### Journals Information

**
Mathematics and Statistics Vol. 10(5), pp. 956 - 970 DOI: 10.13189/ms.2022.100507 Reprint (PDF) (396Kb) **

## Asymptotically Minimax Goodness-of-fit Testing for Single-index Models

**Jean-Philippe Tchiekre ^{1}, Christophe Pouet ^{2}, Armel Fabrice E. Yodé ^{1}^{,*}**

^{1}Department of Mathematics and Computer Science, Félix Houphouët-Boigny University, 22 BP 582 Abidjan 22, Ivory Coast

^{2}Marseille Institute of Mathematics, Marseille Central School, 38 Rue Frédéric Joliot-Curie, 13013 Marseille, France

**ABSTRACT**

In the context of non parametric multivariate regression model, we are interested in goodness-of-fit testing for the single-index models. These models are dimension reduction models and are therefore useful in multidimensional nonparametric statistics because of the well-known phenomenon called the curse of dimensionality. Fan and Li [5] have proposed the first consistent test for goodness-of-fit testing of the single-index by using nonparametric kernel estimation method and a central limit theorem for degenerate -statistics of order higher than two. Since then, the minimax properties of this test have not been investigated. Following this work, we use here the asymptotic minimax approach. We are interested in finding the asymptotic minimax rate of testing which gives the minimal distance between the null and alternative hypotheses such that a successful testing is possible. We propose a test procedure of level which can tend to zero when the sample size tends to infinity. We have established the minimax asymptotic properties of our test procedure by showing that it reaches the asymptotic minimax rate for the dimension and there is no test of level reaching this rate for . Because of its minimax asymptotic properties, our test is able to distinguish the null hypothesis of the closest possible alternative. The results obtained were possible thanks to a large deviation result that we established for a degenerate U-statistic of order two appearing in our decision variable.

**KEYWORDS**

Single-index Model, Asymptotically Minimax Hypothesis Testing, Asymptotics of Errors Probabilities, Large Deviation, -statistics

**Cite This Paper in IEEE or APA Citation Styles**

(a). IEEE Format:

[1] Jean-Philippe Tchiekre , Christophe Pouet , Armel Fabrice E. Yodé , "Asymptotically Minimax Goodness-of-fit Testing for Single-index Models," Mathematics and Statistics, Vol. 10, No. 5, pp. 956 - 970, 2022. DOI: 10.13189/ms.2022.100507.

(b). APA Format:

Jean-Philippe Tchiekre , Christophe Pouet , Armel Fabrice E. Yodé (2022). Asymptotically Minimax Goodness-of-fit Testing for Single-index Models. Mathematics and Statistics, 10(5), 956 - 970. DOI: 10.13189/ms.2022.100507.