Extreme values and kernel estimates of point processes boundaries

We present a method for estimating the edge of a two-dimensional bounded set, given a finite random set of points drawn from the interior. The estimator is based both on a Parzen-Rosenblatt kernel and extreme values of point processes. We give conditions for various kinds of convergence and asymptotic normality. We propose a method of reducing the negative bias and edge effects, illustrated by a simulation

Frontier estimation with local polynomials and high power-transformed data

We present a new method for estimating the frontier of a sample. The estimator is based on a local polynomial regression on the power-transformed data. We assume that the exponent of the transformation goes to infinity while the bandwidth goes to zero. We give conditions on these two parameters to obtain almost complete convergence. The asymptotic conditional bias and variance of the estimator are provided and its good performance is illustrated on some finite sample situations

Central limit theorems for smoothed extreme value estimates of Poisson point processes boundaries

In this paper, we give sufficient conditions to establish central limit theorems for boundary estimates of Poisson point processes. The considered estimates are obtained by smoothing some bias corrected extreme values of the point process. We show how the smoothing leads Gaussian asymptotic distributions and therefore pointwise confidence intervals. Some new unidimensional and multidimensional examples are provided

Smoothed extreme value estimators of non-uniform point processes boundaries with application to star-shaped supports estimation

We address the problem of estimating the edge of a bounded set in R^d given a random set of points drawn from the interior. Our method is based on a transformation of estimators dedicated to uniform point processes and obtained by smoothing some of its bias corrected extreme points. An application to the estimation of star-shaped supports is presented

Estimation procedures for a semiparametric family of bivariate copulas

In this paper, we propose simple estimation methods dedicated to a semiparametric family of bivariate copulas. These copulas can be simply estimated through the estimation of their univariate generating function. We take profit of this result to estimate the associated measures of association as well as the high probability regions of the copula. These procedures are illustrated on simulations and on real data

Symmetry and dependence properties within a semiparametric family of bivariate copulas

In this paper, we study a semiparametric family of bivariate copulas. The family is generated by an univariate function, determining the symmetry (radial symmetry, joint symmetry) and dependence property (quadrant dependence, total positivity, ...) of the copulas. We provide bounds on different measures of association (such as Kendall's Tau, Spearman's Rho) for this family and several choices of generating functions allowing to reach these bounds

Projection estimates of point processes boundaries

We present a method for estimating the edge of a two-dimensional bounded set, given a finite random set of points drawn from the interior. The estimator is based both on projections on C^1 bases and on extreme points of the point process. We give conditions on the Dirichlet's kernel associated to the C^1 bases for various kinds of convergence and asymptotic normality. We propose a method for reducing the negative bias and illustrate it by a simulation

Estimation of the Weibull tail-coefficient with linear combination of upper order statistics

We present a new family of estimators of the Weibull tail-coefficient. The Weibull tail-coefficient is defined as the regular variation coefficient of the inverse failure rate function. Our estimators are based on a linear combination of log-spacings of the upper order statistics. Their asymptotic normality is established and illustrated for two particular cases of estimators in this family. Their finite sample performances are presented on a simulation study

Auto-associative models, nonlinear Principal component analysis, manifolds and projection pursuit

In this paper, auto-associative models are proposed as candidates to the generalization of Principal Component Analysis. We show that these models are dedicated to the approximation of the dataset by a manifold. Here, the word "manifold" refers to the topology properties of the structure. The approximating manifold is built by a projection pursuit algorithm. At each step of the algorithm, the dimension of the manifold is incremented. Some theoretical properties are provided. In particular, we can show that, at each step of the algorithm, the mean residuals norm is not increased. Moreover, it is also established that the algorithm converges in a finite number of steps. Some particular auto-associative models are exhibited and compared to the classical PCA and some neural networks models. Implementation aspects are discussed. We show that, in numerous cases, no optimization procedure is required. Some illustrations on simulated and real data are presented

A note on extreme values and kernel estimators of sample boundaries

In a previous paper, we studied a kernel estimate of the upper edge of a two-dimensional bounded set, based upon the extreme values of a Poisson point process. The initial paper "Geffroy J. (1964) Sur un probl\`eme d'estimation g\'eom\'etrique.Publications de l'Institut de Statistique de l'Universit\'e de Paris, XIII, 191-200" on the subject treats the frontier as the boundary of the support set for a density and the points as a random sample. We claimed in"Girard, S. and Jacob, P. (2004) Extreme values and kernel estimates of point processes boundaries.ESAIM: Probability and Statistics, 8, 150-168" that we are able to deduce the random sample case fr om the point process case. The present note gives some essential indications to this end, including a method which can be of general interest