pinecone/core-2020-05-10-deduplication huggingface dataset:
Design Adaptive Nearest Neighbor Regression Estimation

AbstractThis paper deals with nonparametric regression estimation under arbitrary sampling with an unknown distribution. The effect of the distribution of the design, which is a nuisance parameter, can be eliminated by conditioning. An upper bound for the conditional mean squared error of k−NN estimates leads us to consider an optimal number of neighbors, which is a random function of the sampling. The corresponding estimate can be used for nonasymptotic inference and is also consistent under a minimal recurrence condition. Some deterministic equivalents are found for the random rate of convergence of this optimal estimate, for deterministic and random designs with vanishing or diverging densities. The proposed estimate is rate optimal for standard designs
Reliability Studies of Bivariate Distributions with Pareto Conditionals

AbstractIn this paper we study Arnold's (1987, Statist. Probab. Lett.5, 263–266) class of bivariate distributions with Pareto conditionals from a reliability point of view. Failure rates and mean residual life function of the marginal distributions and their monotonic properties are studied. The hazard components and their properties are investigated and their relationships with some measures of dependence are established. Finally, the failure rate of the minimum of the two components is examined and its monotonicity is investigated. Some of the results presented here are general and would be useful in studying the dependence structure in other classes of bivariate distributions
Using a Bootstrap Method to Choose the Sample Fraction in Tail Index Estimation

AbstractTail index estimation depends for its accuracy on a precise choice of the sample fraction, i.e., the number of extreme order statistics on which the estimation is based. A complete solution to the sample fraction selection is given by means of a two-step subsample bootstrap method. This method adaptively determines the sample fraction that minimizes the asymptotic mean-squared error. Unlike previous methods, prior knowledge of the second-order parameter is not required. In addition, we are able to dispense with the need for a prior estimate of the tail index which already converges roughly at the optimal rate. The only arbitrary choice of parameters is the number of Monte Carlo replications
ISO* Property of Two-Parameter Compound Poisson Distributions with Applications

AbstractThe ISO* property of noncentrality parameters is derived for the expected value of an ISO* function of independent nonnegative two-parameter compound Poisson random variables and is then applied to unbiasedness of tests and monotonicity of power functions of tests in an order-restricted hypothesis testing problem for the noncentrality parameter. The ISO* property and the Schur convexity are also studied for a class of two-parameter distributions which has the additive property (i.e., is closed under convolution) and contains the one-parameter family with the semigroup property as a special case
On Marginal Quasi-Likelihood Inference in Generalized Linear Mixed Models

AbstractIn view of the cumbersome and often intractable numerical integrations required for a full likelihood analysis, several suggestions have been made recently for approximate inference in generalized linear mixed models (GLMMs). Two closely related approximate methods are the penalized quasi-likelihood (PQL) method and the marginal quasi-likelihood (MQL) method. The PQL approach generally produces biased estimates for the regression effects and the variance component of the random effects. Recently, some corrections have been proposed to remove these biases. But the corrections appear to be satisfactory only when the variance component of the random effects is small. The MQL approach has also been used for inference in the GLMMs. This approach requires the computations of the joint moments of the clustered observations, up to order four. But the derivation of these moments are not easy. Consequently, different “working” formulas have been used, especially for the mean and covariance matrix of the observations, which may not lead to desirable estimates. In this paper, we use a small variance component (of the random effects) approach and develop the MQL estimating equations for the parameters based on the joint moments of order up to four. The proposed approach thus avoids the use of the so-called “working” covariance and higher order moment matrices, leading to better estimates for the regression and the overdispersion parameters, in the sense of efficiency in particular
Stochastic Bounds and Dependence Properties of Survival Times in a Multicomponent Shock Model

AbstractConsider a system that consists of several components. Shocks arrive according to a counting process (which may be non-homogeneous and with correlated interarrival times) and each shock may simultaneously destroy a subset of the components. Shock models of this type arise naturally in reliability modeling in dynamic environments. Due to correlated shock arrivals, individual component lifetimes are statistically dependent, which makes the explicit evaluation of the joint distribution intractable. To facilitate the development of easily computable tight bounds and good approximations, an analytic analysis of the dependence structure of the system is needed. The purpose of this paper is to provide a general framework for studying the correlation structure of shock models in the setup of a multivariate, correlated counting process and to systematically develop upper and lower bounds for its joint component lifetime distribution and survival functions. The thrust of the approach is the interplay between a newly developed notion, majorization with respect to weighted trees, and various stochastic dependence orders, especially orthant dependence orders of random vectors and orthant dependence orders of stochastic processes. It is shown that the dependence nature of the joint lifetime is inherited from spatial dependence and temporal dependence; that is, dependence among various components due to simultaneous arrivals and dependence over different time instants introduced by the shock arrival process. The two types of dependency are investigated separately and their joint impact on the performance of the system is analyzed. The results are used to develop computable bounds for the statistics of the joint component lifetimes, which are tighter than the product-form bounds under certain conditions. The shock model with a non-homogeneous Poisson arrival process is studied as an illustrative example. The result is also applicable to the cumulative damage model with multivariate shock arrival processes
Common Principal Components for Dependent Random Vectors

