pinecone/core-2020-05-10-deduplication huggingface dataset:
Adaptive Semiparametric Estimation of the Memory Parameter

AbstractIn Giraitis, Robinson, and Samarov (1997), we have shown that the optimal rate for memory parameter estimators in semiparametric long memory models with degree of “local smoothness” β is n−r(β), r(β)=β/(2β+1), and that a log-periodogram regression estimator (a modified Geweke and Porter-Hudak (1983) estimator) with maximum frequency m=m(β)≍n2r(β) is rate optimal. The question which we address in this paper is what is the best obtainable rate when β is unknown, so that estimators cannot depend on β. We obtain a lower bound for the asymptotic quadratic risk of any such adaptive estimator, which turns out to be larger than the optimal nonadaptive rate n−r(β) by a logarithmic factor. We then consider a modified log-periodogram regression estimator based on tapered data and with a data-dependent maximum frequency m=m(β), which depends on an adaptively chosen estimator β of β, and show, using methods proposed by Lepskii (1990) in another context, that this estimator attains the lower bound up to a logarithmic factor. On one hand, this means that this estimator has nearly optimal rate among all adaptive (free from β) estimators, and, on the other hand, it shows near optimality of our data-dependent choice of the rate of the maximum frequency for the modified log-periodogram regression estimator. The proofs contain results which are also of independent interest: one result shows that data tapering gives a significant improvement in asymptotic properties of covariances of discrete Fourier transforms of long memory time series, while another gives an exponential inequality for the modified log-periodogram regression estimator
Empirical Likelihood for Partially Linear Models

AbstractIn this paper, we consider the application of the empirical likelihood method to partially linear model. Unlike the usual cases, we first propose an approximation to the residual of the model to deal with the nonparametric part so that Owen's (1990) empirical likelihood approach can be applied. Then, under quite general conditions, we prove that the empirical log-likelihood ratio statistic is asymptotically chi-squared distributed. Therefore, the empirical likelihood confidence regions can be constructed accordingly
Some Remarks on the Supermodular Order

AbstractIn this paper we solve two open problems posed by Joe (1997) concerning the supermodular order. First we give an example which shows that the supermodular order is strictly stronger than the concordance order for dimension d=3. Second we show that the supermodular order fulfils all desirable properties of a multivariate positive dependence order. We especially prove the non-trivial fact that it is closed with respect to weak convergence. This is applied to give a complete characterization of the supermodular order for multivariate normal distributions
Asymptotic Properties of Backfitting Estimators

AbstractWhen additive models with more than two covariates are fitted with the backfitting algorithm proposed by Buja et al. [2], the lack of explicit expressions for the estimators makes study of their theoretical properties cumbersome. Recursion provides a convenient way to extend existing theoretical results for bivariate additive models to models of arbitrary dimension. In the case of local polynomial regression smoothers, recursive asymptotic bias and variance expressions for the backfitting estimators are derived. The estimators are shown to achieve the same rate of convergence as those of univariate local polynomial regression. In the case of independence between the covariates, non-recursive bias and variance expressions, as well as the asymptotically optimal values for the bandwidth parameters, are provided
The Structure of a Linear Model: Sufficiency, Ancillarity, Invariance, Equivariance, and the Normal Distribution

AbstractConsider a general linear model Y=Xβ+Z where CovZ may be known only partially. We investigate carefully the notions of sufficiency, ancillarity, invariance, and equivariance and related notions for projectors in a general linear model. In this way we can prove a Basu type theorem. This result can be used to give the relation between the sufficiency of the generalized least-squares estimator and the assumption that Z is normally distributed. So we can generalize the well-known result that the generalized least-squares estimator is sufficient for β if Z is normally distributed. Further we can solve the converse problem as well
General Properties and Estimation of Conditional Bernoulli Models

AbstractConditional Bernoulli (in short “CB”) models have been recently applied to many statistical fields including survey sampling, logistic regression, case-control studies, lottery, signal processing and Poisson-Binomial distributions. In this paper, we present several general properties of CB models that are necessary for the applications above. We also show the existence and uniqueness of MLE of parameters in CB models and give two efficient algorithms for computing the MLE. General properties of CB models include: (1) mappings between three characterizations of CB models are homeomorphism modulo rescaling and order-preserving; (2) CB variables are unconditionally independent and conditionally negatively correlated; (3) a simple formula relating inclusion probabilities of adjacent orders can be used to ease computational burden and provide important implication on odds-ratio. Asymptotic properties of CB models are also examined. We show that under a mild condition, (1) CB variables are asymptotically independent; (2) covariances of CB variables are asymptotically on a smaller scale than variances of CB variables; and (3) a CB model can be approximated by a multinomial distribution with the same coverage probabilities. The use and implication of each property are illustrated with related statistical applications
On Parameters of Increasing Dimensions

