## Bernoulli

• Bernoulli
• Volume 25, Number 4A (2019), 2597-2619.

### Functional estimation and hypothesis testing in nonparametric boundary models

#### Abstract

Consider a Poisson point process with unknown support boundary curve $g$, which forms a prototype of an irregular statistical model. We address the problem of estimating non-linear functionals of the form $\int\Phi(g(x))\,dx$. Following a nonparametric maximum-likelihood approach, we construct an estimator which is UMVU over Hölder balls and achieves the (local) minimax rate of convergence. These results hold under weak assumptions on $\Phi$ which are satisfied for $\Phi(u)=|u|^{p}$, $p\ge1$. As an application, we consider the problem of estimating the $L^{p}$-norm and derive the minimax separation rates in the corresponding nonparametric hypothesis testing problem. Structural differences to results for regular nonparametric models are discussed.

#### Article information

Source
Bernoulli, Volume 25, Number 4A (2019), 2597-2619.

Dates
Revised: June 2018
First available in Project Euclid: 13 September 2019

https://projecteuclid.org/euclid.bj/1568362037

Digital Object Identifier
doi:10.3150/18-BEJ1064

Mathematical Reviews number (MathSciNet)
MR4003559

Zentralblatt MATH identifier
07110106

#### Citation

Reiß, Markus; Wahl, Martin. Functional estimation and hypothesis testing in nonparametric boundary models. Bernoulli 25 (2019), no. 4A, 2597--2619. doi:10.3150/18-BEJ1064. https://projecteuclid.org/euclid.bj/1568362037

#### References

• [1] Bergh, J. and Löfström, J. (1976). Interpolation Spaces. An Introduction. Berlin: Springer. Grundlehren der Mathematischen Wissenschaften, No. 223.
• [2] Drees, H., Neumeyer, N. and Selk, L. Estimation and hypotheses testing in boundary regression models. Bernoulli. To appear.
• [3] Ibragimov, I.A., Nemirovskii, A.S. and Khas’minskii, R.Z. (1986). Some problems of nonparametric estimation in Gaussian white noise. Teor. Veroyatn. Primen. 31 451–466.
• [4] Ingster, Yu.I. and Suslina, I.A. (2003). Nonparametric Goodness-of-Fit Testing Under Gaussian Models. Lecture Notes in Statistics 169. New York: Springer.
• [5] Jirak, M., Meister, A. and Reiß, M. (2014). Adaptive function estimation in nonparametric regression with one-sided errors. Ann. Statist. 42 1970–2002.
• [6] Kallenberg, O. (2002). Foundations of Modern Probability, 2nd ed. Probability and Its Applications (New York). New York: Springer.
• [7] Karr, A.F. (1991). Point Processes and Their Statistical Inference, 2nd ed. Probability: Pure and Applied 7. New York: Dekker.
• [8] Korostelëv, A.P. and Tsybakov, A.B. (1993). Minimax Theory of Image Reconstruction. Lecture Notes in Statistics 82. New York: Springer.
• [9] Kutoyants, Yu.A. (1998). Statistical Inference for Spatial Poisson Processes. Lecture Notes in Statistics 134. New York: Springer.
• [10] Last, G. and Penrose, M. (2018). Lectures on the Poisson Process. Institute of Mathematical Statistics Textbooks 7. Cambridge: Cambridge Univ. Press.
• [11] Lepski, O., Nemirovski, A. and Spokoiny, V. (1999). On estimation of the $L_{r}$ norm of a regression function. Probab. Theory Related Fields 113 221–253.
• [12] Lepski, O.V. and Spokoiny, V.G. (1999). Minimax nonparametric hypothesis testing: The case of an inhomogeneous alternative. Bernoulli 5 333–358.
• [13] Meister, A. and Reiß, M. (2013). Asymptotic equivalence for nonparametric regression with non-regular errors. Probab. Theory Related Fields 155 201–229.
• [14] Nemirovski, A. (2000). Topics in non-parametric statistics. In Lectures on Probability Theory and Statistics (Saint-Flour, 1998). Lecture Notes in Math. 1738 85–277. Berlin: Springer.
• [15] Reiß, M. and Selk, L. (2017). Efficient estimation of functionals in nonparametric boundary models. Bernoulli 23 1022–1055.
• [16] Tsybakov, A.B. (2009). Introduction to Nonparametric Estimation. Springer Series in Statistics. New York: Springer.