AbstractLet the kp-variate random vector X be partitioned into k subvectors Xi of dimension p each, and let the covariance matrix Ψ of X be partitioned analogously into submatrices Ψij. The common principal component (CPC) model for dependent random vectors assumes the existence of an orthogonal p by p matrix β such that βtΨijβ is diagonal for all (i, j). After a formal definition of the model, normal theory maximum likelihood estimators are obtained. The asymptotic theory for the estimated orthogonal matrix is derived by a new technique of choosing proper subsets of functionally independent parameters
Characteristic Functions of L1-Spherical and L1-Norm Symmetric Distributions and Their Applications

AbstractIn this article we obtain the characteristic functions (c.f.'s) for L1-spherical distributions and simplify that of the L1-norm symmetric distributions to an expression of a finite sum. These forms of c.f.'s can be used to derive the probability density functions (p.d.f.'s) of linear combinations of variables. We shall show that this gives a unified approach to the treatment of the linear function of i.i.d. random variables and their order statistics associated with double-exponential (i.e., Laplace), exponential, and uniform distributions. Some applications in reliability prediction, random weighting, and serial correlation are also shown
Admissibility of Confidence Estimators in the Regression Model

AbstractIn the regression model, we assume that the independent variables are random instead of fixed. Consider the problem of estimating the coverage function of a usual confidence interval for the unknown intercept parameter. In this paper, we consider a case in which the number of unknown parameters is smaller than 5. We show that the usual constant coverage probability estimator is admissible in the usual sense in this case. Note that this estimator is inadmissible in the usual sense in the other case where the number of unknown parameters is greater than 4
Analysis of Variance in Nonparametric Regression Models

AbstractIn a nonparametric regression model with a multivariate explanatory variable we consider the problem of testing the hypothesis that specific interactions in a canonical decomposition of the model vanish. A simple consistent test is developed which is based on the difference between the corresponding sum of squares of the ordinary ANOVA for the multi-factor case and a nonparametric variance estimator. This presents an analogue of the classical ANOVA in the nonparametric regression setup. Asymptotic normality of the introduced test statistic is derived under the null hypothesis and under fixed alternatives. The finite sample behaviour of the proposed procedures is illustrated by a small simulation study and a data example
MU-Estimation and Smoothing

AbstractIn the M-estimation theory developed by Huber (1964, Ann. Math. Statist.43, 1449–1458), the parameter under estimation is the value of θ which minimizes the expectation of what is called a discrepancy measure (DM) δ(X, θ) which is a function of θ and the underlying random variable X. Such a setting does not cover the estimation of parameters such as the multivariate median defined by Oja (1983) and Liu (1990), as the value of θ which minimizes the expectation of a DM of the type δ(X1, …, Xm, θ) where X1, …, Xm are independent copies of the underlying random variable X. Arcones et al. (1994, Ann. Statist.22, 1460–1477) studied the estimation of such parameters. We call such an M-type MU-estimation (or μ-estimation for convenience). When a DM is not a differentiable function of θ, some complexities arise in studying the properties of estimators as well as in their computation. In such a case, we introduce a new method of smoothing the DM with a kernel function and using it in estimation. It is seen that smoothing allows us to develop an elegant approach to the study of asymptotic properties and possibly apply the Newton–Raphson procedure in the computation of estimators
Invariant Tests for Covariance Structures in Multivariate Linear Model

AbstractThe null hypothesis that the error vectors in a multivariate linear model are independent is tested against the alternative hypothesis that they are dependent in some specified manner. This dependence is assumed to be due to common random components or autocorrelation over time. The testing problem is solved by classical invariance arguments under multinormality. The most powerful invariant test usually depends on the particular alternative and may even lack a closed form expression. Then the locally best test is derived. The power is maximized at the null hypothesis in the direction of some alternative. In most applications the direction where the maximization is performed does not enter the test. Then the locally uniformly best test exists. Several applications are outlined
Efficient Variable Screening for Multivariate Analysis

