-mixing processes. However, this family of β-mixing processes is quite restrictive. We expand some of those results, for the finite-dimensional marginal distributions (fidis), to a more general dependent processes family, known as weakly dependent processes in the sense of Doukhan P. and Louhichi S. [A new weak dependence condition and applications to moment inequalities. Stoch. Proc. Appl. 1999;84:313–342]. In this context, the CLT for the fidis of EPCF is sufficient in some applications. For instance, we prove the convergence without mixing conditions of the extremogram estimator, including a small example with simulation of the extremogram of a weakly dependent random process but nonmixing, in order to confirm the efficacy of our result.     

A Euclidean Likelihood Estimator for Bivariate Tail Dependence

The spectral measure plays a key role in the statistical modeling of multivariate extremes. Estimation of the spectral measure is a complex issue, given the need to obey a certain moment condition. We propose a Euclidean likelihood-based estimator for the spectral measure which is simple and explicitly defined, with its expression being free of Lagrange multipliers. Our estimator is shown to have the same limit distribution as the maximum empirical likelihood estimator of Einmahl and Segers (2009 Einmahl , J. H. J. , Segers , J. ( 2009 ). Maximum empirical likelihood estimation of the spectral measure of an extreme-value distribution . Ann. Statist. 37 ( 5B ): 2953 – 2989 .[Crossref], [Web of Science ®] , [Google Scholar]). Numerical experiments suggest an overall good performance and identical behavior to the maximum empirical likelihood estimator. We illustrate the method in an extreme temperature data analysis.     

On integral functionals of a density

ABSTRACT

Estimation of a non linear integral functional of probability distribution density and its derivatives is studied. The truncated plug-in-estimator is taken for the estimation. The integrand function can be unlimited, but it cannot exceed polynomial growth. Consistency of the estimator is proved and the convergence order is established. Aversion of the central limit theorem is proved. As an example an extended Fisher information integral and generalized Shannon's entropy functional are considered.     

Kernel Estimators for Distribution Functions on Dependent Random Fields

Consider observations (representing lifelengths) taken on a random field indexed by lattice points. Estimating the distribution function F(x) = P(X _i  ≤ x) is an important problem in survival analysis. We propose to estimate F(x) by kernel estimators, which take into account the smoothness of the distribution function. Under some general mixing conditions, our estimators are shown to be asymptotically unbiased and consistent. In addition, the proposed estimator is shown to be strongly consistent and sharp rates of convergence are obtained.     

Asymptotics for a class of generalized multicast autoregressive processes

We propose a new class of generalized multicast autoregressive (GMCAR, for short, hereafter) models indexed by a multi-casting tree where each individual produces exactly the same number of offspring. This class includes standard bifurcating autoregressive processes (BAR, cf. Cowan and Staudte (1986)) and multicast autoregressive (MCAR, cf. Hwang and Choi (2009)) models as special cases. Accommodating non-Gaussian, non-negative and count data, the class includes various models such as nonlinear autoregression, conditionally heteroscedastic process and conditional exponential family. The pathwise stationarity of the GMCAR model is discussed. A law of large numbers and a central limit theorem are established which are in turn used to derive asymptotic distributions associated with martingale estimating functions.     

Wavelet-based estimation of regression function with strong mixing errors under fixed design

We consider wavelet-based non linear estimators, which are constructed by using the thresholding of the empirical wavelet coefficients, for the mean regression functions with strong mixing errors and investigate their asymptotic rates of convergence. We show that these estimators achieve nearly optimal convergence rates within a logarithmic term over a large range of Besov function classes B^s_{p, q}. The theory is illustrated with some numerical examples.

A new ingredient in our development is a Bernstein-type exponential inequality, for a sequence of random variables with certain mixing structure and are not necessarily bounded or sub-Gaussian. This moderate deviation inequality may be of independent interest.     

Asymptotics for Tail Probability of Random Sums with a Heavy-Tailed Number and Dependent Increments

This article obtains the asymptotics for the tail probability of random sums, where the random number and the increments are all heavy tailed, and the increments follow a certain wide dependence structure. This dependence structure can contain some commonly used negatively dependent random variables as well as some positively dependent random variables.     

Consistency of minimizing a penalized density power divergence estimator for mixing distribution

In this paper, we study the MDPDE (minimizing a density power divergence estimator), proposed by Basu et al. (Biometrika 85:549–559, 1998), for mixing distributions whose component densities are members of some known parametric family. As with the ordinary MDPDE, we also consider a penalized version of the estimator, and show that they are consistent in the sense of weak convergence. A simulation result is provided to illustrate the robustness. Finally, we apply the penalized method to analyzing the red blood cell SLC data presented in Roeder (J Am Stat Assoc 89:487–495, 1994). This research was supported (in part) by KOSEF through Statistical Research Center for Complex Systems at Seoul National University.     

