pinecone/core-2020-05-10-deduplication huggingface dataset:
Optimal Transportation Plans and Convergence in Distribution

AbstractExplicit expression of mappings optimal transportation plans for the Wasserstein distance in Rp,p>1, are not generally available. Therefore, it is of great interest to provide results which justify the practical use of simulation techniques to obtain approximate optimal transportation plans. This is done in this paper, where we obtain the consistency of the empirical optimal transportation plans. Our results can also be employed to justify a definition of multidimensional complete dependence
Bivariate Distribution and Hazard Functions When a Component is Randomly Truncated

AbstractIn random truncation models one observes the i.i.d. pairs (Ti⩽Yi),i=1, …, n. IfYis the variable of interest, thenTis another independent variable which prevents the complete observation ofYand random left truncation occurs. Such a type of incomplete data is encountered in medical studies as well as in economy, astronomy, and insurance applications. Let (Y, Y) be a bivariate vector of random variables with joint distribution functionF(y, x) and suppose the variableYis randomly truncated from the left. In this study, nonparametric estimators for the bivariate distribution and hazard functions are considered. A nonparametric estimator forF(y, x) is proposed and an a.s. representation is obtained. This representation is used to establish the consistency and the weak convergence of the empirical process. An expression for the variance of the asymptotic distribution is presented and an estimator is proposed. Bivariate “diverse-hazard” vector is introduced whic h captures the individual and joint failure behaviors of the random variables in opposite “time” directions. Estimators for this vector are presented and the large sample properties are discussed. Possible applications and a moderate size simulation study are also presented
Dependence and Order in Families of Archimedean Copulas

AbstractThe copula for a bivariate distribution functionH(x, y) with marginal distribution functionsF(x) andG(y) is the functionCdefined byH(x, y)=C(F(x), G(y)).Cis called Archimedean ifC(u, v)=ϕ−1(ϕ(u)+ϕ(v)), whereϕis a convex decreasing continuous function on (0, 1] withϕ(1)=0. A copula has lower tail dependence ifC(u, u)/uconverges to a constantγin (0, 1] asu→0+; and has upper tail dependence ifC(u, u)/(1−u) converges to a constantδin (0, 1] asu→1−whereCdenotes the survival function corresponding toC. In this paper we develop methods for generating families of Archimedean copulas with arbitrary values ofγandδ, and present extensions to higher dimensions. We also investigate limiting cases and the concordance ordering of these families. In the process, we present answers to two open problems posed by Joe (1993,J. Multivariate Anal.46262–282)
Smooth Tests of Goodness-of-Fit for Directional and Axial Data

AbstractIn this paper we develop, for directional and axial data, smooth tests of goodness-of-fit for rotationally symmetric distributions against general families of embedding alternatives constructed from complete orthonormal bases of functions. These families generalize a proposal of Beran (1979) based on spherical harmonics. Combined with Rao's score test, our alternatives yield simple test strategies. We present a method for constructing an orthonormal basis adapted to the case where the alternatives are first assumed to be rotationally symmetric and then for more general situations. As an example of the versatility of our method, the results are applied to the problem of testing goodness-of-fit for the uniform, the von Mises–Fisher–Langevin, and the Scheiddegger–Dimroth–Watson distributions. It is shown that the proposed test strategy encompasses and generalizes many of the approaches that have so far been proposed for these distributions. Moreover, our method allows for easy adaptation to more complex alternatives than those previously available. In addition, the test statistic can be broken into parts that may be used to detect specific departures from the null hypothesis
The Probability that a Random Real Gaussian Matrix haskReal Eigenvalues, Related Distributions, and the Circular Law

