pinecone/core-2020-05-10-deduplication huggingface dataset:
Strongly Consistent Nonparametric Forecasting and Regression for Stationary Ergodic Sequences

Let {(Xi, Yi)} be a stationary ergodic time series with (X, Y) values in the product space Rd ⊗ R. This study offers what is believed to be the first strongly consistent (with respect to pointwise, least-squares, and uniform distance) algorithm for inferring m(x) = E[Y0 | X0 = x] under the presumption that m(x) is uniformly Lipschitz continuous. Auto-regression, or forecasting, is an important special case, and as such our work extends the literature of nonparametric, nonlinear forecasting by circumventing customary mixing assumptions. The work is motivated by a time series model in stochastic finance and by perspectives of its contribution to the issues of universal time series estimation
On the Underfitting and Overfitting Sets of Models Chosen by Order Selection Criteria

AbstractFor a general class of order selection criteria, we establish analytic and non-asymptotic evaluations of both the underfitting and overfitting sets of selected models. These evaluations are further specified in various situations including regressions and autoregressions with finite or infinite variances. We also show how upper bounds for the misfitting probabilities and hence conditions ensuring the weak consistency can be derived from the given evaluations. Moreover, it is demonstrated how these evaluations, combined with a law of the iterated logarithm for some relevant statistic, can provide conditions ensuring the strong consistency of the model selection criterion used
Application of the Limit of Truncated Isotonic Regression in Optimization Subject to Isotonic and Bounding Constraints

An isotonic regression truncated by confining its domain to a union of its level sets is the isotonic regression in the reduced function space. When some of the weights with which the inner product system is defined go to infinity, the truncated isotonic regression converges. This limit can be used in discribing the projection onto the set of vectors which satisfy an order restriction and have one or more of its coordinates bounded above and/or below. Through this characterization, two inequalities associated with the projection are established and found useful in order and bound restricted statistical inference. The results obtained show that for an exponential distribution family the inequalities lead to the linkage of the order and bound restricted MLE with the projection of the unrestricted MLE, and the dominance of the order and bound restricted MLE over the unrestricted MLE with respect to two classes of loss functions and risks as well as Bayes risks
Normal Approximation Rate of the Kernel Smoothing Estimator in a Partial Linear Model

AbstractBy establishing the asymptotic normality for the kernel smoothing estimatorβnof the parametric componentsβin the partial linear modelY=X′β+g(T)+ε, P. Speckman (1988,J. Roy. Statist. Soc. Ser. B50, 413–456) proved that the usual parametric raten−1/2is attainable under the usual “optimal” bandwidth choice which permits the achievement of the optimal nonparametric rate for the estimation of the nonparametric componentg. In this paper we investigate the accuracy of the normal approximation forβnand find that, contrary to what we might expect, the optimal Berry–Esseen raten−1/2is not attainable unlessgis undersmoothed, that is, the bandwidth is chosen with faster rate of tending to zero than the “optimal” bandwidth choice
Stochastic Comparisons for Multivariate Shock Models

Consider two devices subjected to shocks arriving according to two identically defined counting processes. Let N1 and N2 be the random numbers of shocks until failure of the two devices, respectively, and let T1 and T2 be their random lifetimes. Conditions such that stochastic orders between N1 and N2 are preserved by T1 and T2 have been investigated in recent literature. Here we study this problem in a multivariate setting, considering systems of non-independent components, and we extend some known results to multivariate stochastic orders. Two kinds of multi-variate generalizations are considered; the case that each one of the components is subjected to its own fond of shocks and the case that all the components of the same system are subjected to a common font of shocks
Spatial Sampling Design Based on Stochastic Complexity

A new methodology is introduced for spatial sampling design when the variable of interest cannot be directly observed, but information on it can be obtained by sampling a related variable, and estimation of the underlying model is required. An approach based on entropy has been proposed by Bueso, Angulo, and Alonso (1998, Environ. Ecol. Statist. 5, No. 1, 29–14) in the case where a model for the involved variables is given. However, in some cases a predetermined structure modelling the behaviour of the variables cannot be assumed. In this context, we derive criteria for solving the design problem based on the stochastic complexity theory and on the philosophy of the EM algorithm. For applying the proposed criteria a computational procedure is developed based on the supplemented EM algorithms. The methodology is illustrated with a numerical example
Moment Estimator for Random Vectors with Heavy Tails

If a set of independent, identically distributed random vectors has heavy tails, so that the covariance matrix does not exist, there is no reason to expect that the sample covariance matrix conveys useful information. On the contrary, this paper shows that the eigenvalues and eigenvectors of the sample covariance matrix contain detailed information about the probability tails of the data. The eigenvectors indicate a set of marginals which completely determine the moment behavior of the data, and the eigenvalues can be used to estimate the tail thickness of each marginal. The paper includes an example application to a data set from finance
Asymptotic Normality of a Combined Regression Estimator

