On the additivity of preference aggregation methods

The paper reviews some axioms of additivity concerning ranking methods used for generalized tournaments with possible missing values and multiple comparisons. It is shown that one of the most natural properties, called consistency, has strong links to independence of irrelevant comparisons, an axiom judged unfavourable when players have different opponents. Therefore some directions of weakening consistency are suggested, and several ranking methods, the score, generalized row sum and least squares as well as fair bets and its two variants (one of them entirely new) are analysed whether they satisfy the properties discussed. It turns out that least squares and generalized row sum with an appropriate parameter choice preserve the relative ranking of two objects if the ranking problems added have the same comparison structure.Comment: 24 pages, 9 figure

An invitation to quantum tomography (II)

The quantum state of a light beam can be represented as an infinite dimensional density matrix or equivalently as a density on the plane called the Wigner function. We describe quantum tomography as an inverse statistical problem in which the state is the unknown parameter and the data is given by results of measurements performed on identical quantum systems. We present consistency results for Pattern Function Projection Estimators as well as for Sieve Maximum Likelihood Estimators for both the density matrix of the quantum state and its Wigner function. Finally we illustrate via simulated data the performance of the estimators. An EM algorithm is proposed for practical implementation. There remain many open problems, e.g. rates of convergence, adaptation, studying other estimators, etc., and a main purpose of the paper is to bring these to the attention of the statistical community.Comment: An earlier version of this paper with more mathematical background but less applied statistical content can be found on arXiv as quant-ph/0303020. An electronic version of the paper with high resolution figures (postscript instead of bitmaps) is available from the authors. v2: added cross-validation results, reference

Zero-inflated generalized Poisson models with regression effects on the mean, dispersion and zero-inflation level applied to patent outsourcing rates

This paper focuses on an extension of zero-inflated generalized Poisson (ZIGP) regression models for count data. We discuss generalized Poisson (GP) models where dispersion is modelled by an additional model parameter. Moreover, zero-inflated models in which overdispersion is assumed to be caused by an excessive number of zeros are discussed. In addition to ZIGP regression introduced by Famoye and Singh (2003), we now allow for regression on the overdispersion and zero-inflation parameters. Consequently, we propose tools for an exploratory data analysis on the dispersion and zero-inflation level. An application dealing with outsourcing of patent filing processes will be used to compare these nonnested models. The model parameters are fitted by maximum likelihood. Asymptotic normality of the ML estimates in this non-exponential setting is proven. Standard errors are estimated using the asymptotic normality of the estimates. Appropriate exploratory data analysis tools are developed. Also, a model comparison using AIC statistics and Vuong tests (see Vuong (1989)) is carried out. For the given data, our extended ZIGP regression model will prove to be superior over GP and ZIP models and even ZIGP models with constant overall dispersion and zero-inflation parameters demonstrating the usefulness of our proposed extensions

Fast, Exact Bootstrap Principal Component Analysis for p>1 million

Many have suggested a bootstrap procedure for estimating the sampling variability of principal component analysis (PCA) results. However, when the number of measurements per subject ($p$) is much larger than the number of subjects ($n$), the challenge of calculating and storing the leading principal components from each bootstrap sample can be computationally infeasible. To address this, we outline methods for fast, exact calculation of bootstrap principal components, eigenvalues, and scores. Our methods leverage the fact that all bootstrap samples occupy the same $n$-dimensional subspace as the original sample. As a result, all bootstrap principal components are limited to the same $n$-dimensional subspace and can be efficiently represented by their low dimensional coordinates in that subspace. Several uncertainty metrics can be computed solely based on the bootstrap distribution of these low dimensional coordinates, without calculating or storing the $p$-dimensional bootstrap components. Fast bootstrap PCA is applied to a dataset of sleep electroencephalogram (EEG) recordings ($p=900$, $n=392$), and to a dataset of brain magnetic resonance images (MRIs) ($p\approx$ 3 million, $n=352$). For the brain MRI dataset, our method allows for standard errors for the first 3 principal components based on 1000 bootstrap samples to be calculated on a standard laptop in 47 minutes, as opposed to approximately 4 days with standard methods.Comment: 25 pages, including 9 figures and link to R package. 2014-05-14 update: final formatting edits for journal submission, condensed figure

Lognormal Distributions and Geometric Averages of Positive Definite Matrices

This article gives a formal definition of a lognormal family of probability distributions on the set of symmetric positive definite (PD) matrices, seen as a matrix-variate extension of the univariate lognormal family of distributions. Two forms of this distribution are obtained as the large sample limiting distribution via the central limit theorem of two types of geometric averages of i.i.d. PD matrices: the log-Euclidean average and the canonical geometric average. These averages correspond to two different geometries imposed on the set of PD matrices. The limiting distributions of these averages are used to provide large-sample confidence regions for the corresponding population means. The methods are illustrated on a voxelwise analysis of diffusion tensor imaging data, permitting a comparison between the various average types from the point of view of their sampling variability.Comment: 28 pages, 8 figure

A survey on pairwise comparison matrices over abelian linearly ordered groups

In this paper, we provide a survey of our results about the pairwise comparison matrices defined over abelian linearly ordered groups