L 1 bounds for asymptotic normality of random sums of independent random variables

2013 ◽  
Vol 53 (4) ◽  
pp. 438-447 ◽  
Author(s):  
Jonas Kazys Sunklodas
Keyword(s):  
Asymptotic Normality ◽  
Random Variables ◽  
Independent Random Variables ◽  
Random Sums

On the asymptotic normality of Hill's estimator

1995 ◽  
Vol 118 (2) ◽  
pp. 375-382 ◽  
Author(s):  
Sándor Csörgő ◽  
László Viharos
Keyword(s):  
Distribution Function ◽  
Asymptotic Normality ◽  
Order Statistics ◽  
Random Variables ◽  
Fixed Number ◽  
Distribution Functions ◽  
Independent Random Variables ◽  
Hill’S Estimator ◽  
Common Distribution ◽  
Common Distribution Function

Let X, X1, X2, …, be independent random variables with a common distribution function F(x) = P {X ≤ x}, x∈ℝ, and for each n∈ℕ, let X1, n ≤ … ≤ Xn, n denote the order statistics pertaining to the sample X1, …, Xn. We assume that 1–F(x) = x−1/cl(x), 0 < x < ∞, where l is some function slowly varying at infinity and c > 0 is any fixed number. The class of all such distribution functions will be denoted by .

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

2020 ◽  
Vol 18 (1) ◽  
pp. 930-947
Author(s):  
Yu Zhang ◽  
Xinsheng Liu ◽  
Yuncai Yu ◽  
Hongchang Hu
Keyword(s):  
Asymptotic Normality ◽  
Law Of Large Numbers ◽  
Random Variables ◽  
Least Square ◽  
Independent Random Variables ◽  
Errors In Variables ◽  
Unknown Parameters ◽  
Strong Law ◽  
Large Numbers ◽  
Nsd Random Variables

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.

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