AbstractLetAbe annbynmatrix whose elements are independent random variables with standard normal distributions. Girko's (more general) circular law states that the distribution of appropriately normalized eigenvalues is asymptotically uniform in the unit disk in the complex plane. We derive the exact expected empirical spectral distribution of the complex eigenvalues for finiten, from which convergence in the expected distribution to the circular law for normally distributed matrices may be derived. Similar methodology allows us to derive a joint distribution formula for the real Schur decomposition ofA. Integration of this distribution yields the probability thatAhas exactlykreal eigenvalues. For example, we show that the probability thatAhas all real eigenvalues is exactly 2−n(n−1)/4
On Sequential Fixed-Size Confidence Regions for the Mean Vector

AbstractIn order to construct a fixed-size confidence region for the mean vector of an unknown distribution functionF, a new purely sequential sampling strategy is proposed first. For this new procedure, under some regularity conditions onF, the coverage probability is shown (Theorem 2.1) to be at least (1−α)−Bα2d2+o(d2) asd→0, where (1−α) is the preassigned level of confidence,Bis an appropriate functional ofF, and 2dis the preassigned diameter of the proposed spherical confidence region for the mean vector ofF. An accelerated version of the stopping rule is also provided with the analogous second-order characteristics (Theorem 3.1). In the special case of ap-dimensional normal random variable, analogous purely sequential and accelerated sequential procedures as well as a three-stage procedure are briefly introduced together with their asymptotic second-order characteristics
Multivariate Gini Indices

AbstractTwo extensions of the univariate Gini index are considered:RD, based on expected distance between two independent vectors from the same distribution with finite meanμ∈Rd; andRV, related to the expected volume of the simplex formed fromd+1 independent such vectors. A new characterization ofRDas proportional to a univariate Gini index for a particular linear combination of attributes relates it to the Lorenz zonoid. TheLorenz zonoidwas suggested as a multivariate generalization of the Lorenz curve.RVis, up to scaling, the volume of the Lorenz zonoid plus a unit cube of full dimension. Whend=1, bothRDandRVequal twice the area between the usual Lorenz curve and the line of zero disparity. Whend>1, they are different, but inherit properties of the univariate Gini index and are related via the Lorenz zonoid:RDis proportional to the average of the areas of some two-dimensioned projections of the lift zonoid, whileRVis the average of the volumes of projections of the Lorenz zonoid over all coordinate subspaces
Variable Selection for the Growth Curve Model

AbstractIn this paper we consider the problem of selecting the covariables within individuals in the growth curve model. We propose two modifications ofAICandMIC(Cp-static), which have improvements on the bias properties. Asymptotic distributions of variable slection criteria are derived under a general situation where a polynomial growth curve of degreej0is approximately suitable. A simulation study is also given to gain some understanding on the small sample properties of these variable selection criteri
Supermodular Stochastic Orders and Positive Dependence of Random Vectors

AbstractThe supermodular and the symmetric supermodular stochastic orders have been cursorily studied in previous literature. In this paper we study these orders more thoroughly. First we obtain some basic properties of these orders. We then apply these results in order to obtain comparisons of random vectors with common values, but with different levels of multiplicity. Specifically, we show that if the vectors of the levels of multiplicity are ordered in the majorization order, then the associated random vectors are ordered in the symmetric supermodular stochastic order. In the non-symmetric case we obtain bounds (in the supermodular stochastic order sense) on such random vectors. Finally, we apply the results to problems of optimal assembly of reliability systems, of optimal allocation of minimal repair efforts, and of optimal allocation of reliability items
A Cross-Validation Bandwidth Choice for Kernel Density Estimates with Selection Biased Data

AbstractThis paper studies the risks and bandwidth choices of a kernel estimate of the underlying density when the data are obtained fromsindependent biased samples. The main results of this paper give the asymptotic representation of the integrated squared errors and the mean integrated squared errors of the estimate and establish a cross-validation criterion for bandwidth selection. This kernel density estimate is shown to be asymptotically superior to many other intuitive kernel density estimates. The data-driven cross-validation bandwidth is shown to be asymptotically optimal in the sense of Stone (1984,Ann. Statist.12, 1285–1297). The finite sample properties of the cross-validation bandwidth are investigated through a Monte Carlo simulation
Efficiency Comparisons in Multivariate Multiple Regression with Missing Outcomes

