pinecone/core-2020-05-10-deduplication huggingface dataset:
On Distribution-Free Tests for the Multivariate Two-Sample Location-Scale Model

AbstractIn this paper, we propose simple exact procedures for testing both a location shift and/or a scale change between two multivariate distributions. Our tests are strictly distribution-free and can be made either scale invariant or rotation invariant. Our approach combines a generalization of the Wilcoxon test based on projections of the data onto the first principal component, a generalization of the Siegel–Tukey test based on the concept of data depth, and a bivariate test for the location problem proposed by K. V. Mardia (1967, J. Roy. Statist. Soc. Ser. B29, 320–342). In addition, we show that the limiting null distribution of a test statistic proposed by R. Y. Liu and K. Singh (1993, J. Amer. Statist. Assoc.88, 252–260) does not depend on the depth considered
Likelihood-Based Local Polynomial Fitting for Single-Index Models

AbstractThe parametric generalized linear model assumes that the conditional distribution of a response Y given a d-dimensional covariate X belongs to an exponential family and that a known transformation of the regression function is linear in X. In this paper we relax the latter assumption by considering a nonparametric function of the linear combination βTX, say η0(βTX). To estimate the coefficient vector β and the nonparametric component η0 we consider local polynomial fits based on kernel weighted conditional likelihoods. We then obtain an estimator of the regression function by simply replacing β and η0 in η0(βTX) by these estimators. We derive the asymptotic distributions of these estimators and give the results of some numerical experiments
Minimax Hierarchical Empirical Bayes Estimation in Multivariate Regression

AbstractThe multivariate normal regression model, in which a vector y of responses is to be predicted by a vector x of explanatory variables, is considered. A hierarchical framework is used to express prior information on both x and y. An empirical Bayes estimator is developed which shrinks the maximum likelihood estimator of the matrix of regression coefficients across rows and columns to nontrivial subspaces which reflect both types of prior information. The estimator is shown to be minimax and is applied to a set of chemometrics data for which it reduces the cross-validated predicted mean squared error of the maximum likelihood estimator by 38%
An Adaptive Design for Multi-Arm Clinical Trials

AbstractThe randomized play-the-winner (RPW) rule is very useful in clinical trials for patient allocation with two treatments. L. J. Wei (1979, Ann. Statist.7, 291–296) introduces the generalized Friedman's urn (GFU) model to clinical trials of K treatments (as an extension of the RPW). In this paper, we propose a new adaptive design for multi-arm clinical trials. The proposed adaptive design proportionally depends on the success rates of each treatment, so that a treatment which is doing well is more likely to be assigned in future trials than a treatment which is doing poorly. The new design is more reasonable although it is no longer a GFU model. In the paper we show that the new design has some desirable asymptotic properties and that it has wider and easier applications in practice. Some simulations also support this new design
Set-Indexed Conditional Empirical and Quantile Processes Based on Dependent Data

AbstractWe consider a conditional empirical distribution of the form Fn(C∣x)=∑nt=1ωn(Xt−x)I{Yt∈C} indexed by C∈C, where {(Xt, Yt), t=1, …, n} are observations from a strictly stationary and strong mixing stochastic process, {ωn(Xt−x)} are kernel weights, and C is a class of sets. Under the assumption on the richness of the index class C in terms of metric entropy with bracketing, we have established uniform convergence and asymptotic normality for Fn(·∣x). The key result specifies rates of convergences for the modulus of continuity of the conditional empirical process. The results are then applied to derive Bahadur–Kiefer type approximations for a generalized conditional quantile process which, in the case with independent observations, generalizes and improves earlier results. Potential applications in the areas of estimating level sets and testing for unimodality (or multimodality) of conditional distributions are discussed
Asymptotic Expansions and Bootstrap Approximations in Factor Analysis

