Showing papers in "Journal of Nonparametric Statistics in 2000"
TL;DR: In this article, the authors deal with the simultaneous nonparametric estimations of n curves or observations of a random process corrupted by noise in which sample paths belong to a finite dimension functional subspace.
Abstract: This study deals with the simultaneous nonparametric estimations of n curves or observations of a random process corrupted by noise in which sample paths belong to a finite dimension functional subspace. The estimation, by means of B-splines, leads to a new kind of functional principal components analysis. Asymptotic rates of convergence are given for the mean and the eigenelements of the empirical covariance operator. Heuristic arguments show that a well chosen smoothing parameter may improve the estimation of the subspace which contains the sample path of the process. Finally, simulations suggest that the estimation method studied here is advantageous when there are a small number of design points.
110 citations
TL;DR: In this article, the authors proposed a new and intuitive method of removing boundary effects in density estimation, which replaces the unwanted terms in the bias expansion by their estimators, offers new ways of constructing boundary kernels and optimal endpoint kernels.
Abstract: Boundary effects are well known to occur in nonparametric density estimation when the support of the density has a finite endpoint. The usual kernel density estimators require modifications when estimating the density near endpoints of the support. In this paper, we propose a new and intuitive method of removing boundary effects in density estimation. Our idea, which replaces the unwanted terms in the bias expansion by their estimators, offers new ways of constructing boundary kernels and optimal endpoint kernels. We also discuss the choice of bandwidth variation functions at the boundary region. The performance of our results are numerically analyzed in a Monte Carlo study.
46 citations
TL;DR: In this paper, the scaling characteristics of stochastic processes can be examined using wavelet cross-co- variances for jointly stationary but generally non-Gaussian linear processes.
Abstract: Scaling characteristics of stochastic processes can be examined using wavelet cross-co- variances For jointly stationary but generally non-Gaussian linear processes, the asymptotic properties of the resulting wavelet cross-covariance estimator are derived The linear processes are assumed to have only a square-summable weight sequence, so that the class of processes includes long-memory processes The variance of the estimator can in each case be expressed as a spectrum value at zero frequency, and can hence be readily estimated A simulation experiment is reported which demonstrates the utility of this approach A comparison of estimated standard deviations of wavelet autocovar-iances and cross-covariances for Beaufort Sea albedo/temperature data under both Gaussian and non-Gaussian assumptions, illustrates the necessity of developing the non-Gaussian theory
37 citations
TL;DR: In this paper, a scale-by-scale decomposition of the usual cross-covariance in terms of scale by scale wavelet crosscovariances is presented.
Abstract: Many scientific studies require a thorough understanding of the scaling characteristics of observed processes. We derive and justify a decomposition of the usual cross-covariance in terms of scale-by-scale wavelet cross-covariances, and describe estimators of both quantities along with asymptotic equivalences. The scale-by-scale decomposition of auto-covariance and cross-covariance sequences are illustrated via a simulation study. A detailed scale analysis of the surface albedo and temperature of pack ice in the Beaufort Sea shows that the differing resolutions of the two sensors can be seen in the wavelet cross-correlations and autocorrelations at small scales, while at larger scales correlation patterns emerged suggesting fractal-like behaviour over a range of scales.
31 citations
TL;DR: In this paper, the authors propose estimating density functions by means of a constrained optimization problem whose criterion function is the maximum likelihood function, and whose constraints model any prior information that might be available.
Abstract: We propose estimating density functions by means of a constrained optimization problem whose criterion function is the maximum likelihood function, and whose constraints model any (prior) information that might be available. The asymptotic justification for such an approach relies on the theory of epi-convergence. A simple numerical example is used to signal the potential of such an approach.
25 citations
TL;DR: In this article, the local likelihood approach has been used for spline smoothing in varying-coefficient models with univariate response and the results for continuous effect modifiers and asymptotically optimal rates of smoothing are derived.
Abstract: Varying coefficient models result from generalized linear models by allowing the parameter of the linear predictor to vary across some additional explanatory quantity called effect modifier. While Hastie and Tibshirani (1993) have used spline smoothing techniques in varying-coefficient models with univariate response here the local likelihood approach is considered within the framework of multivariate generalized models. The local likelihood approach has several advantages. It allows the derivation of asymptotic properties under weak assumptions, consistency and asymptotic normality of the estimates are shown under rather general conditions. The estimation procedure may be performed with standard software. This holds even for the additive bias reduction method which is proposed. The results are given for continuous effect modifiers and asymptotically optimal rates of smoothing are derived. An alternative normalization of weights is proposed which corresponds to the augmentation of the information supplied...
