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## Recovering the shape of a point cloud in the plane

### TEST (2013-03-01) 22: 19-45 , March 01, 2013

In this work we deal with the problem of estimating the support *S* of a probability distribution under shape restrictions. The shape restriction we deal with is an extension of the notion of convexity named *α*-convexity. Instead of assuming, as in the convex case, the existence of a separating hyperplane for each exterior point of *S*, we assume the existence of a separating open ball with radius *α*. Given an *α*-convex set *S*, the *α*-convex hull of independent random points in *S* is the natural estimator of the set. If *α* is unknown the *r*_{n}-convex hull of the sample can be considered being *r*_{n} a sequence of positive numbers. We analyze the asymptotic properties of the *r*_{n}-convex hull estimator in the bidimensional case and obtain the convergence rate for the expected distance in measure between the set and the estimator. The geometrical complexity of the estimator and its dependence on *r*_{n} are also obtained via the analysis of the expected number of vertices of the *r*_{n}-convex hull.

## Maxiset in sup-norm for kernel estimators

### TEST (2008-04-15) 18: 475-496 , April 15, 2008

In the Gaussian white noise model, we study the estimation of an unknown multidimensional function *f* in the uniform norm by using kernel methods. We determine the sets of functions that are well estimated at the rates (log *n*/*n*)^{β/(2β+d)} and *n*^{−β/(2β+d)} by kernel estimators. These sets are called maxisets. Then, we characterize the maxisets associated to kernel estimators and to the Lepski procedure for the rate of convergence (log *n*/*n*)^{β/(2β+d)} in terms of Besov and Hölder spaces of regularity *β*. Using maxiset results, optimal choices for the bandwidth parameter of kernel rules are derived. Performances of these rules are studied from the numerical point of view.

## Weighted Moment Estimators for the Second Order Scale Parameter

### Methodology and Computing in Applied Probability (2012-09-01) 14: 753-783 , September 01, 2012

We consider the estimation of the scale parameter appearing in the second order condition when the distribution underlying the data is of Pareto-type. Inspired by the work of Goegebeur et al. (J Stat Plan Inference 140:2632–2652, 2010) on the estimation of the second order rate parameter, we introduce a flexible class of estimators for the second order scale parameter, which has weighted sums of scaled log spacings of successive order statistics as basic building blocks. Under the second order condition, some conditions on the weight functions, and for appropriately chosen sequences of intermediate order statistics, we establish the consistency of our class of estimators. Asymptotic normality is achieved under a further condition on the tail function 1 − *F*, the so-called third order condition. As the proposed estimator depends on the second order rate parameter, we also examine the effect of replacing the latter by a consistent estimator. The asymptotic performance of some specific examples of our proposed class of estimators is illustrated numerically, and their finite sample behavior is examined by a small simulation experiment.

## Bias-corrected and robust estimation of the bivariate stable tail dependence function

### TEST (2016-11-19): 1-24 , November 19, 2016

The stable tail dependence function gives a full characterisation of the extremal dependence between two or more random variables. In this paper, we propose an estimator for this function which is robust against outliers in the sample. The estimator is derived from a bivariate second-order tail model together with a proper transformation of the bivariate observations, and its asymptotic properties are studied under some suitable regularity conditions. Our estimation procedure depends on two parameters: $$\alpha $$ , which controls the trade-off between efficiency and robustness of the estimator, and a second-order parameter $$\tau $$ , which can be replaced by a fixed value or by an estimate. In case where $$\tau $$ has been replaced by the true value or by an external consistent estimator, our robust estimator is asymptotically unbiased, whereas in case where $$\tau $$ is mis-specified, one loses this property, but still our estimator performs quite well with respect to bias. The finite sample performance of our robust and bias-corrected estimator of the stable tail dependence function is examined on a simulation study involving uncontaminated and contaminated samples. In particular, its behavior is illustrated for different values of the pair $$(\alpha , \tau )$$ and is compared with alternative estimators from the extreme value literature.

## U-tests for variance components in linear mixed models

### TEST (2013-11-01) 22: 580-605 , November 01, 2013

We propose a *U*-statistics-based test for null variance components in linear mixed models and obtain its asymptotic distribution (for increasing number of units) under mild regularity conditions that include only the existence of the second moment for the random effects and of the fourth moment for the conditional errors. We employ contiguity arguments to derive the distribution of the test under local alternatives assuming additionally the existence of the fourth moment of the random effects. Our proposal is easy to implement and may be applied to a wide class of linear mixed models. We also consider a simulation study to evaluate the behaviour of the *U*-test in small and moderate samples and compare its performance with that of exact *F*-tests and of generalized likelihood ratio tests obtained under the assumption of normality. A practical example in which the normality assumption is not reasonable is included as illustration.

## Kernel type smoothed quantile estimation under long memory

### Statistical Papers (2008-01-05) 51: 57-67 , January 05, 2008

This paper studies nonparametric kernel type (smoothed) estimation of quantiles for long memory stationary sequences. The uniform strong consistency and asymptotic normality of the estimates with rates are established. Finite sample behaviors are investigated in a small Monte Carlo simulation study.

## Recent and classical tests for exponentiality: a partial review with comparisons

### Metrika (2005-02-01) 61: 29-45 , February 01, 2005

### Abstract.

A wide selection of classical and recent tests for exponentiality are discussed and compared. The classical procedures include the statistics of Kolmogorov-Smirnov and Cramér-von Mises, a statistic based on spacings, and a method involving the score function. Among the most recent approaches emphasized are methods based on the empirical Laplace transform and the empirical characteristic function, a method based on entropy as well as tests of the Kolmogorov-Smirnov and Cramér-von Mises type that utilize a characterization of exponentiality via the mean residual life function. We also propose a new goodness-of-fit test utilizing a novel characterization of the exponential distribution through its characteristic function. The finite-sample performance of the tests is investigated in an extensive simulation study.

## Strong approximations for dependent competing risks with independent censoring

### TEST (2009-05-01) 18: 76-95 , May 01, 2009

We deal with the problem of dependent competing risks in presence of independent right-censoring. The Aalen–Johansen estimator for the cause-specific subdistribution functions is considered. We obtain strong approximations by Gaussian processes which are valid up to a certain order statistic of the observations. We derive two LIL-type results and asymptotic confidence bands.

## Asymptotic efficiency of new exponentiality tests based on a characterization

### Metrika (2016-02-01) 79: 221-236 , February 01, 2016

Two new tests for exponentiality, of integral- and Kolmogorov-type, are proposed. They are based on a recent characterization and formed using appropriate V-statistics. Their asymptotic properties are examined and their local Bahadur efficiencies against some common alternatives are found. A class of locally optimal alternatives for each test is obtained. The powers of these tests, for some small sample sizes, are compared with different exponentiality tests.

## Testing skew normality via the moment generating function

### Mathematical Methods of Statistics (2010-03-01) 19: 64-72 , March 01, 2010

In this paper, goodness-of-fit tests are constructed for the skew normal law. The proposed tests utilize the fact that the moment generating function of the skew normal variable satisfies a simple differential equation. The empirical counterpart of this equation, involving the empiricalmoment generating function, yields appropriate test statistics. The consistency of the tests is investigated under general assumptions, and the finite-sample behavior of the proposed method is investigated via a parametric bootstrap procedure.