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## 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.

## The cost of not knowing the radius

### Statistical Methods and Applications (2008-02-01) 17: 13-40 , February 01, 2008

Robust Statistics considers the quality of statistical decisions in the presence of deviations from the ideal model, where deviations are modelled by neighborhoods of a certain size about the ideal model. We introduce a new concept of optimality (radius-minimaxity) if this size or radius is not precisely known: for this notion, we determine the increase of the maximum risk over the minimax risk in the case that the optimally robust estimator for the false neighborhood radius is used. The maximum increase of the relative risk is minimized in the case that the radius is known only to belong to some interval [*r*_{l},*r*_{u}]. We pursue this minmax approach for a number of ideal models and a variety of neighborhoods. Also, the effect of increasing parameter dimension is studied for these models. The minimax increase of relative risk in case the radius is completely unknown, compared with that of the most robust procedure, is 18.1% versus 57.1% and 50.5% versus 172.1% for one-dimensional location and scale, respectively, and less than 1/3 in other typical contamination models. In most models considered so far, the radius needs to be specified only up to a factor
$$\rho\le \frac{1}{3}$$
, in order to keep the increase of relative risk below 12.5%, provided that the radius–minimax robust estimator is employed. The least favorable radii leading to the radius–minimax estimators turn out small: 5–6% contamination, at sample size 100.

## 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.

## Assessing the error in bootstrap estimates with dependent data

### Test (2000-12-01) 9: 471-486 , December 01, 2000

Bootstrap estimates, like most random variables, are subject to sampling variation. Efron and Tibshirani (1993) studied the variability in bootstrap estimates with independent data. Efron (1992) proposed the jackknife-after-bootstrap, a method for estimating the variability from the bootstrap samples themselves. We address the issue of studying the variability in bootstrap estimates for dependent data. We modify Efron's method to render it suitable to operate through the block bootstrap. A simulation study is carried out to investigate the consistency of the modified method. The performance of this method is judged by using the same setting as that used by Efron and Tibshirani (1993). Our results confirm that this method is reliable and has an advantage in the context of dependent data.

## 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.

## Minimax pointwise estimation of an anisotropic regression function with unknown density of the design

### Mathematical Methods of Statistics (2011-03-01) 20: 30-57 , March 01, 2011

Our aim in this paper is to estimate *with best possible accuracy* an unknown multidimensional regression function at a given point where the design density is also unknown. To reach this goal, we will follow the minimax approach: it will be assumed that the regression function belongs to a known anisotropic Hölder space. In contrast to the parameters defining the Hölder space, the density of the observations is assumed to be unknown and will be treated as a nuisance parameter. New minimax rates are exhibited as well as local polynomial estimators which achieve these rates. As these estimators depend on a tuning parameter, the problem of its selection is also discussed.

## Infinitesimally Robust estimation in general smoothly parametrized models

### Statistical Methods & Applications (2010-08-01) 19: 333-354 , August 01, 2010

The aim of the paper is to give a coherent account of the robustness approach based on shrinking neighborhoods in the case of i.i.d. observations, and add some theoretical complements. An important aspect of the approach is that it does not require any particular model structure but covers arbitrary parametric models if only smoothly parametrized. In the meantime, equal generality has been achieved by object-oriented implementation of the optimally robust estimators. Exponential families constitute the main examples in this article. Not pretending a complete data analysis, we evaluate the robust estimates on real datasets from literature by means of our R packages *ROptEst* and *RobLox*.

## Application of the delta method to functions of the sample mean when observations are dependent

### Statistical Papers (2017-12-01) 58: 957-986 , December 01, 2017

Let
$$\mu $$
be the expected value of a random variable and
$$\bar{X}_n$$
the corresponding sample mean of *n* observations. If the transformed expectation
$$f(\mu )$$
is to be estimated by
$$f\left( \bar{X}_n\right) $$
then the delta method is a widely used tool to describe the asymptotic behaviour of
$$f\left( \bar{X}_n\right) $$
. Regarding bias and variance, however, conventional theorems require independent observations as well as boundedness conditions of *f* being violated even by “simple” functions such as roots or logarithms. It is shown that asymptotic expansions for bias and variance still hold if restrictive boundedness conditions are replaced by considerably weaker requirements upon the global growth of *f*. Moreover, observations are allowed to be dependent.

## Inferences for odds ratio with dependent pairs

### TEST (2008-05-01) 17: 101-119 , May 01, 2008

In familial data analyses quantifying familial association is scientifically important. As analogies of the intraclass and interclass correlations of a normally distributed trait, we study intraclass and interclass (log) odds ratios for a binary trait. We propose non-parametric estimators of the odds ratios and derive the asymptotic variances of the estimators under the assumptions of exchangeability and closure of multivariate binary distributions under marginals. These estimators are straightforward, except for the consideration of how to weight by family size. The relative efficiencies of the non-parametric estimators are studied for some parametric models. It shows that our estimators are highly efficient, and that weighting by family size is recommended for the intraclass odds ratio. The computations of the estimators and their standard errors are illustrated with two examples.