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Optimal detection of Fechnerasymmetry
 J. Statist. Plann. Inference
, 2008
"... We consider a general class of skewed univariate densities introduced by Fechner (1897), and derive optimal testing procedures for the null hypothesis of symmetry within that class. Locally and asymptotically optimal (in the Le Cam sense) tests are obtained, both for the case of symmetry with respec ..."
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Cited by 7 (1 self)
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We consider a general class of skewed univariate densities introduced by Fechner (1897), and derive optimal testing procedures for the null hypothesis of symmetry within that class. Locally and asymptotically optimal (in the Le Cam sense) tests are obtained, both for the case of symmetry with respect to a specified location as for the case of symmetry with respect to some unspecified location. Signedrank based versions of these tests are also provided. The efficiency properties of the proposed procedures are investigated by a derivation of their asymptotic relative efficiencies with respect to the corresponding Gaussian parametric tests based on the traditional PearsonFisher coefficient of skewness. Smallsample performances under several types of asymmetry are investigated via simulations.
ASYMPTOTIC EQUIVALENCE AND ADAPTIVE ESTIMATION FOR ROBUST NONPARAMETRIC REGRESSION
, 2009
"... Asymptotic equivalence theory developed in the literature so far are only for bounded loss functions. This limits the potential applications of the theory because many commonly used loss functions in statistical inference are unbounded. In this paper we develop asymptotic equivalence results for rob ..."
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Cited by 6 (4 self)
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Asymptotic equivalence theory developed in the literature so far are only for bounded loss functions. This limits the potential applications of the theory because many commonly used loss functions in statistical inference are unbounded. In this paper we develop asymptotic equivalence results for robust nonparametric regression with unbounded loss functions. The results imply that all the Gaussian nonparametric regression procedures can be robustified in a unified way. A key step in our equivalence argument is to bin the data and then take the median of each bin. The asymptotic equivalence results have significant practical implications. To illustrate the general principles of the equivalence argument we consider two important nonparametric inference problems: robust estimation of the regression function and the estimation of a quadratic functional. In both cases easily implementable procedures are constructed and are shown to enjoy simultaneously a high degree of robustness and adaptivity. Other problems such as construction of confidence sets and nonparametric hypothesis testing can be handled in a similar fashion.
ON NONPARAMETRIC TESTS OF POSITIVITY/MONOTONICITY/CONVEXITY
, 2002
"... We consider the problem of estimating the distance from an unknown signal, observed in a whitenoise model, to convex cones of positive/monotone/convex functions. We show that, when the unknown function belongs to a Hölder class, the risk of estimating the Lrdistance, 1 ≤ r<∞, from the signal to ..."
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Cited by 4 (0 self)
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We consider the problem of estimating the distance from an unknown signal, observed in a whitenoise model, to convex cones of positive/monotone/convex functions. We show that, when the unknown function belongs to a Hölder class, the risk of estimating the Lrdistance, 1 ≤ r<∞, from the signal to a cone is essentially the same (up to a logarithmic factor) as that of estimating the signal itself. The same risk bounds hold for the test of positivity, monotonicity and convexity of the unknown signal. We also provide an estimate for the distance to the cone of positive functions for which risk is, by a logarithmic factor, smaller than that of the “plugin ” estimate.
Prospects of nonparametric modeling
 Journal of the American Statistical Association
, 2000
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Thresholding for weighted χ2
 Statist. Sinica
, 2001
"... Abstract: Given data from a spherical Gaussian distribution with unknown mean vector θ, estimates of quadratic functionals are constructed by thresholding. Mean squared error bounds are derived via a comparison with those already available for a suitable noncentral χ 2 variate. By way of illustratio ..."
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Abstract: Given data from a spherical Gaussian distribution with unknown mean vector θ, estimates of quadratic functionals are constructed by thresholding. Mean squared error bounds are derived via a comparison with those already available for a suitable noncentral χ 2 variate. By way of illustration, the resulting inequalities are used to yield an optimal rate adaptivity result for estimation of integrated squared derivatives in the white noise model of nonparametric function estimation. Key words and phrases: Adaptive estimation, integrated squared derivative, noncentral χ 2, quadratic functional. 1.
