On the Baum–Katz theorem for randomly weighted sums of negatively associated random variables with general normalizing sequences and applications in some random design regression models

In this paper, we develop Jajte’s technique, which is used in the proof of strong laws of large numbers, to prove complete convergence for randomly weighted sums of negatively associated random variables. Based on a general normalizing function that satisfies some specific conditions, we give some g...

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Vydáno v:Statistical papers (Berlin, Germany) Ročník 65; číslo 3; s. 1869 - 1900
Hlavní autoři: Ta Cong, Son, Tran Manh, Cuong, Bui Khanh, Hang, Le Van, Dung
Médium: Journal Article
Jazyk:angličtina
Vydáno: Berlin/Heidelberg Springer Berlin Heidelberg 01.05.2024
Springer Nature B.V
Témata:
Convergence
Economic Theory/Quantitative Economics/Mathematical Methods
Economics
Estimators
Finance
Insurance
Management
Mathematics and Statistics
Operations Research/Decision Theory
Probability Theory and Stochastic Processes
Random variables
Regression analysis
Regression models
Regular Article
Statistics
Statistics for Business
Sums
Theorems
Weighting functions
62G05
Complete convergence
Simple linear regression model with random design
Non parametric regression model with random design
Randomly weighted sum
62G20
60F15
ISSN:0932-5026, 1613-9798
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Popis
Shrnutí:In this paper, we develop Jajte’s technique, which is used in the proof of strong laws of large numbers, to prove complete convergence for randomly weighted sums of negatively associated random variables. Based on a general normalizing function that satisfies some specific conditions, we give some general results on complete convergence for randomly weighted sums of random variables. The Baum–Katz theorem for randomly weighted sums with general normalizing sequences is also presented. Our results have an interesting connection with the theory of regularly varying functions. These results are applied to simple linear regression models as well as nonparametric regression models with random design. Furthermore, simulations to study the numerical performance of the consistency for nearest neighbor weight function estimators in nonparametric regression and least-squares estimators in a simple linear regression with random design are given.
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ISSN:0932-5026
1613-9798
DOI:10.1007/s00362-023-01483-4