On the Attainment of the Cramer-Rao Bound in $\mathbb{L}_r$-Differentiable Families of Distributions

Müller-Funk, Ulrich; Pukelsheim, Friedrich; Witting, Hermann

doi:10.1214/aos/1176347392

Cited by 12 publications

(7 citation statements)

References 7 publications

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“…The (second) derivative of on the whole of now exists as a generalized function on , i.e., the second derivative of may contain a summation of Dirac's delta functions on and derivatives of the delta function. The CRB can still be calculated when the appropriate rules for delta functions are applied (for a more general version of the CRB, see [6]).…”

Section: Main Theoremsmentioning

confidence: 99%

“…The derivative of the white Gaussian distribution is given by (see also [2]) (5) and further On the other hand, by differentiating (3) with respect to , we obtain These two equations imply that However, since the left-hand side does not depend on and is invertible, must be some constant matrix independent of so that (6) Since has full rank in at least one point and since is invertible, it follows that must have full rank .…”

Section: Main Theoremsmentioning

confidence: 99%

“…In that case, is the efficient estimator, which turns out to be the maximum likelihood estimator (MLE) [1], [2]. It is well known that (1) is equivalent to the condition that should be a member of an exponential family ([3, Example IV.C.4], [6]). However, this fact will not be used here as it seems more efficient to use (1) directly in the proofs.…”

Section: Introductionmentioning

confidence: 99%

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On the existence of efficient estimators

Høst-Madsen

2000

IEEE Trans. Signal Process.

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Section: Main Theoremsmentioning

confidence: 99%

Section: Main Theoremsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the existence of efficient estimators

Høst-Madsen

2000

IEEE Trans. Signal Process.

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“…Let us give some finite sample properties for the estimation (7). We show that the sample-wise estimators of (9) are efficient estimators of (9); this is not a surprise, but comes from properties of exponential families [13].…”

Section: Theorem 1 Suppose ψ Strictly Convex Differentiable the Trumentioning

confidence: 95%

Boosting Nearest Neighbors for the Efficient Estimation of Posteriors

D'Ambrosio

Nock

Ali

et al. 2012

Machine Learning and Knowledge Discovery in Databases

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It is an admitted fact that mainstream boosting algorithms like AdaBoost do not perform well to estimate class conditional probabilities. In this paper, we analyze, in the light of this problem, a recent algorithm, unn, which leverages nearest neighbors while minimizing a convex loss. Our contribution is threefold. First, we show that there exists a subclass of surrogate losses, elsewhere called balanced, whose minimization brings simple and statistically efficient estimators for Bayes posteriors. Second, we show explicit convergence rates towards these estimators for unn, for any such surrogate loss, under a Weak Learning Assumption which parallels that of classical boosting results. Third and last, we provide experiments and comparisons on synthetic and real datasets, including the challenging SUN computer vision database. Results clearly display that boosting nearest neighbors may provide highly accurate estimators, sometimes more than a hundred times more accurate than those of other contenders like support vector machines.

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“…The result has been proved under different conditions (Wijsman (1973), Fabian and Hannan (1977), Müller-Funk, Pukelsheim and Witting (1989)). An analogous result for prediction appears in Bosq and Blanke (2007) in the one-dimensional case and in Onzon (2011) in the multidimensional case.…”

mentioning

confidence: 86%

Efficient prediction in $L^2$-differentiable families of distributions

Onzon¹

2013

Preprint

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A proof of the Cramér-Rao inequality for prediction is presented under conditions of L 2 -differentiability of the family of distributions of the model. The assumptions and the proof differ from those of Miyata ( 2001) who also proved this inequality under L 2differentiability conditions. It is also proved that an efficient predictor (i.e. which risk attains the bound) exists if and only if the family of distributions is of a special form which can be seen as an extension of the notion of exponential family. This result is also proved under L 2 -differentiability conditions.

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On the Attainment of the Cramer-Rao Bound in $\mathbb{L}_r$-Differentiable Families of Distributions

Cited by 12 publications

References 7 publications

On the existence of efficient estimators

On the existence of efficient estimators

Boosting Nearest Neighbors for the Efficient Estimation of Posteriors

Efficient prediction in $L^2$-differentiable families of distributions

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