25 citations
TL;DR: In this article, the authors proposed a new method to estimate nonparametrically a univariate time varying coefficients model, which allows to incorporate both, seasonal and smoothness constraints.
Abstract: This paper proposes a new method to estimate nonparametrically a univariate time varying coefficients model This estimation procedure allows to incorporate both, seasonal and smoothness constraints The resulting estimator nests as particular cases many other estimators proposed in the literature We derive its asymptotic bounds and we also show consistency and the asymptotic distribution Finally, we illustrate its performance by estimating the Spanish money multiplier and we provide a data driven method to compute the smoothing parameters
24 citations
TL;DR: In this article, a uniform convergence rate of almost sure convergence over compact subsets of R d is established for strongly mixing processes in the Besov space, using orthonormal wavelet bases.
Abstract: Let (Y, X) = {Y i , X i } be real-valued jointly stationary processes and let ρ be a Borel measurable function on the real line. Let be a d-dimensional regression function. For regression functions in the Besov space B s,p,q we estimate g using orthonormal wavelet bases. Uniform rates of almost sure convergence over compact subsets of R d are established for strongly mixing processes.
23 citations
TL;DR: In this paper, a simple transformation-based approach is proposed to estimate the density of a density f on the basis of a random sample from a weighted distribution G with density g given by, where w(u) > 0 for all u and.
Abstract: In this paper we consider the estimation of a density f on the basis of random sample from a weighted distribution G with density g given by ,where w(u) > 0 for all u and . A special case of this situation is that of length-biased sampling, where w(x) = x. In this paper we examine a simple transformation-based approach to estimating the density f. The approach is motivated by the form of the nonparametric estimator maximum likelihood of f in the same context and under a monotonicity constraint. Since the method does not depend on the specific density estimate used (only the transformation), it can be used to construct both simple density estimates (histograms or frequency polygons) and more complex methods with favorable properties (e.g., local or penalized likelihood estimates). Monte Carlo simulations indicate that transformation-based density estimation can outperform the kernel-based estimator of Jones (1991) depending on the weight function w, and leads to much better estimation of monotone densities...
20 citations
TL;DR: In this article, a generalization of the classical Mann-Whitney-Wilcoxon (MWW) statistic for testing stochastic ordering between two distributions is presented, and the asymptotic normality of the generalization is established.
Abstract: We present a generalization of the classical Mann-Whitney-Wilcoxon (MWW) statistic for testing stochastic ordering between two distributions. Our class of statistics includes as special cases the generalizations of the MWW statistic by Kochar (1978); Deshpande and Kochar (1980); Stephenson and Ghosh (1985); Shetty and Govindarajulu (1988); Shetty and Bhat (1993); Ahmad (1996); Kumar (1997) and Priebe and Cowen (1999). We establish the asymptotic normality and demonstrate the admissibility of the generalization in the sense of Pitman's asymptotic efficacy. Corresponding distribution-free confidence intervals are derived and a generalization of the Wilcoxon signed rank statistic for testing the center of a symmetric, univariate, continuous distribution is obtained.
19 citations
TL;DR: The rates of convergence are given and an improved estimator is also proposed which achieves the desirable root-n rate of convergence.
Abstract: Transformation from a parametrized family can be combined with kernel density estimation for improved effectiveness. Pilot estimators had been proposed for the parameter that gives the optimal transformation, yet their rates of convergence had not been resolved. In this paper, the rates of convergence are given. An improved estimator is also proposed which achieves the desirable root-n rate of convergence.
TL;DR: In this paper, it was shown that the kernel estimators of the second-order product density of the point process of exposed tangent points associated with a stationary d-dimensional Boolean model with convex compact grains are asymptotically normally distributed with constant variance.
Abstract: We investigate asymptotic properties Including MSE of kernel estimators of the second-order product density of the point process of ‘exposed tangent points’ (for given direction ) associated with a stationary d-dimensional Boolean model with convex compact grains. Under minimal conditions on the typical grain we prove that the square root of the kernel estimators is asymptotically normally distributed with constant variance which only depends on the chosen kernel function. In the particular case of spherical grains, as first shown by Molchanov and Stoyan (1994), the diameter distribution function F(t) is just equal to the product of and some function of t>=0 which can be estimated by standard methods. Using this fact we are able to derive a multivariate CLT for a suitably defined empirical diameter distribution function. Owing to this result we suggest a χ2 ‐ goodness-of-fit-test for testing a hypothetical diameter distribution.
