Locally minimax efficiency of nonparametric density estimators for χ2-type losses

Radavičius, Marijus; Rudzkis, Rimantas

doi:10.1007/bf02465851

Cited by 2 publications

(4 citation statements)

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“…As it is shown in [30], adaptive estimators can not achieve the asymptotic minimax risk for the sup-norm losses on the scale of Holder classes (Korestelev's case). The presented paper is closely related to [33]. Although [33] deals with a more general weighted mean-square risk, for simplicity, we discuss the main result only in a special case of usual L 2 -losses Q(i/j,f) = \\ф -f\\l-Then it would be natural to define the neighborhoods W a , a G A, as contracting balls in L 2 .…”

Section: (Rw a )~6(w A )mentioning

confidence: 99%

“…The presented paper is closely related to [33]. Although [33] deals with a more general weighted mean-square risk, for simplicity, we discuss the main result only in a special case of usual L 2 -losses Q(i/j,f) = \\ф -f\\l-Then it would be natural to define the neighborhoods W a , a G A, as contracting balls in L 2 . However, since the estimating accuracy heavily depends upon the smoothness of the unknown d.d., the definition of neighborhoods under consideration should also include certain smoothness conditions.…”

Section: (Rw a )~6(w A )mentioning

confidence: 99%

“…/ onto the linear subspace span {u 0 ,.. •, u k }, and >>0asiV->oo. In paper [33], under quite general conditions for the orthobasis и and the set of sequences 6, it is established that for each b G 83 there exists an estimator /*(•) = /*(-,Ь, X) possessing the property of local asymptotic minimaxity:…”

Section: (Rw a )~6(w A )mentioning

confidence: 99%

“…Proceeding from [33], conceptually it is not difficult to build up an adap tive estimator /*(•) = /*(-,X) possessing the same asymptotic efficiency without prior knowledge of the sequence b. However, the proof of this fact is rather complicate and cumbersome.…”

Section: (Rw a )~6(w A )mentioning

confidence: 99%

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Adaptive estimation of distribution density in the basis of algebraic polynomials

Rudzkis¹,

Radavicius²

2004

Teor. Veroyatnost. i Primenen.

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Статья посвящена задаче адаптивного статистического оце нивания плотности распределения, определенной на конечном ин тервале. Рассматриваются оценки проективного типа, основан ные на многочленах Якоби. Построена адаптивная статистическая оценка, являющаяся асимптотически минимаксной в смысле сред-неквадратических потерь для всех множеств из семейства сжима ющихся множеств функций различной гладкости. Условия глад кости формулируются в терминах Z/2-норм остатков при аппрок симации плотностей распределений линейными комбинациями ко нечного числа первых многочленов Якоби. Обсуждается также обобщение этого результата на другие ортонормированные базисы, обладающие некоторыми естественными свойствами регулярности.Ключевые слова и фразы: адаптивное оценивание, локально ми нимаксное оценивание, многочлены Якоби, оценки проективного типа, среднеквадратические потери. Introduction.In the statistical estimation of an unknown parame ter (finite-or infinite-dimensional) one seeks to use optimal estimators. For this a mathematical definition of optimality is needed. In the Bayes setting this problem is solved, however, practical realization of this solution runs into another problem: to select a prior distribution of the parameter. In the recent decades the notion of minimax estimators has found increasing favour. From the viewpoint of applications, the main criticism is that their construction is guided by the worst (in the sense of identification accuracy) case of possible models. This results in a strong dependence of the estimat ing quality on a priori information. Significantly more flexible are adaptive estimators which have the same construction as minimax estimators, but they use more exact prior information specified on the basis of the sample. These estimators are far less sensitive to a priori information yet retaining the asymptotic minimax optimality.Let us introduce necessary definitions. Let X = {Х г ,..., X^} be a sam ple of independent and identically distributed (i.i.d.) random variables Xi

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Section: (Rw a )~6(w A )mentioning

confidence: 99%

Section: (Rw a )~6(w A )mentioning

confidence: 99%

Section: (Rw a )~6(w A )mentioning

confidence: 99%

Section: (Rw a )~6(w A )mentioning

confidence: 99%

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