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This work is concerned with the properties of the ridge regression where the number of predictors p is proportional to the sample size n . Asymptotic properties of the means square error (MSE) of the estimated mean vector using ridge regression is investigated when the design matrix X may be non-random or random. Approximate asymptotic expression of the MSE is derived under fairly general conditions on the decay rate of the eigenvalues of X T X when the design matrix is nonrandom. The value of the optimal MSE provides conditions under which the ridge regression is a suitable method for estimating the mean vector. In the random design case, similar results are obtained when the eigenvalues of E [ X T X ] satisfy a similar decay condition as in the non-random case.

We revisit a singular value decomposition (SVD) algorithm given in Chen et al. (Psychometrika 84:124–146, 2019b) for exploratory item factor analysis (IFA). This algorithm estimates a multidimensional IFA model by SVD and was used to obtain a starting point for joint maximum likelihood estimation in Chen et al. (2019b). Thanks to the analytic and computational properties of SVD, this algorithm guarantees a unique solution and has computational advantage over other exploratory IFA methods. Its computational advantage becomes significant when the numbers of respondents, items, and factors are all large. This algorithm can be viewed as a generalization of principal component analysis to binary data. In this note, we provide the statistical underpinning of the algorithm. In particular, we show its statistical consistency under the same double asymptotic setting as in Chen et al. (2019b). We also demonstrate how this algorithm provides a scree plot for investigating the number of factors and provide its asymptotic theory. Further extensions of the algorithm are discussed. Finally, simulation studies suggest that the algorithm has good finite sample performance.

Correlation matrices play a key role in many multivariate methods (e.g., graphical model estimation and factor analysis). The current state-of-the-art in estimating large correlation matrices focuses on the use of Pearson's sample correlation matrix. Although Pearson's sample correlation matrix enjoys various good properties under Gaussian models, it is not an effective estimator when facing heavy-tailed distributions. As a robust alternative, Han and Liu [J. Am. Stat. Assoc. 109 (2015) 275-287] advocated the use of a transformed version of the Kendall's tau sample correlation matrix in estimating high dimensional latent generalized correlation matrix under the transelliptical distribution family (or elliptical copula). The transelliptical family assumes that after unspecified marginal monotone transformations, the data follow an elliptical distribution. In this paper, we study the theoretical properties of the Kendall's tau sample correlation matrix and its transformed version proposed in Han and Liu [J. Am. Stat. Assoc. 109 (2015) 275-287] for estimating the population Kendall's tau correlation matrix and the latent Pearson's correlation matrix under both spectral and restricted spectral norms. With regard to the spectral norm, we highlight the role of \"effective rank\" in quantifying the rate of convergence. With regard to the restricted spectral norm, we for the first time present a \"sign sub-Gaussian condition\" which is sufficient to guarantee that the rank-based correlation matrix estimator attains the fast rate of convergence. In both cases, we do not need any moment condition.

In this paper we derive the asymptotic properties of within groups (WG), GMM, and LIML estimators for an autoregressive model with random effects when both T and N tend to infinity. GMM and LIML are consistent and asymptotically equivalent to the WG estimator. When T/N → 0 the fixed T results for GMM and LIML remain valid, but WG, although consistent, has an asymptotic bias in its asymptotic distribution. When T/N tends to a positive constant, the WG, GMM, and LIML estimators exhibit negative asymptotic biases of order 1/T, 1/N, and 1/(2N - T), respectively. In addition, the crude GMM estimator that neglects the autocorrelation in first differenced errors is inconsistent as T/N → c > 0, despite being consistent for fixed T. Finally, we discuss the properties of a random effects pseudo MLE with unrestricted initial conditions when both T and N tend to infinity.

The double asymptotic homogenization method originated for analyzing physical systems containing two or more length scales was adopted to predict the characteristic of 1-3 type cement-based piezoelectric composites for the first time. The piezoelectric properties of 1-3 type cement-based piezoelectric composites were measured and comparisons between the experimental data and predicted values validate the effectiveness of the present analytical model. Moreover, numerical discussions and experiments show that one should choose proper volume fraction of constituents to achieve the best performance of the 1-3 type cement-based piezoelectric composites.

