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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
Ahmed, Hanan; Einmahl, John H. J.
Heavy tailed phenomena are naturally analyzed by extreme value statistics. A crucial step in such an analysis is the estimation of the extreme value index, which describes the tail heaviness of the underlying probability distribution. We consider the situation where we have next to the n observations of interest another n + m observations of one or more related variables, like, e.g., financial losses due to earthquakes and the related amounts of energy released, for a longer period than that of the losses. Based on such a data set, we present an adapted version of the Hill estimator. For this adaptation the tail dependence between the variable of interest and the related variable(s) plays an important role. We establish the asymptotic normality of this new estimator. It shows greatly improved behavior relative to the Hill estimator, in particular the asymptotic variance is substantially reduced, whereas we can keep the asymptotic bias the same. A simulation study confirms the substantially improved performance of our adapted estimator. We also present an application to the aforementioned earthquake losses.
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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)
5 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
Mahdizadeh, M. ; Zamanzade, Ehsan
This article concerns estimation of a symmetric distribution function under multistage ranked set sampling. A nonparametric estimator is developed and its theoretical properties are explored. Performance of the suggested estimator is further evaluated using numerical studies.
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