TL;DR: In this paper, an optimal ranked set sample allocation scheme for a two-sample median test for a location shift between X- and Y-sample distributions F and G is presented.
Abstract: This paper presents an optimal ranked set sample allocation scheme for a two-sample ranked set sample median test for a location shift between X- and Y-sample distributions F and G. It is shown that the optimal design based on maximizing the asymptotic Pitman efficiency quantifies only the middle observation(s) in a set of X- and Y-samples uniformly over all probability models..The exact probability distribution of the ranked set two-sample median test is derived and tabulated for selected sample and set sizes.
TL;DR: In this article, the authors apply the empirical likelihood method to find estimates and confidence intervals for the treatment effect in the two-sample problem, making use of auxiliary information about a control group.
Abstract: In this paper, we apply the empirical likelihood method to find estimates and confidence intervals for the treatment effect in the two-sample problem. These estimates and confidence intervals make use of auxiliary information about a control group. A simulation study is presented to assess the finite sample performance of the proposed estimates.
TL;DR: In this article, nonparametric empirical Bayes (EB) solutions to two-tail test in the exponential family, under the standard product loss function which is proportional to (θ-θ1) (η-δ2) for incorrectly accepting H1.
Abstract: This paper provides nonparametric empirical Bayes (EB) solutions to two-tail test in the exponential family , under the standard product loss function which is proportional to (θ-θ1) (θ-θ2) for incorrectly accepting H1. Based on empirical data X1,…X n , and the present data X from nonparametric (in the sense that G is completely unknown and unspecified) EB test procedures are proposed. These procedures are asymptotically optimal (a.o.) whenever Further, for every integer s > 0 a class of non-parametric EB test procedures is proposed. These procedures are shown to be a.o. with rates for 0 < λ ≤ 2 satisfying certain conditions. Examples of exponential families and gamma densities are given where these conditions reduce to some simple moment conditions on G. No assumption on the smoothness of the function u(.), (and hence of the density function of X), is made at all for any of the results of this paper. By an example of a family of distributions, it is demonstrated that the rates arbitrarily close to o(n -1...
TL;DR: In this paper, the authors investigated Bahadur representations of the empirical likelihood algorithm under the constrained estimation model and the selection biis model and showed that tie additional model information improves the quantile estimation in large samples.
Abstract: The empirical likelihood method was introduced by Art B. Owen about a decade ago to construct confidence intervals as a nonparametric technique. It is shown that the empirical likelihood method has very general application. One of the advantages of the empirical likelihood method is that it can readily take into account the model information available for statistical analysis. In this paper we investigate Bahadur representations of the empirical likelihood auantiles under the commonlv used constrained estimation model and the selection biis model. It is shown that tie additional model information improves the quantile estimation in large samples.
TL;DR: In this article, the authors considered empirical Bayes estimation when the i-th component has a density, given the parameter θ i, that is proportional to a given function on (0,θ i ) which may be different for different i >= 1.
Abstract: We consider empirical Bayes estimation when the i-th component has a density, given the parameter θ i , that is proportional to a given function on (0,θ i ) which may be different for different i >=1. The common prior distribution of the 0 is estimated using nonparametric function estimation via a representation of it in terms of an average marginal density and a locally weighted average of the marginal distribution function of the data. The estimated prior is used to construct the empirical Bayes estimator. The asymptotic properties of the estimators are established using empirical processes methodology. The finite sample performance of the estimators are studied using a computer simulation. Extensions of the methods to families with threshold parameters are also provided.
TL;DR: In this paper, the problem of testing linearity of the regression function and homoscedasticness of the distribution of the error e is considered in the multivariate nonparametric regression model.
Abstract: In the multivariate nonparametric regression model Y = gt(t)+∑ the problem of testing linearity of the regression function g and homoscedasticity of the distribution of the error e is considered. F...
TL;DR: An information-theoretic based approach is used to provide consistent estimators of the intensity function of a point process in the Aalen model and shows that the statistical risk is bounded by an index of resolvability, which produces rates of convergence for specific families of candidate functions.