Distributed estimation methods have recently been used to compute the maximum likelihood estimate of the precision matrix for large graphical Gaussian models. Our aim, in this article, is to give a Bayesian estimate of the precision matrix for large graphical Gaussian models with, additionally, symmetry constraints imposed by an underlying graph which is coloured. We take the sample posterior mean of the precision matrix as our estimate. We study its asymptotic behaviour under the regular asymptotic regime when the number of variables p is fixed and under the double asymptotic regime when both p and n grow to infinity. We show in particular that when the number of parameters of the local models is uniformly bounded the standard convergence rate of our estimate of the precision matrix to its true value, in the Frobenius norm, compares well with the rates in the current literature for the maximum likelihood estimate. Des méthodes distribuées d’estimation ont récemment été utilisées pour calculer l’estimateur au maximum de vraisemblance de la matrice de précision pour de grands modèles graphiques gaussiens. Les auteures proposent un estimateur bayésien de cette matrice dans le cas où une contrainte de symétrie est imposée par un graphique coloré sous-jacent. Elles utilisent comme estimateur la moyenne a posteriori de la matrice de précision dont elles étudient le comportement asymptotique sous le régime habituel où le nombre de variables p est fixe, mais également sous le régime double où n et p tendent tous les deux vers l’infini. Lorsque le nombre de paramètres des modèles locaux est uniformément borné, les auteures montrent que le taux de convergence dans la norme de Frobenius de leur estimateur vers sa vraie valeur se compare avantageusement aux taux pour l’estimateur au maximum de vraisemblance trouvés dans la littérature courante.

High-dimensional data generally refer to data in which the number of variables is larger than the sample size. Analyzing such datasets poses great challenges for classical statistical learning because the finite-sample performance of methods developed within classical statistical learning does not live up to classical asymptotic premises in which the sample size unboundedly grows for a fixed dimensionality of observations. Much work has been done in developing mathematical-statistical techniques for analyzing high-dimensional data. Despite remarkable progress in this field, many practitioners still utilize classical methods for analyzing such datasets. This state of affairs can be attributed, in part, to a lack of knowledge and, in part, to the ready-to-use computational and statistical software packages that are well developed for classical techniques. Moreover, many scientists working in a specific field of high-dimensional statistical learning are either not aware of other existing machineries in the field or are not willing to try them out. The primary goal in this work is to bring together various machineries of high-dimensional analysis, give an overview of the important results, and present the operating conditions upon which they are grounded. When appropriate, readers are referred to relevant review articles for more information on a specific subject.

The generalized fuzzy valued$\\theta$ -Choquet integrals will be established for the given$\\mu$ -integrable fuzzy valued functions on a general fuzzy measure space, and the convergence theorems of this kind of fuzzy valued integral are being discussed. Furthermore, the whole of integrals is regarded as a fuzzy valued set function on measurable space, the double-null asymptotic additivity and pseudo-double-null asymptotic additivity of the fuzzy valued set functions formed are studied when the fuzzy measure satisfies autocontinuity from above (below).\\\

We provide a fundamental theorem that can be used in conjunction with Kolmogorov asymptotic conditions to derive the first moments of well-known estimators of the actual error rate in linear discriminant analysis of a multivariate Gaussian model under the assumption of a common known covariance matrix. The estimators studied in this paper are plug-in and smoothed resubstitution error estimators, both of which have not been studied before under Kolmogorov asymptotic conditions. As a result of this work, we present an optimal smoothing parameter that makes the smoothed resubstitution an unbiased estimator of the true error. For the sake of completeness, we further show how to utilize the presented fundamental theorem to achieve several previously reported results, namely the first moment of the resubstitution estimator and the actual error rate. We provide numerical examples to show the accuracy of the succeeding finite sample approximations in situations where the number of dimensions is comparable or even larger than the sample size.

This article proposes asymptotically unbiased estimators of autocovariances and autocorrelations for panel data with both individual and time effects. We show that the conventional autocovariance estimators suffer from the bias caused by the elimination of individual and time effects. The bias related to individual effects is proportional to the long-run variance, and it related to time effects is proportional to the value of the estimated autocovariance. For the conventional autocorrelation estimators, the elimination of time effects does not cause a bias while the elimination of individual effects does. We develop methods to estimate the long-run variance and propose bias-corrected estimators based on the proposed long-run variance estimator. We also consider the half-panel jackknife estimation for bias correction. The theoretical results are given by employing double asymptotics under which both the number of observations and the length of the time series tend to infinity. Monte Carlo simulations show that the asymptotic theory provides a good approximation to the actual bias and that the proposed bias-correction methods work well.