Small confidence sets for the mean of a spherically symmetric distribution

Summary.  Suppose that X has a k -variate spherically symmetric distribution with mean vector θ and identity covariance matrix. We present two spherical confidence sets for θ , both centred at a positive part Stein estimator     . In the first, we obtain the radius by approximating the upper α -point of the sampling distribution of     by the first two non-zero terms of its Taylor series about the origin. We can analyse some of the properties of this confidence set and see that it performs well in terms of coverage probability, volume and conditional behaviour. In the second method, we find the radius by using a parametric bootstrap procedure. Here, even greater improvement in terms of volume over the usual confidence set is possible, at the expense of having a less explicit radius function. A real data example is provided, and extensions to the unknown covariance matrix and elliptically symmetric cases are discussed.     

Structural learning with time-varying components: tracking the cross-section of financial time series

Summary.  When modelling multivariate financial data, the problem of structural learning is compounded by the fact that the covariance structure changes with time. Previous work has focused on modelling those changes by using multivariate stochastic volatility models. We present an alternative to these models that focuses instead on the latent graphical structure that is related to the precision matrix. We develop a graphical model for sequences of Gaussian random vectors when changes in the underlying graph occur at random times, and a new block of data is created with the addition or deletion of an edge. We show how a Bayesian hierarchical model incorporates both the uncertainty about that graph and the time variation thereof.     

Local Influence in Generalized Estimating Equations

Abstract.  We investigate the influence of subjects or observations on regression coefficients of generalized estimating equations (GEEs) using local influence. The GEE approach does not require the full multivariate distribution of the response vector. We extend the likelihood displacement to a quasi-likelihood displacement, and propose local influence diagnostics under several perturbation schemes. An illustrative example in GEEs is given and we compare the results using the local influence and deletion methods.     

Empirical Bayes confidence intervals shrinking both means and variances

Summary.  We construct empirical Bayes intervals for a large number p of means. The existing intervals in the literature assume that variances     are either equal or unequal but known. When the variances are unequal and unknown, the suggestion is typically to replace them by unbiased estimators     . However, when p is large, there would be advantage in 'borrowing strength' from each other. We derive double-shrinkage intervals for means on the basis of our empirical Bayes estimators that shrink both the means and the variances. Analytical and simulation studies and application to a real data set show that, compared with the t -intervals, our intervals have higher coverage probabilities while yielding shorter lengths on average. The double-shrinkage intervals are on average shorter than the intervals from shrinking the means alone and are always no longer than the intervals from shrinking the variances alone. Also, the intervals are explicitly defined and can be computed immediately.     

On generalized moment identity and its applications: a unified approach

In this paper, we obtain a generalized moment identity for the case when the distributions of the random variables are not necessarily purely discrete or absolutely continuous. The proposed identity is useful to find the generator which has been used for the approximation of distributions by Stein's method. Apparently, a new approach is discussed for the approximation of distributions by Stein's method. We bring the characterization based on the relationship between conditional expectations and hazard measure in our unified framework. As an application, a new lower bound to the mean-squared error is obtained and it is compared with Bayesian Cramer–Rao bound.     

Hybrid Samplers for Ill-Posed Inverse Problems

Abstract.  In the Bayesian approach to ill-posed inverse problems, regularization is imposed by specifying a prior distribution on the parameters of interest and Markov chain Monte Carlo samplers are used to extract information about its posterior distribution. The aim of this paper is to investigate the convergence properties of the random-scan random-walk Metropolis (RSM) algorithm for posterior distributions in ill-posed inverse problems. We provide an accessible set of sufficient conditions, in terms of the observational model and the prior, to ensure geometric ergodicity of RSM samplers of the posterior distribution. We illustrate how these conditions can be checked in an application to the inversion of oceanographic tracer data.     

Identities and inequalities for the expected values of random matrices and loewner's theory

We consider the identity associated with the expected values of the matrix-valued quadratic forms of two random matrices. As a special case the identity yields the matrix-valued inequality of the Schwarz type. The most important application is for Loewner's theory in the context of the operator-monotone function, and the inequalities based on the concavityconvexity of the matrix-valued functions are derived.     

