General Moment Conditions Research Articles

On the strong laws of large numbers for pairwise NQD random variables

Let { X , X n , n ≥ 1 } be a sequence of pairwise NQD identically distributed random variables and { b n , n ≥ 1 } be a sequence of positive constants. In this article, we study the strong laws of large numbers for the sequence { X , X n , n ≥ 1 } , under the general moment condition ∑ n = 1 ∞ P ( | X | > b n / log n ) < ∞ , which improve some known results.

Properties of eigenvalues and eigenvectors of large-dimensional sample correlation matrices

This paper is to study the properties of eigenvalues and eigenvectors of high-dimensional sample correlation matrices. We first improve the result of Jiang (Sankhyā 66 (2004) 35–48), Xiao and Zhou (J. Theoret. Probab. 23 (2010) 1–20) and the Theorem 1 of El Karoui (Ann. Appl. Probab. 19 (2009) 2362–2405), both concerning the limiting spectral distribution and the extreme eigenvalues of sample correlation matrices, by allowing a more general fourth moment condition. Then, we establish a central limit theorem (CLT) for the linear statistics of the eigenvectors of large sample correlation matrices. We discover that the difference between the functional CLT of the sample covariance matrix and the sample correlation matrix is fundamentally influenced by the direction of a nonrandom projection vector. In the special case where the square root of the correlation matrix is identity, the difference will be determined by the sum of the fourth powers of the entries of the projection vector. These results also indicate that the eigenmatrix of sample correlation matrices is not asymptotically Haar if the underlying distribution is Gaussian. In other words, the normalization based on the sample variances affects the asymptotic properties of the eigenmatrix of the Wishart matrix. Furthermore, we establish a theorem concerning CLT for the linear statistics of the eigenvectors of large sample covariance matrices. This theorem improves the main results in Bai, Miao and Pan (Ann. Probab. 35 (2007) 1532–1572), which requires the assumption that the fourth moment of the underlying variable matches the one of Gaussian distribution, as well as Theorem 1.3 in Pan and Zhou (Ann. Appl. Probab. 18 (2008) 1232–1270), which relaxed the Gaussian like fourth moment requirement but assumes the maximum entries of the projection vector converge to 0 uniformly. We illustrate the usefulness of the theoretical results through an application in communications.

On the complete convergence and strong law for dependent random variables with general moment conditions

On the complete convergence and strong law for dependent random variables with general moment conditions

Estimation of nonparametric additive models with high order spatial autoregressive errors

AbstractIn this article, we propose nonparametric generalized method of moments estimation for nonparametric additive models with high order spatial autoregressive dependence. The estimation procedure is derived in three steps by combining a spline‐backfitting method with generalized moment conditions that relieve correlations within the dependent variables. Consistency and asymptotic normality are demonstrated under mild conditions. Specifically, compared with estimators of nonparametric functions that ignore cross‐sectional dependence in errors, the resultant estimators that consider the error term are asymptotically more efficient and achieve the well‐known oracle properties. Simulation studies investigating the finite sample performance of the estimation procedure confirm the validity of our asymptotic theory. An application to the Boston housing data serves as a practical illustration.

On the convergence for PNQD sequences with general moment conditions

Let {X, Xn, n ≥ 1} be a sequence of identically distributed pairwise negative quadrant dependent (PNQD) random variables and {an, n ≥ 1} be a sequence of positive constants with an = f(n) and f(θk)/f(θk−1) ≥ β for all large positive integers k, where 1 0 (x ≥ 1) is a non-decreasing function on [b, +∞) for some b ≥ 1. In this paper, we obtain the strong law of large numbers and complete convergence for the sequence {X, Xn, n ≥ 1}, which are equivalent to the general moment condition $$\sum\nolimits_{n = 1}^\infty {P\left( {\left| X \right| > {a_n}} \right) < \infty }$$. Our results extend and improve the related known works in Baum and Katz [1], Chen at al. [3], and Sung [14].

Weak and strong laws of large numbers for sub-linear expectation

In this paper, we derive a new form of weak laws of large numbers for sub-linear expectation and establish the equivalence relation among this new form and the other two forms of weak laws of large numbers for sub-linear expectation. Moreover, we obtain the strong laws of large numbers for sub-linear expectation under a general moment condition by applying our new weak laws of large numbers.

Strong laws for weighted sums of $$\rho ^*$$ ρ ∗ -mixing random variables under general, moment conditions

An analog of the Marcinkiewicz–Zygmund strong law of large numbers for weighted sums of $$\rho ^*$$ -mixing random variables is obtained under a general moment condition, which improves and extends a result of Bai and Cheng (Stat Probab Lett 46:105–112, 2000) and among others.

UNIFORMLY VALID POST-REGULARIZATION CONFIDENCE REGIONS FOR MANY FUNCTIONAL PARAMETERS IN Z-ESTIMATION FRAMEWORK.

