Minimax nonparametric hypothesis testing for ellipsoids and Besov bodies

Yuri I. Ingster; Irina A. Suslina

ESAIM: Probability and Statistics (2010)

  • Volume: 4, page 53-135
  • ISSN: 1292-8100

Abstract

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We observe an infinitely dimensional Gaussian random vector x = ξ + v where ξ is a sequence of standard Gaussian variables and v ∈ l2 is an unknown mean. We consider the hypothesis testing problem H0 : v = 0versus alternatives H ε , τ : v V ε for the sets V ε = V ε ( τ , ρ ε ) l 2 . The sets Vε are lq-ellipsoids of semi-axes ai = i-s R/ε with lp-ellipsoid of semi-axes bi = i-r pε/ε removed or similar Besov bodies Bq,t;s (R/ε) with Besov bodies Bp,h;r (pε/ε) removed. Here τ = ( κ , R ) or τ = ( κ , h , t , R ) ; κ = ( p , q , r , s ) are the parameters which define the sets Vε for given radii pε → 0, 0 < p,q,h,t ≤ ∞, -∞ ≤ r,s ≤ ∞, R > 0; ε → 0 is the asymptotical parameter. We study the asymptotics of minimax second kind errors β ε ( α ) = β ( α , V ε ( τ , ρ ε ) ) and construct asymptotically minimax or minimax consistent families of tests ψ α ; ε , τ , ρ ε , if it is possible. We describe the partition of the set of parameters κ into regions with different types of asymptotics: classical, trivial, degenerate and Gaussian (of various types). Analogous rates have been obtained in a signal detection problem for continuous variant of white noise model: alternatives correspond to Besov or Sobolev balls with Besov or Sobolev balls removed. The study is based on an extension of methods of constructions of asymptotically least favorable priors. These methods are applicable to wide class of “convex separable symmetrical" infinite-dimensional hypothesis testing problems in white Gaussian noise model. Under some assumptions these methods are based on the reduction of hypothesis testing problem to convex extreme problem: to minimize specially defined Hilbert norm over convex sets of sequences π ¯ of measures πi on the real line. The study of this extreme problem allows to obtain different types of Gaussian asymptotics. If necessary assumptions do not hold, then we obtain other types of asymptotics.

How to cite

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Ingster, Yuri I., and Suslina, Irina A.. "Minimax nonparametric hypothesis testing for ellipsoids and Besov bodies." ESAIM: Probability and Statistics 4 (2010): 53-135. <http://eudml.org/doc/197739>.

@article{Ingster2010,
abstract = { We observe an infinitely dimensional Gaussian random vector x = ξ + v where ξ is a sequence of standard Gaussian variables and v ∈ l2 is an unknown mean. We consider the hypothesis testing problem H0 : v = 0versus alternatives $H_\{\varepsilon,\tau\}:v\in V_\{\varepsilon\}$ for the sets $V_\{\varepsilon\}=V_\{\varepsilon\}(\tau,\rho_\{\varepsilon\})\subset l_2$. The sets Vε are lq-ellipsoids of semi-axes ai = i-s R/ε with lp-ellipsoid of semi-axes bi = i-r pε/ε removed or similar Besov bodies Bq,t;s (R/ε) with Besov bodies Bp,h;r (pε/ε) removed. Here $\tau =(\kappa,R)$ or $\tau =(\kappa,h,t,R);\ \ \kappa=(p,q,r,s)$ are the parameters which define the sets Vε for given radii pε → 0, 0 < p,q,h,t ≤ ∞, -∞ ≤ r,s ≤ ∞, R > 0; ε → 0 is the asymptotical parameter. We study the asymptotics of minimax second kind errors $\beta_\{\varepsilon\}(\alpha)=\beta(\alpha, V_\{\varepsilon\}(\tau,\rho_\{\varepsilon\}))$ and construct asymptotically minimax or minimax consistent families of tests $\psi_\{\alpha;\varepsilon,\tau,\rho_\{\varepsilon\}\}$, if it is possible. We describe the partition of the set of parameters κ into regions with different types of asymptotics: classical, trivial, degenerate and Gaussian (of various types). Analogous rates have been obtained in a signal detection problem for continuous variant of white noise model: alternatives correspond to Besov or Sobolev balls with Besov or Sobolev balls removed. The study is based on an extension of methods of constructions of asymptotically least favorable priors. These methods are applicable to wide class of “convex separable symmetrical" infinite-dimensional hypothesis testing problems in white Gaussian noise model. Under some assumptions these methods are based on the reduction of hypothesis testing problem to convex extreme problem: to minimize specially defined Hilbert norm over convex sets of sequences $\bar\{\pi\}$ of measures πi on the real line. The study of this extreme problem allows to obtain different types of Gaussian asymptotics. If necessary assumptions do not hold, then we obtain other types of asymptotics. },
author = {Ingster, Yuri I., Suslina, Irina A.},
journal = {ESAIM: Probability and Statistics},
keywords = {Nonparametric hypotheses testing; minimax hypotheses testing; asymptotics of error probabilities.; asymptotics of error probabilities},
language = {eng},
month = {3},
pages = {53-135},
publisher = {EDP Sciences},
title = {Minimax nonparametric hypothesis testing for ellipsoids and Besov bodies},
url = {http://eudml.org/doc/197739},
volume = {4},
year = {2010},
}

