pinecone/core-2020-05-10-deduplication huggingface dataset:
Universally Consistent Regression Function Estimation Using Hierarchial B-Splines

AbstractEstimation of multivariate regression functions from i.i.d. data is considered. We construct estimates by empiricalL2-error minimization over data-dependent spaces of polynomial spline functions. For univariate regression function estimation these spaces are spline spaces with data-dependent knot sequences. In the multivariate case, we use so-called hierarchical spline spaces which are defined as linear span of tensor product B-splines with nested knot sequences. The knot sequences of the chosen B-splines depend locally on the data. We show the strongL2-consistency of the estimators without any condition on the underlying distribution. The estimators are similar to histogram regression estimators using data-dependent partitions and partitioning regression estimators based on local polynomial fits. The main difference is that the estimators considered here are smooth functions, which seems to be desirable especially in the case that the regression function to be estimated is smooth
Dynamic Linkages for Multivariate Distributions with Given Nonoverlapping Multivariate Marginals

AbstractOne of the most useful tools for handling multivariate distributions with givenunivariatemarginals is the copula function. Using it, any multivariate distribution function can be represented in a way that emphasizes the separate roles of the marginals and of the dependence structure. Liet al.(1996) introduced an analogous tool, called linkage, which is useful for handling multivariate distributions with givenmultivariatemarginals. The goal of the present paper is to introduce a new kind of linkage, called thedynamic linkage, which can usefully handle multivariate life distributions (that is, distributions of non-negative random variables) by taking advantage of the time dynamics of the underlying lifetimes. Like the linkages of Liet al.(1996), the new dynamic linkage can be used for the study of multivariate distributions with given multivariate marginals by emphasizing the separate roles of the dependence structureamongthe given multivariate marginals and the dependence structurewithineach of the nonoverlapping marginals. Preservation of some setwise positive dependence properties, from the dynamic linkage functionLto the joint distributionFand vice versa, are studied. When two different distribution functions are associated with the same dynamic linkage (that is, have the same setwise dependence structure) we show that the cumulative hazard order among the corresponding multivariate marginal distributions implies an overall stochastic dominance between the two underlying distribution functions
A Nonsymmetric Correlation Inequality for Gaussian Measure

AbstractLetμbe a Gaussian measure (say, onRn) and letK,L⊆Rnbe such thatKis convex,Lis a “layer” (i.e.,L={x:a⩽〈x,u〉⩽b} for somea,b∈Randu∈Rn), and the centers of mass (with respect toμ) ofKandLcoincide. Thenμ(K∩L)⩾μ(K)·μ(L). This is motivated by the well-known “positive correlation conjecture” for symmetric sets and a related inequality of Sidak concerning confidence regions for means of multivariate normal distributions. The proof uses the estimateΦ(x)> 1−((8/π)1/2/(3x+(x2+8)1/2))e−x2/2,x>−1, for the (standard) Gaussian cumulative distribution function, which is sharper than the classical inequality of Komatsu
Geometrical Aspects of Discrimination by Multilayer Perceptrons

AbstractWe investigate some geometrical aspects of the discriminant functions of the kindfp(x)=∑pk=1ckτ(a′kx) for suitable constantsak,ckwhereτis a sigmoidal transformation. This function is realized by a multilayer perceptron with one hidden layer. These results are applied in the analysis of the discriminating power offp. In particular, we prove that the class of finite populationsΩ1andΩ2that can be distinguished byfpis monotonically increasing inpand we give a minimal sufficientpleading to a complete separation betweenΩ1andΩ2
Screening among Multivariate Normal Data

AbstractThis paper considers the problem of screeningkmultivariate normal populations (secondary data) with respect to a control population (primary data) in terms of covariance structure. A screening procedure, developed based upon statistical ranking and selection theory, is designed to include in the selected subset those populations which have the same (or similar) covariance structure as the control population, and exclude those populations which differ significantly. Formulas for computing the probability of a correct selection and the least favorable configuration are developed. The sample size required to achieve a specific probability requirement is also developed, with results presented in tabular form. This secondary data selection procedure is illustrated via an example with applications to radar signal processing
Improving on the Best Affine Equivariant Estimator of the Ratio of Generalized Variances