Abstract: (2000). Convergence rates for the minimum complexity estimator of counting process intensities. Journal of Nonparametric Statistics: Vol. 12, No. 5, pp. 611-643.
TL;DR: In this article, the authors proposed a threshold criterion (TIC) and criteria with datadependent penalty terms P I C using the idea of thresholding, which are more efficient in choosing the correct order in the sense of consistency and convergence rate.
Abstract: Tests of no effect in a regression context have been developed using the idea of nonparametric function estimation techniques. & error tests based on the difference between the estimated function and a constant perform well as omnibus tests. & error tests can be considered as data-driven Neyman smooth tests. In this case, an order (or truncation point) plays an important role in the performance of test statistics and is usually determined by an AIC type criterion (Kuchibhatla and Hart, 1996) or Schwarz's BIC criterion (Ledwina, 1994). When the sample size is small and the underlying function has high frequency behavior, AIC as well as BIC often fail to select a good order for the Fourier series estimator. The failure of correct order selection results in low power for the & error tests. To prevent such failure, we propose a Threshold criterion (TIC) and criteria with datadependent penalty terms P I C ) using the idea of thresholding. These new criteria are more efficient in choosing the correct order in the sense of consistency and convergence rate. We propose new datadriven Neyman smooth tests based on DIC.
TL;DR: In this paper, the authors compared the maximal achievable efficacies in the defined classes of generalized Jonckheere-type and Tryon-Hettmansperger-type tests.
Abstract: For the c-sample location problem with ordered alternatives the test of Jonckheere (1954) and Terpstra (1952) is a well-known competitor to the parametric test introduced by Barlow et al. (1972, p. 184). Generalizations of the Jonckheere-Terpstra test are proposed e.g., by Puri (1965), Tryon and Hettmansperger (1973) and Buning and Kossler (1996). All these tests are based on the pairwise ranking method. In the present paper their efficacies are compared. The Jonckheere-type and Tryon-Hettmansperger-type test statistics can be further generalized by introducing weight coefficients to the substatistics. For the case of a specified alternative these weights are determined to obtain maximal efficacies. It is shown that the maximal achievable efficacies in the defined classes of generalized Jonckheere-type tests and Tryon -Hettmansperger-type tests always are equal.
TL;DR: This paper developed recursive kernel estimators for the probability density and the regression function of nonlinear and nonstationary time series, characterized by two smoothing coefficients (the bandwidth and the discounting rate of observations) that may be selected with a prediction error criterion.
Abstract: This paper develops recursive kernel estimators for the probability density and the regression function of nonlinear and nonstationary time series. The resulting method is characterized by two smoothing coefficients (the bandwidth and the discounting rate of observations) that may be selected with a prediction error criterion. Statistical properties are investigated under a null hypothesis of stationarity and asymptotic elimination of the discounting. Simulation experiments on complex processes show the ability of the method in estimating time-varying nonlinear regression functions.
TL;DR: In this article, the authors examined smoothed bootstrap consistency of the sample mean, regular functions of the sampled mean and multiple regression models and used Mallows metric convergence of the empirical distribution, plugged-in by the bootstrap method, towards the theoretical distribution of the data.
Abstract: This article examines smoothed bootstrap consistency of the sample mean, regular functions of the sample mean and multiple regression models. We use Mallows metric convergence of the empirical distribution, plugged-in by the bootstrap method, towards the theoretical distribution of the data. We demonstrate convergence in Mallows metric for kernel and kernel shrunk estimates, and also for histograms with fixed partition. A class of locally adaptive smooth estimators is introduced and its convergence in Mallows metric is proved.
TL;DR: In this article, a general framework for simultaneous estimation of a smooth multivariate cumulative distribution function and its partial derivatives is presented by means of a multi-variate version of the popular local polynomial fitting technique.
Abstract: We present a general framework for simultaneous estimation of a smooth multivariate cumulative distribution function and its partial derivatives. This achieved by means of a multi-variate version of the popular local polynomial fitting technique. We investigate some asymptotic properties of the obtained estimators.
TL;DR: The minimax kernels are shown to possess not only the minimax property, but also have higher asymptotic efficiency in the conventional, non-minimax sense.
