pinecone/core-2020-05-10-deduplication huggingface dataset:
Highly Robust Estimation of Dispersion Matrices

AbstractIn this paper, we propose a new componentwise estimator of a dispersion matrix, based on a highly robust estimator of scale. The key idea is the elimination of a location estimator in the dispersion estimation procedure. The robustness properties are studied by means of the influence function and the breakdown point. Further characteristics such as asymptotic variance and efficiency are also analyzed. It is shown in the componentwise approach, for multivariate Gaussian distributions, that covariance matrix estimation is more difficult than correlation matrix estimation. The reason is that the asymptotic variance of the covariance estimator increases with increasing dependence, whereas it decreases with increasing dependence for correlation estimators. We also prove that the asymptotic variance of dispersion estimators for multivariate Gaussian distributions is proportional to the asymptotic variance of the underlying scale estimator. The proportionality value depends only on the underlying dependence. Therefore, the highly robust dispersion estimator is among the best robust choice at the present time in the componentwise approach, because it is location-free and combines small variability and robustness properties such as high breakdown point and bounded influence function. A simulation study is carried out in order to assess the behavior of the new estimator. First, a comparison with another robust componentwise estimator based on the median absolute deviation scale estimator is performed. The highly robust properties of the new estimator are confirmed. A second comparison with global estimators such as the method of moment estimator, the minimum volume ellipsoid, and the minimum covariance determinant estimator is also performed, with two types of outliers. In this case, the highly robust dispersion matrix estimator turns out to be an interesting compromise between the high efficiency of the method of moment estimator in noncontaminated situations and the highly robust properties of the minimum volume ellipsoid and minimum covariance determinant estimators in contaminated situations
Censored Partial Linear Models and Empirical Likelihood

AbstractConsider the partial linear model Yi=Xτiβ+g(Ti)+εi, i=1, …, n, where β is a p×1 unknown parameter vector, g is an unknown function, Xi's are p×1 observable covariates, Ti's are other observable covariates in [0, 1], and Yi's are the response variables. In this paper, we shall consider the problem of estimating β and g and study their properties when the response variables Yi are subject to random censoring. First, the least square estimators for β and kernel regression estimator for g are proposed and their asymptotic properties are investigated. Second, we shall apply the empirical likelihood method to the censored partial linear model. In particular, an empirical log-likelihood ratio for β is proposed and shown to have a limiting distribution of a weighted sum of independent chi-square distributions, which can be used to construct an approximate confidence region for β. Some simulation studies are conducted to compare the empirical likelihood and normal approximation-based method
U-Statistics for Change under Alternatives

AbstractAsymptotic distributions of U-statistics to test for possible changes in the distribution will be derived when the change occurred. We will show that for all possible types of kernels, symmetric, antisymmetric, degenerate, non-degenerate, the test statistics are asymptotically normally distributed. We also study the distribution of the estimator of the time of change. Its large sample behaviour is approximately that of the maximum of a two-sided random walk. The terms in these random walks explain the exact nature of bias in the change-point estimator. Several examples will be given as illustration
Characterization of the Spectra of Periodically Correlated Processes

AbstractA complete characterization of the spectrum of a locally square integrable periodically correlated (PC) processes is obtained. The result makes use of the author's recent theorem establishing a one to one correspondence between PC processes and a certain class on infinite dimensional stationary processes. In terms of distributions it is proved that the Fourier transform of a positive definite distribution on the plane which is the sum of complex uniformly bounded measures supported on equidistant lines parallel to diagonal is a locally square integrable function
Local Power Properties of Kernel Based Goodness of Fit Tests

AbstractIf (Xi, i∈Z) is a strictly stationary process with marginal density function f, we are interested in testing the hypothesis H0:{f=f0}, where f0 is given. We consider different test statistics based on integrated quadratic forms measuring the proximity between fn, a kernel estimator of f, and f0, or between fn and its expected value computed under H0. We study the asymptotic local power properties of the testing procedures under local alternatives. This study generalizes to the multidimensional case in a context of dependence the corresponding one made by P. J. Bickel and M. Rosenblatt in 1973 (Ann. Statist.1, 1071–1095)
Efficiency and Robustness of a Resampling M-Estimator in the Linear Model

AbstractIn the literature, there are basically two kinds of resampling methods for least squares estimation in linear models; the E-type (the efficient ones like the classical bootstrap), which is more efficient when error variables are homogeneous, and the R-type (the robust ones like the jackknife), which is more robust for heterogeneous errors. However, for M-estimation of a linear model, we find a counterexample showing that a usually E-type method is less efficient than an R-type method when error variables are homogeneous. In this paper, we give sufficient conditions under which the classification of the two types of the resampling methods is still true
Estimating a Distribution Function for Censored Time Series Data