AbstractWe consider a follow-up study in which an outcome variable is to be measured at fixed time points and covariate values are measured prior to start of follow-up. We assume that the conditional mean of the outcome given the covariates is a linear function of the covariates and is indexed by occasion-specific regression parameters. In this paper we study the asymptotic properties of several frequently used estimators of the regression parameters, namely the ordinary least squares (OLS), the generalized least squares (GLS), and the generalized estimating equation (GEE) estimators when the complete vector of outcomes is not always observed, the missing data patterns are monotone and the data are missing completely at random (MCAR) in the sense defined by Rubin [11]. We show that when the covariance of the outcome given the covariates is constant, as opposed to the nonmissing data case: (a) the GLS estimator is more efficient than the OLS estimator, (b) the GLS estimator is inefficient, and (c) the semiparametric efficient estimator in a model that imposes linear restrictions only on the conditional mean of the last occasion regression can be less efficient than the efficient estimator in a model that imposes linear restrictions on the conditional means of all the outcomes. We provide formulae and calculations of the asymptotic relative efficiencies of the considered estimators in three important cases: (1) for the estimators of the occasion-specific means, (2) for estimators of occasion-specific mean differences, and (3) for estimators of occasion-specific dose-response model parameters
On the Asymptotics of Quantizers in Two Dimensions

AbstractWhen the mean square distortion measure is used, asymptotically optimal quantizers of uniform bivariate random vectors correspond to the centers of regular hexagons (Newman, 1982), and if the random vector is non-uniform, asymptotically optimal quantizers are the centers of piecewise regular hexagons where the sizes of the hexagons are determined by a properly chosen density function (Su and Cambanis, 1996). This paper considers bivariate random vectors with finiteγth (γ>0) moment. If theγth mean distortion measure is used, a complete characterization of the asymptotically optimal quantizers is given. Furthermore, it is shown that the procedure introduced by Su and Cambanis (1996) is also asymptotically optimal for everyγ>0. Examples with a normal distribution and a Pearson type VII distribution are considered
Wishart and Chi-Square Distributions Associated with Matrix Quadratic Forms

AbstractFor a normally distributed random matrixYwith a general variance–covariance matrixΣY, and for a nonnegative definite matrixQ, necessary and sufficient conditions are derived for the Wishartness ofY′QY. The conditions resemble those obtained by Wong, Masaro, and Wang (1991,J. Multivariate Anal.39, 154–174) and Wong and Wang (1993,J. Multivariate Anal.44, 146–159), but are verifiable and are obtained by elementary means. An explicit characterization is also obtained for the structure ofΣYunder which the distribution ofY′QYis Wishart. AssumingΣYpositive definite, a necessary and sufficient condition is derived for every univariate quadratic fromlY′QYlto be distributed as a multiple of a chi-square. For the caseQ=In, the corresponding structure ofΣYis identified. An explicit counterexample is constructed showing that Wishartness ofY′Yneed not follow when, for every vectorl, l′Y′Ylis distributed as a multiple of a chi-square, complementing the well-known counterexample by Mitra (1969,Sankhyā31, 19–22). Application of the results to multivariate components of variance models is briefly indicated
LAD Regression for Detecting Outliers in Response and Explanatory Variables

AbstractLeast absolute deviations regression resists outliers in the response variable but is relatively sensitive to outlying observations in the explanatory variables. In this paper a simple solution is proposed to overcome this problem. This is achieved by minimizing the absolute values of vertical and horizontal deviations in turn. Two algorithms are proposed: one for the simple and one for the multiple regression case. The methods presented have been tested on a variety of data and have proven to be quite effective
Combining Independent Information in a Multivariate Calibration Problem

