Limiting behavior of the maximum of the partial sum for asymptotically negatively associated random variables under residual Cesàro alpha-integrability assumption

Asymptotically negative association is a special dependence structure. By relating such dependence condition to residual Cesàro alpha-integrability and to strongly residual Cesàro alpha-integrability, some L_p-convergence and complete convergence results of the maximum of the partial sum are derived, respectively. In addition, some of these conclusions are based on a new Rosenthal type inequality concerning asymptotically negatively associated random variables, which is of independent interest.     

On Vervaat and Vervaat-error-type processes for partial sums and renewals

We study the asymptotic behaviour of stochastic processes that are generated by sums of partial sums of i.i.d. random variables and their renewals. We conclude that these processes cannot converge weakly to any nondegenerate random element of the space D[0,1]$D[0,1]$. On the other hand, we show that their properly normalized integrals as Vervaat-type stochastic processes converge weakly to a squared Wiener process. Moreover, we also deal with the asymptotic behaviour of the deviations of these processes, the so-called Vervaat-error-type processes.     

Shortening the distance between Edgeworth and Berry–Esseen in the classical case

We obtain near optimal Berry–Esseen bounds for standardized sums of independent identically distributed random variables. This is achieved by distinguishing the lattice and the non-lattice cases, as one-term Edgeworth expansions do. The main tool is an easy inequality involving the usual second modulus of continuity, in substitution of Esseen's smoothing inequality. An illustrative example concerning the exponential distribution is also considered.     

Extremal limit theorems for observations separated by random power law waiting times

This paper develops extreme value theory for random observations separated by random waiting times whose exceedence probability falls off like a power law. In the case where the waiting times between observations have an infinite mean, a limit theorem is established, where the limit is comprised of an extremal process whose time index is randomized according to the non-Markovian hitting time process for a stable subordinator. The resulting limit distributions are shown to be solutions of fractional differential equations, where the order of the fractional time derivative coincides with the power law index of the waiting time. The probability that the limit process remains below a threshold is also computed. For waiting times with finite mean but infinite variance, a two-scale argument yields a fundamentally different limit process. The resulting limit is an extremal process whose time index is randomized according to the first passage time of a positively skewed stable Lévy motion with positive drift. This two-scale limit provides a second-order correction to the usual limit behavior.     

Moderate deviations for LS estimator in simple linear EV regression model

In this paper, we consider the following simple linear Errors-in-Variables (EV) regression model _ηi=θ+β_xi+_?i${\eta }_i=\theta +\beta x_i+{?}_i$, _ξi=_xi+_δi${\xi }_i=x_i+{\delta }_i$, 1?i?n$1?i?n$. The moderate deviation principle for the least squares (LS) estimators of the unknown parameters θ$\theta$, β$\beta$ in the model are obtained.     

A generalized drop-the-loser rule for multi-treatment clinical trials

Urn models are popular for response adaptive designs in clinical studies. Among different urn models, Ivanova's drop-the-loser rule is capable of producing superior adaptive treatment allocation schemes. Ivanova [2003. A play-the-winner-type urn model with reduced variability. Metrika 58, 1–13] obtained the asymptotic normality only for two treatments. Recently, Zhang et al. [2007. Generalized drop-the-loser urn for clinical trials with delayed responses. Statist. Sinica, in press] extended the drop-the-loser rule to tackle more general circumstances. However, their discussion is also limited to only two treatments. In this paper, the drop-the-loser rule is generalized to multi-treatment clinical trials, and delayed responses are allowed. Moreover, the rule can be used to target any desired pre-specified allocation proportion. Asymptotic properties, including strong consistency and asymptotic normality, are also established for general multi-treatment cases.     

On functional central limit theorems for dependent,heterogeneous arrays with applications to tail index and tail dependence estimation

We establish invariance principles for a large class of dependent, heterogeneous arrays. The theory equally covers conventional arrays, and inherently degenerate tail arrays popularly encountered in the extreme value theory literature including sample means and covariances of tail events and exceedances. For tail arrays we trim dependence assumptions down to a minimum leaving non-extremes and joint distributions unrestricted, covering geometrically ergodic, mixing, and mixingale processes, in particular linear and nonlinear distributed lags with long or short memory, linear and nonlinear GARCH, and stochastic volatility.     

Berry–Esseen type bounds for trimmed U-statistics

Trimmed U  -statistics can be constructed in two different ways: by basing the statistic on a trimmed sample or by averaging the trimmed set of kernel values. Mild conditions are given to ensure the rate of convergence to normality is O(n^-1/2)$\mathrm{O}(n^{-1/2})$ in both cases.     

Some alternative bivariate Kumaraswamy models

In this article we discuss various strategies for constructing bivariate Kumaraswamy distributions. As alternatives to the Nadarajah et al. (2011) bivariate model, four different models are introduced utilizing a conditional specification approach, a conditional survival function approach, and an Arnold–Ng bivariate beta distribution construction approach. Distributional properties for such bivariate distributions are investigated. Parameter estimation strategies for the models are discussed, as are the consequences of fitting two of the models to a particular data set involving the proportion of foggy days at two different airports in Colombia.     

Variance estimation in the central limit theorem for Markov chains

This article concerns the variance estimation in the central limit theorem for finite recurrent Markov chains. The associated variance is calculated in terms of the transition matrix of the Markov chain. We prove the equivalence of different matrix forms representing this variance. The maximum likelihood estimator for this variance is constructed and it is proved that it is strongly consistent and asymptotically normal. The main part of our analysis consists in presenting closed matrix forms for this new variance. Additionally, we prove the asymptotic equivalence between the empirical and the maximum likelihood estimation (MLE) for the stationary distribution.     

Moderate deviations for parameter estimation in some time inhomogeneous diffusions

We study moderate deviations for the maximum likelihood estimation of some inhomogeneous diffusions. The moderate deviation principle with explicit rate functions is obtained. Moreover, we apply our result to the parameter estimation in α$\alpha$-Wiener bridges.     

A note on asymptotic parametric prediction

This paper is devoted to asymptotic behaviour of plug-in statistical predictors obtained by replacing the unknown parameter in a conditional expectation by a suitable estimator. We derive the L²$L^2$-convergence rate and limit in distribution for the predictors. Applications to ARMA processes and diffusion processes are considered.     

Multivariate spatial U-quantiles: A Bahadur–Kiefer representation,a Theil–Sen estimator for multiple regression,and a robust dispersion estimator

A leading multivariate extension of the univariate quantiles is the so-called “spatial” or “geometric” notion, for which sample versions are highly robust and conveniently satisfy a Bahadur–Kiefer representation. Another extension of univariate quantiles has been to univariate U-quantiles, on the basis of which, for example, the well-known Hodges–Lehmann location estimator has a natural formulation. Generalizing both extensions, we introduce multivariate spatial U-quantiles and develop a corresponding Bahadur–Kiefer representation. New statistics based on spatial U-quantiles are presented for nonparametric estimation of multiple regression coefficients, extending the classical Theil–Sen nonparametric simple linear regression slope estimator, and for robust estimation of multivariate dispersion. Some other applications are mentioned as well.