Negatively Superadditive Dependent Research Articles

On complete moment convergence for the maximal weighted sums of NSD random variables

In this article, the complete moment convergence for weighted sums of non-identically distributed negatively superadditive dependent (NSD) random variables is investigated. Some sufficient conditions of the complete moment convergence and the complete convergence for the maximum of weighted sums of NSD random variables are established. As applications, some equivalent conditions of the complete moment convergence for a weighted version of NSD cases are presented. Our main results extend and improve the corresponding earlier ones in the literature.

Limit distribution for products of sums of partial sums of linear processes generated by NSD sequence

Let be a negatively superadditive dependent (NSD) sequence of positive random variables. In this paper, under suitable conditions, we obtain the limit distribution for products of sums of partial sums of linear processes generated by

Asymptotic normality and mean consistency of LS estimators in the errors-in-variables model with dependent errors

Abstract In this article, an errors-in-variables regression model in which the errors are negatively superadditive dependent (NSD) random variables is studied. First, the Marcinkiewicz-type strong law of large numbers for NSD random variables is established. Then, we use the strong law of large numbers to investigate the asymptotic normality of least square (LS) estimators for the unknown parameters. In addition, the mean consistency of LS estimators for the unknown parameters is also obtained. Some results for independent random variables and negatively associated random variables are extended and improved to the case of NSD setting. At last, two simulations are presented to verify the asymptotic normality and mean consistency of LS estimators in the model.

English

In this work, the complete moment convergence and complete convergence for weighted sums of negatively superadditive dependent (NSD) random variables are studied, and some equivalent conditions of these strong convergences are established. These main results generalize and improve the corresponding theorems of Baum and Katz (1965) and Chow (1988) to weighted sums of NSD random variables without the assumption of identical distribution. As an application, a Marcinkiewicz-Zygmund-type strong law of large numbers for weighted sums of NSD random variables is obtained.

The adaptivity of thresholding wavelet estimators in heteroscedastic nonparametric model with negatively super-additive dependent errors

In this paper, we consider two estimators, a hard thresholding wavelet estimator and a block thresholding wavelet estimator, for the regression function in heteroscedastic nonparametric model with negatively super-additive dependent (NSD) errors. The random design distribution is known or unknown, and the corresponding adaptive properties of these estimators are investigated over Besov spaces, for the $${L^2}$$ risk. The results indicate that the block thresholding estimator is theoretically and computationally superior to the hard thresholding estimator with the former attains the optimal convergence rates, while the later achieves the nearly optimal convergence rates. Thus the block thresholding estimator provides extensive adaptivity to many irregular function classes even though with the presence of heteroscedastic NSD errors.

On adaptivity of wavelet thresholding estimators with negatively super-additive dependent noise

Abstract This paper considers the nonparametric regression model with negatively super-additive dependent (NSD) noise and investigates the convergence rates of thresholding estimators. It is shown that the term-by-term thresholding estimator achieves nearly optimal and the block thresholding estimator attains optimal (or nearly optimal) convergence rates over Besov spaces. Additionally, some numerical simulations are implemented to substantiate the validity and adaptivity of the thresholding estimators with the presence of NSD noise.

Kernel density estimation under negative superadditive dependence and its application for real data

ABSTRACTIn this paper, the kernel density estimator for negatively superadditive dependent random variables is studied. The exponential inequalities and the exponential rate for the kernel estimator of density function with a uniform version, over compact sets are investigated. Also, the optimal bandwidth rate of the estimator is obtained using mean integrated squared error. The results are generalized and used to improve the ones obtained for the case of associated sequences. As an application, FGM sequences that fulfil our assumptions are investigated. Also, the convergence rate of the kernel density estimator is illustrated via a simulation study. Moreover, a real data analysis is presented.

Probability inequalities for sums of NSD random variables and applications

In this paper, we obtain the exponential-type inequalities for maximal partial sums of negatively superadditive dependent (NSD) random variables, which extends the corresponding results for independent and negatively associated (NA) random variables. Using these inequalities, we further investigate the weak convergence of the M-estimators in the generalized linear model with NSD errors, which generalize and improve the corresponding results of the independent random errors to that of NSD random errors.