AbstractThe problem of combining independent information from different sources in a multivariate calibration setup is considered. The dimensions of the response vectors from various sources may be unequal. A linear combination of the classical estimators based on the individual sources is proposed as an estimator for the unknown explanatory variable. It is shown that the combined estimator has finite mean provided the sum of the dimensions of the response vectors exceeds one and has finite mean squared error if it exceeds two. Expressions for asymptotic bias and mean squared error are given
Goodness-of-Fit Tests for a Multivariate Distribution by the Empirical Characteristic Function

AbstractIn this paper, we take the characteristic function approach to goodness-of-fit tests. It has several advantages over existing methods: First, unlike the popular comparison density function approach suggested in Parzen (1979), our approach is applicable to both univariate and multivariate data; Second, in the case where the null hypothesis is composite, the approach taken in this paper yields a test that is superior to tests based on empirical distribution functions such as the Cramér– von Mises test, because on the one hand the asymptotic critical values of our test are easily obtained from the standard normal distribution and are not affected byn-consistent estimation of the unknown parameters in the null hypothesis, and on the other hand, our test extends that in Eubank and LaRiccia (1992) and hence is more powerful than the Cramér–von Mises test for high-frequency alternatives
On Estimated Projection Pursuit-Type Crámer–von Mises Statistics

AbstractThis paper addresses the problem of testing for a multivariate distribution, which belongs to a known parametric distribution family. The estimated Crámer–Von Mises-type test statistics are constructed using projection pursuit technique. Some interested properties of the test statistics, like asymptotics, bootstrap approximations, and the tail behavior of the limits of test statistics are investigated. For computational reasons, an approximation via the number theoretic method to the extreme value and the integral on a super sphere surface is considered
A Theorem on Uniform Convergence of Stochastic Functions with Applications

AbstractIn a variety of statistical problems one needs to manipulate a sequence of stochastic functions involving some unknown parameters. The asymptotic behavior of the estimated parameters often depends on the asymptotic properties of such functions. Especially, the consistency of the estimated parameters follows from the uniform convergence of the sequence of stochastic functions. A theorem on uniform convergence of a sequence of vector valued random functions is presented. The forms of these functions are very general and the assumptions are rather natural. If the sequence of random functions is generated by a sequence of random vectors, these random vectors are only required to be independently distributed and can be of different dimensions. As applications, we consider the consistency of the estimated regression parameters in logistic regression and in M-estimation in a linear model
On Estimating the Dimensionality in Canonical Correlation Analysis

AbstractIn canonical correlation analysis the number of nonzero population correlation coefficients is called the dimensionality. Asymptotic distributions of the dimensionalities estimated by Mallows's criterion and Akaike's criterion are given for nonnormal multivariate populations with finite fourth moments. These distributions have a simple form in the case of elliptical populations, and modified criteria are proposed which adjust for nonzero kurtosis. An estimation method based on a marginal likelihood function for the dimensionality is introduced and the asymptotic distribution of the corresponding estimator is derived for multivariate normal populations. It is shown that this estimator is not consistent, but that a simple modification yields consistency. An overall comparison of the various estimation methods is conducted through simulation studies
Convergence of Dirichlet Measures Arising in Context of Bayesian Analysis of Competing Risks Models

AbstractIn this paper, we study the weak convergence of Dirichlet measures on the class constituted by vectors of subprobability measures such that the sum of its components is a probability measure on a complete separable metric space. This vectorial class of subprobabilities appears in the context of the competing risks theory and the Dirichlet measures are considered as a prior family in a Bayesian approach. The weak convergence results are derived and used to study the convergence of the Bayes estimators of certain parameters in competing risks models
Some New Statistics for Testing Hypotheses in Parametric Models