AbstractWe derive asymptotic expansions for the distributions of the normal theory maximum likelihood estimators of unique variances and uniquenesses (standardized unique variances) in the factor analysis model. Asymptotic expansions are given for the distributions of non-Studentized and also Studentized statistics to construct accurate confidence intervals. In the case of Studentized statistics, we investigate the accuracy of the asymptotic approximations to the exact distributions that are determined by Monte Carlo simulations. The results show that, compared with normal approximations, the asymptotic expansions generally improve the accuracy of the approximations in the tail area except for the cases of the uniqueness estimators whose true values are close to their upper bounds unity. We also compare three types of confidence intervals that are based on the distributions of the Studentized statistics; each of which employs normal approximation, asymptotic expansion of the percentile points of the Studentized statistic, and further modification using the bootstrap. The results show that while the first type of confidence intervals were far from equal-tailed, the latter two achieved better balance in both sides
Graph-Theoretic Procedures for Dimension Identification

AbstractWe consider the problem of identifying the dimension in which a sample of data points lives, when only their interpoint distances are known. We study as a random variable the average “reach” of vertices in the k-nearest-neighbors graph associated to the interpoint distance matrix, and we show how this variable can be used to accurately (from a probabilistic viewpoint) identify the unknown dimension at low computational cost. We discuss results that serve as the theoretical foundation for the methodology proposed. We illustrate how our method can help in dimension reduction procedures
Spatially Adaptive Splines for Statistical Linear Inverse Problems

AbstractThis paper introduces a new nonparametric estimator based on penalized regression splines for linear operator equations when the data are noisy. A local roughness penalty that relies on local support properties of B-splines is introduced in order to deal with spatial heterogeneity of the function to be estimated. This estimator is shown to be consistent under weak conditions on the asymptotic behaviour of the singular values of the linear operator. Furthermore, in the usual nonparametric settings, it is shown to attain optimal rates of convergence. Then its good performances are confirmed by means of a simulation study
A Characterization of Poisson–Gaussian Families by Convolution-Stability

AbstractIf the convolution of natural exponential families on Rd is still a natural exponential family, then the families are all Poisson–Gaussian, up to affinity. This statement is a generalization of the one-dimensional versions proved by G. Letac (1992, “Lectures on Natural Exponential Functions and Their Variance Functions,” Instituto de Matemática pura e aplicada: Monografias de matemática, 50, Rı́o de Janeiro) in the case of two families, and by D. Pommeret (1999, C. R. Acad. Sci. Ser. I328, 929–933) for more than two families
On the Dependence Structure of Certain Multi-dimensional Ito Processes and Corresponding Hitting Times

AbstractA direct approach to derive dependence properties among components of multi-dimensional stochastic processes has been discussed by N. Ebrahimi (1994, J. Multivariate Anal.50, 55–67). Dependence properties among hitting times involving multi-dimensional stochastic processes has been initiated by N. Ebrahimi (1987, J. Appl. Probab.24, 115–122) and explored further by N. Ebrahimi and T. Ramalingham (1988, J. Appl. Probab.25, 355–362; 1989, J. Appl. Probab.26, 287–295). Let X(t)=(X1(t), …, Xk(t)) be an Ito process assuming values in Rk. In this article, under certain conditions, we show that the process X(t) has a certain dependence structure. We also consider the first passage problem involving X(t) and discuss the dependence structure among hitting times of X1(t), …,  and Xk(t)
The Deepest Regression Method

AbstractDeepest regression (DR) is a method for linear regression introduced by P. J. Rousseeuw and M. Hubert (1999, J. Amer. Statis. Assoc.94, 388–402). The DR method is defined as the fit with largest regression depth relative to the data. In this paper we show that DR is a robust method, with breakdown value that converges almost surely to 1/3 in any dimension. We construct an approximate algorithm for fast computation of DR in more than two dimensions. From the distribution of the regression depth we derive tests for the true unknown parameters in the linear regression model. Moreover, we construct simultaneous confidence regions based on bootstrapped estimates. We also use the maximal regression depth to construct a test for linearity versus convexity/concavity. We extend regression depth and deepest regression to more general models. We apply DR to polynomial regression and show that the deepest polynomial regression has breakdown value 1/3. Finally, DR is applied to the Michaelis–Menten model of enzyme kinetics, where it resolves a long-standing ambiguity
R-estimation in Autoregression with Square-Integrable Score Function

