Covariance structure of wavelet coefficients: theory and models in a Bayesian perspective

We present theoretical results on the random wavelet coefficients covariance structure. We use simple properties of the coefficients to derive a recursive way to compute the within- and across-scale covariances. We point out a useful link between the algorithm proposed and the two-dimensional discrete wavelet transform. We then focus on Bayesian wavelet shrinkage for estimating a function from noisy data. A prior distribution is imposed on the coefficients of the unknown function. We show how our findings on the covariance structure make it possible to specify priors that take into account the full correlation between coefficients through a parsimonious number of hyperparameters. We use Markov chain Monte Carlo methods to estimate the parameters and illustrate our method on bench-mark simulated signals.     

A model selection approach for the identification of quantitative trait loci in experimental crosses

Summary. We consider the problem of identifying the genetic loci (called quantitative trait loci (QTLs)) contributing to variation in a quantitative trait, with data on an experimental cross. A large number of different statistical approaches to this problem have been described; most make use of multiple tests of hypotheses, and many consider models allowing only a single QTL. We feel that the problem is best viewed as one of model selection. We discuss the use of model selection ideas to identify QTLs in experimental crosses. We focus on a back-cross experiment, with strictly additive QTLs, and concentrate on identifying QTLs, considering the estimation of their effects and precise locations of secondary importance. We present the results of a simulation study to compare the performances of the more prominent methods.     

Generalized Score Test of Homogeneity Based on Correlated Random Effects Models

We present a family of tests based on correlated random effects models which provides a synthesis and a generalization of recent work on homogeneity testing. In these models each subject has a particular random effect, but the random effects between subjects are correlated. We derive the general form of the score statistic for testing that the random effects have a variance equal to 0. We apply this result to both parametric and semi-parametric models. In both cases we show that under certain conditions the score statistic has an asymptotic normal distribution. We consider several applications of this theory, including overdispersion, heterogeneity between groups, spatial correlations and genetic linkage.     

Non-parametric estimation of the long-range dependence exponent for Gaussian processes

We consider a class of long-range-dependent Gaussian processes defined in a semiparametric framework. We propose a new estimator of the long-range dependence parameter, based on the integration of the periodogram in two windows. We show that it is asymptotically Gaussian and calculate the rate of convergence. We optimise parameters defining the window function for the minimum mean-square-error criterion. In a Monte-Carlo study, we compare the proposed estimator with previously studied estimators.     

Finding an unknown number of multivariate outliers

Summary.  We use the forward search to provide robust Mahalanobis distances to detect the presence of outliers in a sample of multivariate normal data. Theoretical results on order statistics and on estimation in truncated samples provide the distribution of our test statistic. We also introduce several new robust distances with associated distributional results. Comparisons of our procedure with tests using other robust Mahalanobis distances show the good size and high power of our procedure. We also provide a unification of results on correction factors for estimation from truncated samples.     

A new ranked set sample estimator of variance

Summary. We develop an unbiased estimator of the variance of a population based on a ranked set sample. We show that this new estimator is better than estimating the variance based on a simple random sample and more efficient than the estimator based on a ranked set sample proposed by Stokes. Also, a test to determine the effectiveness of the judgment ordering process is proposed.     

Testing Supremacy or Inferiority of Multinomial Cell Probabilities with Application to Biting Preferences of Loggerhead Marine Turtles

We propose exact tests for uniform superiority or uniform inferiority of a multinomial cell probability. We extend the approach to testing supremacy or inferiority in multivariate settings. We also show how to perform superiority or inferiority tests in presence of covariates, and discuss the different interpretations of the proposed tests. The tests are illustrated on an original data set on biting behavior of loggerhead marine turtles.     

