Consistency of LS estimators in the EV regression model with martingale difference errors

Asymptotic properties of LS estimators in the errors-in-variables model with MD errors

“…The last one is the complete convergence for weighted sums of martingale difference, which was obtained by Miao et al (2015).…”

“…Under the case that the errors are sequences of dependent random variables, Fazekas and Kukush (1997) studied the asymptotic properties of an estimator in nonlinear functional EV models with α-mixing error terms; Baran (2004) considered the consistency of the corrected score estimator in the EV regression model with α-mixing errors. Fan et al (2010) established the consistency and the asymptotic normality for the LS estimators of θ and β with stationary α-mixing error; Miao et al (2013) studied the strong consistency of LS estimators in the EV regression model with negatively associated (NA, in short) errors; Wang et al (2015) extended and improved the results of Miao et al (2013) for NA random variables to the case of negatively superadditive dependent (NSD, in short) random variables, and established the complete consistency for the the LS estimators of θ and β with NSD errors; Miao et al (2015) obtained the strong consistency and weak consistency for the the LS estimators of θ and β with martingale difference errors, and so on. The main purpose of the paper is to investigate the complete consistency for the LS estimators of θ and β with martingale difference errors.…”

On Consistency of Ls Estimators in the Errors-in-Variable Regression Model

“…For the case that the errors are independent random variables, one can refer to Amemiya and Fuller [1], Cui and Chen [4], Gleser [8], Hslao, Wang, and Wang [9], Lai, Robbins, and Wei [12], Liu and Chen [14] among others. Under the case that the errors are dependent random variables, Fan et al [7] and Miao, Wang, and Zheng [16] Miao et al [17] studied the consistency of LS estimators in the EV regression model with stationary α-mixing errors, negatively associated (NA, in short) errors and martingale difference errors, respectively; Wang et al [29] established the complete convergence for weighted sums of negatively superadditive dependent (NSD) random variables and gave its application in the EV regression model, and so on. For more details about the EV regression model and applications, one can refer to Carroll et al [2], Dagenais and Dagenais [5], Jung [11], Li [13] and Taupin [26] among others.…”

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

“…In the case of independent random errors, the consistency of LS estimators in the linear EV model was established by Liu and Chen [2] and Chen et al [3]; Miao et al [4] obtained the central limit theorem for the LS estimators in the simple linear EV regression model; Xu and Li [5] studied the consistency of LS estimators in the linear EV regression model with replicate observations; Miao et al [6] established some limit behaviors of estimators in the simple linear EV regression model; Miao and Yang [7] obtained the loglog law for the LS estimators in the EV regression model; Miao et al [8] investigated the consistency and asymptotic normality of LS estimators in the simple linear EV regression model, and so on. In the case of dependent random errors, Fazekas and Kukush [9] obtained the consistency of the regression parameter of the nonlinear functional EV models under mixing conditions; Fan et al [10] established the asymptotic properties for the LS estimators of the unknown parameters in the simple linear EV regression model with stationary-mixing errors; Miao et al [11] derived the asymptotic normality and strong consistency of the estimators in the simple linear EV model with negatively associated (NA) errors; Miao et al [12] obtained the weak consistency and strong consistency of LS estimators for the unknown parameters with martingale difference errors; Shen [13] studied some asymptotic properties of estimators in the EV model with martingale difference errors, and so forth. In this article, we consider model (1.1) under the assumptions that the random errors are negatively superadditive dependent (NSD) random variables whose concept was proposed by Hu [14] as follows.…”