AbstractThe paper deals with simple and composite hypotheses in statistical models with i.i.d. observations and with arbitrary families dominated byσ-finite measures and parametrized by vector-valued variables. It introducesφ-divergence testing statistics as alternatives to the classical ones: the generalized likelihood ratio and the statistics of Wald and Rao. It is shown that, under the assumptions of standard type about hypotheses and model densities, the results about asymptotic distribution of the classical statistics established so far for the counting and Lebesgue dominating measures (discrete and continuous models) remain true also in the general case. Further, these results are extended to theφ-divergence statistics with smooth convex functionsφ. The choice ofφ-divergence statistics optimal from the point of view of power is discussed and illustrated by several examples
On the Estimation of a Support Curve of Indeterminate Sharpness

AbstractWe propose nonparametric methods for estimating the support curve of a bivariate density, when the density decreases at a rate which might vary along the curve. Attention is focused on cases where the rate of decrease is relatively fast, this being the most difficult setting. It demands the use of a relatively large number of bivariate order statistics. By way of comparison, support curve estimation in the context of slow rates of decrease of the density may be addressed using methods that employ only a relatively small number of order statistics at the extremities of the point cloud. In this paper we suggest a new type of estimator, based on projecting onto an axis those data values lying within a thin rectangular strip. Adaptive univariate methods are then applied to the problem of estimating an endpoint of the distribution on the axis. The new method is shown to have theoretically optimal performance in a range of settings. Its numerical properties are explored in a simulation study
On a Conjecture of Krishnamoorthy and Gupta

AbstractWe consider the problem of estimating the precision matrix (Σ−1) under a fully invariant convex loss. Suppose that there exists a minimax constant risk estimatorΦ(say) for this problem. K. Krishnamoorthy and A. K. Gupta have proposed an operation which transforms this estimator into an orthogonally invariant estimatorΦ* (say) and they have a conjecture saying thatΦ* is minimax as well. This paper contains two parts. In the first part, we present counterexamples. In the second part, we elaborate a technique which can be used to prove that certain estimators are minimax. This technique is then applied successfully to some of the estimators proposed in the Krishnamoorthy and Gupta paper
A New Approach to the BHEP Tests for Multivariate Normality

AbstractLetX1, …, Xnbe i.i.d. randomd-vectors,d⩾1, with sample meanXand sample covariance matrixS. For testing the hypothesisHdthat the law ofX1is some nondegenerate normal distribution, there is a whole class of practicable affine invariant and universally consistent tests. These procedures are based on weighted integrals of the squared modulus of the difference between the empirical characteristic function of the scaled residualsYj=S−1/2(Xj−X) and its almost sure pointwise limit exp(−‖t‖2/2) underHd. The test statistics have an alternative interpretation in terms ofL2-distances between a nonparametric kernel density estimator and the parametric density estimator underHd, applied toY1, …, Yn. By working in the Fréchet space of continuous functions on Rd, we obtain a new representation of the limiting null distributions of the test statistics and show that the tests have asymptotic power against sequences of contiguous alternatives converging toHdat the raten−1/2, independent ofd
Classification of Binary Vectors by Stochastic Complexity

AbstractStochastic complexity is treated as a tool of classification, i.e., of inferring the number of classes, the class descriptions, and the class memberships for a given data set of binary vectors. The stochastic complexity is evaluated with respect to the family of statistical models defined by finite mixtures of multivariate Bernoulli distributions obtained by the principle of maximum entropy. It is shown that stochastic complexity is asymptotically related to the classification maximum likelihood estimate. The formulae for stochastic complexity have an interpretation as minimum code lengths for certain universal source codes for storing the binary data vectors and their assignments into the classes in a classification. There is also a decomposition of the classification uncertainty in a sum of an intraclass uncertainty, an interclass uncertainty, and a special parsimony term. It is shown that minimizing the stochastic complexity amounts to maximizing the information content of the classification. An algorithm of alternating minimization of stochastic complexity is given. We discuss the relation of the method to the AUTOCLASS system of Bayesian classification. The application of classification by stochastic complexity to an extensive data base of strains ofEnterobacteriaceaeis described