pinecone/core-2020-05-10-deduplication huggingface dataset:
New Multivariate Product Density Estimators

AbstractLet X be an Rd-valued random variable with unknown density f. Let X1, …, Xn be i.i.d. random variables drawn from f. The objective is to estimate f(x), where x=(x1, …, xd). We study the pointwise convergence of two new density estimates, the Hilbert product kernel estimated!n∑i=1n∏j=1d12logn|xj−Xij|, where Xi=(Xi1, …, Xid), and the Hilbert k-nearest neighbor estimatek(d−1)!2dnlogd−1(n/(k(d−1)!))∏dj=1|xj−X(k)j|, where X(k)=(X(k)1, …, X(k)d), and X(k) is the kth nearest neighbor of x when points are ordered by increasing values of the product ∏dj=1|xj−X(k)j|, and k=o(logn), k→∞. The auxiliary results needed permit us to formulate universal consistency results (pointwise and in L1) for product kernel estimates with different window widths for each coordinate, and for rectangular partitioning and tree estimates. In particular, we show that locally adapted smoothing factors for product kernel estimates may make the kernel estimate inconsistent even under standard conditions on the bandwidths
Large Sample Properties of Mixture Models with Covariates for Competing Risks

AbstractWe study the large-sample properties of a class of parametric mixture models with covariates for competing risks. The models allow general distributions for the survival times and incorporate the idea of long-term survivors. Asymptotic results are obtained under a commonly assumed independent censoring mechanism and some modest regularity conditions on the survival distributions. The existence, consistency, and asymptotic normality of maximum likelihood estimators for the parameters of the model are rigorously derived under general sufficient conditions. Specific conditions for particular models can be derived from the general conditions for ready check. In addition, a likelihood-ratio statistic is proposed to test various hypotheses of practical interest, and its asymptotic distribution is provided
Variance Estimation for High-Dimensional Regression Models

AbstractThe paper is concerned with the problem of variance estimation for a high-dimensional regression model. The results show that the accuracy n−1/2 of variance estimation can be achieved only under some restrictions on smoothness properties of the regression function and on the dimensionality of the model. In particular, for a two times differentiable regression function, the rate n−1/2 is achievable only for dimensionality smaller or equal to 8. For a higher dimensional model, the optimal accuracy is n−4/d which is worse than n−1/2. The rate optimal estimating procedure is presented
Measurement Error Models with Nonconstant Covariance Matrices

AbstractIn this paper we consider measurement error models when the observed random vectors are independent and have mean vector and covariance matrix changing with each observation. The asymptotic behavior of the sample mean vector and the sample covariance matrix are studied for such models. Using the derived results, we study the case of the elliptical multiplicative error-in-variables models, providing formal justification for the asymptotic distribution of consistent slope parameter estimators. The model considered extends a normal model previously considered in the literature. Asymptotic relative efficiencies comparing several estimators are also reported
Linear Least Squares Estimation of Regression Models for Two-Dimensional Random Fields

AbstractWe consider the problem of estimating regression models of two-dimensional random fields. Asymptotic properties of the least squares estimator of the linear regression coefficients are studied for the case where the disturbance is a homogeneous random field with an absolutely continuous spectral distribution and a positive and piecewise continuous spectral density. We obtain necessary and sufficient conditions on the regression sequences such that a linear estimator of the regression coefficients is asymptotically unbiased and mean square consistent. For such regression sequences the asymptotic covariance matrix of the linear least squares estimator of the regression coefficients is derived
Principal Component Analysis from the Multivariate Familial Correlation Matrix

AbstractThis paper considers principal component analysis (PCA) in familial models, where the number of siblings can differ among families. S. Konishi and C. R. Rao (1992, Biometrika79, 631–641) used the unified estimator of S. Konishi and C. G. Khatri (1990, Ann. Inst. Statist. Math.42, 561–580) to develop a PCA derived from the covariance matrix. However, because of the lack of invariance to componentwise change of scale, an analysis based on the correlation matrix is often preferred. The asymptotic distribution of the estimated eigenvalues and eigenvectors of the correlation matrix are derived under elliptical sampling. A Monte Carlo simulation shows the usefulness of the asymptotic expressions for samples as small as N=25 families
More on Stochastic Comparisons and Dependence among Concomitants of Order Statistics

AbstractFor a sample of iid observations {(Xi, Yi)} from an absolutely continuous distribution, the multivariate dependence of concomitants Y[]=(Y[1], Y[2], …, Y[n]) and the stochastic order of subsets of Y[] are studied. If (X, Y) is totally positive dependent of order 2, Y[] is multivariate totally positive dependent of order 2. If the conditional hazard rate function of Y given X, hY∣X(y∣x), is decreasing in x for every y, Y[] is multivariate right corner set increasing. And if Y is stochastically increasing in X, the concomitants are increasing in multivariate stochastic order
A Formula for the Tail Probability of a Multivariate Normal Distribution and Its Applications

