In this paper, we propose a class of general pretest estimators for the univariate normal mean. The main mathematical idea proposed is adaptation randomized tests, where randomization probability related to shrinkage parameter. Consequently, includes many existing estimators, such as pretest, shrinkage, Bayes, and empirical Bayes special cases. Furthermore, can be easily tuned users by adjustin...

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 ...