AbstractIn statistical analyses the complexity of a chosen model is often related to the size of available data. One important question is whether the asymptotic distribution of the parameter estimates normally derived by taking the sample size to infinity for a fixed number of parameters would remain valid if the number of parameters in the model actually increases with the sample size. A number of authors have addressed this question for the linear models. The component-wise asymptotic normality of the parameter estimate remains valid if the dimension of the parameter space grows more slowly than some root of the sample size. In this paper, we consider M-estimators of general parametric models. Our results apply to not only linear regression but also other estimation problems such as multivariate location and generalized linear models. Examples are given to illustrate the applications in different settings
Bivariate Density Estimation with Randomly Truncated Data

AbstractIn this study bivariate kernel density estimators are considered when a component is subject to random truncation. In bivariate truncation models one observes the i.i.d. samples from the triplets (T, Y, X) only if T⩽Y. In this set-up, Y is said to be left truncated by T and T is right truncated by Y. We consider the estimation of the bivariate density function of (Y, X) via nonparametric kernel methods where Y is the variable of interest and X a covariate. We establish an i.i.d. representation of the bivariate distribution function estimator and show that the remainder term achieves an improved order of O(n−1lnn), which is desirable for density estimation purposes. Expressions are then provided for the bias and the variance of the estimators. Finally some simulation results are presented
The Construction of Multivariate Distributions from Markov Random Fields

AbstractWe address the problem of constructing and identifying a valid joint probability density function from a set of specified conditional densities. The approach taken is based on the development of relations between the joint and the conditional densities using Markov random fields (MRFs). We give a necessary and sufficient condition on the support sets of the random variables to allow these relations to be developed. This condition, which we call the Markov random field support condition, supercedes a common assumption known generally as the positivity condition. We show how these relations may be used in reverse order to construct a valid model from specification of conditional densities alone. The constructive process and the role of conditions needed for its application are illustrated with several examples, including MRFs with multiway dependence and a spatial beta process
Optimal Tests for the General Two-Sample Problem

AbstractIn this paper we consider the classical problem of testing whether two samples of observations are from the same distribution. Since in many situations the data are multivariate or even of functional type, classical methodology is not applicable. In our approach we conceive a difference in distribution as the occurrence of a change-point problem, where the change-point is known in advance. This point of view enables us to construct new tests which are distribution-free under the null hypothesis for general sample spaces. The power function of the tests is studied under local and global alternatives. Finally some optimality results are provided
Stochastic Comparisons and Dependence among Concomitants of Order Statistics

AbstractLet (Xi, Yi) i=1, 2, …, n be n independent and identically distributed random variables from some continuous bivariate distribution. If X(r) denotes the rth ordered X-variate then the Y-variate, Y[r], paired with X(r) is called the concomitant of the rth order statistic. In this paper we obtain new general results on stochastic comparisons and dependence among concomitants of order statistics under different types of dependence between the parent random variables X and Y. The results obtained apply to any distribution with monotone dependence between X and Y. In particular, when X and Y are likelihood ratio dependent, it is shown that the successive concomitants of order statistics are increasing according to likelihood ratio ordering and they are TP2 dependent in pairs. If we assume that the conditional hazard rate of Y given X=x is decreasing in x, then the concomitants are increasing according to hazard rate ordering and are dependent according to the right corner set increasing property. Finally, it is proved that if Y is stochastically increasing in X, then the concomitants of order statistics are stochastically increasing and are associated. Analogous results are obtained when the variables X and Y are negatively dependent. We also prove that if the hazard rate of the conditional distribution of Y given X=x is decreasing in x and y, then the concomitants have DFR (decreasing failure rate) distributions and are ordered according to dispersive ordering
Variable Bandwidth Selection in Varying-Coefficient Models

AbstractThe varying-coefficient model is an attractive alternative to the additive and other models. One important method in estimating the coefficient functions in this model is the local polynomial fitting approach. In this approach, the choice of bandwidth is crucial. If the unknown curve is spatial homogeneous, a constant bandwidth is sufficient. However, for estimating curves with a more complicated structure, a variable bandwidth is needed. The present article focuses on a variable bandwidth selection procedure, and provides the conditional bias and the conditional variance of the estimator, the convergence rate of the bandwidth, and the asymptotic distribution of its error relative to the theoretical optimal variable bandwidth
Combining Different Procedures for Adaptive Regression

