Electronic Journal of Statistics

On the asymptotics of $Z$-estimators indexed by the objective functions

François Portier

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We study the convergence of $Z$-estimators $\widehat{\theta}(\eta)\in \mathbb{R}^{p}$ for which the objective function depends on a parameter $\eta$ that belongs to a Banach space $\mathcal{H}$. Our results include the uniform consistency over $\mathcal{H}$ and the weak convergence in the space of bounded $\mathbb{R}^{p}$-valued functions defined on $\mathcal{H}$. When $\eta$ is a tuning parameter optimally selected at $\eta_{0}$, we provide conditions under which $\eta_{0}$ can be replaced by an estimated $\widehat{\eta}$ without affecting the asymptotic variance. Interestingly, these conditions are free from any rate of convergence of $\widehat{\eta}$ to $\eta_{0}$ but require the space described by $\widehat{\eta}$ to be not too large in terms of bracketing metric entropy. In particular, we show that Nadaraya-Watson estimators satisfy this entropy condition. We highlight several applications of our results and we study the case where $\eta$ is the weight function in weighted regression.

Article information

Electron. J. Statist., Volume 10, Number 1 (2016), 464-494.

Received: September 2015
First available in Project Euclid: 24 February 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62F12: Asymptotic properties of estimators 62F35: Robustness and adaptive procedures
Secondary: 62G20: Asymptotic properties

Asymptotic theory empirical process semiparametric estimation weighted regression $Z$-estimation


Portier, François. On the asymptotics of $Z$-estimators indexed by the objective functions. Electron. J. Statist. 10 (2016), no. 1, 464--494. doi:10.1214/15-EJS1097. https://projecteuclid.org/euclid.ejs/1456322682

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