1991
DOI: 10.1214/aos/1176348252
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Approximation of Density Functions by Sequences of Exponential Families

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Cited by 174 publications
(218 citation statements)
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“…In recent years, theory has been developed in which a parametric family is not restricted to a given size, but rather the dimension of the family is increased at a certain rate as a function of the sample size, so as to get the smallest possible total risk, uniformly over classes of smooth functions, (see Cox, 1988, Stone, 1990, Barren and Sheu, 1991. A surprising aspect of this work is that the same rates of convergence of the total risk that are achievable by nonparametric estimators can be achieved by sequences of parametric families.…”
Section: Introductionmentioning
confidence: 99%
“…In recent years, theory has been developed in which a parametric family is not restricted to a given size, but rather the dimension of the family is increased at a certain rate as a function of the sample size, so as to get the smallest possible total risk, uniformly over classes of smooth functions, (see Cox, 1988, Stone, 1990, Barren and Sheu, 1991. A surprising aspect of this work is that the same rates of convergence of the total risk that are achievable by nonparametric estimators can be achieved by sequences of parametric families.…”
Section: Introductionmentioning
confidence: 99%
“…One approach, based on the Kullback-Leibler divergence, was first considered by Barron and Sheu [10]. Basically, it uses the following well-known Pythagorean property (see Lemma 3 in [10]):…”
Section: Consistency and Generalization Bounds Of Estimation Errormentioning
confidence: 99%
“…The restriction, λ ∈ Ω, will guarantee that the maximum likelihood estimate is an interior point of the set of λ's for which p λ (x) is defined. The optimal solution, pλ(x), of Equation (1) or Equation (3) is called the information projection [10,17] of p 0 (x) to the exponential family, E(x).…”
Section: Introductionmentioning
confidence: 99%
“…In order to impose the non-negative condition, a novel transformation approach is proposed using a log function. This is partially inspired by the method presented in [23].…”
Section: Introductionmentioning
confidence: 99%