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We study a certain class circle maps which are constant on one interval (called flat piece), and such that the degrees of the singularities at the boundary of the flat piece are different. In this paper, we show that if the topological conjugacy between two maps of my class is a bi-Lipschitz homeomorphism, then it is a $C^1$ diffeomorphism; that is, the bi-Lipschitz homeomorphism class and $C^1$ diffeomorphism class of a map in our class are equivalent., Comment: 9 pages

In this paper, we propose an estimator of the generalized maximum mean discrepancy between several distributions, constructed by modifying a naive estimator. Asymptotic normality is obtained for this estimator both under equality of these distributions and under the alternative hypothesis.

We consider the smoothed version of sliced average variance estimation (SAVE) dimension reduction method for dealing with spatially dependent data that are observations of a strongly mixing random field. We propose kernel estimators for the interest matrix and the effective dimension reduction (EDR) space, and show their consistency.

In this paper we deal with the problem of testing for the equality of $k$ probability distributions defined on $(\mathcal{X},\mathcal{B})$, where $\mathcal{X}$ is a metric space and $\mathcal{B}$ is the corresponding Borel $\sigma$-field. We introduce a test statistic based on reproducing kernel Hilbert space embeddings and derive its asymptotic distribution under the null hypothesis. Simulations show that the introduced procedure outperforms known methods., Comment: 30 pages, 4 figures

In this paper we deal with the problem of testing for the quality of $k$ probability distributions. We introduce a generalization of the maximum mean discrepancy that permits to characterize the null hypothesis. Then, an estimator of it is proposed as test statistic, and its asymptotic distribution under the null hypothesis is derived. Simulations show that the introduced procedure outperforms classical ones., Comment: 20 pages, 4 figures

We propose an estimation method that we call functional average variance estimation (FAVE), for estimating the EDR space in functional semiparametric regression model, based on kernel estimates of density and regression. Consistency results are then established for the estimator of the interest operator, and for the directions of EDR space. A simulation study that shows that the proposed approach performs as well as traditional ones is presented., Comment: 35 pages,14 figures

Recently in [1, 2], Ali-Akbar Bromideh introduced the Kullback-Leibler Divergence (KLD) test statistic in discrim- inating between two models. It was found that the Ratio Minimized Kulback-Leibler Divergence (RMKLD) works better than the Ratio of Maximized Likelihood (RML) for small sample size. The aim of this paper is to generalize the works of Ali-Akbar Bromideh by proposing a hypothesis testing based on Bregman divergence in order to improve the process of choice of the model. Our aproach differs from him. After observing n data points of unknown density f ; we firstly measure the closness between the bias reduced kernel density estimator and the first estimated candidate model. Secondly between the bias reduced kernel density estimator and the second estimated candidate model. In these two cases Bregman Divergence (BD) and the bias reduced kernel estimator [3] focuses on improving the con- vergence rates of kernel density estimators are used. Our testing procedure for model selection is thus based on the comparison of the value of model selection test statistic to critical values from a standard normal table. We establish the asymptotic properties of Bregman divergence estimator and approximations of the power functions are deduced. The multi-step MLE process will be used to estimate the parameters of the models. We explain the applicability of the BD by a real data set and by the data generating process (DGP). The Monte Carlo simulation and then the numerical analysis will be used to interpret the result.

Let (M,g) be a compact Riemannian manifold of hyperbolic type, i.e M is a manifold admitting another metric of strictly negative curvature. In this paper we study the geodesic flow restricted to the set of geodesics which are minimal on the universal covering. In particular for surfaces we show that the topological entropy of the minimal geodesics coincides with the volume entropy of (M, g) generalizing work of Freire and Mane., Comment: 16 pages