Search

We study the problem of loss estimation that involves for an observable X∼fθthe choice of a first-stage estimator γ^of γ(θ), incurred loss L=L(θ,γ^), and the choice of a second-stage estimator L^of L. We consider both: (i) a sequential version where the first-stage estimate and loss are fixed and optimization is performed at the second-stage level, and (ii) a simultaneous version with a Rukhin-type loss function designed for the evaluation of (γ^,L^)as an estimator of (γ,L). We explore various Bayesian solutions and provide minimax estimators for both situations (i) and (ii). The analysis is carried out for several probability models, including multivariate normal models Nd(θ,σ2Id)with both known and unknown σ2, Gamma, univariate and multivariate Poisson, and negative binomial models, and relates to different choices of the first-stage and second-stage losses. The minimax findings are achieved by identifying a least favourable sequence of priors and depend critically on particular Bayesian solution properties, namely situations where the second-stage estimator L^(x)is constant as a function of x.

Distributional results are obtained for runs of length 2 in Bernoulli arrays X k , j with multinomial distributed rows. These include a multivariate Poisson mixture representation with Dirichlet mixing for the joint distribution of the number of runs in each column. [ABSTRACT FROM AUTHOR]

For estimating a bounded normal mean with known variance, we exhibit situations of pronounced discrepancy between the credibility of Bayes credible regions and frequentist coverage. Analogously, frequentist confidence intervals are shown to have credibility one in some cases. [ABSTRACT FROM AUTHOR]

We consider minimax shrinkage estimation of location for spherically symmetric distributions under a concave function of the usual squared error loss. Scale mixtures of normal distributions and losses with completely monotone derivatives are featured. [ABSTRACT FROM AUTHOR]

For normal models with $X \sim N_p(\theta, \sigma^{2} I_{p}), \;\; S^{2} \sim \sigma^{2}\chi^{2}_{k}, \;\mbox{independent}$, we consider the problem of estimating ?under scale invariant squared error loss ||d?-??||2/s2, when it is known that the signal-to-noise ratio ${\left\|\theta\right\|}/{\sigma}$is bounded above by m. Risk analysis is achieved by making use of a conditional risk decomposition and we obtain in particular sufficient conditions for an estimator to dominate either the unbiased estimator dUB(X)?=?X, or the maximum likelihood estimator $\delta_{ML}(X,S^2)$, or both of these benchmark procedures. The given developments bring into play the pivotal role of the boundary Bayes estimator dBU,0associated with a prior on (?,s2) such that ?|s2is uniformly distributed on the (boundary) sphere of radius msand a non-informative 1/s2prior measure is placed marginally on s2. With a series of technical results related to dBU,0; which relate to particular ratios of confluent hypergeometric functions; we show that, whenever $m \leq \sqrt{p}$and p?=?2, dBU,0dominates both dUBand dML. The finding can be viewed as both a multivariate extension of p?=?1 result due to Kubokawa (Sankhya 67:499–525, 2005) and an unknown variance extension of a similar dominance finding due to Marchand and Perron (Ann Stat 29:1078–1093, 2001). Various other dominance results are obtained, illustrations are provided and commented upon. In particular, for $m \leq \sqrt{p/2}$, a wide class of Bayes estimators, which include priors where ?|s2is uniformly distributed on the ball of radius mscentered at the origin, are shown to dominate dUB.

We introduce and motivate the study of (n+ 1) × rarrays Xwith Bernoulli entries Xk,jand independently distributed rows. We study the distribution of which denotes the number of consecutive pairs of successes (or runs of length 2) when reading the array down the columns and across the rows. With the case r= 1 having been studied by several authors, and permitting some initial inferences for the general case r> 1, we examine various distributional properties and representations of Snfor the case r= 2, and, using a more explicit analysis, the case of multinomial and identically distributed rows. Applications are also given in cases where the array Xarises from a Pólya sampling scheme.