AbstractIt is shown how known algorithms for the comparison of all variables subsets in regression analysis can be adapted to subset comparisons in multivariate analysis, according to any index based on Wilks, Lawley–Hotelling, or Bartllet–Pillai statistics and, in some special cases, according to any function of the sample squared canonical correlations. The issues regarding the choice of an appropriate comparison criterion are discussed. The computational effort of the proposed algorithms is studied, and it is argued that, for a moderate number of variables, they should be preferred to stepwise selection methods. A software implementation of the methods discussed is freely available and can be downloaded from the Internet
Exact Misclassification Probabilities for Plug-In Normal Quadratic Discriminant Functions. I. The Equal-Means Case

AbstractWe consider the problem of discriminating, on the basis of random “training” samples, between two independent multivariate normal populations, Np(μ, Σ1) and Np(μ, Σ2), which have a common mean vector μ and distinct covariance matrices Σ1 and Σ2. Using the theory of Bessel functions of the second kind of matrix argument developed by Herz (1955, Ann. Math.61, 474–523), we derive stochastic representations for the exact distributions of the “plug-in” quadratic discriminant functions for classifying a newly obtained observation. These stochastic representations involve only chi-squared and F-distributions, hence we obtain an efficient method for simulating the discriminant functions and estimating the corresponding probabilities of misclassification. For some special values of p, Σ1 and Σ2 we obtain explicit formulas and inequalities for the probabilities of misclassification. We apply these results to data given by Stocks (1933, Ann. Eugen.5, 1–55) in a biometric investigation of the physical characteristics of twins, and to data provided by Rencher (1995, “Methods of Multivariate Analysis,” Wiley, New York) in a study of the relationship between football helmet design and neck injuries. For each application we estimate the exact probabilities of misclassification, and in the case of Stocks' data we make extensive comparisons with previously published estimates
Normal Approximation Rate and Bias Reduction for Data-Driven Kernel Smoothing Estimator in a Semiparametric Regression Model

AbstractAccuracy of the normal approximation for Speckman's kernel smoothing estimator of the parametric component β in the semiparametric regression model y=xτβ+g(t)+e is studied when the bandwidth used in the estimator is selected by a general data-based method which includes such commonly used bandwidth selectors as (delete-one-out) CV, GCV, and Mallows' CL criterion. We find that, contrary to what we might expect, this data-driven estimator cannot attain the optimal Berry–Esseen rate n−1/2. Consequently, the confidence region of β based on this normal approximation is not first-order accurate. The reason for this is that the bias of Speckman's estimator is still of nonparametric order at the data-driven bandwidth choice. We then propose a resmoothing method to reduce the bias and show that the proposed estimator can achieve the optimal Berry–Esseen rate. A simulation study shows a slightly better small-sample performance of the proposed estimator
An Expectation Formula for the Multivariate Dirichlet Distribution

AbstractSuppose that the random vector (X1, …, Xq) follows a Dirichlet distribution on Rq+ with parameter (p1, …, pq)∈Rq+. For f1, …, fq>0, it is well-known that E(f1X1+…+fqXq)−(p1+…+pq)=f−p11…f−pqq. In this paper, we generalize this expectation formula to the singular and non-singular multivariate Dirichlet distributions as follows. Let Ωr denote the cone of all r×r positive-definite real symmetric matrices. For x∈Ωr and 1⩽j⩽r, let detjx denote the jth principal minor of x. For s=(s1, …, sr)∈Rr, the generalized power function of x∈Ωr is the function Δs(x)=(det1x)s1−s2(det2x)s2−s3…(detr−1x)sr−1−sr(detrx)sr; further, for any t∈R, we denote by s+t the vector (s1+t, …, sr+t). Suppose X1, …, Xq∈Ωr are random matrices such that (X1, …, Xq) follows a multivariate Dirichlet distribution with parameters p1, …, pq. Then we evaluate the expectation E[Δs1(X1)…Δsq(Xq)Δs1+…+sq+p((a+f1X1+…+fqXq)−1)], where a∈Ωr, p=p1+…+pq, f1, …, fq>0, and s1, …, sq each belong to an appropriate subset of Rr+. The result obtained is parallel to that given above for the univariate case, and remains valid even if some of the Xj's are singular. Our derivation utilizes the framework of symmetric cones, so that our results are valid for multivariate Dirichlet distributions on all symmetric cones
Influence Function of Halfspace Depth

