We consider iid homogeneous Poisson processes that are each stopped at some random time L i, independent of the history of the process. We observe at a fixed time to whether or not the process has been stopped, the number of events in (O,t O) and the time of the last event. We want to predict the termination times, and estimate the distribution of the L i. A method for unbi~sed prediction of L i is derived. This yields an inconsistent estimate of the distribution. A general consistent method for estimating the distribution is presented. In a special case it yields a maximum likehood estimate. The consistent estimate is not always preferrable in finite samples. KEYWORDS: Stopping time, prediction, minimum discrepancy estimation.

The two standard methods of obtaining a least-squares optimal estimator are (1) Bayesian estimation, in which one assumes a prior distribution on the true values and combines this with a model of the measurement process to obtain an optimal estimator, and (2) supervised regression, in which one optimizes a parametric estimator over a training set containing pairs of corrupted measurements and their associated true values. But many real-world systems do not have access to either supervised training examples or a prior model. Here, we study the problem of obtaining an optimal estimator given a measurement process with known statistics, and a set of corrupted measurements of random values drawn from an unknown prior. We develop a general form of nonparametric empirical Bayesian estimator that is written as a direct function of the measurement density, with no explicit reference to the prior. We study the observation conditions under which such “prior-free ” estimators may be obtained, and we derive specific forms for a variety of different corruption processes. Each of these prior-free estimators may also be used to express the mean squared estimation error as an expectation over the measurement density, thus generalizing Stein’s unbiased risk estimator (SURE) which provides such an expression for the additive Gaussian noise case. Minimizing this expression over measurement samples provides an “unsupervised

Abstract not found

We consider a joint processing of n independent sparse regression problems. Eachis based on a sample (yi1,xi1)...,(yim,xim) of m i.i.d. observationsfrom yi1 = x T i1 βi+εi1, yi1 ∈ R, xi1 ∈ R p, i = 1,...,n, and εi1 ∼ N(0,σ 2), say. p is large enough so that the empirical risk minimizer is not consistent. We consider three possible extensions of the lasso estimator to deal with this problem, the lassoes, the group lasso and the RING lasso, each utilizing a different assumption how these problems are related. For each estimator we give a Bayesian interpretation, and we present both persistency analysis and non-asymptotic error bounds based on restricted eigenvalue- type assumptions. “...and only a star or two set sparsedly in the vault of heaven; and you will find a sight as stimulating as the hoariest summit of the Alps. ” R. L. Stevenson 1

This paper presents a novel framework for denoising magnetic resonance images. The framework relies on adaptive Markov-random-field (MRF) image models that we infer nonparametrically from the corrupted input data itself. The proposed denoising method produces an optimal reconstruction based on principles in empirical-Bayesian estimation and information theory. Given the corrupted input data and the knowledge of the Rician noise model, the Bayesian-denoising method bootstraps itself by estimating the uncorrupted-signal Markov statistics, by optimizing an informationtheoretic metric using the expectation-maximization (EM) algorithm. It then employs the inferred uncorrupted-signal Markov statistics as an adaptive prior in a Bayesian-denoising process at each pixel. Furthermore, it proposes a novel Bayesian-inference algorithm on MRFs incorporating entropy reduction, namely iterated conditional entropy reduction (ICER). The results demonstrate that the method denoises conservatively while ensuring the preservation of most of the important features in brain-MR images. Qualitative and quantitative comparisons with the state of the art clearly depict the advantages of the proposed method.

Dissertation submitted to the University of Cambridge

User authentication and intrusion detection differ from standardclassificationproblemsinthatwhilewe havedata generatedfrom legitimateusers, impostororintrusiondata is scarce or non-existent. We review existing techniques for dealing with this problem and propose a novel alternative based on a principled statistical decision-makingview point. Weexaminethetechniqueonatoyproblemandvalidateitoncomplexreal-worlddatafromanRFIDbasedaccess control system. The results indicate that it can significantlyoutperformtheclassicalworldmodelapproach. The method could be more generally useful in other decisionmakingscenarioswherethere isalackof adversarydata. 1

• Observe zi ∼ N(µi, 1) for i = 1, 2,..., N • Select the m biggest ones: z(1)> z(2)> z(3)> · · ·> z(m) • Question: µ values? What can we say about their corresponding • Selection Bias selected z’s. The µ’s will usually be smaller than the

with several members of the “Economic Analysis and Forecasting Department ” at the Banco de España. We would also like to thank, without implicating, Samuel Hurtado, Gabriel Perez-Quiros and Alberto Urtasun for comments and suggestions at the earliest stage of the work. DISCLAIMER: The views expressed in this paper are the author’s, not those of Banco de España. Documentos de Trabajo. N.º 1037 2010 The Working Paper Series seeks to disseminate original research in economics and fi nance. All papers have been anonymously refereed. By publishing these papers, the Banco de España aims to contribute to economic analysis and, in particular, to knowledge of the Spanish economy and its international environment. The opinions and analyses in the Working Paper Series are the responsibility of the authors and, therefore, do not necessarily coincide with those of the Banco de España or the Eurosystem. The Banco de España disseminates its main reports and most of its publications via the INTERNET at the following website:

Statistical decision making for authentication and intrusion detection