Stein Shrinkage with Penalization and Second Order Efficiency in Semiparametrics
, 2005
"... The problem of estimating the centre of symmetry of an unknown periodic function observed in Gaussian white noise is considered. Using the penalized blockwise Stein method, a smoothing filter allowing to define the penalized profile likelihood is proposed. The estimator of the centre of symmetry is ..."
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The problem of estimating the centre of symmetry of an unknown periodic function observed in Gaussian white noise is considered. Using the penalized blockwise Stein method, a smoothing filter allowing to define the penalized profile likelihood is proposed. The estimator of the centre of symmetry is then the maximizer of this penalized profile likelihood. This estimator is shown to be semiparametrically adaptive and efficient. Moreover, the second order term of its risk expansion is proved to behave at least as well as the second order term for the best possible estimator using monotone smoothing filter. Under mild assumptions, this estimator is shown to be second order minimax sharp adaptive over the whole scale of Sobolev balls with smoothness β > 1. Thus, these results improve on Dalalyan, Golubev and Tsybakov (2006), where β ≥ 2 is required.
Rates of Convergence for the Preasymptotic Substitution Bandwidth Selector
, 1996
"... : An effective bandwidth selection method for local linear regression is proposed in Fan and Gijbels (1995). The method is based on the idea of the preasymptotic substitution and has been tested extensively. This paper investigates the rate of convergence of this method. In particular, we show that ..."
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: An effective bandwidth selection method for local linear regression is proposed in Fan and Gijbels (1995). The method is based on the idea of the preasymptotic substitution and has been tested extensively. This paper investigates the rate of convergence of this method. In particular, we show that the relative rate of convergence is of order n \Gamma2=7 if the locally cubic fitting is used in the pilot stage, and the rate of convergence is n \Gamma2=5 when the local polynomial of degree 5 is used in the pilot fitting. The study also reveals a marked difference between the bandwidth selection for nonparametric regression and that for density estimation: The plugin approach for the latter case can admit the rootn rate of convergence while for the former case the best rate is of order n \Gamma2=5 . Key words and phrases: Bandwidth selection, convergence rate, local polynomial regression, kernel density estimation. Abbreviated title: Bandwidth selection. 1 Introduction Local ...
Minimax Estimation of a Bounded Squared Mean
"... Consider a normal model with unknown mean bounded by a known constant. This paper deals with minimax estimation of the squared mean. We establish an expression for the asymptotic minimax risk. This result is applied in nonparametric estimation of quadratic functionals. 1 ..."
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Consider a normal model with unknown mean bounded by a known constant. This paper deals with minimax estimation of the squared mean. We establish an expression for the asymptotic minimax risk. This result is applied in nonparametric estimation of quadratic functionals. 1
The Annals of Statistics SEMIPARAMETRICALLY EFFICIENT RANKBASED INFERENCE FOR SHAPE I. OPTIMAL RANKBASED TESTS FOR SPHERICITY
"... We propose a class of rankbased procedures for testing that the shape matrix V of an elliptical distribution (with unspecified center of symmetry, scale, and radial density) has some fixed value V0; this includes, for V0 = Ik, the problem of testing for sphericity as an important particular case. T ..."
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We propose a class of rankbased procedures for testing that the shape matrix V of an elliptical distribution (with unspecified center of symmetry, scale, and radial density) has some fixed value V0; this includes, for V0 = Ik, the problem of testing for sphericity as an important particular case. The proposed tests are invariant under translations, monotone radial transformations, rotations, and reflections with respect to the estimated center of symmetry. They are valid without any moment assumption. For adequately chosen scores, they are locally asymptotically maximin (in the Le Cam sense) at given radial densities. They are strictly distributionfree when the center of symmetry is specified, and asymptotically so, when it has to be estimated. The multivariate ranks used throughout are those of the distances—in the metric associated with the null value V0 of the shape matrix—between the observations and the (estimated) center of the distribution. Local powers (against elliptical alternatives) and