In this paper, we develop procedures to construct simultaneous confidence bands for potentially infinite-dimensional parameters after model selection for general moment condition models where is potentially much larger than the sample size of available data, n. This allows us to cover settings with functional response data where each of the parameters is a function. The procedure is based on the construction of score functions that satisfy Neyman orthogonality condition approximately. The proposed simultaneous confidence bands rely on uniform central limit theorems for high-dimensional vectors (and not on Donsker arguments as we allow for ). To construct the bands, we employ a multiplier bootstrap procedure which is computationally efficient as it only involves resampling the estimated score functions (and does not require resolving the high-dimensional optimization problems). We formally apply the general theory to inference on regression coefficient process in the distribution regression model with a logistic link, where two implementations are analyzed in detail. Simulations and an application to real data are provided to help illustrate the applicability of the results.

On simultaneous confidence interval estimation for the difference of paired mean vectors in high-dimensional settings

To test whether two populations have the same mean vector in a high-dimensional setting, Chen and Qin (2010, Ann. Statist.) derived an unbiased estimator of the squared Euclidean distance between the mean vectors and proved the asymptotic normality of this estimator under local assumptions about the mean vectors. In this study, their results are extended without assumptions about the mean vectors. In addition, asymptotic normality is established in the class of general statistics including Chen and Qin’s statistics and other important statistics under general moment conditions that cover both Chen and Qin’s moment condition and elliptical distributional assumption. These asymptotic results are applied to the construction of simultaneous intervals for all pair-wise differences between mean vectors of k≥2 groups. The finite-sample and dimension performance of the proposed methods is also studied via Monte Carlo simulations. The methodology is illustrated using microarray data.

Convergence rates in the strong law of large numbers for negatively orthant dependent random variables with general moment conditions

Convergence rates in the strong law of large numbers for negatively orthant dependent random variables with general moment conditions

Robust adaptive efficient estimation for semi-Markov nonparametric regression models

We consider the nonparametric robust estimation problem for regression models in continuous time with semi-Markov noises. An adaptive model selection procedure is proposed. Under general moment conditions on the noise distribution a sharp non-asymptotic oracle inequality for the robust risks is obtained and the robust efficiency is shown. It turns out that for semi-Markov models the robust minimax convergence rate may be faster or slower than the classical one.

A strong law of large numbers for sub-linear expectation under a general moment condition

In this paper, we derive a strong law of large numbers for sub-linear expectation under a general moment condition. The result can be reduced to the classical strong law of large numbers when the sub-linear expectation coincides with the classical linear expectation. Moreover, we illustrate that this moment condition for strong law of large numbers in sub-linear situation is the weakest.

CLT for Linear Spectral Statistics of Hermitian Wigner Matrices with General Moment Conditions

In this paper, we study the Hermitian Wigner matrix $W_n=(x_{ij})_{1\le i,j\le n}$ with independent (up to symmetry) mean zero variance one entries. Under some Lindeberg type condition on the fourth moments of the entries, we establish a central limit theorem for the linear eigenvalue statistics of $W_n$. Our result extends the previous results on this topic to a more general case without the assumption ${\bf E}\,x_{ij}^2=0$ for $1\le i<j\le n$. Instead, we only assume that the real part and imaginary part of the upper-diagonal entry are uncorrelated. More precisely, we require ${\bf E}\,x_{ij}^2$ to be real and homogeneous for all $1\le i<j\le n$. The limiting normal distribution of the central limit theorem is shown to depend on the parameter ${\bf E}\,x_{ij}^2\in[-1,1]$.

An extension of the Baum-Katz theorem to i.i.d. random variables with general moment conditions

For a sequence of i.i.d. random variables $\{X, X_{n}, n\ge1\}$ and a sequence of positive real numbers $\{a_{n}, n\ge1\}$ with $0< a_{n}/n^{1/p}\uparrow$ for some $0< p<2$ , the Baum-Katz complete convergence theorem is extended to the $\{X, X_{n}, n\ge1\}$ with the general moment condition $\sum^{\infty}_{n=1}n^{r-1}P\{|X|>a_{n}\}<\infty$ , where $r\ge1$ . The relationship between the complete convergence and the strong law of large numbers is established.

Asymptotic Mutual Information Statistics of MIMO Channels and CLT of Sample Covariance Matrices

In this paper, we consider the fluctuation of mutual information statistics of a multiple input multiple output channel communication systems without assuming that the entries of the channel matrix have zero pseudovariance. To this end, we also establish a central limit theorem of the linear spectral statistics for sample covariance matrices under general moment conditions by removing the restrictions imposed on the second moment and fourth moment on the matrix entries in Bai and Silverstein (2004).

