pinecone/core-2020-05-10-deduplication huggingface dataset:
Functional Central Limit Theorems for Triangular Arrays of Function-Indexed Processes under Uniformly Integrable Entropy Conditions

AbstractFunctional central limit theorems for triangular arrays of rowwise independent stochastic processes are established by a method replacing tail probabilities by expectations throughout. The main tool is a maximal inequality based on a preliminary version proved by P. Gaenssler and Th. Schlumprecht. Its essential refinement used here is achieved by an additional inequality due to M. Ledoux and M. Talagrand. The entropy condition emerging in our theorems was introduced by K. S. Alexander, whose functional central limit theorem for so-calledmeasure-like processeswill be also regained. Applications concern, in particular, so-calledrandom measure processeswhich include function-indexed empirical processes and partial-sum processes (with random or fixed locations). In this context, we obtain generalizations of results due to K. S. Alexander, M. A. Arcones, P. Gaenssler, and K. Ziegler. Further examples include nonparametric regression and intensity estimation for spatial Poisson processes
Wishart and Pseudo-Wishart Distributions and Some Applications to Shape Theory

AbstractSuppose thatX∽NN×m(μ,Σ,Θ). An expression for the density function is given whenΣ⩾0 and/orΘ:⩾0. An extension of Uhlig's result (Uhlig [17]) is expanded for the singular value decomposition of a matrixZof orderN×mwhen the rank (Z)=q⩽min(N,m). This paper fills an important gap in unifying, for the first time, all Wishart and pseudo-Wishart distributions, whether central or noncentral, whether singular or nonsingular, and applying them in shape analysis. In particular, the shape density and the size-and-shape cone density are obtained for the singular general case
On the Effect of Inliers on the Spatial Median

AbstractWe point out that inliers adversely affect performance of the spatial median and its generalization due to Gentleman. They are most deleterious in the case of the median itself, and in the important setting of two dimensions. There, the second term in a stochastic expansion of the median has a component with a Cauchy limiting distribution, and does not have any finite moments. This term is substantially determined by a small number of extreme, inlying data values. The implications for bootstrap methods are significant, since the bootstrap is notoriously poor in capturing properties of extremes. Indeed, the bootstrap does not accurately approximate second-order features of the distribution of the two-dimensional spatial median. We suggest a Winsorizing device for alleviating the effects of inliers. The issue of outliers is also discussed
Multivariate Hazard Rates under Random Censorship

AbstractThe data consists of multivariate failure times under right random censorship. By the kernel smoothing technique, convolutions of cumulative multivariate hazard functions suggest estimators of the so-called multivariate hazard functions. We establish strong i.i.d. representations and uniform bounds of the remainder terms on some compact sets of the underlying space. Thus asymptotic normality and uniform consistency on such sets are obtained. The asymptotic mean squared error gives an optimal bandwidth by the plug-in method. Simulations assess the performance of our estimators
Unbiased Estimation for a Multivariate Exponential whose Components have a Common Shift

AbstractIt is shown that for independent and identically distributed random vectors, for which the components are independent and exponentially distributed with a common shift, we can construct unbiased estimators of their density, derived from the Uniform Minimum Variance Unbiased Estimator (UMVUE) of their distribution function. As direct applications of the UMVUEs of the density functions we present a Chi-square goodness of fit test of the model, and give two tables of the UMVUEs of some commonly used functions of the unknown parameters of the multivariate exponential model considered in this paper
Compact Matrix Expressions for Generalized Wald Tests of Equality of Moment Vectors

AbstractAsymptotic chi-squared test statistics for testing the equality of moment vectors are developed. The test statistics proposed are generalized Wald test statistics that specialize for different settings by inserting an appropriate asymptotic variance matrix of sample moments. Scaled test statistics are also considered for dealing with nonstandard conditions. The specialization will be carried out for testing the equality of multinomial populations, and the equality of variance and correlation matrices for both normal and nonnormal data. When testing the equality of correlation matrices, a scaled version of the normal theory chi-squared statistic is proven to be an asymptotically exact chi-squared statistic in the case of elliptical data
Permutation Tests for Reflected Symmetry

AbstractThe paper presents a permutation procedure for testing reflected (or diagonal) symmetry of the distribution of a multivariate variable. The test statistics are based in empirical characteristic functions. The resulting permutation tests are strictly distribution free under the null hypothesis that the underlying variables are symmetrically distributed about a center. Furthermore, the permutation tests are strictly valid if the symmetric center is known and are asymptotic valid if the center is an unknown point. The equivalence, in the large sample sense, between the tests and their permutation counterparts are established. The power behavior of the tests and their permutation counterparts under local alternative are investigated. Some simulations with small sample sizes (⩽20) are conducted to demonstrate how the permutation tests works
Random Deletion Does Not Affect Asymptotic Normality or Quadratic Negligibility

