The purpose of this paper is to study the asymptotic behaviour of the recursive kernel density estimator. This estimator was introduced and investigated by Amiri (2010) for independent and α -mixing sequences. In this work, we are interested in η -weak dependence, which is different from the notion of α -mixing. We provide the variance and the mean squared error of this estimator. The asymptotic normality is also discussed. A simulation study for two η -dependent models which are not necessarily α -mixing shows the advantage in time computation of considering the recursive kernel estimation rather than the Parzen–Rosenblatt one.

Recursive kernel estimation of the density under η-weak dependence

In this article, we investigate the asymptotic normality of the Hill's estimator of the tail index parameter, when the observations are weakly dependent in the sense of Doukhan and Louhichi (1999) and are drawn from a strictly linear process. We show that the previous result on Hill estimator obtained by Rootzen et al. (1990) and Resnick and Starica (1997) for strong mixing can be extended to weak dependence.

Hill's estimator under weak dependence

Accurate estimation for extent of cross{sectional dependence in large panel data analysis is paramount to further statistical analysis on the data under study. Grouping more data with weak relations (cross{sectional dependence) together often results in less efficient dimension reduction and worse forecasting. This paper describes cross-sectional dependence among a large number of objects (time series) via a factor model and parameterizes its extent in terms of strength of factor loadings. A new joint estimation method, benefiting from unique feature of dimension reduction for high dimensional time series, is proposed for the parameter representing the extent and some other parameters involved in the estimation procedure. Moreover, a joint asymptotic distribution for a pair of estimators is established. Simulations illustrate the effectiveness of the proposed estimation method in the finite sample performance. Applications in cross-country macro-variables and stock returns from S&P 500 are studied.

Estimation of Cross-Sectional Dependence in Large Panels

Accurate estimation for extent of cross-sectional dependence in large panel data analysis is paramount to further statistical analysis on the data under study. Grouping more data with weak relations (cross--sectional dependence) together often results in less efficient dimension reduction and worse forecasting. This paper describes cross-sectional dependence among a large number of objects (time series) via a factor model and parametrizes its extent in terms of strength of factor loadings. A new joint estimation method is proposed for the parameter representing such extent and some other parameters involved in the estimation procedure. In particular, a marginal estimator is also proposed as the related other parameters are observed. Asymptotic distributions for the joint estimator and marginal estimator are both established. Various simulation designs illustrate the effectiveness of the proposed estimation method in the finite sample performance. Applications in cross-country macro-variables and stock returns from S&P 500 are studied.

Extent Pursuit for Cross-Sectional Dependence in Large Panels

Identifying a model by the penalized contrast procedure, we give an analytical estimation of misfitting subsets in the specific case of a least squares contrast. Then, specifying the statistical model, this allows to determine penalization rates ensuring a consistent identification. Applications are given to time series and geostatistical identification.

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Misparametrization subsets for penalized least squares model selection

We prove a general functional central limit theorem for weak dependent time series. A very large variety of models, for instance, causal or non causal linear, ARCH(1), bilinear, Volterra processes, satisfies this theorem. Moreover, it provides numerous application as well for bounding the distance between the empirical mean and the Gaussian measure than for obtaining central limit theorem for sample moments and cumulants.

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C. Jose Rafael Leon

Papers

A functional limit theorem for η-weakly dependent processes and its applications