AbstractIn this paper, we propose a combined regression estimator by using a parametric estimator and a nonparametric estimator of the regression function. The asymptotic distribution of this estimator is obtained for cases where the parametric regression model is correct, incorrect, and approximately correct. These distributional results imply that the combined estimator is superior to the kernel estimator in the sense that it can never do worse than the kernel estimator in terms of convergence rate and it has the same convergence rate as the parametric estimator in the case where the parametric model is correct. Unlike the parametric estimator, the combined estimator is robust to model misspecification. In addition, we also establish the asymptotic distribution of the estimator of the weight given to the parametric estimator in constructing the combined estimator. This can be used to construct consistent tests for the parametric regression model used to form the combined estimator
A Determinant Representation for the Distribution of a Generalised Quadratic Form in Complex Normal Vectors

AbstractConsider the quadratic form Z=YH(XLXH)−1Y where Y is a p×m complex Gaussian matrix, X is an independent p×n complex Gaussian matrix, L is a Hermitian positive definite matrix, and m⩽p⩽n. The distribution of Z has been studied for over 30 years due to its importance in certain multivariate statistics but no satisfactory numerical methods for computing this distribution appear to be available. Hence this paper deals with a representation for the density function of Z in terms of a ratio of determinants which is shown to be more amenable to numerical work than previous representations, at least for small values of p. Also for m=1 this work has applications in digital mobile radio for a specific channel where p antennas are used to receive a signal with n interferers. Some of these applications in radio communication systems are discussed
Universal Inadmissibility of Least Squares Estimator

AbstractFor a p-dimensional normal distribution with mean vector θ and covariance matrix Ip, it is known that the maximum likelihood estimator θ of θ with p⩾3 is inadmissible under the squared loss. The present paper considers possible extensions of the result to the case where the loss is a member of a general class of losses of the form L(|δ−θ|Q), where L is nondecreasing and |δ−θ|Q denotes the Mahalanobis distance [(δ−θ)tQ(δ−θ)]1/2 with respect to a given positive definite matrix Q, which, without loss of generality, may be assumed to be diagonal, i.e., Q=diag(q1, …, qp), q1>q2⩾q3⩾…⩾qp>0. For the case where q1>q2=q3=…=qp>0, L. D. Brown and J. T. Hwang (1989, Ann. Statist.17, 252–267) showed that there exists an estimate of θ universally dominates θ if and only if p⩾4. This paper further extends Brown and Hwang's result to the case in which q1>q2 and at least there are two equal elements among q2, …, qp−1; namely, we show that, for this case, there exists an estimate of θ which universally dominates θ if and only if p⩾4. For a general Q, we gives a lower bound on p that implies the least squares estimators is universally inadmissible
Functional Principal Components Analysis by Choice of Norm

AbstractThe functional principal components analysis (PCA) involves new considerations on the mechanism of measuring distances (the norm). Some properties arising in functional framework (e.g., smoothing) could be taken into account through an inner product in the data space. But this proposed inner product could make, for example, interpretational or (and) computational abilities worse. The results obtained in this paper establish equivalences between the PCA with the proposed inner product and certain PCA with a given well-suited inner product. These results have been proved in the theoretical framework given by Hilbert valued random variables, in which multivariate and functional PCAs appear jointly as particular cases
Bivariate Distributions with Given Extreme Value Attractor

AbstractA new class of bivariate distributions is introduced and studied, which encompasses Archimedean copulas and extreme value distributions as special cases. Its dependence structure is described, its maximum and minimum attractors are determined, and an algorithm is given for generating observations from any member of this class. It is also shown how it is possible to construct distributions in this family with a predetermined extreme value attractor. This construction is used to study via simulation the small-sample behavior of a bivariate threshold method suggested by H. Joe, R. L. Smith, and I. Weissman (1992, J. Roy. Statist. Soc. Ser. B54, 171–183) for estimating the joint distribution of extremes of two random variates
Properties of Prior and Posterior Distributions for Multivariate Categorical Response Data Models

AbstractIn this article, we model multivariate categorical (binary and ordinal) response data using a very rich class of scale mixture of multivariate normal (SMMVN) link functions to accommodate heavy tailed distributions. We consider both noninformative as well as informative prior distributions for SMMVN-link models. The notation of informative prior elicitation is based on available similar historical studies. The main objectives of this article are (i) to derive theoretical properties of noninformative and informative priors as well as the resulting posteriors and (ii) to develop an efficient Markov chain Monte Carlo algorithm to sample from the resulting posterior distribution. A real data example from prostate cancer studies is used to illustrate the proposed methodologies
Properties of Likelihood Inference for Order Restricted Models

