# 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

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topIngster, 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 -

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