AbstractThis paper develops an asymptotic theory for R-estimation based on a square-integrable, not necessarily bounded, score function in the pth order stationary autoregressive model. Asymptotic uniform linearity of a class of linear rank statistics is established and the asymptotic normality of the corresponding R-estimators is derived. This paper thus solves a long-standing problem in the development of the asymptotics for rank estimators under the autoregressive setup. The proofs use a combination of the approximation technique, the contiguity technique and the weak convergence technique of Hájek, Jurečková and Koul, respectively
Geometric Generalization of the Exponential Law

AbstractFor the multivariate ℓ1-norm symmetric distributions, which are generalizations of the n-dimensional exponential distribution with independent marginals, a geometric representation formula is given, together with some of its basic properties. This formula can especially be applied to a new developed and statistically well motivated system of sets. From that the distribution of a t-statistic adapted for the two-parameter exponential distribution and its generalizations is determined. Asymptotic normality of this adapted t-statistic is shown under certain conditions
Asymptotic Approximations for the Distributions of the Multinomial Goodness-of-Fit Statistics under Local Alternatives

AbstractN. Cressie and T. R. C. Read (1984, J. Roy. Statist. Soc. B46, 440–464) introduced a class of multinomial goodness-of-fit statistics Ra based on power divergence. All Ra have the same chi-square limiting distribution under null hypothesis and have the same noncentral chi-square limiting distribution under local alternatives. In this paper, we investigate asymptotic approximations for the distributions of Ra under local alternatives. We obtain an expression of approximation for the distribution of Ra under local alternatives. The expression consists of continuous and discontinuous terms. Using the continuous term of the expression, we propose a new approximation of the power of Ra. We call the approximation AE approximation. By numerical investigation of the accuracy of the AE approximation, we present a range of sample size n that the omission of the discontinuous term exercises only slight influence on power approximation of Ra. We find that the AE approximation is effective for a much wider range of the value of a than the other power approximations, except for an approximation method which requires high computer performance
Testing for Ordered Trends of Binary Responses between Contingency Tables

AbstractIn this article, likelihood ratio tests (LRTs) are developed for detecting that stochastic trends of binary responses are ordered between 2×k contingency tables. We provide a simple iterative algorithm for the maximum likelihood estimators under the order restriction and construct the LRTs using those estimators. All the distributional results of these tests are based on the large sampling theory. The finite-sample behaviors of these tests are investigated through a simulation study. As an illustration of these tests, we analyze a set of data on wheeziness of smoking coalminers
On the n-Coupling Problem

AbstractIn this paper we obtain based on an idea of M. Knott and C. S. Smith (1994, Linear Algebra Appl.199, 363–371) characterizations of solutions of three-coupling problems by reduction to the construction of optimal couplings of each of the variables to the sum. In the case of normal distributions this leads to a complete solution. Under a technical condition this idea also works for general distributions and one obtains explicit results. We extend these results to the n-coupling problem and derive a characterization of optimal n-couplings by several 2-coupling problems. This leads to some constructive existence results for Monge solutions
Fixed Width Confidence Region for the Mean of a Multivariate Normal Distribution