AbstractThe most common mode of inference for order restricted models is likelihood inference. See T. Robertson, F. T. Wright, and R. L. Dykstra (1988, “Order Restricted Statistical Inference,” Wiley, New York) for an excellent treatment of inference in such models. In this paper we demonstrate that maximum likelihood estimation and likelihood ratio testing are prone to behavior that is somewhat unintuitive and unappealing to our sensibilities for many order restricted models. From a practical standpoint this behavior sometimes is cause enough for us to seek alternative procedures. For restrictions to a simple order cone, likelihood inference is satisfactory. However, if, for example, the restrictions are of the tree order type, umbrella order, star-shaped order, or stochastic order, then likelihood methods seem to have some shortcomings. General results will identify when likelihood methods are potentially wanting
Dependent Hazards in Multivariate Survival Problems

AbstractA new class of bivariate survival distributions is constructed from a given family of survival distributions. The properties of these distributions are analyzed. It is shown that the same bivariate survival function can be derived using two radically different concepts: one involves transformation of the well-known bivariate survival function; the other involves correlated stochastic hazards. The new conditions that guarantee negative associations of life spans are derived. An exponential representation of the survival function for two related individuals is derived in terms of the conditional distribution of the stochastic hazards among survivors. Versions of the multivariate correlated gamma-frailty model are investigated
Restricted Regression Quantiles

AbstractRegression quantiles can be used as prediction intervals for the response variable. But such applications are often hampered by the problem of quantile crossing in finite sample cases. This article examines the efficiency properties of restricted regression quantiles that are proposed by X. He (1997, Amer. Statist.51, 186–192) to overcome the crossing problem of the usual regression quantiles of R. Koenker and G. Bassett (1978, Econometrica46, 33–50) for linear models. An example using esterase assay data is presented to illustrate the use of restricted regression quantiles in constructing calibration intervals
On the Loss of Information Due to Nonrandom Truncation

AbstractIt is assumed that observations among an iid sample falling into certain subsets of the sample space cannot be observed directly, but only through their frequencies. Bounds for the corresponding loss of information are established, which are based on the Hellinger distance between the empirical point process Nn of the complete set of observations and an empirical process N*n that aims at restoring Nn. An application of these bounds to parametric models generalizes and quantifies results for locally asymptotically Gaussian experiments. When applied to extreme value models, this approach generalizes the peaks-over-threshold method for modeling the exceedances over high thresholds in an iid sample
Noninformative Priors for Multivariate Linear Calibration

AbstractThis paper derives a class of first order probability matching priors and a complete catalog of the reference priors for the general multivariate linear calibration problem. In an important special case, a complete characterization of first order probability matching priors is given, and a fairly general class of second order probability matching priors is also provided. Orthogonal transformations (1987, D. R. Cox and N. Reid, J. Roy. Statist. Soc. Ser. B49, 1–18) are found to facilitate the derivations. It turns out that under orthogonal parameterization, reference priors (including Jeffreys' prior) are first order probability matching priors for unidimensional multivariate linear calibration. Also, in univariate linear calibration, the prior of W. G. Hunter and W. F. Lamboy (1981, Technometrics23, 323–350) is a second order probability matching prior
Canonical Analysis Applied to Multivariate Analysis of Variance

AbstractIn this paper it is shown how a generalized form of canonical analysis can be useful to reveal which parametric functions of a MANOVA model, for instance treatment contrasts or combinations of observed variables, are responsible for rejection of a general linear hypothesis on these functions. For the decomposition in successive canonical terms the choice of a matrix norm is crucial. It is shown that the norm derived from the standardization of the least squares estimators of the parametric functions involved is equal to the Lawley–Hotelling statistic for testing the hypothesis under investigation. Thus, some useful interpretations based on canonical variates can be given in terms of the contributions of the various parametric functions to the overall test statistic or to statistics relevant to specific subhypotheses. Corresponding to these possibilities for interpretation, three different types of biplot are proposed. As an example, an agricultural block design experiment is thoroughly analyzed
Multivariate Normal Distributions Parametrized as a Riemannian Symmetric Space

AbstractThe construction of a distance function between probability distributions is of importance in mathematical statistics and its applications. The distance function based on the Fisher information metric has been studied by a number of statisticians, especially in the case of the multivariate normal distribution (Gaussian) on Rn. It turns out that, except in the case n=1, where the Fisher metric describes the hyperbolic plane, it is difficult to obtain an exact formula for the distance function (although this can be achieved for special families with fixed mean or fixed covariance). We propose to study a slightly different metric on the space of multivariate normal distributions on Rn. Our metric is based on the fundamental idea of parametrizing this space as the Riemannian symmetric space SL(n+1)/SO(n+1). Symmetric spaces are well understood in Riemannian geometry, allowing us to compute distance functions and other relevant geometric data
Transformations with Improved Chi-Squared Approximations