CLT for linear spectral statistics of Hermitian Wigner matrices with general moment conditions

Изучается эрмитова матрица Вигнера $W_n=(x_{ij})_{1\leq i,j\leq n}$ с независимыми (с точностью до симметрии) внедиагональными элементами, имеющими нулевое среднее и единичную дисперсию. При некотором условии типа Линдеберга на четвертые моменты элементов матрицы доказана центральная предельная теорема для линейных статистик собственных значений матрицы $W_n$. Наш результат распространяет предыдущие результаты подобного рода на более общий случай, когда отсутствует ограничение $\bf E x^2_{ij}=0$ для $1\leq i &lt; j \leq n$. Вместо этого мы предполагаем только, что вещественная и мнимая части наддиагональных элементов некоррелированы. Более точно, мы требуем, чтобы математические ожидания $\bf E x^2_{ij}$ были вещественными и однородными для всех $1\leq i &lt; j \leq n$. Показано, что предельное нормальное распределение в центральной предельной теореме зависит от параметра $\bf E x^2_{ij}\in [-1,1]$.

Efficient semiparametric estimation in generalized partially linear additive models for longitudinal/clustered data

We consider efficient estimation of the Euclidean parameters in a generalized partially linear additive models for longitudinal/clustered data when multiple covariates need to be modeled nonparametrically, and propose an estimation procedure based on a spline approximation of the nonparametric part of the model and the generalized estimating equations (GEE). Although the model in consideration is natural and useful in many practical applications, the literature on this model is very limited because of challenges in dealing with dependent data for nonparametric additive models. We show that the proposed estimators are consistent and asymptotically normal even if the covariance structure is misspecified. An explicit consistent estimate of the asymptotic variance is also provided. Moreover, we derive the semiparametric efficiency score and information bound under general moment conditions. By showing that our estimators achieve the semiparametric information bound, we effectively establish their efficiency in a stronger sense than what is typically considered for GEE. The derivation of our asymptotic results relies heavily on the empirical processes tools that we develop for the longitudinal/clustered data. Numerical results are used to illustrate the finite sample performance of the proposed estimators.

MARGINAL EMPIRICAL LIKELIHOOD AND SURE INDEPENDENCE FEATURE SCREENING.

We study a marginal empirical likelihood approach in scenarios when the number of variables grows exponentially with the sample size. The marginal empirical likelihood ratios as functions of the parameters of interest are systematically examined, and we find that the marginal empirical likelihood ratio evaluated at zero can be used to differentiate whether an explanatory variable is contributing to a response variable or not. Based on this finding, we propose a unified feature screening procedure for linear models and the generalized linear models. Different from most existing feature screening approaches that rely on the magnitudes of some marginal estimators to identify true signals, the proposed screening approach is capable of further incorporating the level of uncertainties of such estimators. Such a merit inherits the self-studentization property of the empirical likelihood approach, and extends the insights of existing feature screening methods. Moreover, we show that our screening approach is less restrictive to distributional assumptions, and can be conveniently adapted to be applied in a broad range of scenarios such as models specified using general moment conditions. Our theoretical results and extensive numerical examples by simulations and data analysis demonstrate the merits of the marginal empirical likelihood approach.

On the strong law of large numbers for pairwise i.i.d. random variables with general moment conditions

Let {an,n≥1} be a sequence of positive constants with an/n↑ and let {X,Xn,n≥1} be a sequence of pairwise independent identically distributed random variables. In this paper, we obtain the strong law of large numbers and complete convergence for the sequence {X,Xn,n≥1}, which are equivalent to the general moment condition ∑n=1∞P(|X|>an)<∞. We obtain, as a corollary, the strong law of large numbers due to Kruglov [Kruglov, V.M., 2008. A strong law of large numbers for pairwise independent identically distributed random variables with infinite means. Statist. Probab. Lett. 78, 890–895].

Efficient robust nonparametric estimation in a semimartingale regression model

Dans cette article nous considérons le problème d’estimation robuste d’une fonction périodique dans un modèle de régression en temps continu avec un bruit dépendant décrit par une semi martingale carrée intégrable de distribution inconnue. Un exemple de ce bruit est un processus d’Ornstein–Uhlenbeck non gaussien avec sauts (voir (J. R. Stat. Soc. Ser. B Stat. Methodol. 63 (2001) 167–241), (Ann. Appl. Probab. 18 (2008) 879–908)). Nous proposons une procédure adaptative de sélection de modèle basée sur les estimateurs des moindres carrés pondérés. Sous des conditions générales sur les deux premiers moments de la distribution du bruit, des inégalités d’Oracle non asymptotiques pointues pour des risques quadratiques robustes sont obtenues et l’efficacité robuste est établie. Nous avons établi aussi que dans le cas du processus d’Ornstein–Uhlenbeck non Gaussian, la borne inférieure pour le risque quadratique robuste est donnée par la limite de l’intensité du bruit quand la fréquence tend vers l’infini. Nous donnons un exemple d’un modèle de régression avec un bruit martingale où la vitesse de convergence du risque quadratique devient plus lente si l’intensité du bruit tend vers l’infini.