For nonparametric estimation of a smooth regression function, local linear fitting is a widely-used method. The goal of this paper is to briefly review how to use this method when the unknown curve possibly has some irregularities, such as jumps or peaks, at unknown locations. It is then explained how the same basic method can be used when estimating unsmooth probability densities and conditional variance functions.
An adaptive estimator (of a slope parameter) based on rank statistics is constructed and its asymptotic optimality is studied. A complete orthonormal system is incorporated in the adaptive determination of the score generating function. The proposed sequential procedure is based on a suitable stopping rule. Various properties of the sequential adaptive procedure and the stopping rule are studied. Asymptotic linearity results of linear rank statistics are also studied and some rates of the convergence...
After recalling previous work on probability generating functions for real valued random variables we extend to these random variables uniform laws of large numbers and functional limit theorem for the empirical probability generating function. We present an application to the study of continuous laws, namely, estimation of parameters of Gaussian, gamma and uniform laws by means of a minimum contrast estimator that uses the empirical probability generating function of the sample. We test the procedure...
We discuss the prediction of a spatial variable of a multivariate mark composed of both dependent and explanatory variables. The marks are location-dependent and they are attached to a point process. We assume that the marks are assigned independently, conditionally on an unknown underlying parametric field. We compare (i) the classical non-parametric Nadaraya-Watson kernel estimator based on the dependent variable (ii) estimators obtained under an assumption of local parametric model where explanatory...
Given a random sample from some unknown density we devise Haar wavelet estimators for with variable resolution levels constructed from localised test procedures (as in Lepski, Mammen and Spokoiny (Ann. Statist.25(1997) 927–947)). We show that these estimators satisfy an oracle inequality that adapts to heterogeneous smoothness of , simultaneously for every point in a fixed interval, in sup-norm loss. The thresholding constants involved in the test procedures can be chosen in practice under...
This article presents a new concept for a statistical fault detection system, including the detection, diagnosis, and prediction of faults. Theoretical material has been collected to provide a complete algorithm making possible the design of a usable system for statistical inference on the basis of the current value of a symptom vector. The use of elements of artificial intelligence enables self-correction and adaptation to changing conditions. The mathematical apparatus is founded on the methodology...
The aim of this paper is to establish a nonparametric estimate of some characteristics of the conditional distribution. Kernel type estimators for the conditional cumulative distribution function and for the successive derivatives of the conditional density of a scalar response variable Y given a Hilbertian random variable X are introduced when the observations are linked with a single-index structure. We establish the pointwise almost complete convergence and the uniform almost complete convergence...
En este trabajo consideramos estimaciones no paramétricas de las funciones de razón de fallo y supervivencia en fiabilidad haciendo uso de suavizaciones no paramétricas de la función de distribución empírica (datos no censurados) y de la distribución de Kaplan-Meier (datos censurados). Se obtienen sesgos, varianzas y distribuciones asintóticas de los estimadores aquí propuestos probándose mediante técnicas de expansiones de segundo orden la eficiencia de éstos respecto de otras estimaciones introducidas...
Se estudian modificaciones de las técnicas de validación cruzada de Kullback-Leibler y mínimos cuadrados para obtener el parámetro de suavización asociado a un estimador general no paramétrico de la función de densidad, a partir de la muestra, en el supuesto de que los datos verifican alguna condición débil de dependencia.Se demuestra que los parámetros obtenidos por estas dos técnicas son asintóticamente óptimos. Y se realiza un estudio de simulación.
Finite mixture modelling of class-conditional distributions is a standard method in a statistical pattern recognition. This paper, using bag-of-words vector document representation, explores the use of the mixture of multinomial distributions as a model for class-conditional distribution for multiclass text document classification task. Experimental comparison of the proposed model and the standard Bernoulli and multinomial models as well as the model based on mixture of multivariate Bernoulli distributions...
This paper concerns the emergence of modern mathematical statistics in France after the First World War. Emile Borel’s achievements are presented, and especially his creation of two institutions where mathematical statistics was developed: the Statistical Institute of Paris University, (ISUP) in 1922 and above all the Henri Poincaré Institute (IHP) in 1928. At the IHP, a new journal Annales de l’Institut Henri Poincaré was created in 1931. We discuss the first papers in that journal dealing with...
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....