AbstractWe consider the problem of decision-theoretic estimation of the ratio of generalized variances of two matrix normal distributions with unknown means under a general loss function. The inadmissibility of the best affine equivariant estimator is established by exhibiting various improved estimators. In particular, under certain conditions on the loss, two classes of improved procedures based onallthe available data are presented. As a preliminary result of independent interest, an improved estimator of an arbitrary power of the generalized variance of a matrix normal distribution with an unknown mean is derived under a general strictly bowl-shaped loss
Predictive Inference for the Elliptical Linear Model

AbstractThis paper derives the prediction distribution of future responses from the linear model with errors having an elliptical distribution with known covariance parameters. For unknown covariance parameters, the marginal likelihood function of the parameters has been obtained and the prediction distribution has been modified by replacing the covariance parameters by their estimates obtained from the marginal likelihood function. It is observed that the prediction distribution with elliptical error has a multivariate Student'st-distribution with appropriate degrees of freedom. The results for some special cases such as the Intra-class correlation model, AR(1), MA(1), and ARMA(1,1) models have been obtained from the general results. As an application, theβ-expectation tolerance region has been constructed. An example has been added
Efficient ML Estimation of the Multivariate Normal Distribution from Incomplete Data

AbstractIt is well known that the maximum likelihood estimates (MLEs) of a multivariate normal distribution from incomplete data with a monotone pattern have closed-form expressions and that the MLEs from incomplete data with a general missing-data pattern can be obtained using the Expectation-Maximization (EM) algorithm. This article gives closed-form expressions, analogous to the extension of the Bartlett decomposition, for both the MLEs of the parameters and the associated Fisher information matrix from incomplete data with a monotone missing-data pattern. For MLEs of the parameters from incomplete data with a general missing-data pattern, we implement EM and Expectation-Constrained-Maximization-Either (ECME), by augmenting the observed data into a complete monotone sample. We also provide a numerical example, which shows that the monotone EM (MEM) and monotone ECME (MECME) algorithms converge much faster than the EM algorithm
Second-Order Properties of a Two-Stage Fixed-Size Confidence Region for the Mean Vector of a Multivariate Normal Distribution

AbstractWe consider the classical fixed-size confidence region estimation problem for the mean vectorμin theNp(μ,Σ) population where Σ is unknown but positive definite. We writeλ1for the largest characteristic root of Σ and assume thatλ1is simple. Moreover, we suppose that, in many practical applications, we will often have available a numberλ*(>0) and that we can assumeλ1>λ*. Given this addi- tional, and yet very minimal, knowledge regardingλ1, the two-stage procedure of Chatterjee (Calcutta Statist. Assoc. Bull.8(1959a), 121–148;9(1959b), 20–28;11(1962), 144–159) is revised appropriately. The highlight in this paper involves the verification ofsecond-order propertiesassociated with such revised two-stage estimation techniques, along with the maintenance of the nominal confidence coefficient
Characterization Theorems when Variables Are Measured with Error

AbstractLinear regression models are studied when variables of interest are observed in the presence of measurement error. Techniques involving Fourier transforms that lead to simple differential equations with unique solutions are used in the context of multiple regression. Necessary and sufficient conditions are proven for a random vector of measurement error of the independent variable to be multivariate normal. One characterization involves the Fisher score of the observed vector. A second characterization involves the Hessian matrix of the observed density
Estimation of Partial Linear Error-in-Variables Models with Validation Data

