Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Asymptotic nonequivalence of density estimation and Gaussian white noise for small densities

Kolyan Ray and Johannes Schmidt-Hieber

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Abstract

It is well-known that density estimation on the unit interval is asymptotically equivalent to a Gaussian white noise experiment, provided the densities are sufficiently smooth and uniformly bounded away from zero. We show that a uniform lower bound, whose size we sharply characterize, is in general necessary for asymptotic equivalence to hold.

Résumé

Il est bien connu que l’estimation de densité sur l’intervalle $[0,1]$ est asymptotiquement équivalente à une expérience de bruit blanc, à condition que les densités soient suffisamment régulières et uniformément bornées loin de $0$. Nous montrons qu’une borne inférieure uniforme, dont on caractérise précisément la valeur, est en général nécessaire pour que cette équivalence asymptotique ait lieu.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 55, Number 4 (2019), 2195-2208.

Dates
Received: 9 February 2018
Revised: 16 August 2018
Accepted: 25 October 2018
First available in Project Euclid: 8 November 2019

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1573203627

Digital Object Identifier
doi:10.1214/18-AIHP946

Mathematical Reviews number (MathSciNet)
MR4029152

Subjects
Primary: 62B15: Theory of statistical experiments
Secondary: 62G10: Hypothesis testing 62G20: Asymptotic properties

Keywords
Asymptotic equivalence Density estimation Gaussian white noise model Small densities

Citation

Ray, Kolyan; Schmidt-Hieber, Johannes. Asymptotic nonequivalence of density estimation and Gaussian white noise for small densities. Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), no. 4, 2195--2208. doi:10.1214/18-AIHP946. https://projecteuclid.org/euclid.aihp/1573203627


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