Abstract: Minimax kernels for nonparametric curve estimation are investigated. They are defined to be the solutions to the kernel variational problem arising from the asymptotic maximum risk of the kernel density or derivative estimation established in this paper. A δ-perturbation method is employed to solve the kernel variational problem. Such a δ-perturbation method can be used in solving other variational problems such as the variational problem of Gasser, Muller and Mammitzsch (1985). We obtain the explicit expressions of the minimax kernels by an algorithm developed in the Appendix and tabulate the asymptotic relative efficiencies among the minimax kernels, optimal kernels and Gaussian-based kernels for further reference. The minimax kernels are shown to possess not only the minimax property, but also have higher asymptotic efficiency in the conventional, non-minimax sense. As a by-product of our study, the asymptotic minimax risks for the kernel density and derivative estimators are also obtained.
TL;DR: In this article, the adaptive location estimate can be obtained by plugging kernel estimates of density and its derivative into the one-step approximation of the parametric maximum likelihood estimate, and the optimal order of bandwidths in terms of estimating the location parameter are established.
Abstract: In the semiparametric location model, an adaptive location estimate can be obtained by plugging kernel estimates of density and its derivative into the one-step approximation of the parametric maximum likelihood estimate. In this paper, we investigate the effect of higher order kernels on second order asymptotics of the adaptive location estimate. The optimal order of bandwidths in terms of estimating the location parameter are established. We also give some simulation results to see the effect of higher order kernels for moderate sample sizes.
TL;DR: In this paper, the authors proved that the B-spline density estimator and all of its nontrivial derivatives converge in mean integrated squared error and the asymptotic rate of convergence was determined.
Abstract: B-Spline density estimators were discussed by Gehringer and Redner in 1992 [9] and later extended to function estimation using partitions of unity over metric spaces in Redner and Gehringer in 1994 [12] to solve problems in computer graphics. In Part I of this paper [11] we returned to the uniform B-Spline density estimators in one dimension. We proved that the B-Spline density estimator and all of its nontrivial derivatives converge in mean integrated squared errorand the asymptotic rate of convergence was determined. In Part II of this paper we establish equivalent results for the asymptotic rates of convergence of the multiple dimensional B-Splinedensity estimator as well as the rates of convergence of all of its partial derivatives.
TL;DR: In this article, the kernel density estimator proposed by Blum and Susarla (1980) is investigated and three inequalities, a L 1 error inequality, a pth, p>=2, order absolute moment inequality and a probability inequality, are established, respectively.
Abstract: In this paper, the kernel density estimator proposed by Blum and Susarla (1980) is investigated. Three inequalities, a L1 error inequality, a pth, p>=2, order absolute moment inequality and a probability inequality, are established, respectively.
TL;DR: In this paper, Mehler's equality and Gebelein's inequality are generalized for bivariate densities having diagonal expansions and review and generalize some of its known properties, in particular, Mehraghdam's equality.
Abstract: We consider bivariate densities having diagonal expansions and review and generalize some of its known properties. In particular, Mehler's equality and Gebelein's inequality are generalized. Moreover, we consider stationary processes with a covariance function r(i) and with bivariate densities of (X1, X1+i) having diagonal form with coefficients a k (i), k = 0,1,… and state general conditions under which sequences subordinated to (X i )∞ i=1 are long-range dependent and obey the reduction principle. Furthermore, in the special case a k (i) = r(i) k , k = 0,1,… estimates based on such sequences enjoy some common asymptotic properties under long-range dependence.
TL;DR: In this paper, the authors consider deconvolution problems where the observations Y are equal in distribution to X+Z with X and Z independent random variables, and the distribution of Z is assumed to be known and X has an unknown probability density that they want to estimate.
Abstract: We consider deconvolution problems where the observations Y are equal in distribution to X+Z with X and Z independent random variables. The distribution of Z is assumed to be known and X has an unknown probability density that we want to estimate. The case where Z has a known Laplace distribution is investigated in detail. We consider an estimator that is the sum of two kernel estimators and investigate the gain to be achieved when we use different bandwidths instead of equal bandwidths. In less detail we review exponential deconvolution and estimation of a linear combination of density derivatives. We derive expansions for the asymptotic mean integrated squared error, asymptotically optimal bandwidths as well as a formula for the ratio of the smallest asymptotic error of the multiple bandwidth and equal bandwidth estimator. The finite sample performance of the multi bandwidth kernel estimators is investigated by computation of the exact mean integrated squared error for several target densities.