Abstract: For estimating an unknown parameter , we introduce and motivate the use of the balanced-type loss function: , where , is a positive weight function, and is a general “target” estimator. Developments and various examples are given with regards to the issues of admissibility, dominance, Bayesianity, and minimaxity. In many cases, as in Dey et al. [1999. On estimation with balanced loss functions. Statist. Probab. Lett. 45, 97–101], we show that results for loss may be inferred directly from corresponding results for weighted squared error loss (i.e., ). Specific issues related to constrained parameter spaces, which include the choice of the target estimator, are addressed. Finally, we derive minimax estimators of a bounded normal mean under loss with being the maximum-likelihood estimator of . [Copyright &y& Elsevier]

Abstract: For a vast class of discrete model families with cdf''s , and for estimating under squared error loss under a constraint of the type , we present a general and unified development concerning the minimaxity of a boundary supported prior Bayes estimator. While the sufficient conditions obtained are of the expected form , the approach presented leads, in many instances, to both necessary and sufficient conditions, and/or explicit values for . Finally, the scope of the results is illustrated with various examples that, not only include several common distributions (e.g., Poisson, Binomial, Negative Binomial), but many others as well. [Copyright &y& Elsevier]

Abstract: For location families with densitiesf 0(x−θ), we study the problem of estimating θ for location invariant lossL(θ,d)=ρ(d−θ), and under a lower-bound constraint of the form θ≥a. We show, that for quite general (f 0, ρ), the Bayes estimator δ U with respect to a uniform prior on (a, ∞) is a minimax estimator which dominates the benchmark minimum risk equivariant (MRE) estimator. In extending some previous dominance results due to Katz and Farrell, we make use of Kubokawa'sIERD (Integral Expression of Risk Difference) method, and actually obtain classes of dominating estimators which include, and are characterized in terms of δ U . Implications are also given and, finally, the above dominance phenomenon is studied and extended to an interval constraint of the form θ∈[a, b].

For estimating under squared-error loss the mean of a p-variate normal distribution when this mean lies in a ball of radius m centered at the origin and the covariance matrix is equal to the identity matrix, it is shown that the Bayes estimator with respect to a uniformly distributed prior on the boundary of the parameter space (δBU) is minimax whenever m⩽√ of p. Further descriptions of the cutoff points of small enough radiuses (i.e., m⩽m0(p)) for δBU to be minimax are given. These include lower bounds and the large dimension p limiting behaviour of m0(p)/p. Further descriptions of the cutoff points of small enough radiuses (i.e., m⩽m0(p)) for δBU to be minimax are given. These include lower bounds and the large dimension p limiting behaviour of m0(p)/√ of p. Finally, implications for the associated minimax risk are described. [Copyright &y& Elsevier]

We propose in this paper an active vision approach for performing the 3D reconstruction of static scenes. The perception-action cycles are handled at various levels: from the definition of perception strategies for scene exploration downto the automatic generation of camera motions using visual servoing. To perform the reconstruction, we use a structure from controlled motion method which allows an optimal estimation of geometrical primitive parameters. As this method is based on particular camera motions, perceptual strategies able to appropriately perform a succession of such individual primitive reconstructions are proposed in order to recover the complete spatial structure of the scene. Two algorithms are proposed to ensure the exploration of the scene. The former is an incremental reconstruction algorithm based on the use of a prediction/verification scheme managed using decision theory and Bayes nets. It allows the visual system to get a high level description of the observed part of the scene. The latter, based on the computation of new viewpoints ensures the complete reconstruction of the scene. Experiments carried out on a robotic cell have demonstrated the validity of our approach.

For Y = ‖ a Z + θ ‖ 2 , a > 0 , Z ∼ N p (θ , I p) , θ ≠ { 0 } , we show that the distribution of Y is not stochastically ordered in a > 0. We provide extensions to spherically symmetric, elliptically symmetric, and skew-normal distributions, as well as to other quadratic forms. [ABSTRACT FROM AUTHOR]