AbstractConsider the partial linear models of the formY=Xτβ+g(T)+e, where thep-variate explanatoryXis erroneously measured, and bothTand the responseYare measured exactly. LetXbe the surrogate variable forXwith measurement error. Let the primary data set be that containing independent observations on (Y,X,T) and the validation data set be that containing independent observations on (X,X,T), where the exact observations onXmay be obtained by some expensive or difficult procedures for only a small subset of subjects enrolled in the study. In this paper, without specifying any structure equation and the distribution assumption ofXgivenX, a semiparametric method with the primary data is employed to obtain the estimators ofβandg(·) based on the least-squares criterion with the help of validation data. The proposed estimators are proved to be strongly consistent. The asymptotic representation and the asymptotic normality of the estimator ofβare derived, respectively. The rate of the weak consistency of the estimator ofg(·) is also obtained
An Extension of the Bivariate Method of Polynomials and a Reduction Formula for Bonferroni-Type Inequalities

AbstractLetA1,A2,…,ANandB1,B2,…,BMbe two sequences of events, and letνN(A) andνM(B) be the number of thoseAiandBj, respectively, that occur. We prove that Bonferroni-type inequalities forP(νN(A)⩾u,νM(B)⩾v), whereuandvare positive integers, are valid if and only if they are valid for a two dimensional triangular array of independent eventsAiandBj, withP(Ai)=p1andP(Bj)=p2for alliandj. This result allows to derive a formula from which arbitrary Bonferroni-type inequalities of the above type are reduced to the special case of no events occurring. Such methods for proof and similar reduction formula were so far available only for the case of exactlyuandvevents occurring. Several new inequalities are obtained by using our results
Radial Positive Definite Functions Generated by Euclid's Hat

AbstractRadial positive definite functions are of importance both as the characteristic functions of spherically symmetric probability distributions, and as the correlation functions of isotropic random fields. The Euclid's hat functionhn(‖x‖),x∈Rn, is the self-convolution of an indicator function supported on the unit ball in Rn. This function is evidently radial and positive definite, and so are its scale mixtures that form the classHn. Our main results characterize the classesHn,n⩾1, andH∞=∩n⩾1Hn. This leads to an analogue of Pólya's criterion for radial functions on Rn,n⩾2: Ifϕ:[0,∞)→R is such thatϕ(0)=1,ϕ(t) is continuous, limt→∞ϕ(t)=0, and(−1)kdkdtk[−ϕ′(t)]is convex fork=[(n−2)/2], the greatest integer less than or equal to (n−2)/2, thenϕ(‖x‖) is a characteristic function in Rn. Along the way, side results on multiply monotone and completely monotone functions occur. We discuss the relations ofHnto classes of radial positive definite functions studied by Askey (Technical Report No. 1262, Math. Res. Center, Univ. of Wisconsin–Madison), Mittal (Pacific J. Math.64(1976), 517–538), and Berman (Pacific J. Math.78(1978), 1–9), and close with hints at applications in geostatistics
On Confidence Intervals in Nonparametric Binary Regression via Edgeworth Expansions

AbstractLocal confidence intervals for regression function with binary response variable are constructed. These intervals are based on both theoretical and “plug-in” normal asymptotic distribution of a usual statistic. In the plug-in approach, two ways of estimating bias are proposed; for them we obtain the mean squared error and deduce an expression of an optimal bandwidth. The rate of convergence of theoretical distributions to their limits is obtained by means of Edgeworth expansions. Likewise, these expansions allow us to deduce properties about the coverage probability of the confidence intervals. Theoretic approximations to that probability are compared in a simulation study with the corresponding coverage rates
Nonparametric Empirical Bayes Estimation of the Matrix Parameter of the Wishart Distribution

AbstractWe consider independent pairs (X1, Σ1), (X2, Σ2), …, (Xn, Σn), where eachΣiis distributed according to some unknown density functiong(Σ) and, givenΣi=Σ,Xihas conditional density functionq(x∣Σ) of the Wishart type. In each pair the first component is observable but the second is not. After the (n+1)th observationXn+1is obtained, the objective is to estimateΣn+1corresponding toXn+1. This estimator is called the empirical Bayes (EB) estimator ofΣ. An EB estimator ofΣis constructed without any parametric assumptions ong(Σ). Its posterior mean square risk is examined, and the estimator is demonstrated to be pointwise asymptotically optimal
A Strong Representation of the Product-Limit Estimator for Left Truncated and Right Censored Data