AbstractSuppose a number of points are deleted from a sample of random vectors in Rd. The number of deleted points may depend on the sample sizen, and on any other sample information, provided only that it is bounded in probability asn→∞. In particular, “extremes” of the sample, however defined, may be deleted. We show that this operation has no effect on the asymptotic normality of the sample sum, in the sense that the sum of the deleted sample is asymptotically normal, after norming and centering, if and only if the sample sum itself is asymptotically normal with the same norming and centering as the deleted sum. That is, the sample must be drawn from a distribution in the domain of attraction of the multivariate normal distribution. The domain of attraction concept we employ uses general operator norming and centering, as developed by Hahn and Klass. We also show that random deletion has no effect on the “quadratic negligibility” of the sample. These are conditions that are important in the robust analysis of multivariate data and in regression problems, for example
On Inconsistency of the Jackknife-after-Bootstrap Bias Estimator for Dependent Data

AbstractB. Efron introducedjackknife-after-bootstrapas a computationally efficient method for estimating standard errors of bootstrap estimators. In a recent paper consistency of the jackknife-after-bootstrap variance estimators has been established for different bootstrap quantities for independent and dependent data. In this paper, it is shown that in the dependent case, the standard jackknife-after-bootstrap estimator for the bias of block bootstrap quantities is inconsistent for almost any sensible choice of the blocking parameters. Some alternative bias estimators are proposed and shown to be consistent
ML Estimation of the MultivariatetDistribution and the EM Algorithm

AbstractMaximum likelihood estimation of the multivariatetdistribution, especially with unknown degrees of freedom, has been an interesting topic in the development of the EM algorithm. After a brief review of the EM algorithm and its application to finding the maximum likelihood estimates of the parameters of thetdistribution, this paper provides new versions of the ECME algorithm for maximum likelihood estimation of the multivariatetdistribution from data with possibly missing values. The results show that the new versions of the ECME algorithm converge faster than the previous procedures. Most important, the idea of this new implementation is quite general and useful for the development of the EM algorithm. Comparisons of different methods based on two datasets are presented
A New Class of Consistent Estimators for Stochastic Linear Regressive Models

AbstractIn this paper we propose a new approach for estimating the unknown parameter in the stochastic linear regressive model with stationary ergodic sequence of covariates. Under mild conditions on the joint distribution of the covariate and the error, the estimator constructed is shown to be strongly consistent in two important special cases: (1) The sequence of (variate, covariate) is independent identically distributed (i.i.d.), and (2) the sequence of variates is a stationary autoregressive series. The asymptotical normality is also discussed under more assumptions on the distribution of the covariate
On Domains of Attraction of Multivariate Extreme Value Distributions under Absolute Continuity

AbstractThe paper gives sufficient conditions for domains of attraction of multivariate extreme value distributions. Under the assumption of absolute continuity of a multivariate distribution, the criteria enable one to examine, by using limits of some rescaled conditional densities, whether the distribution belongs to the domain of attraction of some multivariate extreme value distribution. If this is the case, the criteria also determine how to construct such an extreme value distribution. Unlike the criterion given by de Haan and Resnick [1987,Stochastic Process. Appl.2583–93], the criteria are easily applicable even when the marginal tails are not Pareto-like
Best Attainable Rates of Convergence for Estimators of the Stable Tail Dependence Function

AbstractIt is well known that a bivariate distribution belongs to the domain of attraction of an extreme value distribution G if and only if the marginals belong to the domain of attraction of the univariate marginal extreme value distributions and the dependence function converges to the stable tail dependence function of G. Hall and Welsh (1984,Ann. Statist.12, 1079–1084) and Drees (1997b,Ann. Statist., to appear) addressed the problem of finding optimal rates of convergence for estimators of the extreme value index of an univariate distribution. The present paper deals with the corresponding problem for the stable tail dependence function. First an upper bound on the rate of convergence for estimators of the stable tail dependence function is established. Then it is shown that this bound is sharp by proving that it is attained by the tail empirical dependence function. Finally, we determine the limit distribution of this estimator if the dependence function satisfies a certain second-order condition
Modeling Probabilistic Networks of Discrete and Continuous Variables

AbstractIn this paper we show how discrete and continuous variables can be combined using parametric conditional families of distributions and how the likelihood weighting method can be used for propagating uncertainty through the network in an efficient manner. To illustrate the method we use, as an example, the damage assessment of reinforced concrete structures of buildings and we formalize the steps to be followed when modeling probabilistic networks. We start with one set of conditional probabilities. Then, we examine this set for uniqueness, consistency, and parsimony. We also show that cycles can be removed because they lead to redundant probability information. This redundancy may cause inconsistency, hence the probabilities must be checked for consistency. This examination may require a reduction to an equivalent set instandard canonicalform from which one can always construct a Bayesian network, which is the most convenient model. We also perform a sensitivity analysis, which shows that the model is robust
The Generalized Integer Gamma Distribution—A Basis for Distributions in Multivariate Statistics

