Minimax Detection of a Signal In a Gaussian White Noise

Ermakov, Michael Sergeevich

doi:10.1137/1135098

Cited by 59 publications

(86 citation statements)

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“…The same holds for V ε = l 2 \ E p,r (ρ) for any ρ > 0 and r ≥ r p (see Ingster [12], Th. 2.5; for p = 2 it follows from Ermakov [6]). …”

Section: Trivial Type (T)mentioning

confidence: 83%

“…The problem of sharp asymptotics for ellipsoids were studied by Ermakov [6], Ingster [11][12][13] and by Suslina [26,27] for different values of τ, s > r. In Ermakov [6] the case p = q = 2 had been investigated. In Ingster [11,12] the results for the cases 0 < p = q < ∞ and q ≤ p = ∞ had been obtained.…”

Section: Discussionmentioning

confidence: 99%

“…The asymptotics of type G 1 arise in Ermakov [6] for p = q = 2, in Ingster [11,12] for p = q ≤ 2, in Suslina [26] for p ≤ 2, q > p. The asymptotical minimax families of tests ψ ε,α = 1 {Lε,τ >Tα} in these cases are based on the statistics…”

Section: Gaussian Asymptoticsmentioning

confidence: 99%

“…We use an approach which is close to Pinsker [22], Donoho and Johnstone [4] in estimation problems. In hypothesis testing problems this approach had been used by Ermakov [6], Ingster [11−14].…”

Section: Extreme Problemmentioning

confidence: 99%

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Minimax nonparametric hypothesis testing for ellipsoids and Besov bodies

Ingster

Suslina²

2000

ESAIM: PS

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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 = 0 versus alternatives Hε,τ : v ∈ Vε for the sets Vε = Vε(τ, ρε) ⊂ l2. The sets Vε are lq-ellipsoids of semi-axes ai = i −s R/ε with lp-ellipsoid of semi-axes bi = i −r ρε/ε removed or similar Besov bodies Bq,t;s(R/ε) with Besov bodies B p,h;r (ρε/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 ρε → 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 βε(α) = β (α, Vε(τ, ρε)) and construct asymptotically minimax or minimax consistent families of tests ψα;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 sequencesπ 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.

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“…The same holds for V ε = l 2 \ E p,r (ρ) for any ρ > 0 and r ≥ r p (see Ingster [12], Th. 2.5; for p = 2 it follows from Ermakov [6]). …”

Section: Trivial Type (T)mentioning

confidence: 83%

Section: Discussionmentioning

confidence: 99%

Section: Gaussian Asymptoticsmentioning

confidence: 99%

Section: Extreme Problemmentioning

confidence: 99%

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Minimax nonparametric hypothesis testing for ellipsoids and Besov bodies

Ingster

Suslina²

2000

ESAIM: PS

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“…This is a standard procedure. If we test the hypothesis versus sets of alternatives defined in terms of series of ortogonal functions (see Ingster and Suslina [22], Lepskii and Spokoiny [25], Ermakov [7]), the tests statistics are also based on the first Fourier coefficients and estimates of these coefficients. The Fourier coefficients of higher orders are ignored both for the hypothesis and alternatives.…”

Section: K T − S H S(s)dsmentioning

confidence: 99%

On Asymptotic Minimaxity of Kernel-based Tests

Ermakov

2003

ESAIM: PS

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Abstract. In the problem of signal detection in Gaussian white noise we show asymptotic minimaxity of kernel-based tests. The test statistics equal L2-norms of kernel estimates. The sets of alternatives are essentially nonparametric and are defined as the sets of all signals such that the L2-norms of signal smoothed by the kernels exceed some constants ρ > 0. The constant ρ depends on the power of noise and ρ → 0 as → 0. Similar statements are proved also if an additional information on a signal smoothness is given. By theorems on asymptotic equivalence of statistical experiments these results are extended to the problems of testing nonparametric hypotheses on density and regression. The exact asymptotically minimax lower bounds of type II error probabilities are pointed out for all these settings. Similar results are also obtained for the problems of testing parametric hypotheses versus nonparametric sets of alternatives.Mathematics Subject Classification. 62G10, 62G20.

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