On the Rates of Asymptotic Normality for Bernstein Polynomial Estimators in a Triangular Array
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On the Rates of Asymptotic Normality for Bernstein Polynomial Estimators in a Triangular Array Dawei Lu1,2
· Lina Wang1
Received: 17 July 2020 / Revised: 24 September 2020 / Accepted: 28 September 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract It is well known that the empirical distribution function has superior properties as an estimator of the underlying distribution function F . However, considering its jump discontinuities, the estimator is limited when F is continuous. Mixtures of the binomial probabilities relying on Bernstein polynomials lead to good approximation properties for the resulting estimator of F . In this paper, we establish the rates of (pointwise) asymptotic normality for Bernstein estimators by the Berry-Esseen Theorem in the case that the observations are in a triangular array. Particularly, the (asymptotic) absence of the boundary bias and the asymptotic behaviors of the variance are investigated. Besides, numerical simulations are presented to verify the validity of our main results. Keywords Bernstein polynomials · Distribution function estimator · Asymptotic normality · Berry-Esseen Theorem · Triangular array Mathematics Subject Classification (2010) 60F05 · 62G05 · 62G30
1 Introduction and Assumptions 1.1 Berry-Esseen Bound The Central Limit Theorem is one of the most fundamental and profound results in probability and statistics. In its classical form, it states that the distribution of the sample mean of independent and identically distributed (i.i.d) observations approximates a normal distribution as the sample size becomes sufficiently large, under a very mild condition on the observations (finite variance). Precisely, let Wn denote the standardization of Sn = Dawei Lu
ludawei [email protected] Lina Wang Linawang [email protected] 1
School of Mathematical Sciences, Dalian University of Technology, Dalian, 116023, China
2
Key Laboratory for Computational Mathematics and Data Intelligence of Liaoning Province Dalian, Dalian University of Technology, Dalian, 116024, China
Methodology and Computing in Applied Probability
n
i=1 Zi , where Z1 , Z2 , . . . , Zn are i.i.d random variables with mean μ and variance then Wn converges in distribution to a standard normal random variable Z, namely
σ 2,
d Wn = (Sn − nμ) / nσ 2 → Z,
d
where “ → ” denotes the convergence in distribution. A large and growing body of literature has investigated the asymptotic normality for various estimators. For instance, (Leblanc 2012a) obtained the asymptotic normality for the Bernstein estimator of a distribution function by checking the Lindeberg condition. Recently, La¨ıb and Louani (2019) used the techniques of martingale difference devices and a sequence of projections on appropriate σ -fields to prove the asymptotic normality for the kernel density function estimator in the setting of continuous time, stationary and dependent data. Makigusa and Naito (2020) proposed a consistent estimator of the maximum mean discrepancy between a specified distribution and an u
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