pinecone/core-2020-05-10-deduplication huggingface dataset:
Asymptotic Normality for a Vector Stochastic Difference Equation with Applications in Stochastic Approximation

AbstractIn this paper, we consider an asymptotic normality problem for a vector stochastic difference equation of the formUn+1=(I+an(B+En))Un+an(un+en), whereBis a stable matrix, andEn→n0,anis a positive real step size sequence withan→n0, ∑∞n=1an=∞, anda−1n+1−a−1n→nλ⩾0,unis an infinite-term moving average process, and[formula]. Obviously,anhere is a quite general step size sequence and includes (logn)β/nα, 12<α<1, orα=1 withβ⩾0 as special cases. It is well known that the problem of an asymptotic normality for a vector stochastic approximation algorithm is usually reduced to the above problem. We prove that[formula]converges in distribution to a zero mean normal random vector with covariance ∫∞0e(B+(1/2)λI)tRe(Bτ+(1/2)λI)tdt, where matrixRdepends only on some stochastic properties ofun, which implies that the asymptotic distributions for both the vector stochastic difference equation and vector stochastic approximation algorithm do not depend on the specific choices ofandirectly but onλ, the limit ofa−1n+1−a−1n
Resampling Permutations in Regression without Second Moments

AbstractFor given observations we consider the resampling distribution obtained by permuting residuals versus thedistribution of errors conditional on their order statistics. We observe thatwith high probabilityboth are approximately equal oncontrast vectorsafter suitable normalization. No moment/distribution assumptions on the underlying errors other than exchangeability are required. These results remain valid if we consider reduced regression models derived from the original one by removing data for which the corresponding residuals take extreme values. We obtain general conditions under which confidence regions produced by such methods are accurate. In the absence of finite second moments randomness of the limiting bootstrap distributions has been considered as a failure of the traditional bootstrap and no practical meaning has been given to the phenomenon. For our method this type of randomness is still present in the limit but it has clear probabilistic interpretation as a conditional distribution which can be used, e.g., to obtain conditional confidence sets. We study this phenomenon in detail for the case of independent errors with distribution from the domain of attraction of stable laws
Projection Method for Moment Bounds on Order Statistics from Restricted Families I. Dependent Case

AbstractWe present a method of projections onto convex cones for establishing the sharp bounds in terms of the first two moments for the expectations ofL-estimates based on samples from restricted families. In this part, we consider the case of possibly dependent identically distributed parent random variables. For the classes of decreasing failure probability, DFR, and symmetric unimodal marginal distributions, we first determine parametric subclasses which contain the distributions attaining the extreme expectations for allL-estimates. Then we derive the bounds for single order statistics. The results provide some new characterizations of uniform and exponential distribution
Order Determination for Multivariate Autoregressive Processes Using Resampling Methods

AbstractLetX1, …, Xnbe observations from a multivariate AR(p) model with unknown orderp. A resampling procedure is proposed for estimating the orderp. The classical criteria, such as AIC and BIC, estimate the orderpas the minimizer of the function[formula]wherenis the sample size,kis the order of the fitted model, Σ2kis an estimate of the white noise covariance matrix, andCnis a sequence of specified constants (for AIC,Cn=2m2/n, for Hannan and Quinn's modification of BIC,Cn=2m2(lnlnn)/n, wheremis the dimension of the data vector). A resampling scheme is proposed to estimate an improved penalty factorCn. Conditional on the data, this procedure produces a consistent estimate ofp. Simulation results support the effectiveness of this procedure when compared with some of the traditional order selection criteria. Comments are also made on the use of Yule–Walker as opposed to conditional least squares estimations for order selection
Estimation of the Location of the Maximum of a Regression Function Using Extreme Order Statistics