AbstractSrivastava gave an asymptotically efficient and consistent sequential procedure to obtain a fixed-width confidence region for the mean vector of any p-dimensional random vector with finite second moments. For normally distributed random vectors, Srivastava and Bhargava showed that the specified coverage probability is attained independent of the width, the mean vector, and the covariance matrix by taking a finite number of observations over and above T prescribed by the sequential rule. However, the problem of showing that E(T−n0) is bounded, where n0 is the sample size required if the covariance matrix were known, has not been available. In this paper, we not only show that it is bounded but obtain sharper estimates of E(T) on the lines of Woodroofe. We also give an asymptotic expansion of the coverage probability. Similar results have recently been obtained by Nagao under the assumption that the covariance matrix Σ=∑ki=1σiAi and ∑ki=1Ai=I, where Ai's are known matrices. We obtain the results of this paper under the sole assumption that the largest latent root of Σ is simple
A Statistic for Testing the Null Hypothesis of Elliptical Symmetry

AbstractWe present and study a procedure for testing the null hypothesis of multivariate elliptical symmetry. The procedure is based on the averages of some spherical harmonics over the projections of the scaled residual (1978, N. J. H. Small, Biometrika65, 657–658) of the d-dimensional data on the unit sphere of Rd. We find, under mild hypothesis, the limiting null distribution of the statistic presented, showing that, for an appropriate choice of the spherical harmonics included in the statistic, this distribution does not depend on the parameters that characterize the underlying elliptically symmetric law. We describe a bivariate simulation study that shows that the finite sample quantiles of our statistic converge fairly rapidly, with sample size, to the theoretical limiting quantiles and that our procedure enjoys good power against several alternatives
Asymptotics for the Tukey Median

AbstractThe asymptotic distribution of the Tukey median has recently been obtained by Nolan in a bivariate setting and by Bai and He in the general multivariate case. To establish their theorem, these authors made a strong symmetry hypothesis on the distribution and, in the case of Bai and He, assumed the existence of a density function with a gradient satisfying a finite-moment condition. This paper extends the above result by deriving the asymptotic distribution of the above location estimate without making any symmetry or differentiability assumption on the distribution
Bayesian Object Identification: Variants

AbstractWe present a Bayesian theory of object identification. Here, identifying an object means selecting a particular observation from a group of observations (variants), this observation (the regular variant) being characterized by a distributional model. In this sense, object identification means assigning a given model to one of several observations. Often, it is the statistical model of the regular variant, only, that is known. We study an estimator which relies essentially on this model and not on the characteristics of the “irregular” variants. In particular, we investigate under what conditions this variant selector is optimal. It turns out that there is a close relationship with exchangeability and Markovian reversibility. We finally apply our theory to the case of irregular variants generated from the regular variant by a Gaussian linear model
Some Extensions of Tukey's Depth Function

AbstractAs the extensions of Tukey's depth, a family of affine invariant depth functions are introduced for multivariate location and dispersion. The location depth functions can be used for the purpose of multivariate ordering. Such kind ordering can retain more information from the original data than that based on Tukey's depth. The dispersion depth functions provide some additional view of the dispersion of the data set. It is shown that these sample depth functions converge to their population versions uniformly on any compact subset of the parameter space. The deepest points of these depth functions are affine equivariant estimates of multivariate location and dispersion. Under some general conditions these estimates are proved to have asymptotic breakdown points at least 1/3 and convergence rates of 1/n. Their asymptotic distributions are also obtained under some regularity conditions. A new algorithm based on the idea of thresholding is presented for computing these kinds of estimates and realized in the bivariate case. Simulations indicate that some of them could have the empirical mean squared errors smaller than those based on Tukey's depth function or Donoho's depth function
Local Polynomial Fitting in Semivarying Coefficient Model

AbstractVarying coefficient models are useful extensions of the classical linear models. Under the condition that the coefficient functions possess about the same degrees of smoothness, the model can easily be estimated via simple local regression. This leads to the one-step estimation procedure. In this paper, we consider a semivarying coefficient model which is an extension of the varying coefficient model, which is called the semivarying-coefficient model. Procedures for estimation of the linear part and the nonparametric part are developed and their associated statistical properties are studied. The proposed methods are illustrated by some simulation studies and a real example
Two-Step Likelihood Estimation Procedure for Varying-Coefficient Models

