**1. Introduction**

A moment condition model is a family <sup>M</sup><sup>1</sup> of probability measures, all defined on the same measurable space (R*m*, <sup>B</sup>(R*m*)), such that

$$\int \mathcal{g}(\mathbf{x}, \theta) dQ(\mathbf{x}) = 0, \text{ for all } \mathcal{Q} \in \mathcal{M}^1. \tag{1}$$

The parameter *<sup>θ</sup>* belongs to <sup>Θ</sup> <sup>⊂</sup> <sup>R</sup>*d*; the function *<sup>g</sup>* := (*g*1, ... , *gl*) is defined on <sup>R</sup>*<sup>m</sup>* <sup>×</sup> <sup>Θ</sup>, each of the *gi*'s being real-valued, *l* ≥ *d*, and the functions *g*1, ... , *gl* and **1**<sup>X</sup> are supposed to be linearly independent. Denote by *<sup>M</sup>*<sup>1</sup> the set of all probability measures on (R*m*, <sup>B</sup>(R*m*)) and

$$\mathcal{M}^1\_\theta := \{ \mathbb{Q} \in M^1 : \int \mathbb{g}(\mathfrak{x}, \theta) d\mathbb{Q}(\mathfrak{x}) = 0 \},\tag{2}$$

such that

$$\mathcal{M}^1 = \bigcup\_{\theta \in \Theta} \mathcal{M}^1\_{\theta}. \tag{3}$$

Let *X*1, ... , *Xn* be an i.i.d. sample on the random vector *X* with unknown probability distribution *P*0. We considered the problem of the estimation of the parameter *θ*<sup>0</sup> for which the constraints of the model are satisfied:

$$\int \lg(\mathfrak{x}, \theta\_0)dP\_0(\mathfrak{x}) = 0. \tag{4}$$

We supposed that *θ*<sup>0</sup> is the unique solution of Equation (4). Thus, we assumed that information about *θ*<sup>0</sup> and *P*<sup>0</sup> is available in the form of *l* ≥ *d* functionally independent unbiased estimating functions, and we used this information to estimate *θ*0.

Among the best-known estimation methods for moment condition models, we mention the generalized method of moments (GMM) [1], the continuous updating (CU) estimator [2],

**Citation:** Toma, A. Robust Z-Estimators for Semiparametric Moment Condition Models. *Entropy* **2023**, *25*, 1013. https://doi.org/ 10.3390/e25071013

Academic Editor: Leandro Pardo

Received: 8 May 2023 Revised: 28 June 2023 Accepted: 29 June 2023 Published: 30 June 2023

**Copyright:** © 2023 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

the empirical likelihood (EL) estimator [3,4], the exponential tilting (ET) estimator [5], and the generalized empirical likelihood (GEL) estimators [6]. Although the EL estimator is superior to other estimators in terms of higher-order asymptotic properties, these properties hold only under the correct specification of the moment conditions. In [7] was proposed the exponentially tilted empirical likelihood (ETEL) estimator, which has the same higher-order property as the EL estimator under the correct specification, while maintaining the usual asymptotic properties such as the consistency and asymptotic normality under misspecification. The so-called information and entropy econometric techniques have been proposed to improve the finite sample performance of the GMM-estimators and tests (see, e.g., [4,5]).

Some recent methods for the estimation and testing of moment condition models are based on using divergences. Divergences between probability measures are widely used in statistics and data science in order to perform inference in models of various kinds, parametric or semiparametric. Statistical methods based on divergence minimization extend the likelihood paradigm and often have the advantage of providing a trade-off between efficiency and robustness [8–11]. A general methodology for the estimation and testing of moment condition models was developed in [12]. This approach is based on minimizing divergences in their dual form and allows the asymptotic study of the estimators, called minimum empirical divergence estimators, and of the associated test statistics, both under the model and under misspecification of the model. The approach based on minimizing dual forms of divergences was initially used in the case of parametric models, the results being published in a series of articles [13–16]. The broad class of minimum empirical divergence estimators contains in particular the EL estimator, the CU estimator, as well as the ET estimator mentioned above. Using the influence function as the robustness measure, it has been shown that the minimum empirical divergence estimators are not robust, because the corresponding influence functions are generally not bounded [17]. On the other hand, the minimum empirical divergence estimators have the same efficiency of first order, and moreover, the EL estimator, which belong to this class, is superior in higher-order efficiency. Therefore, proposing robust versions of the minimum empirical divergence estimators would bring a trade-off between robustness and efficiency. These aspects motivated our studies in the present paper.

Some robust estimation methods for moment condition models have been proposed in the literature, for example in [18–22]. In the present paper, we introduce a class of robust Z-estimators for moment condition models. These new estimators can be seen as robust alternatives for the minimum empirical divergence estimators. By using the multidimensional Huber function, we first define robust estimators of the element that realizes the supremum in the dual form of the divergence. A linear relationship between the influence function of a minimum empirical divergence estimator and the influence function of the estimator of the element that realizes the supremum in the dual form of the divergence led to the idea of defining new Z-estimators for the parameter of the model, by using robust estimators in the dual form of the divergence. The asymptotic properties of the proposed estimators were proven, including here the consistency and their asymptotic normality. Then, the influence functions of the estimators were derived, and their robustness is demonstrated.

The paper is organized as follows. In Section 2, we briefly recall the context and the definitions of the minimum empirical divergence estimators, these being necessary for defining the new estimators. In Section 3, the new Z-estimators for moment condition models are defined. The asymptotic properties of these estimators were proven, including here the consistency and their asymptotic normality. Then, the influence functions of the estimators were derived, and their robustness is demonstrated. The proofs of the theoretical results are deferred in Appendix A.