AbstractConsider a long term study, where a series of dependent and possibly censored failure times is observed. Suppose that the failure times have a common marginal distribution function, but they exhibit a mode of time series structure such as α-mixing. The inference on the marginal distribution function is of interest to us. The main results of this article show that, under some regularity conditions, the Kaplan–Meier estimator enjoys uniform consistency with rates, and a stochastic process generated by the Kaplan–Meier estimator converges weakly to a certain Gaussian process with a specified covariance structure. Finally, an estimator of the limiting variance of the Kaplan–Meier estimator is proposed and its consistency is established
The Law of the Iterated Logarithm and Central Limit Theorem for L-Statistics

AbstractThe Chung–Smirnov law of the iterated logarithm and the Finkelstein functional law of the iterated logarithm for empirical processes are used to establish new results on the central limit theorem, the law of the iterated logarithm, and the strong law of large numbers for L-statistics with certain bounded and smooth weight functions. These results are used to obtain necessary and sufficient conditions for almost sure convergence and for convergence in distribution of some well-known L-statistics and U-statistics, including Gini's mean difference statistic. A law of the logarithm for weighted sums of order statistics is also presented
Inadmissibility of the Maximum Likekihood Estimator of Normal Covariance Matrices with the Lattice Conditional Independence

AbstractLattice conditional independence (LCI) models introduced by S. A. Andersson and M. D. Perlman (1993, Ann. Statist.21, 1318–1358) have the pleasant feature of admitting explicit maximum likelihood estimators and likelihood ratio test statistics. This is because the likelihood function and parameter space for a LCI model can be factored into products of conditional likelihood functions and parameter spaces, where the standard multivariate techniques can be applied. In this paper we consider the problem of estimating the covariance matrices under LCI restriction in a decision theoretic setup. The Stein loss function is used in this study and, using the factorization mentioned above, minimax estimators are obtained. Since the maximum likelihood estimator has constant risk and is different from the minimax estimator, this shows that the maximum likelihood estimator under LCI restriction inadmissible. These results extend those obtained by W. James and C. Stein (1960, in “Proceedings of the Fourth Berkeley Symposium on Mathematics, Statistics, and Probability,” Vol. 1, pp. 360–380, Univ. of California Press, Berkeley, CA) and D. K. Dey and C. Srinivasan (1985, Ann. Statist.13, 1581–1591) for estimating normal covariance matrices to the LCI models
Consecutive Collapsibility of Odds Ratios over an Ordinal Background Variable

AbstractG. R. Ducharme and Y. Lepage (1986, J. Roy. Statist. Soc. Ser. B48, 197–205) presented the strong collapsibility of odds ratio in 2×2×K tables. However, the concept is not suitable for an ordinal background variable since it is meaningless to pool nonadjacent levels in this case. In this paper, we introduce the consecutive collapsibility of odds ratios over an ordinal background in an I×J×K table and present necessary and sufficient conditions for consecutive collapsibility of odds ratios. The conditions are important for guiding the possible merging of consecutive levels of the background
A Martingale Approach to the Copula-Graphic Estimator for the Survival Function under Dependent Censoring

AbstractThe product limit estimator is arguably the most popular method of estimating survival probabilities in homogeneous samples. When the survival time and the censoring time are dependent, the product-limit estimator is an inconsistent estimator of the marginal survival function. Recently M. Zheng and J. P. Klein (1995, Biometrika82, 127–138) proposed a copula-graphic estimator that models the dependency between censoring and survival using a copula function. This work investigates their proposal. First it derives a closed form expression for the copula-graphic estimator when the joint survival function is modeled with an Archimedean copula. The copula-graphic estimator is then shown to be uniformly consistent and asymptotically normal. It is also equivalent to the usual product-limit estimator when the survival and censoring times are assumed to be independent. A sensitivity analysis of the specification of the copula model for the dependency is also presented
Semiparametric Mixtures in Case-Control Studies

AbstractWe consider likelihood based inference in a class of logistic models for case- control studies with a partially observed covariate. The likelihood is a combination of a nonparametric mixture, a parametric likelihood, and an empirical likelihood. We prove the asymptotic normality of the maximum likelihood estimator for the regression slope, the asymptotic chi-squared distribution of the likelihood ratio statistic, and the consistency of the observed information, in both the prospective and the retrospective model
Multivariate Stability and Strong Limiting Behaviour of Intermediate Order Statistics