TY - JOUR
AU - Ingster, Yuri I.
AU - Suslina, Irina A.
TI - Minimax nonparametric hypothesis testing for ellipsoids and Besov bodies
JO - ESAIM: Probability and Statistics
DA - 2010/3//
PB - EDP Sciences
VL - 4
SP - 53
EP - 135
AB - We observe an infinitely dimensional Gaussian random vector x = ξ + v where ξ is a sequence of standard Gaussian variables and v ∈ l2 is an unknown mean. We consider the hypothesis testing problem H0 : v = 0versus alternatives $H_{\varepsilon,\tau}:v\in V_{\varepsilon}$ for the sets $V_{\varepsilon}=V_{\varepsilon}(\tau,\rho_{\varepsilon})\subset l_2$. The sets Vε are lq-ellipsoids of semi-axes ai = i-s R/ε with lp-ellipsoid of semi-axes bi = i-r pε/ε removed or similar Besov bodies Bq,t;s (R/ε) with Besov bodies Bp,h;r (pε/ε) removed. Here $\tau =(\kappa,R)$ or $\tau =(\kappa,h,t,R);\ \ \kappa=(p,q,r,s)$ are the parameters which define the sets Vε for given radii pε → 0, 0 < p,q,h,t ≤ ∞, -∞ ≤ r,s ≤ ∞, R > 0; ε → 0 is the asymptotical parameter. We study the asymptotics of minimax second kind errors $\beta_{\varepsilon}(\alpha)=\beta(\alpha, V_{\varepsilon}(\tau,\rho_{\varepsilon}))$ and construct asymptotically minimax or minimax consistent families of tests $\psi_{\alpha;\varepsilon,\tau,\rho_{\varepsilon}}$, if it is possible. We describe the partition of the set of parameters κ into regions with different types of asymptotics: classical, trivial, degenerate and Gaussian (of various types). Analogous rates have been obtained in a signal detection problem for continuous variant of white noise model: alternatives correspond to Besov or Sobolev balls with Besov or Sobolev balls removed. The study is based on an extension of methods of constructions of asymptotically least favorable priors. These methods are applicable to wide class of “convex separable symmetrical" infinite-dimensional hypothesis testing problems in white Gaussian noise model. Under some assumptions these methods are based on the reduction of hypothesis testing problem to convex extreme problem: to minimize specially defined Hilbert norm over convex sets of sequences $\bar{\pi}$ of measures πi on the real line. The study of this extreme problem allows to obtain different types of Gaussian asymptotics. If necessary assumptions do not hold, then we obtain other types of asymptotics.
LA - eng
KW - Nonparametric hypotheses testing; minimax hypotheses testing; asymptotics of error probabilities.; asymptotics of error probabilities
UR - http://eudml.org/doc/197739
ER -

References

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