Choice between Semi-parametric Estimators of Markov and Non-Markov Multi-state Models from Coarsened Observations

Abstract.  We consider models based on multivariate counting processes, including multi-state models. These models are specified semi-parametrically by a set of functions and real parameters. We consider inference for these models based on coarsened observations, focusing on families of smooth estimators such as produced by penalized likelihood. An important issue is the choice of model structure, for instance, the choice between a Markov and some non-Markov models. We define in a general context the expected Kullback–Leibler criterion and we show that the likelihood-based cross-validation (LCV) is a nearly unbiased estimator of it. We give a general form of an approximate of the leave-one-out LCV. The approach is studied by simulations, and it is illustrated by estimating a Markov and two semi-Markov illness–death models with application on dementia using data of a large cohort study.     

The Burr XII System of densities: properties,regression model and applications

We introduce a new class of distributions called the Burr XII system of densities with two extra positive parameters. We provide a comprehensive treatment of some of its mathematical properties. We estimate the model parameters by maximum likelihood. We assess the performance of the maximum likelihood estimators in terms of biases and mean squared errors by means of a simulation study. We also introduce a new family of regression models based on this system of densities. The usefulness of the proposed models is illustrated by means of three real data sets.     

Consistent procedures for multiclass classification of discrete diffusion paths

The recent advent of modern technology has generated a large number of datasets which can be frequently modeled as functional data. This paper focuses on the problem of multiclass classification for stochastic diffusion paths. In this context we establish a closed formula for the optimal Bayes rule. We provide new statistical procedures which are built either on the plug-in principle or on the empirical risk minimization principle. We show the consistency of these procedures under mild conditions. We apply our methodologies to the parametric case and illustrate their accuracy with a simulation study through examples.     

Fractions of permutations. An application to Sudoku

We study how to simplify fractional factorial design generation by exploiting the a priori knowledge that can be derived from the orthogonality constraints that the fractional factorial design itself must satisfy. We work on Sudoku puzzles that can be considered as a special case of Latin squares in the class of gerechte designs. We prove that the generation of a Sudoku is equivalent to that of a fraction of a proper set of permutations. We analyse both the 4×4 and the 9×9 Sudoku types.     

Concurrent processing of heteroskedastic vector-valued mixture density models

We introduce a combined two-stage least-squares (2SLS)–expectation maximization (EM) algorithm for estimating vector-valued autoregressive conditional heteroskedasticity models with standardized errors generated by Gaussian mixtures. The procedure incorporates the identification of the parametric settings as well as the estimation of the model parameters. Our approach does not require a priori knowledge of the Gaussian densities. The parametric settings of the 2SLS_EM algorithm are determined by the genetic hybrid algorithm (GHA). We test the GHA-driven 2SLS_EM algorithm on some simulated cases and on international asset pricing data. The statistical properties of the estimated models and the derived mixture densities indicate good performance of the algorithm. We conduct tests on a massively parallel processor supercomputer to cope with situations involving numerous mixtures. We show that the algorithm is scalable.     

Targeted estimation of state occupation probabilities for the non-Markov illness-death model

We use semi-parametric efficiency theory to derive a class of estimators for the state occupation probabilities of the continuous-time irreversible illness-death model. We consider both the setting with and without additional baseline information available, where we impose no specific functional form on the intensity functions of the model. We show that any estimator in the class is asymptotically linear under suitable assumptions about the estimators of the intensity functions. In particular, the assumptions are weak enough to allow the use of data-adaptive methods, which is important for making the identifying assumption of coarsening at random plausible in realistic settings. We suggest a flexible method for estimating the transition intensity functions of the illness-death model based on penalized Poisson regression. We apply this method to estimate the nuisance parameters of an illness-death model in a simulation study and a real-world application.     

Estimation of the characteristics of a Lévy process

We consider a Lévy process that is e.g. used in finance to model stock price developments. We want to estimate the characteristics of that process, based on historical data where we assume that we have discrete, high frequency observations. We introduce a threshold estimation method and show consistency and in the case of finite activity asymptotic normality of these estimators.     

