Komarova, Tatiana; Hidalgo, Javier Testing nonparametric shape restrictions. (English) Zbl 1539.62113 Ann. Stat. 51, No. 6, 2299-2317 (2023). Summary: We describe and examine a test for a general class of shape constraints, such as signs of derivatives, U-shape, quasi-convexity, log-convexity, among others, in a nonparametric framework using partial sums empirical processes. We show that, after a suitable transformation, its asymptotic distribution is a functional of a Brownian motion index by the c.d.f. of the regressor. As a result, the test is distribution-free and critical values are readily available. However, due to the possible poor approximation of the asymptotic critical values to the finite sample ones, we also describe a valid bootstrap algorithm. MSC: 62G08 Nonparametric regression and quantile regression 62H15 Hypothesis testing in multivariate analysis Keywords:monotonicity; convexity; concavity; U-shape; quasi-convexity; log-convexity; convexity in means; B-splines; CUSUM transformation; distribution-free estimation Software:FITPACK × Cite Format Result Cite Review PDF Full Text: DOI arXiv Link References: [1] ABREVAYA, J. and JIANG, W. (2005). A nonparametric approach to measuring and testing curvature. J. Bus. Econom. Statist. 23 1-19. Digital Object Identifier: 10.1198/073500104000000316 Google Scholar: Lookup Link MathSciNet: MR2108688 · doi:10.1198/073500104000000316 [2] AGARWAL, G. G. and STUDDEN, W. J. (1980). Asymptotic integrated mean square error using least squares and bias minimizing splines. Ann. Statist. 8 1307-1325. MathSciNet: MR0594647 · Zbl 0522.62032 [3] Akakpo, N., Balabdaoui, F. and Durot, C. (2014). Testing monotonicity via local least concave majorants. Bernoulli 20 514-544. 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