AbstractSuppose that a nonnegative statistic T is asymptotically distributed as a chi-squared distribution with f degrees of freedom, χ2f, as a positive number n tends to infinity. Bartlett correction T was originally proposed so that its mean is coincident with the one of χ2f up to the order O(n−1). For log-likelihood ratio statistics, many authors have shown that the Bartlett corrections are asymptotically distributed as χ2f up to O(n−1), or with errors of terms of O(n−2). Bartlett-type corrections are an extension of Bartlett corrections to other statistics than log-likelihood ratio statistics. These corrections have been constructed by using their asymptotic expansions up to O(n−1). The purpose of the present paper is to propose some monotone transformations so that the first two moments of transformed statistics are coincident with the ones of χ2f up to O(n−1). It may be noted that the proposed transformations can be applied to a wide class of statistics whether their asymptotic expansions are available or not. A numerical study of some test statistics that are not a log-likelihood ratio statistic is discribed. It is shown that the proposed transformations of these statistics give a larger improvement to the chi-squared approximation than do the Bartlett corrections. Further, it is seen that the proposed approximations are comparable with the approximation based on an Edgeworth expansion
Singularly Perturbed Markov Chains: Convergence and Aggregation

AbstractAsymptotic properties of singularly perturbed Markov chains having measurable and/or continuous generators are developed in this work. The Markov chain under consideration has a finite-state space and is allowed to be nonstationary. Its generator consists of a rapidly varying part and a slowly changing part. The primary concerns are on the properties of the probability vectors and an aggregated process that depend on the characteristics of the fast varying part of the generators. The fast changing part of the generators can either consist of l recurrent classes, or include also transient states in addition to the recurrent classes. The case of inclusion of transient states is examined in detail. Convergence of the probability vectors under the weak topology of L2 is obtained first. Then under slightly stronger conditions, it is shown that the convergence also takes place pointwise. Moreover, convergence under the norm topology of L2 is derived. Furthermore, a process with aggregated states is obtained which converges to a Markov chain in distribution
Inferences on Correlation Coefficients in Some Classes of Nonnormal Distributions

AbstractCorrelation coefficients have many applications for studying the relationship among multivariate observations. Classical inferences on correlation coefficients are mainly based on the normality assumption. This assumption is hardly realistic in the real world, which implies that the procedures on correlation coefficients used in many statistical software packages may not be relevant to most data sets in practice. However, we show that the classical procedures, possibly after simple corrections, are also valid in classes of distributions with large skewnesses and heterogeneous marginal kurtoses. A useful class of nonnormal distributions is identified for each of several types of correlation coefficients. The marginals of these distributions may include a variety of univariate distributions with different shapes. The results generalize the classical procedures to much larger classes of distributions than previously known and give a better understanding of the historical controversy regarding the behavior of the sample correlation coefficient. An implication is that one need not be worried so much by the nonnormality of data sets when using these classical procedures, providing simple corrections are evaluated and possibly undertaken
Estimation of Densities and Derivatives of Densities with Directional Data

AbstractEstimating the density function of a random vector taking values on the d-dimensional unit sphere is considered. Also the estimation of the Laplacian of the density and estimation of other types of derivatives is considered. Fast convergence rate theory is developed for pointwise, L1, and L2 error, extending some results of Hall, Watson and Cabrera (1987). It is also proved that asymptotically the plug-in method is as good as using the asymptotically optimal deterministic smoothing parameter sequence
Error Bounds for Asymptotic Approximations of the Linear Discriminant Function When the Sample Sizes and Dimensionality are Large

AbstractTheoretical accuracies are studied for asymtotic approximations of the expected probabilities of misclassification (EPMC) when the linear discriminant function is used to classify an observation as coming from one of two multivariate normal populations with a common covariance matrix. The asymptotic approximations considered are the ones under the situation where both the sample sizes and the demensionality are large. We give explicit error bounds for asymptotic approximations of EPMC, based on a general approximation result. We also discuss with a method of obtaining asymptotic expansions for EPMC and their error bounds
On Positive Definiteness of Some Functions

AbstractLet ρ be a nonnegative homogeneous function on Rn. General structure of the set of numerical pairs (δ, λ), for which the function (1−ρλ(x))δ+ is positive definite on Rn is investigated; a criterion for positive definiteness of this function is given in terms of completely monotonic functions; a connection of this problem with the Schoenberg problem on positive definiteness of the function exp(−ρλ(x)) is found. We also obtain a general sufficient condition of Polya type for a function f(ρ(x)) to be positive definite on Rn