In this paper we, firstly, present a recursive formula of the empirical estimator of the semi-Markov kernel. Then a non-parametric estimator of the expected cumulative operational time for semi-Markov systems is proposed. The asymptotic properties of this estimator, as the uniform strongly consistency and normality are given. As an illustration example, we give a numerical application.
The least absolute shrinkage and selection operator (LASSO) is a popular technique for simultaneous estimation and model selection. There have been a lot of studies on the large sample asymptotic distributional properties of the LASSO estimator, but it is also well-known that the asymptotic results can give a wrong picture of the LASSO estimator's actual finite-sample behaviour. The finite sample distribution of the LASSO estimator has been previously studied for the special case of orthogonal models....
Let be the mode of a probability density and its kernel estimator. In the case is nondegenerate, we first specify the weak convergence rate of the multivariate kernel mode estimator by stating the central limit theorem for . Then, we obtain a multivariate law of the iterated logarithm for the kernel mode estimator by proving that, with probability one, the limit set of the sequence suitably normalized is an ellipsoid. We also give a law of the iterated logarithm for the norms, , of ....
Let θ be the mode of a probability density and θn its kernel estimator. In the case θ is nondegenerate, we first specify the weak convergence rate of the multivariate kernel mode estimator by stating the central limit theorem for θn - θ. Then, we obtain a multivariate law of the iterated logarithm for the kernel mode estimator by proving that, with probability one, the limit set of the sequence θn - θ suitably normalized is an ellipsoid. We also give a law of the iterated logarithm for the...
Output least squares stability for the diffusion coefficient in an elliptic equation in dimension two is analyzed. This guarantees Lipschitz stability of the solution of the least squares formulation with respect to perturbations in the data independently of their attainability. The analysis shows the influence of the flow direction on the parameter to be estimated. A scale analysis for multi-scale resolution of the unknown parameter is provided.
Output least squares stability for the diffusion coefficient in an elliptic equation in dimension two is analyzed. This guarantees Lipschitz stability of the solution of the least squares formulation with respect to perturbations in the data independently of their attainability. The analysis shows the influence of the flow direction on the parameter to be estimated. A scale analysis for multi-scale resolution of the unknown parameter is provided.
We deal with a subject in the interplay between nonparametric statistics and geometric measure theory. The measure L0(G) of the boundary of a set G ⊂ ℝd (with d ≥ 2) can be formally defined, via a simple limit, by the so-called Minkowski content. We study the estimation of L0(G) from a sample of random points inside and outside G. The sample design assumes that, for each sample point, we know (without error) whether or not that point belongs to G. Under this design we suggest a simple nonparametric...
We consider a model selection estimator of the covariance of a random process. Using the Unbiased Risk Estimation (U.R.E.) method, we build an estimator of the risk which allows to select an estimator in a collection of models. Then, we present an oracle inequality which ensures that the risk of the selected estimator is close to the risk of the oracle. Simulations show the efficiency of this methodology.
Usually the problem of drift estimation for a diffusion process is considered under the hypothesis of ergodicity. It is less often considered under the hypothesis of null-recurrence, simply because there are fewer limit theorems and existing ones do not apply to the whole null-recurrent class. The aim of this paper is to provide some limit theorems for additive functionals and martingales of a general (ergodic or null) recurrent diffusion which would allow us to have a somewhat unified approach...
We consider the high order moments estimator of the frontier of a random pair, introduced by [S. Girard, A. Guillou and G. Stupfler, J. Multivariate Anal. 116 (2013) 172–189]. In the present paper, we show that this estimator is strongly uniformly consistent on compact sets and its rate of convergence is given when the conditional cumulative distribution function belongs to the Hall class of distribution functions.
In many practical situations sample sizes are not sufficiently large and estimators based on such samples may not be satisfactory in terms of their variances. At the same time it is not unusual that some auxiliary information about the parameters of interest is available. This paper considers a method of using auxiliary information for improving properties of the estimators based on a current sample only. In particular, it is assumed that the information is available as a number of estimates based...
In many practical situations sample sizes are not sufficiently large and estimators based on such samples may not be satisfactory in terms of their variances. At the same time it is not unusual that some auxiliary information about the parameters of interest is available. This paper considers a method of using auxiliary information for improving properties of the estimators based on a current sample only. In particular, it is assumed that the information is available as a number of estimates based...