AbstractWe investigate the almost sure stability of intermediate order statistics of a sample from a multivariate distribution. It is shown that the ordering of the multivariate sample—based on an increasing family of conditional quantile surfaces—enables us to use a univariate approach. In the univariate case, we characterise the strong limiting behaviour of weighted uniform quantile processes along an intermediate sequence satisfying suitable conditions of growth and regularity. From this characterisation a necessary and sufficient condition for the almost sure stability is deduced
Double k-Class Estimators in Regression Models with Non-spherical Disturbances

AbstractIn this paper, we consider a family of feasible generalised double k-class estimators in a linear regression model with non-spherical disturbances. We derive the large sample asymptotic distribution of the proposed family of estimators and compare its performance with the feasible generalized least squares and Stein-rule estimators using the mean squared error matrix and risk under quadratic loss criteria. A Monte-Carlo experiment investigates the finite sample behaviour of the proposed family of estimators
Influence Diagnostics in Common Principal Components Analysis

AbstractIn principal components analysis, the influence function and local influence approaches have been well established as important diagnostic tools. In this article, we first review the generalized local influence approach in the restricted likelihood framework. We then apply the restricted likelihood local influence diagnostic in the common principal components analysis. One special part of this local influence result is an elliptical norm of the empirical influence function, which is comparable to the deletion diagnostic scaled by the same matrix which requires iterative solutions for parameter estimates with every case deleted. Local influence diagnostics are constructed by some basic building blocks that are obtained directly from the maximum likelihood estimates of the parameters, and which are based on the original data and thus require less computation. A numerical example illustrates the technique and some joint influence effects are identified by the proposed method
Multiway Dependence in Exponential Family Conditional Distributions

AbstractConditionally specified statistical models are frequently constructed from one-parameter exponential family conditional distributions. One way to formulate such a model is to specify the dependence structure among random variables through the use of a Markov random field (MRF). A common assumption on the Gibbsian form of the MRF model is that dependence is expressed only through pairs of random variables, which we refer to as the “pairwise-only dependence” assumption. Based on this assumption, J. Besag (1974, J. Roy. Statist. Soc. Ser. B36, 192–225) formulated exponential family “auto-models” and showed the form that one-parameter exponential family conditional densities must take in such models. We extend these results by relaxing the pairwise-only dependence assumption, and we give a necessary form that one-parameter exponential family conditional densities must take under more general conditions of multiway dependence. Data on the spatial distribution of the European corn borer larvae are fitted using a model with Bernoulli conditional distributions and several dependence structures, including pairwise-only, three-way, and four-way dependencies
Optimal Spherical Deconvolution

AbstractThis paper addresses the issue of optimal deconvolution density estimation on the 2-sphere. Indeed, by using the transitive group action of the rotation matrices on the 2-dimensional unit sphere, rotational errors can be introduced analogous to the Euclidean case. The resulting density turns out to be convolution in the Lie group sense and so the statistical problem is to recover the true underlying density. This recovery can be done by deconvolution; however, as in the Euclidean case, the difficulty of the deconvolution turns out to depend on the spectral properties of the rotational error distribution. This therefore leads us to define smooth and super-smooth classes and optimal rates of convergence are obtained for these smoothness classes
A Test of Multivariate Independence Based on a Single Factor Model

AbstractA test of the independence of two sets of variables is developed to have high power against a special family of dependence. In this each set of variables has the structure of a single factor model and the dependence is solely via the correlation γ between the underlying latent variables. This is a model with only one nonzero canonical correlation. It is shown that a test based on the maximum likelihood estimate of γ is appreciably more powerful than that based on r1, the largest sample canonical correlation. If, however, the model is used, not just as a family of alternatives but as the basis for interpretation, and if substantial cross-correlation is present then the procedure is essentially equivalent to the use of r1
Asymptotic Limit of the Bayes Actions Set Derived from a Class of Loss Functions

AbstractSimilarly to the determination of a prior in Bayesian Decision theory, an arbitrarily precise determination of the loss function is unrealistic. Thus, analogously to global robustness with respect to the prior, one can consider a set of loss functions to describe the imprecise preferences of the decision maker. In this paper, we investigate the asymptotic behavior of the Bayes actions set derived from a class of loss functions. When the collection of additional observations induces a decrease in the range of the Bayes actions, robustness is improved. We give sufficient conditions for the convergence of the Bayes actions set with respect to the Hausdorff metric and we also give the limit set. Finally, we show that these conditions are satisfied when the set of decisions and the set of states of nature are subsets of Rp
Prediction from Randomly Right Censored Data