AbstractIn this paper we consider the TJW product-limit estimatorFn(x) of an unknown distribution functionFwhen the data are subject to random left truncation and right censorship. An almost sure representation of PL-estimatorFn(x) is derived with an improved error bound under some weaker assumptions. We obtain the strong approximation ofFn(x)−F(x) by Gaussian processes and the functional law of the iterated logarithm is proved for maximal derivation of the product-limit estimator toF. A sharp rate of convergence theorem concerning the smoothed TJW product-limit estimator is obtained. Asymptotic properties of kernel estimators of density function based on TJW product-limit estimator is given
Permutation Tests for Multivariate Location Problems

AbstractThe paper presents some permutation test procedures for multivariate location. The tests are based on projected univariate versions of multivariate data. For one-sample cases, the tests are affine invariant and strictly distribution-free for the symmetric null distribution with elliptical direction and their permutation counterparts are conditionally distribution-free when the underlying null distribution of the sample is angularly symmetric. For multi-sample cases, the tests are also affine invariant and permutation counterparts of the tests are conditionally distribution-free for any null distribution with certain continuity. Hence all of the tests in this paper are exactly valid. Furthermore, the equivalence, in the large sample sense, between the tests and their permutation counterparts are established. The power behavior of the tests and of their permutation counterparts under local alternative are investigated. A simulation study shows the tests to perform well compared with some existing tests in the literature, particularly when the underlying null distribution is symmetric whether light-tailed or heavy-tailed. For revealing the influence of data sparseness on the effect of the test, some simulations with different dimensions are also performed
A Characterization of Quasi-copulas

AbstractThe notion of quasi-copula was introduced by C. Alsina, R. B. Nelsen, and B. Schweizer (Statist. Probab. Lett.(1993), 85–89) and was used by these authors and others to characterize operations on distribution functions that can or cannot be derived from operations on random variables. In this paper, the concept of quasi-copula is characterized in simpler operational terms and the result is used to show that absolutely continuous quasi-copulas are not necessarily copulas, thereby answering in the negative an open question of the above mentioned authors
A Central Limit Theorem for Local Polynomial Backfitting Estimators

AbstractAdditive models based on backfitting estimators are among the most important recent contributions to modern statistical modelling. However, the statistical properties of backfitting estimators have received relatively little attention. Recently, J.-D. Opsomer and D. Ruppert (1997,Ann. Statist.25, 186-211; 1998,J. Amer. Statist. Assoc.93, 605–619) and J.-D. Opsomer (1997, preprint 96-12, Department of statistics, Iowa State University) derived their mean squared error properties in the case of local polynomial smoothers. In this paper the asymptotic distributional behaviour of backfitting estimators is investigated
Nonparametric Regression with Singular Design

AbstractTheories of nonparametric regression are usually based on the assumption that the design density exists. However, in some applications such as those involving high-dimensional or chaotic time series data, the design measure may be singular and may be likely to have a fractal (nonintegral) dimension. In this paper, the popular Nadaraya–Watson estimator is studied under the general setup that the continuity of the design measure is governed by the local or pointwise dimension. It will be shown in the iid setup that the nonparametric regression estimator achieves a convergence rate which is dependent only on the pointwise dimension. The case of time series data is also studied. For the latter case, a new mixing condition is introduced, and an assumption of marginal or joint density is completely avoided. Three examples, a fractal regression and two applications for predicting chaotic time series, are used to illustrate the implications of the obtained results
Bayesian Estimation for Spherically Symmetric Distributions

AbstractThe theory of Bayesian least squares is developed for a general and more tangible notion of conjugacy than in models which make the more conventional assumption of normality. This paper is primarily concerned with extending the results of classical conjugate normal-normal Bayesian analysis to the canonical setting of the generalized linear model when, at the same time, the sampling distribution and the prior are spherically symmetric. In order to underline the intrinsic aspect of our results, the approach of multivariate analysis adopted here is coordinate free. Examples which illustrate the theory are also presented
On the Efficiencies of Several Generalized Least Squares Estimators in a Seemingly Unrelated Regression Model and a Heteroscedastic Model