AbstractIn this paper, we consider the problem of approximating the location,x0∈C, of a maximum of a regresion function,θ(x), under certain weak assumptions onθ. HereCis a bounded interval inR. A specific algorithm considered in this paper is as follows. Taking a random sampleX1, …, Xnfrom a distribution overC, we have (Xi, Yi), whereYiis the outcome of noisy measurement ofθ(Xi). Arrange theYi's in nondecreasing order and take the average of therXi's which are associated with therlargest order statistics ofYi. This average,x0, will then be used as an estimate ofx0. The utility of such an algorithm with fixed r is evaluated in this paper. To be specific, the convergence rates ofx0tox0are derived. Those rates will depend on the right tail of the noise distribution and the shape ofθ(·) nearx0
Multivariate Distributions from Mixtures of Max-Infinitely Divisible Distributions

AbstractA class of multivariate distributions that are mixtures of the positive powers of a max-infinitely divisible distribution are studied. A subclass has the property that all weighted minima or maxima belong to a given location or scale family. By choosing appropriate parametric families for the mixing distribution and the distribution being mixed, families of multivariate copulas with a flexible dependence structure and with closed form cumulative distribution functions are obtained. Some dependence properties of the class, as well as some characterizations, are given. Conditions for max-infinite divisibility of multivariate distributions are obtained
Bootstrapping Autoregressive and Moving Average Parameter Estimates of Infinite Order Vector Autoregressive Processes

AbstractWe consider anr-dimensional multivariate time series {yt, t∈Z} which is generated by an infinite order vector autoregressive process. We show that a bootstrap procedure which works by generating time series replicates via an estimated finitek-order vector autoregressive process (k→∞ at an appropriate rate with the sample size) gives asymptotically valid approximations to the joint distribution of the growing set of estimated autoregressive coefficients and to the corresponding set of estimated moving average coefficients (impuls responses)
Covariate Screening in Mixed Linear Models

AbstractWe address the important practical problem of selecting covariates in mixed linear models when the covariance structure is known from the data collection process and there are a possibly large number of covariates available. In particular, we consider procedures which can be considered extensions of the analysis of deviance to mixed linear models. This approach provides an alternative to likelihood ratio test methodology which can be applied in the case that the components of variance are estimated by restricted maximum likelihood (REML), thus resolving the open question of how to proceed in this context. Moreover, it is simple to robustify and allows us to consider a wider class of procedures than those which fit into the simple likelihood ratio test framework. The key insights are that the deviance should be specified by the procedure used to estimate the fixed effects and that the estimated covariance matrix should be held fixed across different models for the fixed effects
Bayesian Local Influence for the Growth Curve Model with Rao's Simple Covariance Structure

AbstractIn this paper, the Bayesian local influence approach is employed to diagnose the adequacy of the growth curve model with Rao's simple covariance structure, based on the Kullback–Leibler divergence. The Bayesian Hessian matrices of the model are investigated in detail under an abstract perturbation scheme. For illustration, covariance-weighted perturbation is considered particularly and used to analyze two real-life biological data sets, which shows that the criteria presented in this article are useful in practice
Nonparametric Estimation of the Bivariate Survival Function with Truncated Data

AbstractRandomly left or right truncated observations occur when one is concerned with estimation of the distribution of time between two events and when one only observes the time if one of the two events falls in a fixed time-window, so that longer survivial times have higher probability to be part of the sample than short survival times. In important AIDS-applications the time between seroconversion and AIDS is only observed if the person did not die before the start of the time-window. Hence, here the time of interest is truncated if another related time-variable is truncated. This problem is a special case of estimation of the bivariate survival function based on truncation by a bivariate truncation time, the problem covered in this paper; in the AIDS-application one component of the bivariate truncation time- vector is alway zero. In this application the bivariate survival function is of interest itself in order to study the relation between time till AIDS and time between AIDS and death. We provide a quick algorithm for computation of the NPMLE. In particular, it is shown that the NPMLE is explicit for the special case when one of the truncation times is zero, as in the aids-application above. We prove that the NPMLE is consistent under the minimal condition that ∫dF/G<∞. Moreover, we prove asymptotic normality under a tail assumption at the origin which is an empirical analoque of ∫dF/G<∞. The condition holds in particular if the truncation distribution has an atom at zero. We provide an algorithm for estimation of its limiting variance. By simply plugging in one of the several proposals for estimation of the bivariate survival function based on right-censored data in the estimating equation we obtain an estimator based on right-censored randomly truncated data. Here, substitution of an estimator which handles the right-censoring efficiently leads to an efficient estimator
Incorporating Extra Information in Nonparametric Smoothing