AbstractGiven any countable collection of regression procedures (e.g., kernel, spline, wavelet, local polynomial, neural nets, etc.), we show that a single adaptive procedure can be constructed to share their advantages to a great extent in terms of global squared L2 risk. The combined procedure basically pays a price only of order 1/n for adaptation over the collection. An interesting consequence is that for a countable collection of classes of regression functions (possibly of completely different characteristics), a minimax-rate adaptive estimator can be constructed such that it automatically converges at the right rate for each of the classes being considered. A demonstration is given for high-dimensional regression, for which case, to overcome the well-known curse of dimensionality in accuracy, it is advantageous to seek different ways of characterizing a high-dimensional function (e.g., using neural nets or additive modelings) to reduce the influence of input dimension in the traditional theory of approximation (e.g., in terms of series expansion). However, in general, it is difficult to assess which characterization works well for the unknown regression function. Thus adaptation over different modelings is desired. For example, we show by combining various regression procedures that a single estimator can be constructed to be minimax-rate adaptive over Besov classes of unknown smoothness and interaction order, to converge at rate o(n−1/2) when the regression function has a neural net representation, and at the same time to be consistent over all bounded regression functions
Multivariate Dispersion Models

AbstractWe introduce a class of multivariate dispersion models suitable as error distributions for generalized linear models with multivariate non-normal responses. The models preserve some of the main properties of the multivariate normal distribution, and include the elliptically contoured distributions and certain other known distributions as special cases. We give explicit methods for constructing multivariate proper dispersion models. This is exemplified by constructing multivariate gamma, Laplace, hyperbola, and von Mises distributions
M-Estimators Converging to a Stable Limit

AbstractWe discuss the asymptotic linearization of multivariate M-estimators, when the limit distribution is stable. We consider two different types of kernels: VC and bracketing. When applied to the case of normal limits, our work improves the known results to obtain the limit distribution of M-estimators. We give weak conditions for the asymptotic normality of M-estimators over differentiable kernels. To obtain these results, we present an inequality on empirical processes satisfying a bracketing condition with respect to a norm smaller than the L2 norm
On the Conditional Variance for Scale Mixtures of Normal Distributions

AbstractFor a scale mixture of normal vector, X=A1/2G, where X, G∈Rn and A is a positive variable, independent of the normal vector G, we obtain that the conditional variance covariance, Cov(X2∣X1), is always finite a.s. for m⩾2, where X1∈Rn and m<n, and remains a.s. finite even for m=1, if and only if the square root moment of the scale factor is finite. It is shown that the variance is not degenerate as in the Gaussian case, but depends upon a function SA, m(·) for which various properties are derived. Application to a uniform and stable scale of normal distributions are also given
Statistical Aspects of Perpetuities

AbstractFor a distribution μ on the unit interval we define the associated perpetuity Ψ(μ) as the distribution of 1+X1+X1X2+X1X2X3+…, where (Xn)n∈N is a sequence of independent random variables with distribution μ. Such quantities arise in insurance mathematics and in many other areas. We prove the differentiability of the perpetuity functionalψ with respect to integral and supremum norms. These results are then used to investigate the statistical properties of empirical perpetuities, including the behaviour of bootstrap confidence regions
Multivariate Survival Functions with a Min-Stable Property

AbstractThis paper introduces and studies a class of multivariate survival functions with given univariate marginal G0, called min-stable multivariate G0-distributions, which includes min-stable multivariate exponential distributions as a special case. The representation of the form of Pickands (1981) is derived, and some dependence and other properties of the class are given. The functional form of the class is G0(A), where A is a homogeneous function on Rn+. Conditions are obtained for G0 and A so that a proper multivariate survival function obtains. Interesting special cases are studied including the case where G0 is a Gamma distribution
Best Bounds in Doob's Maximal Inequality for Bessel Processes

AbstractLet ((Zt), Pz) be a Bessel process of dimension α>0 started at z under Pz for z⩾0. Then the maximal inequalityEz(max0⩽t⩽τZpt)⩽pp−(2−αp/(2−α)Ez(Zpτ)−pp−(2−α)zpis shown to be satisfied for all stopping times τ for (Zt) with Ez(τp/2)<∞, and all p>(2−α)∨0. The constants (p/(p−(2−α)))p/(2−α) and p/(p−(2−α)) are the best possible. If λ is the greater root of the equation λ1−(2−α)/p−λ=(2−α)/(cp−c(2−α)), the equality is attained in the limit through the stopping timesτλ, p=inf{t>0:Zpt⩽λmax0⩽r⩽tZpr}when c tends to the best constant (p/(p−(2−α)))p/(2−α) from above. Moreover we show that Ez(τq/2λ, p)<∞ if and only if λ>((1−(2−α)/q)∨0)p/(2−α). The proof of the inequality is based upon solving the optimal stopping problemV∗(z)=supτEz(max0⩽t⩽τZpt−cZpτ)by applying the principle of smooth fit and the maximality principle. In addition, the exact formula for the expected waiting time of the optimal strategy is derived by applying the minimality principle. The main emphasis of the paper is on the explicit expressions obtained
Fitting Smooth Histories to Rotation Data

AbstractConsider two tectonic plates diverging at a mid-ocean ridge. Geophysicists are able to estimate the rotation of one plate relative to the other at a discrete sequence of times in the earth's history; also they usually have information as to the likely errors in these rotation estimates. We address the problem of fitting a smooth history to such rotation data. We employ a modification of the method used by Jupp and Kent in their 1987 article dealing with fitting a smooth history to time-labeled points on the surface of the unit sphere in three-dimensional space. They use parallel translation to “unroll” data from the surface of the sphere to a plane. We replace unrolling via parallel translation by unrolling via left group multiplication, using the group structure of SO(3). We explain why our understanding of the errors in tectonic plate reconstructions dictates that left group multiplication is preferable both to parallel translation and to right group multiplication. To choose the smoothing parameter we use the discrepancy method; for the Central Atlantic data set which we consider this method gives considerably better results than cross-validation
Nonparametric Notions of Multivariate “Scatter Measure” and “More Scattered” Based on Statistical Depth Functions

AbstractNonparametric notions of multivariate “scatter measure” and “more scattered,” based on statistical depth functions, are investigated. In particular, notions of “more scattered” based on the “halfspace” depth function are shown to generalize versions introduced by P. J. Bickel and E. L. Lehmann (1976, 1979) in the univariate case and by M. L. Eaton (1982) and H. Oja (1983) in the multivariate case. Scatter measures are also discussed, with emphasis on those based on the halfspace depth. Basic desirable properties established for the previous versions of “more scattered” are shown to carry over to the depth-based notions as well, in both the univariate and the multivariate cases. Further, some properties unique to the depth-based notions are established
Shrinkage Estimation towards a Closed Convex Set with a Smooth Boundary

AbstractWe give James–Stein type estimators of a multivariate normal mean vector by shrinkage towards a closed convex set K with a smooth or piecewise smooth boundary. The rate of shrinkage is determined by the curvature of the boundary of K at the projection point onto K. By considering a sequence of polytopes Kj converging to K, we show that a particular estimator we propose is the limit of a sequence of shrinkage estimators towards Kj given by M. E. Bock (1982). In fact our estimators reduce to the James–Stein estimator and to the Bock estimator when K is a point and a convex polyhedron, respectively. Therefore they can be considered as natural extensions of these estimators. Furthermore we apply the same method to the problem of improving the restricted mle by shrinkage towards the origin in the multivariate normal mean model where the mean vector is restricted to a closed convex cone with a smooth or piecewise smooth boundary. We demonstrate our estimators in two settings, one shrinking to a ball and the other shrinking to the cone of nonnegative definite matrices
Average Regression Surface for Dependent Data

AbstractWe study the estimation of the additive components in additive regression models, based on the weighted sample average of regression surface, for stationary α-mixing processes. Explicit expression of this method makes possible a fast computation and allows an asymptotic analysis. The estimation procedure is especially useful for additive modeling. In this paper, it is shown that the average surface estimator shares the same optimality as the ideal estimator and has the same ability to estimate the additive component as the ideal case where other components are known. Formulas for the asymptotic bias and normality of the estimator are established. A small simulation study is carried out to illustrate the performance of the estimation and a real example is also used to demonstrate our methodology
Asymptotic Expansions for Large Deviation Probabilities of Noncentral Generalized Chi-Square Distributions

AbstractAsymptotic expansions for large deviation probabilities are used to approximate the cumulative distribution functions of noncentral generalized chi-square distributions, preferably in the far tails. The basic idea of how to deal with the tail probabilities consists in first rewriting these probabilities as large parameter values of the Laplace transform of a suitably defined function fk; second making a series expansion of this function, and third applying a certain modification of Watson's lemma. The function fk is deduced by applying a geometric representation formula for spherical measures to the multivariate domain of large deviations under consideration. At the so-called dominating point, the largest main curvature of the boundary of this domain tends to one as the large deviation parameter approaches infinity. Therefore, the dominating point degenerates asymptotically. For this reason the recent multivariate asymptotic expansion for large deviations in Breitung and Richter (1996, J. Multivariate Anal.58, 1–20) does not apply. Assuming a suitably parametrized expansion for the inverse g−1 of the negative logarithm of the density-generating function, we derive a series expansion for the function fk. Note that low-order coefficients from the expansion of g−1 influence practically all coefficients in the expansion of the tail probabilities. As an application, classification probabilities when using the quadratic discriminant function are discussed
On a Matrix Representation Lemma Useful in Determining Maximal Invariance Groups

AbstractBanken (1986, J. Multivariate Anal.19, 156–161) proposed a useful method for determining the group of all affine transformations leaving a multivariate normal testing problem invariant. His main result concerning the derivation of the maximal invariance group is heavily based on a matrix representation lemma which can be considered interesting in its own right. Unfortunately, the proof of this lemma is erroneous and there seems to be no trivial way to correct it. The aim of this note is to show the validity of the assertion