*3.3. Influence Functions and Robustness*

In this section, we derive the influence functions of the estimators 4*t c <sup>θ</sup>* and *θ* 4*c <sup>ϕ</sup>* and prove their B-robustness. The corresponding statistical functionals are defined by (36) and (46), respectively.

Recall that, a map *T*, defined on a set of probability measures and parameter-spacevalued, is a statistical functional corresponding to an estimator *θ* 4 of the parameter *θ*<sup>0</sup> from the model *P*0, if *θ* 4 = *T*(*Pn*), *Pn* being the empirical measure corresponding to the sample. The influence function of *T* at *P*<sup>0</sup> is defined by

$$\text{IF}(\mathfrak{x}; T, P\_0) := \left. \frac{\partial T(\bar{P}\_{\mathfrak{ex}})}{\partial \mathfrak{e}} \right|\_{\mathfrak{e} = 0}$$

,

where *P*\**ε<sup>x</sup>* := (1 − *ε*) *P*<sup>0</sup> + *ε δx*, *δ<sup>x</sup>* being the Dirac measure. An unbounded influence function implies an unbounded asymptotic bias of a statistic under single-point contamination of the model. Therefore, a natural robustness requirement on a statistical functional is the

boundedness of its influence function. Whenever the influence function is bounded with respect to *x*, the corresponding estimator is called B-robust [26].

**Proposition 3.** *For fixed θ, the influence function of the functional t<sup>c</sup> <sup>θ</sup> is given by*

$$\text{IF}(\mathbf{x}; t\_{\boldsymbol{\theta}\prime}^{\boldsymbol{c}} P\_0) = - \left\{ \int \frac{\partial}{\partial t} \psi\_{\boldsymbol{\theta}}(\boldsymbol{y}, t\_{\boldsymbol{\theta}\prime}(P\_0)) dP\_0(\boldsymbol{y}) \right\}^{-1} \cdot \boldsymbol{\psi}\_{\boldsymbol{\theta}}(\boldsymbol{x}, t\_{\boldsymbol{\theta}\prime}(P\_0)). \tag{55}$$

**Proposition 4.** *The influence function of the functional T<sup>c</sup> is given by*

$$\begin{split} \text{IF}(\mathbf{x};T^{c},P\_{0}) &= \left\{ \left[\frac{\partial}{\partial\mathbf{B}}\overline{\mathbf{g}}(\mathbf{y},\theta\_{0})dP\_{0}(\mathbf{y})\right]^{\top} \left[\int \overline{\mathbf{g}}(\mathbf{y},\theta\_{0})\overline{\mathbf{g}}(\mathbf{y},\theta\_{0})^{\top}dP\_{0}(\mathbf{y})\right]^{-1} \left[\frac{\partial}{\partial\mathbf{B}}\overline{\mathbf{g}}(\mathbf{y},\theta\_{0})dP\_{0}(\mathbf{y})\right] \right\}^{-1} \\ &\cdot \left[\frac{\partial}{\partial\mathbf{B}}\overline{\mathbf{g}}(\mathbf{y},\theta\_{0})dP\_{0}(\mathbf{y})\right]^{\top} \frac{1}{\rho^{\tau}(1)}\text{IF}(\mathbf{x};t\_{\theta\_{0}}^{c},P\_{0}). \end{split} \tag{56}$$

On the basis of Propositions 3 and 4, since *x* → *ψθ* (*x*, *t<sup>θ</sup>* (*P*0)) is bounded, all the estimators *θ* 4*c <sup>ϕ</sup>* are B-robust.

### **4. Conclusions**

We introduced 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 truncated functions based on the multidimensional Huber function, we defined robust estimators of the element that realizes the supremum in the dual form of the divergence, as well as new robust estimators for the parameter of the model. The asymptotic properties were proven, including the consistency and the limit laws. The influence functions for all the proposed estimators are bounded; therefore, these estimators are B-robust. The truncated function that we used to define the new robust Z-estimators contains functions implicitly defined, for which analytic forms are not available. The implementation of the estimation method will be addressed in a future research study. The idea of using the multidimensional Huber function, together with a scale matrix and a shift vector, to create a bounded version of the function corresponding to the estimating equation for the parameter of interest, could be considered in other contexts as well and would lead to new robust Z-estimators. As one of the Referees suggested, some other bounded functions could be used to define new robust Z-estimators for moment condition models. For example, the Tukey biweight function used together with a norm inside, in order to be appropriate to be applied to functions with vector values, could also be considered. Again, the original parameter of interest should remain the solution of the estimating equation based on the new bounded function. Such an idea is interesting to be analysed in future studies, in order to provide new robust versions of minimum empirical divergence estimators or robust Z-estimators in other contexts.

**Funding:** This work was supported by a grant of the Ministry of Research, Innovation and Digitization, CNCS CCCDI − UEFISCDI, Project Number PN-III-P4-ID-PCE-2020-1112, within PNCDI III.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** We are very grateful to the Referees for their helpful comments and suggestions.

**Conflicts of Interest:** The author declares no conflict of interest.

#### **Abbreviations**

The following abbreviations are used in this manuscript:

