Showing papers in "Mathematical Methods of Statistics in 2007"

Linear and convex aggregation of density estimators

Abstract: We study the problem of finding the best linear and convex combination of M estimators of a density with respect to the mean squared risk. We suggest aggregation procedures and we prove sharp oracle inequalities for their risks, i.e., oracle inequalities with leading constant 1. We also obtain lower bounds showing that these procedures attain optimal rates of aggregation. As an example, we consider aggregation of multivariate kernel density estimators with different bandwidths. We show that linear and convex aggregates mimic the kernel oracles in asymptotically exact sense. We prove that, for Pinsker’s kernel, the proposed aggregates are sharp asymptotically minimax simultaneously over a large scale of Sobolev classes of densities. Finally, we provide simulations demonstrating performance of the convex aggregation procedure.

Kernel Regression Estimation for Continuous Spatial Processes

Abstract: We investigate here a kernel estimate of the spatial regression function r(x) = E(Y u | X u = x), x ∈ ℝd, of a stationary multidimensional spatial process { Z u = (X u, Y u), u ∈ ℝ N }. The weak and strong consistency of the estimate is shown under sufficient conditions on the mixing coefficients and the bandwidth, when the process is observed over a rectangular domain of ℝN. Special attention is paid to achieve optimal and suroptimal strong rates of convergence. It is also shown that this suroptimal rate is preserved by using a suitable spatial sampling scheme.

Generalized mirror averaging and D-convex aggregation

Abstract: We study the problem of aggregation of estimators. Given a collection of M different estimators, we construct a new estimator, called aggregate, which is nearly as good as the best linear combination over an l 1-ball of ℝM of the initial estimators. The aggregate is obtained by a particular version of the mirror averaging algorithm. We show that our aggregation procedure statisfies sharp oracle inequalities under general assumptions. Then we apply these results to a new aggregation problem: D-convex aggregation. Finally we implement our procedure in a Gaussian regression model with random design and we prove its optimality in a minimax sense up to a logarithmic factor.

Two tests for multivariate normality based on the characteristic function

Abstract: We present two tests for multivariate normality. The presented tests are based on the Levy characterization of the normal distribution and on the BHEP tests. The tests are affine invariant and consistent. We obtain the asymptotic null distribution of the test statistics using some results about generalized one-sample U-statistics, which are of independent interest.

Nonparametric hypothesis testing for intensity of the poisson process

Abstract: We propose a goodness-of-fit test for the hypothesis that the observed Poisson point process has a given periodic intensity function against a nonparametric close alternative of known smoothness. We obtain rate and sharp asymptotics for the errors in the minimax setup.

Deconvolving compactly supported densities

Abstract: This paper addresses the statistical problem of density deconvolution under the condition that the density to be estimated has compact support. We introduce a new estimation procedure, which establishes faster rates of convergence for smooth densities as compared to the optimal rates for smooth densities with unbounded support. This framework also allows us to relax the usual condition of known error density with non-vanishing Fourier transform, so that a nonparametric class of densities is valid; therefore, even the shape of the noise density need not be assumed. These results can also be generalized for fast decaying densities with unbounded support. We prove optimality of the rates in the underlying experiment and study the practical performance of our estimator by numerical simulations.

On the Edgeworth expansion and the M out of N bootstrap accuracy for a Studentized trimmed mean

Abstract: We investigate the second order accuracy of the M out of N bootstrap for a Studentized trimmed mean using the Edgeworth expansion derived in a previous paper. Some simulations, which support our theoretical results, are also given. The effect of extrapolation in conjunction with the M out of N bootstrap for Studentized trimmed means is briefly discussed. As an auxiliary result we obtain a Bahadur’s type representation for an M out of N bootstrap quantile. Our results supplement previous work on (Studentized) trimmed means by Hall and Padmanabhan [13], Bickel and Sakov [7], and Gribkova and Helmers [11].

Minimax estimation of the Wigner function in quantum homodyne tomography with ideal detectors

Abstract: We estimate the quantum state of a light beam from results of quantum homodyne measurements performed on identically prepared pulses. The state is represented through the Wigner function, a “quasi-probability density” on ℝ2 which may take negative values and must satisfy intrinsic positivity constraints imposed by quantum physics. The data consists of n i.i.d. observations from a probability density equal to the Radon transform of the Wigner function. We construct an estimator for the Wigner function and prove that it is minimax efficient for the pointwise risk over a class of infinitely differentiable functions. A similar result was previously derived by Cavalier in the context of positron emission tomography. Our work extends this result to the space of smooth Wigner functions, which is the relevant parameter space for quantum homodyne tomography.

Consistent estimation of a convex density at the origin

Abstract: Motivated by Hampel’s birds migration problem, Groeneboom, Jongbloed, and Wellner [7] established the asymptotic distribution theory for the nonparametric Least Squares and Maximum Likelihood estimators of a convex and decreasing density, g 0, at a fixed point t 0 > 0. However, estimation of the distribution function of the birds’ resting times involves estimation of g′0 at 0, a boundary point at which the estimators are not consistent. In this paper, we focus on the Least Squares estimator, % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC% vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz% ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbb% L8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpe% pae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaam% aaeaqbaaGcbaGafm4zaCMbaGaadaWgaaWcbaGaemOBa4gabeaaaaa!3DC2! $$\tilde g_n $$ . Our goal is to show that consistent estimators of both g 0(0) and g′0(0) can be based solely on % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC% vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz% ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbb% L8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpe% pae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaam% aaeaqbaaGcbaGafm4zaCMbaGaadaWgaaWcbaGaemOBa4gabeaaaaa!3DC2! $$\tilde g_n $$ . Following the idea of Kulikov and Lopuhaa [14] in monotone estimation, we show that it suffices to take % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC% vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz% ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbb% L8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpe% pae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaam% aaeaqbaaGcbaGafm4zaCMbaGaadaWgaaWcbaGaemOBa4gabeaaaaa!3DC2! $$\tilde g_n $$ (n −α ) and % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC% vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz% ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbb% L8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpe% pae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaam% aaeaqbaaGcbaGafm4zaCMbaGGbauaadaWgaaWcbaGaemOBa4gabeaa% aaa!3DCD! $$\tilde g'_n $$ (n −α ), with α ∈ (0, 1/3). We establish their joint asymptotic distributions and show that α = 1/5 should be taken as it yields the fastest rates of convergence.

Testing strict monotonicity in nonparametric regression

Abstract: A new test for strict monotonicity of the regression function is proposed which is based on a composition of an estimate of the inverse of the regression function with a common regression estimate. This composition is equal to the identity if and only if the “true” regression function is strictly monotone, and a test based on an L 2-distance is investigated. The asymptotic normality of the corresponding test statistic is established under the null hypothesis of strict monotonicity.

Stein shrinkage and second-order efficiency for semiparametric estimation of the shift

Abstract: The problem of estimating the shift (or, equivalently, the center of symmetry) of an unknown symmetric and periodic function f observed in Gaussian white noise is considered. Using the blockwise Stein method, a penalized profile likelihood with a data-driven penalization is introduced so that the estimator of the center of symmetry is defined as the maximizer of the penalized profile likelihood. This estimator has the advantage of being independent of the functional class to which the signal f is assumed to belong and, furthermore, is shown to be semiparametrically adaptive and efficient. Moreover, the second-order term of the risk expansion of the proposed estimator is proved to behave at least as well as the second-order term of the risk of 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 extend those of (10), where second-order asymptotic minimaxity is proved for an estimator depending on the functional class containing f and β ≥ 2 is required.

Nonparametric estimation of the purity of a quantum state in quantum homodyne tomography with noisy data.

Abstract: The aim of this paper is to answer an important issue in quantum mechanics, namely to estimate the purity of a quantum state of a light beam. Estimation of the purity is based on the results of quantum homodyne measurements performed on independent identically prepared quantum systems. The quantum state of the light is entirely characterized by the Wigner function, which can take negative values and must satisfy certain constraints of positivity imposed by quantum physics. We estimate the integrated squared Wigner function by a kernel-based second order U — statistic. This quadratic functional is a physical measure of the purity of the state. We also give an adaptive estimator, which does not depend on the smoothness parameters. We establish upper bounds of the minimax risk over a class of infinitely differentiable functions.

Lilliefors test for exponentiality: Large deviations, asymptotic efficiency, and conditions of local optimality

Abstract: We study large deviations and Bahadur efficiency of the Lilliefors statistic for testing of exponentiality. This statistic belongs to the class of Kolmogorov-Smirnov type statistics with estimated parameters. Large deviation asymptotics of such statistic is found for the first time. We show that the test has relatively high local efficiency and construct the alternative for which it is locally optimal.