AbstractAn exact asymptotic formula for the tail probability of a multivariate normal distribution is derived. This formula is applied to establish two asymptotic results for the maximum deviation from the mean: the weak convergence to the Gumbel distribution of a normalized maximum deviation and the precise almost sure rate of growth of the maximum deviation. The latter result gives rise to a diagnostic tool for checking multivariate normality by a simple graph in the plane. Some simulation results are presented
On a Small Sample Adjustment for the Profile Score Function in Semiparametric Smoothing Models

AbstractWe consider the profile score function in models with smooth and parametric components. If local respectively weighted likelihood estimation is used for fitting the smooth component, the resulting profile likelihood estimate for the parametric component is asymptotically efficient as shown in T. A. Severini and W. H. Wong (1992, Ann. Statist.20, 1768–1802). However, as in solely parametric models the profile score function is not unbiased. We propose a small sample bias adjustment which results by extending the correction suggested in P. McCullagh and R. Tibshirani (1990, J. Roy. Statist. Soc. Ser. B52, 325–344) to the framework of semiparametric models
On the Asymptotics of Trimmed Best k-Nets

AbstractTrimmed best k-nets were introduced in J. A. Cuesta-Albertos, A. Gordaliza and C. Matrán (1998, Statist. Probab. Lett.36, 401–413) as a robustified L∞-based quantization procedure. This paper focuses on the asymptotics of this procedure. Also, some possible applications are briefly sketched to motivate the interest of this technique. Consistency and weak limit law are obtained in the multivariate setting. Consistency holds for absolutely continuous distributions without the (artificial) requirement of a trimming level varying with the sample size as in J. A. Cuesta-Albertos, A. Gordaliza and C. Matrán (1998, Statist. Probab. Lett.36, 401–413). The weak convergence will be stated toward a non-normal limit law at a OP(n−1/3) rate of convergence. An algorithm for computing trimmed best k-nets is proposed. Also a procedure is given in order to choose an appropriate number of centers, k, for a given data set
Exact Misclassification Probabilities for Plug-In Normal Quadratic Discriminant Functions II. The Heterogeneous Case

AbstractWe consider the problem of discriminating between two independent multivariate normal populations, Np(μ1, Σ1) and Np(μ2, Σ2), having distinct mean vectors μ1 and μ2 and distinct covariance matrices Σ1 and Σ2. The parameters μ1, μ2, Σ1, and Σ2 are unknown and are estimated by means of independent random training samples from each population. We derive a stochastic representation for the exact distribution of the “plug-in” quadratic discriminant function for classifying a new observation between the two populations. The stochastic representation involves only the classical standard normal, chi-square, and F distributions and is easily implemented for simulation purposes. Using Monte Carlo simulation of the stochastic representation we provide applications to the estimation of misclassification probabilities for the well-known iris data studied by Fisher (Ann. Eugen.7 (1936), 179–188); a data set on corporate financial ratios provided by Johnson and Wichern (Applied Multivariate Statistical Analysis, 4th ed., Prentice–Hall, Englewood Cliffs, NJ, 1998); and a data set analyzed by Reaven and Miller (Diabetologia16 (1979), 17–24) in a classification of diabetic status
The Optimal Classification Using a Linear Discriminant for Two Point Classes Having Known Mean and Covariance

AbstractThe current study provides a simple algorithm for finding the optimal ROC curve for a linear discriminant between two point distributions, given only information about the classes' means and covariances. The method makes no assumptions concerning the exact type of distribution and is shown to provide the best possible discrimination for any physically reasonable measure of the classification error. This very general solution is shown to specialise to results obtained in other papers which assumed multi-dimensional Gaussian distributed classes, or minimised the maximum classification error. Some numerical examples are provided which show the improvement in classification of this method over previously used methods
Asymptotic Properties of HPD Regions in the Discrete Case

AbstractThis paper obtains asymptotic expansions of the frequentist distributions of modified likelihood ratio statistics when the observations are discrete. An upper bound of the uncertainty due to the discrete nature of the observations is obtained, which is slightly larger than Yarnold's result (obtained in the case of elliptic confidence regions). Higher order results are also derived from continuity corrections. They are instrumental in the determination of matching priors for HPD regions to the orders o(n−1) and O(n−3/2)
Change Point Estimation by Local Linear Smoothing

