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PateiroLópez, Beatriz; RodríguezCasal, Alberto
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4 Citations
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 αconvexity. 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 αconvex set S, the αconvex 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.
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Xie, Tianfa; Cao, Ruiyuan; Du, Jiang
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1 Citations
Variable selection has played a fundamental role in regression analysis. Spatial autoregressive model is a useful tool in econometrics and statistics in which context variable selection is necessary but not adequately investigated. In this paper, we consider conducting variable selection in spatial autoregressive models with a diverging number of parameters. Smoothly clipped absolute deviation penalty is considered to obtain the estimators. Moreover the dimension of the covariates are allowed to vary with sample size. In order to attenuate the bias caused by endogeneity, instrumental variable is adopted in the estimation procedure. The proposed method can do parametric estimation and variable selection simultaneously. Under mild conditions, we establish the asymptotic and oracle property of the proposed estimators. Finally, the performance of the proposed estimation procedure is examined via Monte Carlo simulation studies and a data set from a Boston housing price is analyzed as an illustrative example.
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Bertin, Karine; Rivoirard, Vincent
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3 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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de Wet, Tertius; Goegebeur, Yuri; Guillou, Armelle
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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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Nobre, Juvêncio S.; Singer, Julio M.; Sen, Pranab K.
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6 Citations
We propose a Ustatisticsbased 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 Utest in small and moderate samples and compare its performance with that of exact Ftests 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.
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By
Wang, Lihong
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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
Henze, Norbert; Meintanis, Simos G.
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60 Citations
Abstract.
A wide selection of classical and recent tests for exponentiality are discussed and compared. The classical procedures include the statistics of KolmogorovSmirnov and Cramérvon 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 KolmogorovSmirnov and Cramérvon Mises type that utilize a characterization of exponentiality via the mean residual life function. We also propose a new goodnessoffit test utilizing a novel characterization of the exponential distribution through its characteristic function. The finitesample performance of the tests is investigated in an extensive simulation study.
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Mercadier, Cécile; Roustant, Olivier
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All characterizations of nondegenerate multivariate tail dependence structures are both functional and infinitedimensional. Taking advantage of the Hoeffding–Sobol decomposition, we derive new indices to measure and summarize the strength of dependence in a multivariate extreme value analysis. The tail superset importance coefficients provide a pairwise ordering of the asymptotic dependence structure. We then define the tail dependograph, which visually ranks the extremal dependence between the components of the random vector of interest. For the purpose of inference, a rankbased statistic is derived and its asymptotic behavior is stated. These new concepts are illustrated with both theoretical models and real data, showing that our methodology performs well in practice.
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Geffray, Ségolen
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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
Wu, Yi; Wang, Xuejun; Hu, Shuhe; Yang, Lianqiang
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3 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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