AbstractLet X be a random vector taking values in Rd, let Y be a bounded random variable, and let C be a right censoring random variable operating on Y. It is assumed that C is independent of (X, Y), the distribution function of C is continuous, and the support of the distribution of Y is a proper subset of the support of the distribution of C. Given a sample {Xi, min{Yi, Ci}, I[Yi⩽Ci]} and a vector of covariates X, we want to construct an estimate of Y such that the mean squared error is minimized. Without censoring, i.e., for C=∞ almost surely, it is well known that the mean squared error of suitably defined kernel, partitioning, nearest neighbor, least squares, and smoothing spline estimates converges for every distribution of (X, Y) to the optimal value almost surely, if the sample size tends to infinity. In this paper, we modify the above estimates and show that in the random right censoring model described above the same is true for the modified estimates
Inference for the Mean Difference in the Two-Sample Random Censorship Model

AbstractInference for the mean difference in the two-sample random censorship model is an important problem in comparative survival and reliability test studies. This paper develops an adjusted empirical likelihood inference and a martingale-based bootstrap inference for the mean difference. A nonparametric version of Wilks' theorem for the adjusted empirical likelihood is derived, and the corresponding empirical likelihood confidence interval of the mean difference is constructed. Also, it is shown that the martingale-based bootstrap gives a correct first order asymptotic approximation of the corresponding estimator of the mean difference, which ensures that the martingale-based bootstrap confidence interval has asymptotically correct coverage probability. A simulation study is conducted to compare the adjusted empirical likelihood, the martingale-based bootstrap, and Efron's bootstrap in terms of coverage accuracies and average lengths of the confidence intervals. The simulation indicates that the proposed adjusted empirical likelihood and the martingale-based bootstrap confidence procedures are comparable, and both seem to outperform Efron's bootstrap procedure
Bayesian Inference for Multivariate Survival Data with a Cure Fraction

AbstractWe develop Bayesian methods for right censored multivariate failure time data for populations with a cure fraction. We propose a new model, called the multivariate cure rate model, and provide a natural motivation and interpretation of it. To create the correlation structure between the failure times, we introduce a frailty term, which is assumed to have a positive stable distribution. The resulting correlation structure induced by the frailty term is quite appealing and leads to a nice characterization of the association between the failure times. Several novel properties of the model are derived. First, conditional on the frailty term, it is shown that the model has a proportional hazards structure with the covariates depending naturally on the cure rate. Second, we establish mathematical relationships between the marginal survivor functions of the multivariate cure rate model and the more standard mixture model for modelling cure rates. With the introduction of latent variables, we show that the new model is computationally appealing, and novel computational Markov chain Monte Carlo (MCMC) methods are developed to sample from the posterior distribution of the parameters. Specifically, we propose a modified version of the collapsed Gibbs technique (J. S. Liu, 1994, J. Amer. Statist. Assoc.89, 958–966) to sample from the posterior distribution. This development will lead to an efficient Gibbs sampling procedure, which would otherwise be extremely difficult. We characterize the propriety of the joint posterior distribution of the parameters using a class of noninformative improper priors. A real dataset from a melanoma clinical trial is presented to illustrate the methodology
Eigenstructures of Spatial Design Matrices

AbstractIn estimating the variogram of a spatial stochastic process, we use a spatial design matrix. This matrix is the key to Matheron's variogram estimator. We show how the structure of the matrix for any dimension is based on the one-dimensional spatial design matrix, and we compute explicit eigenvalues and eigenvectors for all dimensions. This design matrix involves Kronecker products of second order finite difference matrices, with cosine eigenvectors and eigenvalues. Using the eigenvalues of the spatial design matrix, the statistics of Matheron's variogram estimator are determined. Finally, a small simulation study is performed
Empirical Likelihood Ratio in Terms of Cumulative Hazard Function for Censored Data

AbstractIt has been shown that (with complete data) empirical likelihood ratios can be used to form confidence intervals and test hypotheses about a linear functional of the distribution function just like the parametric case. We study here the empirical likelihood ratios for right censored data and with parameters that are linear functionals of the cumulative hazard function. Martingale techniques make the asymptotic analysis easier, even for random weighting functions. It is shown that the empirical likelihood ratio in this setting can be easily obtained by solving a one parameter monotone equation
Nonnegative Minimum Biased Quadratic Estimation in Mixed Linear Models

AbstractThe problem of nonnegative quadratic estimation of a parametric function γ(β, σ)=β′Fβ+∑ri=1fiσ2i in a general mixed linear model M{y, Xβ, V(σ)=∑ri=1σ2iVi} is discussed. Necessary and sufficient conditions are given for y′A0y to be a minimum biased estimator for γ. It is shown how to formulate the problem of finding a nonnegative minimium biased estimator of γ as a conic optimization problem, which can be efficiently solved using convex optimization techniques. Models with two variance components are considered in detail. Some applications to one-way classification mixed models are given. For these models minimum biased estimators with minimum norms for square of expectation β2 and for σ21 are presented in explicit forms