pinecone/core-2020-05-10-deduplication huggingface dataset:
On Polynomial Estimators of Frontiers and Boundaries

AbstractMotivated by problems of frontier estimation in productivity analysis, and boundary estimation in scatter-point image analysis, we consider polynomial-based estimators of the edge of a distribution. Our aim is to develop methods for correcting polynomial-type estimators of bias, and for constructing simultaneous confidence bands for the data edge. We tackle this problem by first deriving large-sample approximations to distributions of polynomial-based edge estimators, and then developing algorithms for simulating from them so as to produce Monte Carlo approximations to the distribution of the difference between the true edge and its estimator. This involves applying representations for joint extreme value distributions. The majority of attention is focused on the parametric case, but nonparametric problems, where polynomial approximations are fitted locally, are also considered
Conditional Iterative Proportional Fitting for Gaussian Distributions

AbstractA Gaussian version of the iterative proportional fitting procedure (IFP-P) was applied by Speed and Kiiveri to solve the likelihood equations in graphical Gaussian models. The calculation of the maximum likelihood estimates can be seen as the problem to find a Gaussian distribution with prescribed Gaussian marginals. We extend the Gaussian version of the IPF-P so that additionally given conditionals of Gaussian type are taken into account. The convergence of both proposed procedures, called conditional iterative proportional fitting procedures (CIPF-P), is proved
Nonparametric Estimation of the Measurement Error Model Using Multiple Indicators

AbstractThis paper considers the nonparametric estimation of the densities of the latent variable and the error term in the standard measurement error model when two or more measurements are available. Using an identification result due to Kotlarski we propose a two-step nonparametric procedure for estimating both densities based on their empirical characteristic functions. We distinguish four cases according to whether the underlying characteristic functions are ordinary smooth or supersmooth. Using the loglog Law and von Mises differentials we show that our nonparametric density estimators are uniformly convergent. We also characterize the rate of uniform convergence in each of the four cases
Pointwise Improvement of Multivariate Kernel Density Estimates

AbstractMultivariate kernel density estimators are known to systematically deviate from the true value near critical points of the density surface. To overcome this difficulty a method based on Rao–Blackwell's theorem is proposed. Local corrections of kernel density estimators are achieved by conditioning these estimators with respect to locally sufficient statistics. The asymptotic as well as the small sample size behavior of the improved estimators are studied. Asymptotic bias and variance are investigated and weak and complete consistency are derived under mild hypothesis
Testing for Spherical Symmetry of a Multivariate Distribution

AbstractWe consider a test for spherical symmetry of a distribution in Rdwith an unknown center. It is a multivariate version of the tests suggested by Schuster and Barker and by Arcones and Giné. The test statistic is based on the multivariate extension of the distribution and quantile functions, recently introduced by Koltchinskii and Dudley and by Chaudhuri. We study the asymptotic behavior of the sequence of test statistics for large samples and for a fixed spherically asymmetric alternative as well as for a sequence of local alternatives converging to a spherically symmetric distribution. We also study numerically the performance of the test for moderate sample sizes and justify a symmetrized version of bootstrap approximation of the distribution of test statistics
Normal Linear Regression Models With Recursive Graphical Markov Structure

AbstractA multivariate normal statistical model defined by the Markov properties determined by an acyclic digraph admits a recursive factorization of its likelihood function (LF) into the product of conditional LFs, each factor having the form of a classical multivariate linear regression model (≡WMANOVA model). Here these models are extended in a natural way to normal linear regression models whose LFs continue to admit such recursive factorizations, from which maximum likelihood estimators and likelihood ratio (LR) test statistics can be derived by classical linear methods. The central distribution of the LR test statistic for testing one such multivariate normal linear regression model against another is derived, and the relation of these regression models to block-recursive normal linear systems is established. It is shown how a collection of nonnested dependent normal linear regression models (≡Wseemingly unrelated regressions) can be combined into a single multivariate normal linear regression model by imposing a parsimonious set of graphical Markov (≡Wconditional independence) restrictions
Density Estimation on the Stiefel Manifold

AbstractThis paper develops the theory of density estimation on the Stiefel manifoldVk,m, whereVk,mis represented by the set ofm×kmatricesXsuch thatX′X=Ik, thek×kidentity matrix. The density estimation by the method of kernels is considered, proposing two classes of kernel density estimators with small smoothing parameter matrices and for kernel functions of matrix argument. Asymptotic behavior of various statistical measures of the kernel density estimators is investigated for small smoothing parameter matrix and/or for large sample size. Some decompositions of the Stiefel manifoldVk,mplay useful roles in the investigation, and the general discussion is applied and examined for a special kernel function. Alternative methods of density estimation are suggested, using decompositions ofVk,m
Minimax Risk Inequalities for the Location–Parameter Classification Problem

AbstractMinimax risk inequalities are obtained for the location-parameter classification problem. For the classical single observation case with continuous distributions, best possible bounds are given in terms of their Lévy concentration, establishing a conjecture of Hill and Tong (1989). In addition, sharp bounds for the minimax risk are derived for the multiple (i.i.d.) observations case, based on the tail concentration and the Lévy concentration. Some fairly sharp bounds for discontinuous distributions are also obtained
Improved Estimation in Measurement Error Models Through Stein Rule Procedure

AbstractThis paper examines the role of Stein estimation in a linear ultrastructural form of the measurement errors model. It is demonstrated that the application of Stein rule estimation to the matrix of true values of regressors leads to the overcoming of the inconsistency of the least squares procedure and yields consistent estimators of regression coefficients. A further application may improve the efficiency properties of the estimators of regression coefficients. It is observed that the proposed family of estimators under some constraint on the characterizing scalar dominates the conventional consistent estimator with respect to the criterion of asymptotic risk under a specific quadratic loss function. Then the problem of prediction of the values of the study variable within the sample is considered, and it is found that the predictors based on the proposed family of estimators are always more efficient than the predictors based on the conventional estimator according to asymptotic predictive mean squared error criterion, although both are biased
The Probability Content of Cones in Isotropic Random Fields

AbstractThis paper provides computable representations for the evaluation of the probability content of cones in isotropic random fields. A decomposition of quadratic forms in spherically symmetric random vectors is obtained and a representation of their moments is derived in terms of finite sums. These results are combined to obtain the distribution function of quadratic forms in spherically symmetric or central elliptically contoured random vectors. Some numerical examples involving the sample serial covariance are provided. Ratios of quadratic forms are also discussed
The Catline for Deep Regression

AbstractMotivated by the notion of regression depth (Rousseeuw and Hubert, 1996) we introduce thecatline, a new method for simple linear regression. At any bivariate data setZn={(xi,yi);i=1,…,n} its regression depth is at leastn/3. This lower bound is attained for data lying on a convex or concave curve, whereas for perfectly linear data the catline attains a depth ofn. We construct anO(nlogn) algorithm for the catline, so it can be computed fast in practice. The catline is Fisher-consistent at any linear modely=βx+α+ein which the error distribution satisfies med(e|x)=0, which encompasses skewed and/or heteroscedastic errors. The breakdown value of the catline is 1/3, and its influence function is bounded. At the bivariate gaussian distribution its asymptotic relative efficiency compared to theL1line is 79.3% for the slope, and 88.9% for the intercept. The finite-sample relative efficiencies are in close agreement with these values. This combination of properties makes the catline an attractive fitting method
Kernel Density and Hazard Rate Estimation for Censored Dependent Data

AbstractIn some long term studies, a series of dependent and possibly censored failure times may be observed. Suppose that the failure times have a common marginal distribution function having a density, and the nonparametric estimation of density and hazard rate under random censorship is of our interest. In this paper, we establish the asymptotic normality and the uniform consistency (with rates) of the kernel estimators for density and hazard function under a censored dependent model. A numerical study elucidates the behavior of the estimators for moderately large sample sizes
Functional ANOVA Models for Generalized Regression

AbstractThe Functional ANOVA model is considered in the context of generalized regression, which includes logistic regression, probit regression, and Poisson regression as special cases. The multivariate predictor function is modeled as a specified sum of a constant term, main effects, and selected interaction terms. Maximum likelihood estimate is used, where the maximization is taken over a suitably chosen approximating space. The approximating space is constructed from virtually arbitrary linear spaces of functions and their tensor products and is compatible with the assumed ANOVA structure on the predictor function. Under mild conditions, the maximum likelihood estimate is consistent and the components of the estimate in an appropriately defined ANOVA decomposition are consistent in estimating the corresponding components of the predictor function. When the predictor function does not satisfy the assumed ANOVA form, the estimate converges to its best approximation of that form relative to the expected log-likelihood. A rate of convergence result is obtained, which reinforces the intuition that low-order ANOVA modeling can achieve dimension reduction and thus overcome the curse of dimensionality
Multivariate Stable Densities as Functions of One Dimensional Projections

AbstractThe density of a generald-dimensional stable random vectorXis expressed as an integral over the sphere in Rdof a function of the parameters of the one dimensional projections ofX. These formulas give insight into the form of multivariate stable densities and are useful for numerical calculations. Corollaries give simplified expressions for symmetric stable and theα=1 strictly stable densities, relations among the densities in different dimensions, and values of the densities at the location parameter for all cases except theα=1, non-strictly stable ones. Expressions for the densities in the multidimensional analog of Zolotarev's (M) parameterization and a discussion of computational versions of the formulas are also given
Spherical Deconvolution