Adaptive Posterior Mode Estimation of a Sparse Sequence for Model Selection

Abstract.  For the problem of estimating a sparse sequence of coefficients of a parametric or non-parametric generalized linear model, posterior mode estimation with a Subbotin( λ , ν ) prior achieves thresholding and therefore model selection when ν   ∈    [0,1] for a class of likelihood functions. The proposed estimator also offers a continuum between the (forward/backward) best subset estimator ( ν  =  0 ), its approximate convexification called lasso ( ν  =  1 ) and ridge regression ( ν  =  2 ). Rather than fixing ν , selecting the two hyperparameters λ and ν adds flexibility for a better fit, provided both are well selected from the data. Considering first the canonical Gaussian model, we generalize the Stein unbiased risk estimate, SURE( λ , ν ), to the situation where the thresholding function is not almost differentiable (i.e. ν    1 ). We then propose a more general selection of λ and ν by deriving an information criterion that can be employed for instance for the lasso or wavelet smoothing. We investigate some asymptotic properties in parametric and non-parametric settings. Simulations and applications to real data show excellent performance.     

SEMI-PARAMETRIC ANALYSIS OF COVARIANCE UNDER DEPENDENCE CONDITIONS WITHIN EACH GROUP

Consider the problem of covariance analysis based on regression models whose regression function is the sum of a linear and a non-parametric component. We propose a parametric and a non-parametric statistical test to compare the effects of the linear and non-parametric components, respectively, on the response variable in   L ≥ 2  groups. Serially correlated errors within each group are allowed. The first (second) test compares the differences between the estimates of the parametric (non-parametric) components of each group by means of a Mahalanobis  ( L ²)  distance. The asymptotic distribution of each statistic under the null hypothesis is obtained. A modest simulation study and an application to a real data set illustrate our methodology.     

The heat equation and Stein's identity: Connections,applications

This article presents two expectation identities and a series of applications. One of the identities uses the heat equation, and we show that in some families of distributions the identity characterizes the normal distribution. We also show that it is essentially equivalent to Stein's identity. The applications we have presented are of a broad range. They include exact formulas and bounds for moments, an improvement and a reversal of Jensen's inequality, linking unbiased estimation to elliptic partial differential equations, applications to decision theory and Bayesian statistics, and an application to counting matchings in graph theory. Some examples are also given.     

Analysis of familial aggregation in the presence of varying family sizes

Summary.  Family studies are frequently undertaken as the first step in the search for genetic and/or environmental determinants of disease. Significant familial aggregation of disease is suggestive of a genetic aetiology for the disease and may lead to more focused genetic analysis. Of course, it may also be due to shared environmental factors. Many methods have been proposed in the literature for the analysis of family studies. One model that is appealing for the simplicity of its computation and the conditional interpretation of its parameters is the quadratic exponential model. However, a limiting factor in its application is that it is not reproducible , meaning that all families must be of the same size. To increase the applicability of this model, we propose a hybrid approach in which analysis is based on the assumption of the quadratic exponential model for a selected family size and combines a missing data approach for smaller families with a marginalization approach for larger families. We apply our approach to a family study of colorectal cancer that was sponsored by the Cancer Genetics Network of the National Institutes of Health. We investigate the properties of our approach in simulation studies. Our approach applies more generally to clustered binary data.     

Bayesian Geostatistical Design

Abstract.  This paper describes the use of model-based geostatistics for choosing the set of sampling locations, collectively called the design, to be used in a geostatistical analysis. Two types of design situation are considered. These are retrospective design, which concerns the addition of sampling locations to, or deletion of locations from, an existing design, and prospective design, which consists of choosing positions for a new set of sampling locations. We propose a Bayesian design criterion which focuses on the goal of efficient spatial prediction whilst allowing for the fact that model parameter values are unknown. The results show that in this situation a wide range of inter-point distances should be included in the design, and the widely used regular design is often not the best choice.     

An invitation to quantum tomography

Summary.  We describe quantum tomography as an inverse statistical problem in which the quantum state of a light beam is the unknown parameter and the data are given by results of measurements performed on identical quantum systems. The state can be represented as an infinite dimensional density matrix or equivalently as a density on the plane called the Wigner function. We present consistency results for pattern function projection estimators and for sieve maximum likelihood estimators for both the density matrix of the quantum state and its Wigner function. We illustrate the performance of the estimators on simulated data. An EM algorithm is proposed for practical implementation. There remain many open problems, e.g. rates of convergence, adaptation and studying other estimators; a main purpose of the paper is to bring these to the attention of the statistical community.     