AbstractThe sensitivity of halfspace depth values and contours to perturbations of the underlying distribution is investigated. The influence function of the halfspace depth of any point x∈Rp is bounded and discontinuous; it is constant and positive when the perturbing observation z is placed in any optimal halfspace and it is constant and negative when z is placed in any non-optimal halfspace. When the optimal halfspace is unique a von Mises expansion allows an easy derivation of the asymptotic distribution of the sample halfspace depth. In the sampling case, in general, addition of a single observation outside the convex hull of the sample alters all the depth regions but only the outer region can be arbitrarily expanded. To obtain the same effect on the inner regions the size of the perturbation is required to be not less than the depth orders. Numerical illustrations of the results are given
Extended Gauss–Markov Theorem for Nonparametric Mixed-Effects Models

AbstractThe Gauss–Markov theorem provides a golden standard for constructing the best linear unbiased estimation for linear models. The main purpose of this article is to extend the Gauss–Markov theorem to include nonparametric mixed-effects models. The extended Gauss–Markov estimation (or prediction) is shown to be equivalent to a regularization method and its minimaxity is addressed. The resulting Gauss–Markov estimation serves as an oracle to guide the exploration for effective nonlinear estimators adaptively. Various examples are discussed. Particularly, the wavelet nonparametric regression example and its connection with a Sobolev regularization is presented
Identifiability of Modified Power Series Mixtures via Posterior Means

AbstractProblems of specification of discrete bivariate statistical models by a modified power series conditional distribution and a regression function are studied. An identifiability result for a wide class of such mixtures with infinite support is obtained. Also the finite support case within a more specific model is considered. Applications for Poisson, (truncated) geometric, and binomial mixtures are given. From the viewpoint of Bayesian analysis unique determination of the prior by a Bayes estimate of the mean for modified power series mixtures is investigated
Robust Bayesian Inference on Scale Parameters

AbstractWe represent random vectors Z that take values in Rn−{0} as Z=RY, where R is a positive random variable and Y takes values in an (n−1)-dimensional space Y. By fixing the distribution of either R or Y, while imposing independence between them, different classes of distributions on Rn can be generated. As examples, the spherical, lq-spherical, υ-spherical and anisotropic classes can be interpreted in this unifying framework. We present a robust Bayesian analysis on a scale parameter in the pure scale model and in the regression model. In particular, we consider robustness of posterior inference on the scale parameter when the sampling distribution ranges over classes related to those mentioned above. Some links between Bayesian and sampling-theory results are also highlighted
A Class of Robust Principal Component Vectors

AbstractThis paper is concerned with a study of robust estimation in principal component analysis. A class of robust estimators which are characterized as eigenvectors of weighted sample covariance matrices is proposed, where the weight functions recursively depend on the eigenvectors themselves. Also, a feasible algorithm based on iterative reweighting of the covariance matrices is suggested for obtaining these estimators in practice. Statistical properties of the proposed estimators are investigated in terms of sensitivity to outliers and relative efficiency via their influence functions, which are derived with the help of Stein's lemma. We give a simple condition on the weight functions which ensures robustness of the estimators. The class includes, as a typical example, a method by the self-organizing rule in the neural computation. A numerical experiment is conducted to confirm a rapid convergence of the suggested algorithm
Inferences on a Normal Covariance Matrix and Generalized Variance with Monotone Missing Data

AbstractThe problems of testing a normal covariance matrix and an interval estimation of generalized variance when the data are missing from subsets of components are considered. The likelihood ratio test statistic for testing the covariance matrix is equal to a specified matrix, and its asymptotic null distribution is derived when the data matrix is of a monotone pattern. The validity of the asymptotic null distribution and power analysis are performed using simulation. The problem of testing the normal mean vector and a covariance matrix equal to a given vector and matrix is also addressed. Further, an approximate confidence interval for the generalized variance is given. Numerical studies show that the proposed interval estimation procedure is satisfactory even for small samples. The results are illustrated using simulated data
Asymptotics for Homogeneity Tests Based on a Multivariate Random Effects Proportional Hazards Model

AbstractD. Commenges and H. Jacqmin-Gadda (1997, J. Roy. Statist. Soc. B59, 157–171) considered generalized score tests of homogeneity that accommodate parametric and semiparametric regression models arising from multivariate random effects with common variance and known correlations. In this paper we give some sufficient hypotheses under which the asymptotic behavior of the test statistic is standard normal for the proportional hazards model with right censored data. Several examples of applicability are studied
The Vervaat Process in Lp Spaces

AbstractIt is well known that, asymptotically, the appropriately normalized Vervaat process behaves like one half times the squared empirical process. Considering these two processes as elements of the Lp-space, 1⩽p<∞, we give a complete description of the strong and weak asymptotic behaviour of the Lp-distance between them, and thus of the Lp-norm of the Vervaat process as well. The herein obtained results also raise a number of further mathematical and probabilistic problems which we formulate as suggestions for future consideration