AbstractIn this paper the distribution of the sum of independentΓvariables with different scale parameters is obtained by direct integration, without involving a series expansion. One of its particular cases is the distribution of the product of some particular independent Beta variables. Both distributions are obtained in a concise manageable form readily applicable to research work in the area of multivariate statistics distributions. The exact distribution of the generalized Wilks'Λstatistic is then obtained as a direct application of the results
Uniformity of Double Saddlepoint Conditional Probability Approximations

AbstractThis paper presents results showing that the error involved in using the double saddlepoint distribution function approximations of Skovgaard (1987,J. Appl. Probab.24875–887) are uniformly bounded. Particular attention is paid to distributions of sufficient statistics arising from generalized linear models. This work is intended in part to validate the use of the Markov Chain Monte Carlo by Kolassa and Tanner (1994,J. Amer. Statist. Assoc.89697–702) using these conditional distribution function approximations
Onα-Symmetric Multivariate Characteristic Functions

AbstractAnn-dimensional random vector is said to have anα-symmetric distribution,α>0, if its characteristic function is of the formϕ((|u1|α+…+|un|α)1/α). We study the classesΦn(α) of all admissible functionsϕ:[0, ∞)→R. It is known that members ofΦn(2) andΦn(1) are scale mixtures of certain primitivesΩnandωn, respectively, and we show thatωnis obtained fromΩ2n−1byn−1 successive integrations. Consequently, curious relations between 1- and 2- (or spherically) symmetric distributions arise. An analogue of Askey's criterion gives a partial solution to a question of D. St. P. Richards: Ifϕ(0)=1,ϕis continuous, limt→∞ϕ(t)=0, andϕ(2n−2)(t) is convex, thenϕ∈Φn(1). The paper closes with various criteria for the unimodality of anα-symmetric distribution
Maximum Likelihood Estimation of Isotonic Normal Means with Unknown Variances

AbstractTo analyze the isotonic regression problem for normal means, it is usual to assume that all variances are known or unknown but equal. This paper then studies this problem in the case that there are no conditions imposed on the variances. Suppose that we have data drawn fromkindependent normal populations with unknown meansμi's and unknown variancesσ2i's, in which the means are restricted by a given partial ordering. This paper discusses some properties of the maximum likelihood estimates ofμi's andσ2i's under the restriction and proposes an algorithm for obtaining the estimates
Dimensionality in Manova Tested by a Closed Testing Procedure

AbstractThe decision on dimensionality of the space spanned by general linear functions of the parameter matrix of a MANOVA model is considered. This problem is related to the investigation, whether graphically or analytically, of significant empirical departures from the overall null hypothesis on these functions. A closed testing procedure for a sequence of relevant hypotheses is proposed. Unlike the classical procedures based on asymptotic distributions of the likelihood ratio statistics, the proposed method ensures that the Type I familywise error rate does not exceed the nominalα-level. Also, it is consistent with testing the overall null hypothesis, while relying on tests of subsequent linear hypotheses implied by the former. Examples are given to compare the proposed procedure with a classical one
Combining Estimates of Tectonic Plate Rotations: An Extension of Welch's Method to Spherical Regression

AbstractThe relative motion between two diverging tectonic plates is a rotation of the sphere. Given measurements of points on the boundaries of the plates, the rotation can be estimated by minimizing a function which is asymptotically (as the concentration parameter of the data distribution goes to infinity) the sum of squared residuals of a linear regression. The linear approximation permits construction of an asymptotic confidence region for the rotation. To estimate the relative motion between plates that converge, it is necessary to combine two or more rotations of diverging plates, and previous methods required the assumption that separate data sets for all of the rotation estimates have the same concentration parameter. This assumption is frequently contradicted by data, indicating heteroscedasticity in the linear regression model. One successful approach to the problem in the linear model due to Welch, involves sample size asymptotics. A similar solution in the non- linear model thus depends on two kinds of asymptotics. We examine two types of general spherical regression models where the parameter estimate is obtained by maximizing or minimizing a particular function. We establish conditions under which the function converges to a residual sum of squares of linear regression as both concentration parameter and sample size go to infinity. Applying the double-convergence asymptotics to tectonic plate data yields asymptotic confidence regions for combined rotation estimates. The confidence region constructions we propose are appropriate for small to moderate sample sizes. Two kinds of confidence regions are constructed, one of which uses all the data for the separate rotation estimates; the other is a conservative approximation of the first, using only summary statistics from the separate estimates. Simulation runs indicate that both of the new constructions produce confidence regions much more consistent with nominal size, particularly when sample sizes are very different and concentration parameters of the data sets are very unequal
Perturbation Inequalities and Confidence Sets for Functions of a Scatter Matrix