AbstractWe consider the problem of estimating jump points in smooth curves. Observations (Xi,Yi) i=1,…,n from a random design regression function are given. We focus essentially on the basic situation where a unique change point is present in the regression function. Based on local linear regression, a jump estimate process t→γ(t) is constructed. Our main result is the convergence to a compound Poisson process with drift, of a local dilated-rescaled version of γ(t), under a positivity condition regarding the asymmetric kernel involved. This result enables us to prove that our estimate of the jump location converges with exact rate n−1 without any particular assumption regarding the bandwidth hn. Other consequences such as asymptotic normality are investigated and some proposals are provided for an extension of this work to more general situations. Finally we present Monte-Carlo simulations which give evidence for good numerical performance of our procedure
Improved Multivariate Prediction in a General Linear Model with an Unknown Error Covariance Matrix

AbstractThis paper deals with the problem of Stein-rule prediction in a general linear model. Our study extends the work of Gotway and Cressie (1993) by assuming that the covariance matrix of the model's disturbances is unknown. Also, predictions are based on a composite target function that incorporates allowance for the simultaneous predictions of the actual and average values of the target variable. We employ large sample asymptotic theory and derive and compare expressions for the bias vectors, mean squared error matrices, and risks based on a quadratic loss structure of the Stein-rule and the feasible best linear unbiased predictors. The results are applied to a model with first order autoregressive disturbances. Moreover, a Monte-Carlo experiment is conducted to explore the performance of the predictors in finite samples
Generalized Quantile Processes Based on Multivariate Depth Functions, with Applications in Nonparametric Multivariate Analysis

AbstractStatistical depth functions are being used increasingly in nonparametric multivariate data analysis. In a broad treatment of depth-based methods, Liu, Parelius, and Singh (“Multivariate analysis by date depth: Descriptive statistics, graphics and inference (with discussion),” 1999) include several devices for visualizing selected multivariate distributional characteristics by one-dimensional curves constructed in terms of given depth functions. Here we show how these tools may be represented as special depth-based cases of generalized quantile functions introduced by J. H. J. Einmahl and D. M. Mason (1992, Ann. Statist.20, 1062–1078). By specializing results of the latter authors to the depth-based case, we develop an easily applied general result on convergence of sample depth-based generalized quantile processes to a Brownian bridge. As applications, we obtain the asymptotic behavior of sample versions of depth-based curves for “scale” and “kurtosis” introduced by Liu, Parelius and Singh. The kurtosis curve is actually a Lorenz curve designed to measure heaviness of tails of a multivariate distribution. We also obtain the asymptotic distribution of the quantile process of the sample depth values
Bias and Efficiency Loss Due to Categorizing an Explanatory Variable

AbstractIt is a common situation in biomedical research that one or more variables are known to be associated with the outcome of interest. Researchers often discretize some variables and fit a regression model using these discretized variables. Although convenient for illustration purposes, such an approach can be biased and lead to loss of efficiency. In this article, we consider the situation of a regression model with two explanatory variables under an assumption of multivariate normality. We investigate the effect of dichotomizing or categorizing one variable on the estimate of the coefficient of the other continuous variable and on prediction from the models. Algebraic expressions are presented for the asymptotic bias and variance of the coefficient of the continuous explanatory variable and for the residual sum of squares for prediction. Some numerical examples are presented in which we find that the bias of the coefficient of the continuous explanatory variable is always smaller for the categorized model than that for the dichotomized model. The size of the test of a zero coefficient for the continuous variable only depends on the correlations between the response variable, the discretized variable, and the continuous variable. The size of the test for the categorized model is always smaller than for the dichotomized model, however, both can differ substantially from the nominal level if the correlation between the response and the categorical variable or between the two explanatory variables is high. The (predictive) relative efficiency of models also only depends on correlations amongst the three variables. There is a substantial loss of efficiency due to categorization if the correlation between the categorized and response variable is high. The predictive relative efficiency is always higher for the categorized model. The relative predictive efficiency due to dichotomization depends on the choice of cut points, with the least loss of efficency being achieved at the median
Saddlepoint Expansions in Linear Regression

AbstractIn this paper sums of independent but not identically distributed m-dimensional vectors are considered. The summands are generated by a random vector multiplied by a deterministic weight matrix which results in a singular covariance matrix. Under general conditions, given separately for the weight matrix and the random vector, saddlepoint approximations to the distribution of the sum are derived. The results are applied to the least squares estimator, the residual sum of squares, and to an F-statistic in linear regression
A Generalized φ-Divergence for Asymptotically Multivariate Normal Models

