*Article* **Robust Test Statistics Based on Restricted Minimum Rényi's Pseudodistance Estimators**

**María Jaenada †, Pedro Miranda † and Leandro Pardo \*,†**

Department of Statistics and Operation Research, Faculty of Mathematics, Interdisciplinary Mathematical Insititute, Complutense University of Madrid, Plaza Ciencias, 3, 28040 Madrid, Spain; mjaenada@ucm.es (M.J.); pmiranda@mat.ucm.es (P.M.)

**\*** Correspondence: lpardo@mat.ucm.es

† These authors contributed equally to this work.

**Abstract:** The Rao's score, Wald and likelihood ratio tests are the most common procedures for testing hypotheses in parametric models. None of the three test statistics is uniformly superior to the other two in relation with the power function, and moreover, they are first-order equivalent and asymptotically optimal. Conversely, these three classical tests present serious robustness problems, as they are based on the maximum likelihood estimator, which is highly non-robust. To overcome this drawback, some test statistics have been introduced in the literature based on robust estimators, such as robust generalized Wald-type and Rao-type tests based on minimum divergence estimators. In this paper, restricted minimum Rényi's pseudodistance estimators are defined, and their asymptotic distribution and influence function are derived. Further, robust Rao-type and divergence-based tests based on minimum Rényi's pseudodistance and restricted minimum Rényi's pseudodistance estimators are considered, and the asymptotic properties of the new families of tests statistics are obtained. Finally, the robustness of the proposed estimators and test statistics is empirically examined through a simulation study, and illustrative applications in real-life data are analyzed.

**Keywords:** Rényi's pseudodistance; minimum Rényi's pseudodistance estimators; restricted minimum Rényi's pseudodistance estimators; Rao-type tests; divergence-based tests