AbstractThis paper proposes nonparametric deconvolution density estimation overS2. Here we would think of theS2elements of interest being corrupted by randomSO(3) elements (rotations). The resulting density on the observations would be a convolution of theSO(3) density with the trueS2density. Consequently, the methodology, as in the Euclidean case, would be to use Fourier analysis onSO(3) andS2, involving rotational and spherical harmonics, respectively. We especially consider the case where the deconvolution operator is a bounded operator lowering the Sobolev order by a finite amount. Consistency results are obtained with rates of convergence calculated under the expectedS2and Sobolev square norms that are proportionally inverse to some power of the sample size. As an example we introduce the rotational version of the Laplace distribution
The Asymptotic Loss of Information for Grouped Data

AbstractWe study the loss of information (measured in terms of the Kullback–Leibler distance) caused by observing “grouped” data (observing only a discretized version of a continuous random variable). We analyze the asymptotical behaviour of the loss of information as the partition becomes finer. In the case of a univariate observation, we compute the optimal rate of convergence and characterize asymptotically optimal partitions (into intervals). In the multivariate case we derive the asymptotically optimal regular sequences of partitions. Furthermore, we compute the asymptotically optimal transformation of the data, when a sequence of partitions is given. Examples demonstrate the efficiency of the suggested discretizing strategy even for few intervals
Double Shrinkage Estimation of Common Coefficients in Two Regression Equations with Heteroscedasticity

AbstractThe problem of estimating the common regression coefficients is addressed in this paper for two regression equations with possibly different error variances. The feasible generalized least squares (FGLS) estimators have been believed to be admissible within the class of unbiased estimators. It is, nevertheless, established that the FGLS estimators are inadmissible in light of minimizing the covariance matrices if the dimension of the common regression coefficients is greater than or equal to three. Double shrinkage unbiased estimators are proposed as possible candidates of improved procedures
Algebraic Descriptions of Nominal Multivariate Discrete Data

AbstractTraditionally, multivariate discrete data are analyzed by means of log-linear models. In this paper we show how an algebraic approach leads naturally to alternative models, parametrized in terms of the moments of the distribution. Moreover we derive a complete characterization of all meaningful transformations of the components and show how transformations affect the moments of a distribution. It turns out that our models provide the necessary formal description of longitudinal data; moreover in the classical case, they can be considered as an analysis tool, complementary to log-linear models
Prediction and Classification of Non-stationary Categorical Time Series

AbstractPartial likelihood analysis of a general regression model for the analysis of non-stationary categorical time series is presented, taking into account stochastic time dependent covariates. The model links the probabilities of each category to a covariate process through a vector of time invariant parameters. Under mild regularity conditions, we establish good asymptotic properties of the estimator by appealing to martingale theory. Certain diagnostic tools are presented for checking the adequacy of the fit
Some General Characterizations of the Bivariate Gumbel Distribution and the Bivariate Lomax Distribution Based on Truncated Expectations

AbstractRecently attempts have been made to characterize probability distributions via truncated expectations in both univariate and multivariate cases. In this paper we will use a well known theorem of Lau and Rao (1982) to obtain some characterization results, based on the truncated expectations of a functionh, for the bivariate Gumbel distribution, a bivariate Lomax distribution, and a bivariate power distribution. The results of the paper subsume some earlier results appearing in the literature
Kaplan–Meier Estimator under Association

AbstractConsider a long term study, where a series of possibly censored failure times is observed. Suppose the failure times have a common marginal distribution functionF, but they exhibit a mode of dependence characterized by positive or negative association. Under suitable regularity conditions, it is shown that the Kaplan–Meier estimatorFnofFis uniformly strongly consistent; rates for the convergence are also provided. Similar results are established for the empirical cumulative hazard rate function involved. Furthermore, a stochastic process generated byFnis shown to be weakly convergent to an appropriate Gaussian process. Finally, an estimator of the limiting variance of the Kaplan–Meier estimator is proposed and it is shown to be weakly convergent
More Higher-Order Efficiency: Concentration Probability

AbstractBased on concentration probability of estimators about a true parameter, third-order asymptotic efficiency of the first-order bias-adjusted MLE within the class of first-order bias-adjusted estimators has been well established in a variety of probability models. In this paper we consider the class of second-order bias-adjusted Fisher consistent estimators of a structural parameter vector on the basis of an i.i.d. sample drawn from a curved exponential-type distribution, and study the asymptotic concentration probability, about a true parameter vector, of these estimators up to the fifth-order. In particular, (i) we show that third-order efficient estimators are always fourth-order efficient; (ii) a necessary and sufficient condition for fifth-order efficiency is provided; and finally (iii) the MLE is shown to be fifth-order efficient
A Generalization of Rao's Covariance Structure with Applications to Several Linear Models

