**7. Concluding Remarks**

In this paper, we presented for the first time the family of RMRPEs. We derived their asymptotic distribution, and proved some suitable properties as consistency under the parameter restriction and robustness against data contamination. Further, based on these RMRPEs, we generalized two important families of statistics, namely RPTS and Raotype tests, for testing a composite null hypothesis. Moreover, we obtained some explicit expressions of the RMPREs, RPTS and Rao-type test statistics for testing the variance under a normal population with an unknown mean. It was empirically shown that the proposed RPTS and Rao-type test statistics are robust, unlike classical tests based on the MLE, under normal populations. Indeed, the robustness of the tests is controlled by a tuning parameter *τ*, and so larger values of *τ* produce more robust estimators (although less efficient). Finally, some classical numerical examples illustrate the theoretical properties and applicability of the proposed methods.

**Author Contributions:** Conceptualization, M.J., P.M. and L.P.; methodology, M.J., P.M. and L.P.; software, M.J., P.M. and L.P.; validation, M.J., P.M. and L.P.; formal analysis, M.J., P.M. and L.P.; investigation, M.J., P.M. and L.P.; resources, M.J., P.M. and L.P.; data curation, M.J., P.M. and L.P.; writing—original draft preparation, M.J., P.M. and L.P.; writing—review and editing, M.J., P.M. and L.P.; visualization, M.J., P.M. and L.P.; supervision, M.J., P.M. and L.P.; project administration, M.J., P.M. and L.P.; funding acquisition, M.J., P.M. and L.P. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research is supported by the Spanish Grants PGC2018-095 194-B-100 (M. Jaenada, P. Miranda and L. Pardo) and FPU 19/01824 (M.Jaenada). The three authors are members of the Interdisciplinary Mathematics Institute (IMI).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

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

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.