AbstractNonparametric estimation of conditional mean functions has been studied extensively in the literature. This paper addresses the question of how to use extra informations to improve the estimation. Particularly, we consider the situation that the conditional mean functionE(Z|X) is of interest and there is an auxiliary variable available which is correlated with bothXandZ. A two-stage kernel smoother is proposed to incorporate the extra information. We prove that the asymptotic optimal mean squared error of the proposed estimator is smaller than that obtained when using the Nadaraya–Watson estimator directly without the auxiliary variable. A simulation study is also carried out to illustrate the procedure
Estimation of Multinomial Probabilities under a Model Constraint

AbstractIn this paper estimation of the probabilities of a multinomial distribution has been studied. The five estimators considered are: unrestricted estimator (UE), restricted estimator (RE) (under model M), preliminary test estimator (PTE) based on a test of the model M, shrinkage estimator (SE) and the positive-rule shrinkage estimator (PRSE). Asymptotic distributions of these estimators are given under Pitman alternatives and the asymptotic risk under a quadratic loss has been evaluated. The relative performance of the five estimators is then studied with respect to their asymptotic distributional risks (ADR). It is seen that neither of the preliminary test and shrinkage estimators dominates the other, though each fares well relative to the other estimators. However, the positive rule estimator is recommended for use for dimension 3 or more while the PTE is recommended for dimension less than 3
Bivariate Tensor-Product B-Splines in a Partly Linear Model

AbstractIn some applications, the mean or median response is linearly related to some variables but the relation to additional variables are not easily parameterized. Partly linear models arise naturally in such circumstances. Suppose that a random sample {(Ti, Xi, Yi),i=1, 2, …, n} is modeled byYi=XTiβ0+g0(Ti)+errori, whereYiis a real-valued response,Xi∈RpandTiranges over a unit square, andg0is an unknown function with a certain degree of smoothness. We make use of bivariate tensor-product B-splines as an approximation of the functiong0and consider M-type regression splines by minimization of ∑ni=1ρ(Yi−XTiβ−gn(Ti)) for some convex functionρ. Mean, median and quantile regressions are included in this class. We show under appropriate conditions that the parameter estimate ofβachieves its information bound asymptotically and the function estimate ofg0attains the optimal rate of convergence in mean squared error. Our asymptotic results generalize directly to higher dimensions (for the variableT) provided that the functiong0is sufficiently smooth. Such smoothness conditions have often been assumed in the literature, but they impose practical limitations for the application of multivariate tensor product splines in function estimation. We also discuss the implementation of B-spline approximations based on commonly used knot selection criteria together with a simulation study of both mean and median regressions of partly linear models
A Note on the Asymptotic Normality of Sample Autocorrelations for a Linear Stationary Sequence

AbstractWe consider a stationary time series {Xt} given byXt=∑∞k=−∞ψkZt−k, where {Zt} is a strictly stationary martingale difference white noise. Under assumptions that the spectral densityf(λ) of {Xt} is squared integrable andmτ∑|k|⩾mψ2k→0 for someτ>1/2, the asymptotic normality of the sample autocorrelations is shown. For a stationary long memoryARIMA(p, d, q) sequence, the conditionmτ∑|k|⩾mψ2k→0 for someτ>1/2 is equivalent to the squared integrability off(λ). This result extends Theorem 4.2 of Cavazos-Cadena [5], which were derived under the conditionm∑|k|⩾mψ2k→0
On Kendall's Process