AbstractLetΣbe an unknown covariance matrix. Perturbation (in)equalities are derived for various scale-invariant functionals ofΣsuch as correlations (including partial, multiple and canonical correlations) or angles between eigenspaces. These results show that a particular confidence set forΣis canonical if one is interested in simultaneous confidence bounds for these functionals. The confidence set is based on the ratio of the extreme eigenvalues ofΣ−1S, whereSis an estimator forΣ. Asymptotic considerations for the classical Wishart model show that the resulting confidence bounds are substantially smaller than those obtained by inverting likelihood ratio tests
Elliptical Functional Models

AbstractIn this paper, functional models with not replications are investigated within the class of the elliptical distributions. Emphasis is placed on the special case of the Student-t distribution. Main results encompasses consistency and asymptotic normality of the maximum likelihood estimators. Due to the presence of incidental parameters, standard maximum likelihood methodology cannot be used to obtain the main results, which require extensions of some existing results related to elliptical distributions. Asymptotic relative efficiencies are reported which show that the generalized least squares estimator can be highly inefficient when compared with the maximum likelihood estimator under nonnormality
Cyclic Subspace Regression

AbstractBy use of cyclic subspaces, explicit connections between principal component regression (PCR) and partial least squares (PLS) are established that shed light onto why one method works better than the other. These connections clearly identify how both methods make use of calibration data in prediction. Moreover, developments leading to these connections show that they are particular manifestations of a more general easily described and implemented regression/prediction process referred to as cyclic subspace regression (CSR). This process not only contains PCR, PLS, and LS (least squares) as special cases but, also a finite number of other related intermediate or transitional regression techniques. Moreover, CSR shows that PCR, PLS, LS, and the related intermediates can be implemented by the same general procedure and that they differ only in the amount of information used from calibration data matrices. In addition to setting out the CSR procedure, the paper also supplies a robust numerical algorithm for its implementation which is used to show how procedures contained within CSR perform on a chemical data set
Asymptotics of Estimating Equations under Natural Conditions

AbstractIn a variety of statistical problems one needs to solve an equation in order to get an estimator. We consider the large sample properties of such estimators generated from samples that are not necessarily identically distributed. Very general assumptions that lead to the existence, strong consistency, and asymptotic normality of the estimators are given. A number of results that are useful in verifying the general assumptions are given and an example illustrates their use. General applications to maximum likelihood, iteratively reweighted least squares, and robust estimation are discussed briefly
Infinite Divisibility of Random Objects in Locally Compact Positive Convex Cones

AbstractRandom objects taking on values in a locally compact second countable convex cone are studied. The convex cone is assumed to have the property that the class of continuous additive positively homogeneous functionals is separating, an assumption which turns out to imply that the cone is positive. Infinite divisibility is characterized in terms of an analog to the Lévy–Khinchin representation for a generalized Laplace transform. The result generalizes the classical Lévy–Khinchin representation for non-negative random variables and the corresponding result for random compact convex sets inRn. It also gives a characterization of infinite divisibility for random upper semicontinuous functions, in particular for random distribution functions with compact support and, finally, a similar characterization for random processes on a compact Polish space
Model-Building Problem of Periodically Correlatedm-Variate Moving Average Processes

AbstractThe model-building problem of periodically correlatedm-variateq-dependent processes is considered. We show that for a given periodical autocovariance function of anm-variateMA(q) process there are two particular corresponding classes (that may reduce to one class) of periodic (equivalent) models. Furthermore, any other (intermediate) model is not periodic. It is, however, asymptotically periodic. The matrix coefficients of the particular periodic models are given in terms of limits of some periodic matrix continued fractions, which are a generalization of the classical periodic continued fractions (Wall, 1948). These periodic matrix continued fractions are particular solutions of some prospective and/or retrospective recursion equations, arising from the symbolic factorization of the associated linear autocovariance operator. In addition, we establish a procedure to calculate these limits. Numerical examples are given for the simple cases of periodically correlated univariate one- and two-dependent processes
Fixed-Width Simultaneous Confidence Intervals for Multinormal Means in Several Intraclass Correlation Models

AbstractThe problem of constructing fixed-width simultaneous confidence intervals for comparing mean vectors ofk(⩾2) independent multivariate normal distributions is considered when those covariance matrices have the intraclass correlation structures. Two-stage procedures are developed for which the simultaneous confidence levels are shown to beat least1−α, the preassigned nominal value, 0<α<1. Asymptotic efficiency properties are addressed. In the case of fixed and finite initial sample size, efficiency related issues are also discussed