Equivalent conditions of the complete convergence for weighted sums of NSD random variables

In this paper, we consider the complete convergence for weighted sums of negatively superadditive-dependent (NSD) random variables without assumptions of identical distribution. Some sufficient and necessary conditions to prove the complete convergence for weighted sums of NSD random variables are presented, which extend and improve the corresponding ones of Naderi et al. As an application of the main results, the Marcinkiewicz–Zygmund type strong law of large numbers for weighted sums of NSD random variables is also achieved.

M-test in linear models with negatively superadditive dependent errors

This paper is concerned with the testing hypotheses of regression parameters in linear models in which errors are negatively superadditive dependent (NSD). A robust M-test base on M-criterion is proposed. The asymptotic distribution of the test statistic is obtained and the consistent estimates of the redundancy parameters involved in the asymptotic distribution are established. Finally, some Monte Carlo simulations are given to substantiate the stability of the parameter estimates and the power of the test, for various choices of M-methods, explanatory variables and different sample sizes.

Complete moment convergence of widely orthant dependent random variables

ABSTRACTIt is known that the dependence structure of widely orthant dependent (WOD) random variables is weaker than those of negatively associated (NA) random variables, negatively superadditive dependent (NSD) random variables, negatively orthant dependent (NOD) random variables, and extended negatively dependent (END) random variables. In this article, the results of complete moment convergence and complete convergence are presented for WOD sequence under the same moment conditions as independent sequence in classical result (Chow 1988).

On the Strong Consistency of M‐Estimates in Linear Models for Negatively Superadditive Dependent Errors

SummaryIn this paper we discuss the strong consistency of M‐estimates of the regression parameters in a linear model with negatively superadditive dependent (NSD) random errors. The result improves the moment condition and generalises the case of independent random errors to that of NSD random errors.

Complete moment convergence and complete convergence for weighted sums of NSD random variables

In this paper, the equivalent conditions of complete moment convergence of the maximum partial weighted sums for negatively superadditive dependent (NSD) random variables are established without the assumption of identical distribution. As applications, the complete moment convergence, the complete convergence and strong law of large numbers for NSD random variables are obtained. The results obtained in the paper generalize or improve the corresponding ones for weighted sums of independent random variables and negatively associated random variables.

Complete moment convergence for weighted sums of negatively superadditive dependent random variables

In this paper, we investigate the complete moment convergence for maximal partial sum of negatively superadditive dependent (NSD) random variables under some more general conditions. The results obtained in the paper generalize and improve some known ones.

Applications of the Rosenthal-type inequality for negatively superadditive dependent random variables

In this paper, we give some applications of the Rosenthal-type inequality for a sequence of negatively superadditive dependent (NSD) random variables, which includes sequences of negatively associated random variables as a special case. The complete consistency for an estimator of a nonparametric regression model based on NSD errors is investigated. In addition, we extend Feller’s weak law of large numbers for independent and identically distributed random variables to the case of NSD random variables by using the Rosenthal-type inequality.

On the strong convergence and some inequalities for negatively superadditive dependent sequences

Abstract In this paper, we study the Marcinkiewicz-type strong law of large numbers, Hajek-Renyi-type inequality and other inequalities for negatively superadditive dependent (NSD) sequences. As an application, the integrability of supremum for NSD random variables is obtained. Our results extend the corresponding ones of Christofides and Vaggelatou (J. Multivar. Anal. 88:138-151, 2004) and Liu et al. (Stat. Probab. Lett. 43:99-105, 1999). MSC:60F15.

Complete convergence for arrays of rowwise negatively superadditive-dependent random variables and its applications

In this paper, the Rosenthal-type maximal inequalities and Kolmogorov-type exponential inequality for negatively superadditive-dependent (NSD) random variables are presented. By using these inequalities, we study the complete convergence for arrays of rowwise NSD random variables. As applications, the Baum–Katz-type result for arrays of rowwise NSD random variables and the complete consistency for the estimator of nonparametric regression model based on NSD errors are obtained. Our results extend and improve the corresponding ones of Chen et al. [On complete convergence for arrays of rowwise negatively associated random variables. Theory Probab Appl. 2007;52(2):393–397] for arrays of rowwise negatively associated random variables to the case of arrays of rowwise NSD random variables.

Almost sure convergence theorem and strong stability for weighted sums of NSD random variables

In this paper, Kolmogorov-type inequality for negatively superadditive dependent (NSD) random variables is established. By using this inequality, we obtain the almost sure convergence for NSD sequences, which extends the corresponding results for independent sequences and negatively associated (NA) sequences. In addition, the strong stability for weighted sums of NSD random variables is studied.