AbstractLetZ1, …, Znbe a random sample of sizen⩾2 from ad-variate continuous distribution functionH, and letVi, nstand for the proportion of observationsZj,j≠i, such thatZj⩽Zicomponentwise. The purpose of this paper is to examine the limiting behavior of the empirical distribution functionKnderived from the (dependent) pseudo-observationsVi, n. This random quantity is a natural nonparametric estimator ofK, the distribution function of the random variableV=H(Z), whose expectation is an affine transformation of the population version of Kendall's tau in the cased=2. Since the sample version ofτis related in the same way to the mean ofKn, Genest and Rivest (1993,J. Amer. Statist. Assoc.) suggested that[formula]be referred to as Kendall's process. Weak regularity conditions onKandHare found under which this centered process is asymptotically Gaussian, and an explicit expression for its limiting covariance function is given. These conditions, which are fairly easy to check, are seen to apply to large classes of multivariate distributions
Strassen's LIL for the Lorenz Curve

AbstractWe prove Strassen's law of the iterated logarithm for the Lorenz process assuming that the underlying distribution functionFand its inverseF−1are continuous, and the momentEX2+εis finite for someε>0. Previous work in this area is based on assuming the existence of the densityf:=F′ combined with further assumptions onFandf. Being based only on continuity and moment assumptions, our method of proof is different from that used previously by others, and is mainly based on a limit theorem for the (general) integrated empirical difference process. The obtained result covers all those we are aware of on the LIL problem in this area
Bivariate Extension of Lomax and Finite Range Distributions through Characterization Approach

AbstractIn the univariate setup the Lomax distribution is being widely used for stochastic modelling of decreasing failure rate life components. It also serves as a useful model in the study of labour turnover, queueing theory, and biological analysis. A bivariate extension of the Lomax distribution given in Lindley and Singpurwalla (1986) fails to cover the case of independence. Our present attempt is to obtain the unique determination of a bivariate Lomax distribution through characterization results. In this process we also obtain bivariate extensions of the exponential and a finite range distributions. The bivariate Lomax distribution thus obtained is a member of the Arnold (1990) flexible family of Pareto distributions and the bivariate exponential distribution derived here is identical with that of Gumbel (1960). Various properties of the proposed extensions are presented
Likelihood Inference in the Errors-in-Variables Model

AbstractWe consider estimation and confidence regions for the parametersαandβbased on the observations (X1, Y1), …, (Xn, Yn) in the errors-in-variables modelXi=Zi+eiandYi=α+βZi+fifor normal errorseiandfiof which the covariance matrix is known up to a constant. We study the asymptotic performance of the estimators defined as the maximum likelihood estimator under the assumption thatZ1, …, Znis a random sample from a completely unknown distribution. These estimators are shown to be asymptotically efficient in the semi-parametric sense if this assumption is valid. These estimators are shown to be asymptotically normal even in the case thatZ1, Z2, … are arbitrary constants satisfying a moment condition. Similarly we study the confidence regions obtained from the likelihood ratio statistic for the mixture model and show that these are asymptotically consistent both in the mixture case and in the case thatZ1, Z2, … are arbitrary constants
A Paradox Concerning Shrinkage Estimators: Should a Known Scale Parameter Be Replaced by an Estimated Value in the Shrinkage Factor?

AbstractWhen estimating, under quadratic loss, the location parameterθof a spherically symmetric distribution with known scale parameter, we show that it may be that the common practice of utilizing the residual vector as an estimate of the variance is preferable to using the known value of the variance. In the context of Stein-like shrinkage estimators, we exhibit sufficient conditions on the spherical distributions for which this paradox occurs. In particular, we show that it occurs fort-distributions when the dimension of the residual vector is sufficiently large. The main tools in the development are upper and lower bounds on the risks of the James–Stein estimators which are exact atθ=0
On a Constrained Optimal Rule for Classification with Unknown Prior Individual Group Membership

AbstractWe describe a formal approach to constructing the optimal classification rule for classification analysis with unknown prior probabilities ofKmultivariate normal populations membership. This is done by suggesting a balanced design for the classification experiment and by constructing the optimal rule under the balanced design condition. The rule is characterized by a constrained minimization of total risk of misclassification; the constraint of the rule is constructed by a process of equalization among expected utilities ofKpopulation conditional densities. The efficacy of the suggested rule is examined through numerical studies. This indicates that dramatic gains in the accuracy of classification result can be achieved in the case where little is known about the relative population sizes
Multivariate Locally Weighted Polynomial Fitting and Partial Derivative Estimation