AbstractThis paper investigates the efficiencies of several generalized least squares estimators (GLSEs) in terms of the covariance matrix. Two models are analyzed: a seemingly unrelated regression model and a heteroscedastic model. In both models, we define a class of unbiased GLSEs and show that their covariance matrices remain the same even if the distribution of the error term deviates from the normal distributions. The results are applied to the problem of evaluating the lower and upper bounds for the covariance matrices of the GLSEs
Weighting Games in Robust Linear Regression

AbstractIn a number of problems, interest is centered on only a few of the coefficients of the multiple linear regression model, while the remaining parameters are treated as nuisance parameters. At the same time, the experimenter is interested in estimating the parameters robustly. We propose a new weighting scheme which generates estimators for the parameters of interest that are more efficient than their bounded influence counterparts. The new weighting scheme differentially downweights the components of the explanatory variables and produces the estimators as solutions of a set of estimating equations. Moreover, the differential downweighting allows us to maintain a bound on a selected sensitivity while increasing the efficiency of the subvector of parameters of interest. We study the covariance structure of the new estimators and derive conditions which guide us in the weight construction
On Robust Bayesian Analysis for Location and Scale Parameters

AbstractDawid (1973,Biometrika60, 664–666) stated conditions in the univariate location model with known scale parameter needed for there to be either vanishing likelihood or prior influence on the posterior distribution when there is a conflict between likelihood and prior. More recently, Pericchi and Sansó (1995,Biometrika82, 223–225) noted that there are distributions that partially satisfy Dawid's conditions but have bounded rather than vanishing influence on the posterior distribution. In this paper, we present the extension of these results for the location and scale model using the multivariatev-spherical distributions. We show that when thev(·)=‖·‖ function is a norm, the ‖‖-spherical distributions, exponential power, and logistic power provide a robust analysis for the location model with known scale parameter, whereas Student's powertprovides a robust analysis for the location and scale model. Robust analyses are illustrated for normal-gamma prior location and scale models. Numerical computations are implemented via the Gibbs sampler
Rank-Score Tests in Factorial Designs with Repeated Measures

AbstractNonparametric factorial designs for multivariate observations are considered under the framework of general rank-score statistics. Unlike most of the literature, we do not assume the continuity of the underlying distribution functions. The models studied include general repeated measures designs, compound symmetry designs, and designs for longitudinal data. In particular, designs for ordered categorical data are included. The vectors of the multivariate observations may have different lengths. Moreover, our general framework includes missing values and singular covariance matrices which occur quite frequently in practical data analysis problems. The asymptotic properties of the proposed statistics are studied under general nonparametric hypotheses as well as under a sequence of nonparametric contiguous alternatives. L2-consistent estimators for the unknown covariance matrices are given and two types of quadratic forms are considered for testing the nonparametric hypotheses. The results are applied to a two-way mixed model assuming compound symmetry and to a factorial design for longitudinal data. The main idea of the proofs is based on some moment inequalities for empirical distribution functions in mixed models. The details are provided in the Appendix
Shortcomings of Generalized Affine Invariant Skewness Measures

This paper studies the asymptotic behavior of a generalization of Mardia's affine invariant measure of (sample) multivariate skewness. If the underlying distribution is elliptically symmetric, the limiting distribution is a finite sum of weighted independent ξ2-variates, and the weights are determined by three moments of the radial distribution of the corresponding spherically symmetric generator. If the population distribution has positive generalized skewness a normal limiting distribution occurs. The results clarify the shortcomings of generalized skewness measures when used as statistics for testing for multivariate normality. Loosely speaking, normality will be falsely accepted for a short-tailed non-normal elliptically symmetric distribution, and it will be correctly rejected for a long-tailed non-normal elliptically symmetric distribution. The wrong diagnosis in the latter case, however, would be rejection due to positive skewness