AbstractThis paper presents a generalization of Rao's covariance structure. In a general linear regression model, we classify the error covariance structure into several categories and investigate the efficiency of the ordinary least squares estimator (OLSE) relative to the Gauss–Markov estimator (GME). The classification criterion considered here is the rank of the covariance matrix of the difference between the OLSE and the GME. Hence our classification includes Rao's covariance structure. The results are applied to models with special structures: a general multivariate analysis of variance model, a seemingly unrelated regression model, and a serial correlation model
Convergence Rates for Logspline Tomography

AbstractWe consider bivariate logspline density estimation for tomography data. In the usual logspline density estimation for bivariate data, the logarithm of the unknown density function is estimated by tensor product splines, the unknown parameters of which are given by maximum likelihood. In this paper we use tensor product B-splines and the projection-slice theorem to construct the logspline density estimators for tomography data. Rates of convergence are established for log-density functions assumed to belong to a Besov space
A Note on the Comedian for Elliptical Distributions

AbstractThe comedianCOM(X,Y) of random variablesX,Yis a median based robust alternative to the covariance ofXofY. For the bivariate normal case it is known thatCOM(X,Y), standardized by the median absolute deviations ofXandY, is a symmetric, strictly increasing and continuous function of the correlation coefficientρwith range [−1,1] and can therefore serve as a robust alternative toρ. We show that this result, which is not true in general, extends to elliptical distributions even in the case where moments ofX,Ydo not exist
Mean Location and Sample Mean Location on Manifolds: Asymptotics, Tests, Confidence Regions

AbstractIn a previous investigation we studied some asymptotic properties of the sample mean location on submanifolds of Euclidean space. The sample mean location generalizes least squares statistics to smooth compact submanifolds of Euclidean space. In this paper these properties are put into use. Tests for hypotheses about mean location are constructed and confidence regions for mean location are indicated. We study the asymptotic distribution of the test statistic. The problem of comparing mean locations for two samples is analyzed. Special attention is paid to observations on Stiefel manifolds including the orthogonal groupO(p) and spheresSk−1, and special orthogonal groupsSO(p). The results also are illustrated with our experience with simulations
Geometric Ergodicity of Gibbs and Block Gibbs Samplers for a Hierarchical Random Effects Model

AbstractWe consider fixed scan Gibbs and block Gibbs samplers for a Bayesian hierarchical random effects model with proper conjugate priors. A drift condition given in Meyn and Tweedie (1993, Chapter 15) is used to show that these Markov chains are geometrically ergodic. Showing that a Gibbs sampler is geometrically ergodic is the first step toward establishing central limit theorems, which can be used to approximate the error associated with Monte Carlo estimates of posterior quantities of interest. Thus, our results will be of practical interest to researchers using these Gibbs samplers for Bayesian data analysis
Data Driven Smooth Tests for Bivariate Normality

AbstractBased upon the idea of construction of data driven smooth tests for composite hypotheses presented in Inglotet al.(1997) and Kallenberg and Ledwina (1997), two versions of data driven smooth test for bivariate normality are proposed. Asymptotic null distributions are derived, and consistency of the newly introduced tests against every bivariate alternative with marginals having finite variances is proved. Included results of power simulations show that one of the proposed tests performs very well in comparison with other commonly used tests for bivariate normality
Testing for Changes in Multivariate Dependent Observations with an Application to Temperature Changes

AbstractWe develop procedures for testing for changes in the mean of multivariatem-dependent stationary processes. Several test statistics are considered and corresponding limit theorems are derived. These include functional and Darling–Erdo&#x030B;s type limit theorems. The tests are shown to be consistent under alternatives of abrupt and gradual changes in the mean. Finite sample performance is examined by means of a simulation study, and the procedures are applied to the analysis of the average monthly temperatures in Prague
Rates of Convergence for Spline Estimates of Additive Principal Components

AbstractAdditive principal components(APCs) generalize classicalprincipal component analysisto additive nonlinear transformations.Smallest APCsare additive functions of the vectorX=(X1,…,Xp) minimizing the variance under orthogonality constraints and are characterized as eigenfunctions of an operator which is compact under a standard condition on the joint distribution of (X1,…,Xp). As a by-product,smallest APCnearly satisfies the equation ∑jφj(Xj)=0 and then provides powerful tools for regression and data analysis diagnostics. The principal aim of this paper is the estimation of smallest APCs based on a sample from the distribution ofX. This is achieved using additive splines, which have been recently investigated in several functional estimation problems. The rates of convergence are then derived under mild conditions on the component functions. These rates are the same as the optimal rates for a nonparametric estimate of a univariate regression function