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By
Bertin, Karine; Rivoirard, Vincent
4 Citations
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.
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By
de Wet, Tertius; Goegebeur, Yuri; Guillou, Armelle
3 Citations
We consider the estimation of the scale parameter appearing in the second order condition when the distribution underlying the data is of Paretotype. 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 socalled 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.
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By
Rieder, Helmut; Kohl, Matthias; Ruckdeschel, Peter
10 Citations
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 (radiusminimaxity) 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 onedimensional 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.
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By
Wang, Lihong
3 Citations
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.
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By
Mahmoud, Mohamed; Mokhlis, Nahed A.; Ibrahim, Sahar A. N.
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 jackknifeafterbootstrap, 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.
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By
Geffray, Ségolen
3 Citations
We deal with the problem of dependent competing risks in presence of independent rightcensoring. The Aalen–Johansen estimator for the causespecific 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 LILtype results and asymptotic confidence bands.
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By
Guillou, A.; Klutchnikoff, N.
1 Citations
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.
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By
Wu, Yi; Wang, Xuejun; Hu, Shuhe; Yang, Lianqiang
Show all (4)
6 Citations
In this paper, the single index weighted version of Marcinkiewicz–Zygmund type strong law of large numbers and the double index weighted version of Marcinkiewicz–Zygmund type strong law of large numbers are investigated successively for a class of random variables, which extends the classical results for independent and identically distributed random variables. As applications of the results, we further study the strong consistency for the weighted estimator in the nonparametric regression model and the least square estimators in the simple linear errorsinvariables model. Moreover, we also present some numerical study to verify the validity of our results.
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By
Feng, Zhenghui; Zhang, Jun ; Chen, Qian
5 Citations
We consider estimations and hypothesis test for linear regression measurement error models when the response variable and covariates are measured with additive distortion measurement errors, which are unknown functions of a commonly observable confounding variable. In the parameter estimation and testing part, we first propose a residualbased least squares estimator under unrestricted and restricted conditions. Then, to test a hypothesis on the parametric components, we propose a test statistic based on the normalized difference between residual sums of squares under the null and alternative hypotheses. We establish asymptotic properties for the estimators and test statistics. Further, we employ the smoothly clipped absolute deviation penalty to select relevant variables. The resulting penalized estimators are shown to be asymptotically normal and have the oracle property. In the model checking part, we suggest two test statistics for checking the validity of linear regression models. One is a scoretype test statistic and the other is a model adaptive test statistic. The quadratic form of the scaled test statistic is asymptotically chisquared distributed under the null hypothesis and follows a noncentral chisquared distribution under local alternatives that converge to the null hypothesis. We also conduct simulation studies to demonstrate the performance of the proposed procedure and analyze a real example for illustration.
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By
Caeiro, Frederico; Gomes, M. Ivette
5 Citations
In this article, we revisit Feuerverger and Hall’s maximum likelihood estimation of the extreme value index. Based on those estimators we propose new estimators that have the smallest possible asymptotic variance, equal to the asymptotic variance of the Hill estimator. The full asymptotic distributional properties of the estimators are derived under a general thirdorder framework for heavy tails. Applications to a real data set and to simulated data are also presented.
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