Efficiency of a Linear Combination of the Median and the Sample Mean: The Double Truncated Normal Distribution

Abstract.  We characterize all symmetric location models for which a linear combination of the median and the sample mean is an asymptotically efficient estimator of the location parameter. The resulting model can be understood as a symmetrized or double truncated normal distribution. A simple algorithm to estimate the parameters is given and an application is presented.     

Analysis of a long-term study of neurotic disorder, with insights into the process of non-response

Summary.  The paper studies the non-response process in a long-term study of neurotic dis-order by comparing the analysis based on the responses that were collected by the established practice of interviewing the subjects, at dates arranged in advance (appointments), with the analysis of the nearly complete set of responses that were collected by an extensive effort that involved attempts to interview without seeking a prior agreement. The method of multiple imputation is applied, and its properties are explored in a setting that is not perfectly suited for its application: a relatively small sample size, ordinal score outcomes and the likelihood that the outcomes are missing not at random.     

Men matter more: the social class identity of married British women, 1985–1991

Summary.  We investigate the class identity of married women as it relates to their own and their husband's class position. Whereas previous workers have attempted to test whether identity depends solely on the husband's position, not at all on the husband's position or equally on the husband's and wife's position, leaving out all intermediate cases, we estimate new diagonal reference models that quantify the relative weight of each partner's class position on their own class identity. In previous literature, it was also argued that women who work full time should be more likely to adopt a sharing model than other women and in some cases these different types of women were compared. We move beyond this simple dichotomy and more systematically formulate hypotheses about the conditions under which women attach more or less weight to their own class position and less or more weight respectively to that of their husbands. To test these hypotheses, we consider models where the weights are allowed to depend on characteristics of each partner and the couple. Using the British Social Attitudes Survey data for 1985–1991, we find that, when the husband's commitment to the labour force exceeds that of the wife, the husband's weight exceeds the wife's weight but, when the wife's commitment exceeds that of the husband's, the weights are approximately equal. We also find (unexpectedly) that women who hold higher positions than their husbands attach more weight to their husband's position than to their own position.     

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.     

Outlier Detection in a Circular Regression Model Using COVRATIO Statistic

In this article, we model the relationship between two circular variables using the circular regression models, to be called JS circular regression model, which was proposed by Jammalamadaka and Sarma (1993). The model has many interesting properties and is sensitive enough to detect the occurrence of outliers. We focus our attention on the problem of identifying outliers in this model. In particular, we extend the use of the COVRATIO statistic, which has been successfully used in the linear case for the same purpose, to the JS circular regression model via a row deletion approach. Through simulation studies, the cut-off points for the new procedure are obtained and its power of performance is investigated. It is found that the performance improves when the resulting residuals have small variance and when the sample size gets larger. An example of the application of the procedure is presented using a real dataset.     

Measuring agreement between two statistics with applications to age standardization

Summary.  We detail a general method for measuring agreement between two statistics. An application is two ratios of directly standardized rates which differ only by the choice of the standard. If the statistics have a high value for the coefficient of agreement then the expected squared difference between the statistics is small relative to the variance of the average of the two statistics, and inferences vary little by changing statistics. The estimation of a coefficient of agreement between two statistics is not straightforward because there is only one pair of observed values, each statistic calculated from the data. We introduce estimators of the coefficient of agreement for two statistics and discuss their use, especially as applied to functions of standardized rates.     

Non-parametric Estimation of the Death Rate in Branching Diffusions

We consider finite systems of diffusing particles in with branching and immigration. Branching of particles occurs at position dependent rate. Under ergodicity assumptions, we estimate the position-dependent branching rate based on the observation of the particle process over a time interval [0, t ]. Asymptotics are taken as t  → ∞. We introduce a kernel-type procedure and discuss its asymptotic properties with the help of the local time for the particle configuration. We compute the minimax rate of convergence in squared-error loss over a range of Hölder classes and show that our estimator is asymptotically optimal.