AbstractNonparametric regression estimator based on locally weighted least squares fitting has been studied by Fan and Ruppert and Wand. The latter paper also studies, in the univariate case, nonparametric derivative estimators given by a locally weighted polynomial fitting. Compared with traditional kernel estimators, these estimators are often of simpler form and possess some better properties. In this paper, we develop current work on locally weighted regression and generalize locally weighted polynomial fitting to the estimation of partial derivatives in a multivariate regression context. Specifically, for both the regression and partial derivative estimators we prove joint asymptotic normality and derive explicit asymptotic expansions for their conditional bias and conditional convariance matrix (given observations of predictor variables) in each of the two important cases of local linear fit and local quadratic fit
Effects of Smoothing on Multivariate Statistical Properties of the Normal to a Rough Curve or Surface

AbstractMuch of the practical interest attached to curves and surfaces derives from features of roughness, rather than smoothness. For example, considerable attention has been paid to fractal models of curves and surfaces, for which the notions of a normal and curvature are usually not well defined. Nevertheless, these quantities are sometimes measurable, because the device for recording a rough surface (such as a stylus or “compass”) adds its own intrinsic smoothness. In this paper we address the effect of such smoothing operations on the multivariate statistical properties of a normal to the surface. Particular attention is paid to the validity of commonly assumed unimodal approximations to the distribution of the normal. It is shown that the actual distribution may have more than one mode, although in a range of situations the unimodal approximation is valid
An Extension of a Convolution Inequality forG-Monotone Functions and an Approach to Bartholomew's Conjectures

AbstractA variety of convolution inequalities have been obtained since Anderson's theorem. ?In this paper, we extend a convolution theorem forG-monotone functions by weakening the symmetry condition ofG-monotone functions. Our inequalities are described in terms of several orderings obtained from a cone. It is noteworthy that the orderings detect differences in directions. A special case of the orderings induces a majorization-like relation on spheres. Applying our inequality, Bartholomew's conjectures, which concern directions yielding the maximum power and the minimum power of likelihood ratio tests for order-restricted alternatives, are partly settled
Some New Results on Stochastic Comparisons of Spacings from Heterogeneous Exponential Distributions

AbstractSome new results are obtained on stochastic orderings between random vectors of spacings from heterogeneous exponential distributions and homogeneous ones. LetD1, …, Dnbe the normalized spacings associated with independent exponential random variablesX1, …,Xn, whereXihas hazard rateλi,i=1, 2, …, n. LetD*1, …, D*nbe the normalized spacings of a random sampleY1, …, Ynof sizenfrom an exponential distribution with hazard rateλ=∑ni=1λi/n. It is shown that for anyn⩾2, the random vector (D1, …, Dn) is greater than the random vector (D*1, …, D*n) in the sense of multivariate likelihood ratio ordering. It also follows from the results proved in this paper that for anyjbetween 2 andn, the survival function ofXj:n−X1:nis Schur convex
Mixtures of Global and Local Edgeworth Expansions and Their Applications

AbstractEdgeworth expansions which are local in one coordinate and global in the rest of the coordinates are obtained for sums of independent but not identically distributed random vectors. Expansions for conditional probabilities are deduced from these. Both lattice and continuous conditioning variables are considered. The results are then applied to derive Edgeworth expansions for bootstrap distributions, for Bayesian bootstrap distribution, and for the distributions of statistics based on samples from finite populations. This results in a unified theory of Edgeworth expansions for resampling procedures. The Bayesian bootstrap is shown to be second order correct for smooth positive “priors,” whenever the third cumulant of the “prior” is equal to the third power of its standard deviation. Similar results are established for weighted bootstrap when the weights are constructed from random variables with a lattice distribution