Asymptotics for multisample statistics

Consider a family of square-integrable R^d-valued statistics S_k = S_k(X_1,k1; X_2,k2;…; X_m,km), where the independent samples X_i,kj respectively have k_i i.i.d. components valued in some separable metric space X_i. We prove a strong law of large numbers, a central limit theorem and a law of the iterated logarithm for the sequence {S_k}, including both the situations where the sample sizes tend to infinity while m is fixed and those where the sample sizes remain small while m tends to infinity. We also obtain two almost sure convergence results in both these contexts, under the additional assumption that S_k is symmetric in the coordinates of each sample X_i,kj. Some extensions to row-exchangeable and conditionally independent observations are provided. Applications to an estimator of the dimension of a data set and to the Henze-Schilling test statistic for equality of two densities are also presented.     

The Uniform Asymptotics of the Overshoot of a Random Walk with Light-Tailed Increments

Let {S _n : n ≥ 0} be a random walk with light-tailed increments and negative drift, and let τ(x) be the first time when the random walk crosses a given level x ≥ 0. Tang (2007 Tang , Q. ( 2007 ). The overshoot of a random walk with negative drift . Statist. Probab. Lett. 77 : 158 – 165 .[Crossref], [Web of Science ®] , [Google Scholar]) obtained the asymptotics of P(S _τ(x) ? x > y, τ(x) < ∞) as x → ∞, which is uniform for y ≥ f(x) for any positive function f(x) → ∞ as x → ∞. In this article, the uniform asymptotics of P(S _τ(x) ? x > y, τ(x) < ∞) as x → ∞, for 0 ≤ y ≤ N for any positive number N will be given. Using the above two results, the uniform asymptotics of P(S _τ(x) ? x > y, τ(x) < ∞) as x → ∞, for y ≥ 0, is presented.     

Estimation of entropy-type integral functionals

ABSTRACT

Entropy-type integral functionals of densities are widely used in mathematical statistics, information theory, and computer science. Examples include measures of closeness between distributions (e.g., density power divergence) and uncertainty characteristics for a random variable (e.g., Rényi entropy). In this paper, we study U-statistic estimators for a class of such functionals. The estimators are based on ε-close vector observations in the corresponding independent and identically distributed samples. We prove asymptotic properties of the estimators (consistency and asymptotic normality) under mild integrability and smoothness conditions for the densities. The results can be applied in diverse problems in mathematical statistics and computer science (e.g., distribution identification problems, approximate matching for random databases, two-sample problems).     

Complete convergence and complete moment convergence for arrays of rowwise negatively superadditive dependent random variables

In this paper, some complete convergence and complete moment convergence results for arrays of rowwise negatively superadditive dependent (NSD, in short) random variables are studied. The obtained theorems not only extend the result of Gan and Chen (2007 Gan, S. X., and P. Y. Chen. 2007. On the limiting behavior of the maximum partial sums for arrays of rowwise NA random variables. Acta Mathematica Scientia. Series B 27 (2):283–90.[Crossref], [Web of Science ®] , [Google Scholar]) to the case of NSD random variables, but also improve them.     

About estimation of ARIMA process with strong mixing MA part

A central limit theorem is provided for the least squares estimates of the autoregressive parameters in an ARIMA process with strong mixing moving average part.     

On the determination of mixing density in reliability studies

In reliability studies the three quantities (1) the survival function, (2) the failure rate and (3) the mean residual life function are all equivalent in the sense that given one of them, the other two can be determined. In this paper we have considered the class of exponential type distributions and studied its mixture. Given any one of the above mentioned three quantities of the mixture a method is developed for determining the mixing density. Some examples are provided as illustrations. Some well known results follow trivially.     

Convergence of non-linear functionals of smoothed empirical processes and kernel density estimates

We consider regularizations by convolution of the empirical process and study the asymptotic behaviour of non-linear functionals of this process. Using a result for the same type of non-linear functionals of the Brownian bridge, shown in a previous paper [4], and a strong approximation theorem, we prove several results for the p-deviation in estimation of the derivatives of the density. We also study the asymptotic behaviour of the number of crossings of the smoothed empirical process defined by Yukich [17] and of a modified version of the Kullback deviation.