AbstractI. Csiszár's (Magyar. Tud. Akad. Mat. Kutató Int. Közl8 (1963), 85–108) ϕ-divergence, which was considered independently by M. S. Ali and S. D. Silvey (J. R. Statist. Soc. Ser. B28 (1966), 131–142) gives a goodness-of-fit statistic for multinomial distributed data. We define a generalized φ-divergence that unifies the ϕ-divergence approach with that of C. R. Rao and S. K. Mitra (“Generalized Inverse of Matrices and Its Applications,” Wiley, New York, 1971) and derive weak convergence to a χ2 distribution under the assumption of asymptotically multivariate normal distributed data vectors. As an example we discuss the application to the frequency count in Markov chains and thereby give a goodness-of-fit test for observations from dependent processes with finite memory
Continuity of Halfspace Depth Contours and Maximum Depth Estimators: Diagnostics of Depth-Related Methods

AbstractContinuity of procedures based on the halfspace (Tukey) depth (location and regression setting) is investigated in the framework of continuity concepts from set-valued analysis. Investigated procedures are depth contours (upper level sets) and maximum depth estimators. Continuity is studied both as the pointwise continuity of data-analytic functions, and the weak continuity of statistical functionals—the latter having relevance for qualitative robustness. After a real-data example, some general criteria and counterexamples are given, as well as positive results holding for “typical” data. Finally, some consequences for diagnostics and practical use of the depth-based techniques are drawn
Robust Bayesian Inference for Seemingly Unrelated Regressions with Elliptical Errors

AbstractBayesian inference is considered for the seemingly unrelated regressions with an elliptically contoured error distribution. We show that the posterior distribution of the regression parameters and the predictive distribution of future observations under elliptical errors assumption are identical to those obtained under independently distributed normal errors when an improper prior is used. This gives inference robustness with respect to departures from the reference case of independent sampling from the normal distribution
Partial Influence Functions

AbstractIn this paper we extend the definition of the influence function to functionals of more than one distribution, that is, for estimators depending on more than one sample, such as the pooled variance, the pooled covariance matrix, and the linear discriminant analysis coefficients. In this case the appropriate designation should be “partial influence functions,” following the analogy with derivatives and partial derivatives. Some useful results are derived, such as an asymptotic variance formula. These results are then applied to several estimators of the Mahalanobis distance between two populations and the linear discriminant function coefficients
Nonparametric and semiparametric regression model selection

It is known that semiparametric time series regression is often used without checking its suitability and compactness. In theory, this may result in dealing with an 
unnecessarily complicated model. In practice, one may encounter the computational difficulty caused by the spareness of the data. This is partly because the curse of dimensionality problem may still arise from using a semiparametric time series regression model. This paper suggests that in order to provide more precise predictions
we need to choose the most significant regressors for both the parametric and nonparametric time series components. We develop a novel cross-validation based model selection procedure for the choice of both the parametric and nonparametric time series components in semiparametric time series regression, and then establish some asymptotic properties of the proposed model selection procedure. In addition, we demonstrate how to implement the model selection procedure in practice through using both simulated and real examples. Our empirical studies show that the proposed cross-validation selection procedure works 
well numerically.
A Characterization of Joint Distribution of Two-Valued Random Variables and Its Applications

AbstractWe obtain an explicit representation for joint distribution of two-valued random variables with given marginals and for a copula corresponding to such random variables. The results are applied to prove a characterization of r-independent two-valued random variables in terms of their mixed first moments. The characterization is used to obtain an exact estimate for the number of almost independent random variables that can be defined on a discrete probability space and necessary conditions for a sequence of r-independent random variables to be stationary
Empirical Likelihood Semiparametric Regression Analysis under Random Censorship

AbstractThis paper considers large sample inference for the regression parameter in a partly linear model for right censored data. We introduce an estimated empirical likelihood for the regression parameter and show that its limiting distribution is a mixture of central chi-squared distributions. A Monte Carlo method is proposed to approximate the limiting distribution. This enables one to make empirical likelihood-based inference for the regression parameter. We also develop an adjusted empirical likelihood method which only appeals to standard chi-square tables. Finite sample performance of the proposed methods is illustrated in a simulation study
Multivariate Discrete Distributions with a Product-Type Dependence

AbstractA discrete multivariate probability distribution for dependent random variables, which contains the Poisson and Geometric conditionals distributions as particular cases, is characterized by means of conditional expectations of arbitrary one-to-one functions. Independence of the random variables is also characterized in terms of these conditional expectations. For certain exchangeable and partially exchangeable random variables with a joint distribution of this form it is shown that maximum likelihood estimates coincide with the simple method of moments estimates, suggesting that these models offer a pragmatic way to analyze certain dependent data