Mixture Model Analysis of Partially Rank‐Ordered Set Samples: Age Groups of Fish from Length‐Frequency Data

We present a novel methodology for estimating the parameters of a finite mixture model (FMM) based on partially rank‐ordered set (PROS) sampling and use it in a fishery application. A PROS sampling design first selects a simple random sample of fish and creates partially rank‐ordered judgement subsets by dividing units into subsets of prespecified sizes. The final measurements are then obtained from these partially ordered judgement subsets. The traditional expectation–maximization algorithm is not directly applicable for these observations. We propose a suitable expectation–maximization algorithm to estimate the parameters of the FMMs based on PROS samples. We also study the problem of classification of the PROS sample into the components of the FMM. We show that the maximum likelihood estimators based on PROS samples perform substantially better than their simple random sample counterparts even with small samples. The results are used to classify a fish population using the length‐frequency data.     

Monte carlo comparison of estimation methods for additive two-way tables

We considered the problem of estimating effects in the following linear model for data arranged in a two-way table: Response = Common effect + Row effect + Column effect + Residual. This work was occasioned by a project to analyse Federal Aviation Administration (FAA) data on daily temporal deviations from flight plans for commercial US flights, with rows and columns representing origin and destination airports, respectively. We conducted a large Monte Carlo study comparing the accuracy of three methods of estimation: classical least squares, median polish and least absolute deviations (LAD). The experiments included a wide spectrum of tables of different sizes and shapes, with different levels of non-linearity, noise variance, and percentages of empty cells and outliers. We based our comparison on the accuracy of the estimates and on computational speed. We identified factors that significantly affect accuracy and speed, and compared the methods based on their sensitivity to these factors. We concluded that there is no dominant method of estimation and identified conditions under which each method is most attractive.     

Random translation of discrete order statistics

We investigate the existence and uniqueness of a discrete parent distribution supported on the integers whose order statistics are related by a random translation. We also provide some examples using the constructive method that we propose.     

The Risk of a Mortality Catastrophe

We develop a continuous-time model for analyzing and valuing catastrophe mortality contingent claims based on stochastic modeling of the force of mortality. We derive parameter estimates from a 105-year time series of U.S. population mortality data using a simulated maximum likelihood approach based on a particle filter. Relying on the resulting parameters, we calculate loss profiles for a representative catastrophe mortality transaction and compare them to the “official” loss profiles that are provided by the issuers to investors and rating agencies. We find that although the loss profiles are subject to great uncertainties, the official figures fall significantly below the corresponding risk statistics based on our model. In particular, we find that the annualized incidence probability of a mortality catastrophe, defined as a 15% increase in aggregated mortality probabilities, is about 1.4%—compared to about 0.1% according to the official loss profiles.     

The Diffusion of Innovations: A Methodological Reappraisal

Studies of diffusion have traditionally relied on specific distributions, primarily the logistic, to characterize and estimate those processes. We argue that such an approach gives rise to serious problems of comparability and interpretation and may result in large biases in the estimates of the parameters of interest. We propose instead the Gini's expected mean difference as a measure of diffusion speed. We discuss its advantages over the traditional approach, present a nonparametric estimation procedure, and tackle with it the problems of truncated processes and of intergroup comparisons. We also elaborate on the use of the hazard rate and suggest various extensions. The diffusion of computed tomography scanners is presented as an illustration.     

Characterizations of asymptotic distributions of continuous-time Pólya processes

Abstract

We propose an elementary but effective approach to studying a general class of Poissonized tenable and balanced urns on two colors. We characterize the asymptotic behavior of the process via a partial differential equation that governs the process, coupled with the method of moments applied in a bootstrapped manner. We show that the limiting distribution of the process underlying the Bagchi-Pal urn is gamma. We also look into the tenable and balanced processes associated with randomized replacement matrix. Similar results carry over to the process, with minor modifications in the methods of proof, done mutatis mutandis.