Estimation and detection of high-variable functions from Sloan—Woźniakowski space

Abstract: The major difficulty arising in statistics of multi-variable functions is “the curse of dimensionality”: the rates of accuracy in estimation and separation rates in detection problems behave poorly when the number of variables increases. This difficulty arises for most popular functional classes such as Sobolev or Holder balls.

Change point estimation by local linear smoothing under a weak dependence condition

Abstract: We consider a change-point problem in regression estimation. Observations (X i , Y i ), i = 1, ..., n, are governed by the model Y i = m(X i ) + σ(X i )ɛ i , where the (ɛ i )i∈ℤ are independent and identically distributed and independent of (X i )i∈ℤ. The latter sequence satisfies a weak dependence condition proposed by Dedecker and Prieur [4]. We essentially study the basic situation, where the regression function has a unique change point. The construction of the jump estimate process, t → $$\hat \gamma $$ (t), is based on local linear regression. Under a positivity condition regarding the asymmetric kernel involved, we prove the convergence of a local dilated-rescaled version of $$\hat \gamma $$ (t) to a compound Poisson process with an additional drift. We also derive asymptotic normality results.

Methods of Constructing Characterizations by Constancy of Regression on the Sample Mean and Related Problems for NEF's

Abstract: Characterizations of a distribution by zero (or constant) regression properties of arbitrary degree polynomial statistics on the sample mean are discussed. Various practical steps collected from the relevant literature are put together in this framework into a comprehensive guideline for constructing such characterizations. Applications are provided for natural exponential families (NEF’s). In particular, two reciprocal NEF’s associated with the continuous time symmetric Bernoulli random walk are characterized using this guideline. Moreover, a class of infinitely divisible NEF’s having some polynomial variance function structure is discussed in this framework.

Nonparametric estimation for censored lifetimes suffering from unknown selection bias

Abstract: In a population of individuals, where the random variable (r.v.) σ denotes the birth time and X the lifetime, we consider the case, where an individual can be observed only if its life-line $$\mathcal{L}$$ (σ, X) = {(σ + y, y), 0 ≤ y ≤ X} intersects a given Borel set S in ℝ × ℝ+. Denoting by σ S and X S the birth time and lifetime for the observed individuals, we point out that the distribution function (d.f.) F S of the r.v. X S suffers from a selection bias in the sense that F S = ∝ w d F/μ S, where w and μ S depend only on the distribution of σ and on F, the d.f. of X. Assuming in addition that the r.v. X S is randomly right-censored as soon as the individual is selected, we construct a productlimit estimator $$\hat F_\mathcal{S} $$ for the d.f. F S and a nonparametric estimator ŵ for the weight function w. We prove a consistency result for ŵ and a weak convergence result for $$\hat F_\mathcal{S} $$ . We establish in addition an exponential bound for $$\hat F_\mathcal{S} $$ .

A Bayesian approach to the estimation of maps between Riemannian manifolds

Abstract: Let Θ be a smooth compact oriented manifold without boundary, imbedded in a Euclidean space Es, and let γ be a smooth map of Θ into a Riemannian manifold Λ. An unknown state θ ∈ Θ is observed via X = θ + eξ, where e > 0 is a small parameter and ξ is a white Gaussian noise. For a given smooth prior λ on Θ and smooth estimators g(X) of the map γ we derive a second-order asymptotic expansion for the related Bayesian risk. The calculation involves the geometry of the underlying spaces Θ and Λ, in particular, the integration-by-parts formula. Using this result, a second-order minimax estimator of γ is found based on the modern theory of harmonic maps and hypo-elliptic differential operators.

Nonparametric tests for a change in the coefficient of variation

Abstract: We consider the problem of testing the null hypothesis of no change against the alternative of exactly one change point when the change is expressed in terms of the value of the coefficient of variation. We propose a number of nonparametric test statistics for this problem. The asymptotic theory of the proposed tests is developed.

Test for uniformity by empirical Fourier expansion

Abstract: We consider a test for uniformity based on the empirical Fourier coefficients. We establish the asymptotic level, as well as the asymptotic power of this test.