Wavelet analysis has been found to be a powerful tool for the nonparametric estimation of spatially-variable objects. We discuss in detail wavelet methods in nonparametric regression, where the data are modelled as observations of a signal contaminated with additive Gaussian noise, and provide an extensive review of the vast literature of wavelet shrinkage and wavelet thresholding estimators de...

Many multivariate Gaussian models can conveniently be split into independent, block-wise problems. Common settings where this situation arises are balanced ANOVA models, balanced longitudinal models, and certain block-wise shrinkage estimators in nonparametric regression estimation involving orthogonal bases such as Fourier or wavelet bases. It is well known that the standard, least squares est...

The problem of state estimation with stochastic uncertainties in the initial state, model noise, and measurement noise is approached using the restricted risk Bayes approach. It is assumed that the a priori distributions of these quantities are not perfectly known but that some a priori information may be available. While offering robustness, the restricted risk Bayes approach incorporates the ...

This appendix provides some additional discussion and results, supplementing the manuscript of “How to use economic theory to improve estimators.” In Section A of this appendix,we consider two additional applications of our proposed approach, to a general equilibrium model of financial markets, and to structural models of (consumer) preferences. Our theoretical results suggest that the proposed...

In the first part of the course, we focused on optimal inference in the setting of point estimation (see Figure 12.2). We formulated this problem in the framework of decision theory and focused on finite sample criteria of optimality. We immediately discovered that uniform optimality was seldom attainable in practice, and thus, we developed our theory of optimality along two lines: constraining...

In this article we consider the statistical inferences of the unknown parameters of a Weibull distribution when the data are Type-I censored. It is well known that the maximum likelihood estimators do not always exist, and even when they exist, they do not have explicit expressions. We propose a simple fixed point type algorithm to compute the maximum likelihood estimators, when they exist. We ...

The aim of this paper is to study the estimation of Pareto distribution on the basis of progressive type-II censored sample. First, the maximum likelihood estimator (MLE) is derived. Then the Bayes estimator of the unknown parameter of Pareto distribution is derived on the basis of Gamma prior distribution under entropy loss function. Further the empirical Bayes estimator also obtained by using...

let x be a random variable from a normal distribution with unknown mean θ and known variance σ2. in many practical situations, θ is known in advance to lie in an interval, say [−m,m], for some m > 0. as the usual estimator of θ, i.e., x under the linex loss function is inadmissible, finding some competitors for x becomes worthwhile. the only study in the literature considered the problem of min...

The empirical Bayes estimator of the probability of a successful event is deduced from mixed distributions. Specially, binomial and Poisson mixed distributions are analyzed. Some mixing distributions, well known in Reliability, Queuing Theory and other areas of Engineering, are considered. As it will be shown, a family of estimators with interesting characteristics is obtained for different mix...