The Central Limit Theorem for LS Estimator in Simple Linear EV Regression Models

Abstract

In this paper, we obtain the central limit theorems for LS estimator in simple linear errors-in-variables (EV) regression models under some mild conditions. And we also show that those conditions are necessary in some sense.

Asymptotic properties for LS estimators in EV regression model with dependent errors

“…Remark 2.7 The condition A n /(nx 2 n ) → ∞ in Theorem 2.4 has been used by Miao et al (2007). Furthermore, we notice that the normalization ofβ n in Theorem 2.3 is related to √ A n /σ n , however, the normalization ofθ n in Theorem 2.4 is related to √ n/σ 1n .…”

“…random variables, Liu and Chen (2005) discussed the consistency ofβ n andθ n under the moment conditions Ee 2 i = σ 2 1 and Eu 2 i = σ 2 2 , and Miao et al (2007) obtained the asymptotic normality of β n andθ n with E|e i | 2+δ < ∞ and E|u i | 2+δ < ∞ for some δ > 0. The results in Theorems 2.3 and 2.4 also use the moment conditions E|e i | 2+δ < ∞ and E|u i | 2+δ < ∞, which are reasonable in the case of α-mixing assumption.…”

“…Many authors focused on the estimation of the involved parameter in the EV models; see, for example, Lai et al (1979) established strong consistency of the least squares (LS) estimators for the unknown parameter in the multiple EV regression model; Gleser (1981) obtained some large sample results of estimation in a multivariate EV regression model; Amemiya and Fuller (1984) also discussed the estimation for the multivariate EV model; Cui (1997) proved the asymptotic normality of M-estimators in the EV model; Cui and Chen (2003) constructed empirical likelihood confidence regions for the unknown parameters in the linear EV model and to evaluate the coverage accuracy and Bartlett correct ability of the confidence regions. Recently, Miao and Liu (2009) obtained moderate deviation principle for the LS estimators of the unknown parameters in model (1.1); Liu and Chen (2005) gave consistency of the LS estimators for model (1.1) and also proved that the sufficient and necessary condition for β n being strong and weak consistent estimate of β is lim n→∞ n Miao et al (2007) obtained central limit theorem for the LS estimators of unknown parameters in model (1.1) under the condition lim n→∞ n −1 √ A n = ∞. However, all these papers assumed that the errors (e i , u i ) are independent and identically distributed (i.i.d.…”

Convergence Rate for LS Estimator in Simple Linear EV Regression Models

“…Recently, Miao and Yang [12] and Miao et al [13] gave the central limit theorem and the law of iterated logarithm for the LS estimatorsβ n andθ n in [11] obtained the following exponential convergence rate (moderate deviation) for the estimatorŝ β n andθ n under the weaker moment assumptions and the stronger moderate deviation scale conditions by the methods of Puhalskii [14].…”

Central limit theorems for LS estimators in the EV regression model with dependent measurements

“…Recently, Miao and Yang (in press) and Miao, Yang, and Shen (2007) gave the central limit theorem and the law of iterated logarithm for the LS estimatorsβ n andθ n in the simple linear EV regression model (1.1). Miao and Liu (2009) obtained the exponential convergence rate (moderate deviation) for the estimatorsβ n andθ n under the weaker moment assumptions and the stronger moderate deviation scale conditions by the methods of Puhalskii (2001).…”