AbstractOne of the advantages for the varying-coefficient model is to allow the coefficients to vary as smooth functions of other variables and the model can be estimated easily through a simple local quasi-likelihood method. This leads to a simple one-step estimation procedure. We show that such a one-step method cannot be optimal when some coefficient functions possess different degrees of smoothness. This drawback can be attenuated by using a two-step estimation approach. The asymptotic normality and mean-squared errors of the two-step method are obtained and it is also shown that the two-step estimation not only achieves the optimal convergent rate but also shares the same optimality as the ideal case where the other coefficient functions were known. A numerical study is carried out to illustrate the two-step method
Density Deconvolution in the Circular Structural Model

AbstractWe consider deconvolving bivariate irregular densities supported on the circumference of the unit circle. The errors are bivariate, and the observations are available on the plane. Assuming that the estimated density is smooth on the circle, we compute exact asymptotics of the minimax risks and develop asymptotically optimal estimators for the case of normal errors. The proposed estimators are automatically sharp minimax adaptive over a wide collection of smoothness classes. It is shown that the same rates of convergence hold for a variety of different types of error distributions. The interesting feature of the problem is that the optimal rates of convergence do not depend on the error distribution and are determined essentially by the problem geometry
Moment Properties of the Multivariate Dirichlet Distributions

AbstractLet X1, …, Xn be real, symmetric, m×m random matrices; denote by Im the m×m identity matrix; and let a1, …, an be fixed real numbers such that aj>(m−1)/2, j=1, …, n. Motivated by the results of J. G. Mauldon (Ann. Math. Statist.30 (1959), 509–520) for the classical Dirichlet distributions, we consider the problem of characterizing the joint distribution of (X1, …, Xn) subject to the condition that E|Im−∑nj=1TjXj|−(a1+…+an)=∏nj=1|Im−Tj|−aj for all m×m symmetric matrices T1, …, Tn in a neighborhood of the m×m zero matrix. Assuming that the joint distribution of (X1, …, Xn) is orthogonally invariant, we deduce the following results: each Xj is positive-definite, almost surely; X1+…+Xn=Im, almost surely; the marginal distribution of the sum of any proper subset of X1, …, Xn is a multivariate beta distribution; and the joint distribution of the determinants (|X1|, …, |Xn|) is the same as the joint distribution of the determinants of a set of matrices having a multivariate Dirichlet distribution with parameter (a1, …, an). In particular, for n=2 we obtain a new characterization of the multivariate beta distribution
The Meta-elliptical Distributions with Given Marginals

AbstractBased on an analysis of copulas of elliptically contoured distributions, joint densities of continuous variables with given strictly increasing marginal distributions are constructed. A method utilized for this procedure is to embed the spherical distribution quantile transformation of each variable into an elliptically contoured distribution. The new class of distributions is then called meta-elliptical distributions. The corresponding analytic forms of the density, conditional distribution functions, and dependence properties are derived. This new class of distributions has the same Kendall's rank correlation coefficient as meta-Gaussian distributions. As an extension of elliptically contoured distributions, some new classes of distributions are also obtained
Assessing Goodness-of-Fit of Generalized Logit Models Based on Case-Control Data

AbstractWe consider testing the validity of the generalized logit model with I+1 categories based on case-control data. After reparametrization, the assumed logit model is equivalent to an (I+1)-sample semiparametric model in which the Ilog ratios of two unspecified density functions are linear in data. By identifying this (I+1)-sample semiparametric model, which is of intrinsic interest in general (I+1)-sample problems, with a biased sampling model, we propose a weighted Kolmogorov–Smirnov-type statistic to test the validity of the generalized logit model. We establish some asymptotic results associated with the proposed test statistic. We also propose a bootstrap procedure along with some results on simulation and on analysis of three real data sets