multiple and conditional deletion diagnostics for general linear models

In this paper we develop multiple case deletion statistics for the general linear model so that a residual vector and a leverage matrix are identified which have roles analogous to residuals and leverage for ordinary least squares models. We extend the notion of the conditional deletion diagnostic to general linear models. The residuals, leverage and deletion diagnostics are illustrated with data modelled by a linear growth curve.     

A General Approach To Nonparametric Histogram Estimation

We note that some classical functional estimation problems may be reduced to a general unique framework and study an estimator within this general framework that reduces to the classical histogram type estimators in various examples presented. The convergence in probability and the almost complete convergence of this general estimator are studied obtaining convergence conditions which reduce to the classical conditions in each case. Finally, this general framework provides conditions for the convergence of the finite dimensional distributions of the associated empirical process.     

Estimable functions in the nonlinear model

When the method of least squares is used to estimate the parameters in a general model and the generated system of normal equations is linearly dependent, the estimate of the vector of parameters which satisfies the criterion is not unique. However, there exist certain functions of the estimated vector of parameters which are invariant to the least squares solution obtained from the normal equations. We define those invariant functions to be estimable, and present a technique to determine the functions of the parameters which are estimable for the general model. The method results in solving either a linear first order partial differential equation or a system of linear first order partial differential equations corresponding, respectively, to a single or multiple dependency between columns of the Jacobian matrix of the mean of the model. The usual results concerning estimability for linear models are a special case of the general results developed.     

Linear Models with Doubly Exchangeable Distributed Errors

We study the general linear model (GLM) with doubly exchangeable distributed error for m observed random variables. The doubly exchangeable general linear model (DEGLM) arises when the m-dimensional error vectors are “doubly exchangeable,” jointly normally distributed, which is a much weaker assumption than the independent and identically distributed error vectors as in the case of GLM or classical GLM (CGLM). We estimate the parameters in the model and also find their distributions. We show that the tests of intercept and slope are possible in DEGLM as a particular case using parametric bootstrap as well as multivariate Satterthwaite approximation.     

Predicting hospital census using time series regression methods

We first describe the time series modeling problem in a general way. Then some specific assumptions and observations which are pertinent to the application of these models are made. We next propose a specific approach to the modeling problem, one which yields efficient, easily calculated estimators of all parameters (under the stated assumptions). Finally, the technique is applied to the problem of modeling the census of a particular hospital.     

A model for estimating point sources in spatial data

We introduce a point source model which may be useful for estimating point sources in spatial data. It may also be useful for modelling general spatial data, and providing a simple explanatory model for some data, whilst in other cases it may give a parsimonious representation. The model assumes that there are point sources (or sinks), usually at unknown positions, and that the mean value at a site depends on the distance from these sources. We discuss the general form of the model, and some methods for estimating the sources and the regression parameters. We demonstrate the methodology using a simulation study, and apply the model to two real data sets. Some possibilities for further research are outlined.     

Star-shaped distributions and their generalizations

Elliptically contoured distributions can be considered to be the distributions for which the contours of the density functions are proportional ellipsoids. We generalize elliptically contoured densities to “star-shaped distributions” with concentric star-shaped contours and show that many results in the former case continue to hold in the more general case. We develop a general theory in the framework of abstract group invariance so that the results can be applied to other cases as well, especially those involving random matrices.     

Radiocarbon dating with temporal order constraints

Bayesian methods are now widely used for analysing radiocarbon dates. We find that the non-informative priors in use in the literature generate a bias towards wider date ranges which does not in general reflect substantial prior knowledge. We recommend using a prior in which the distribution of the difference between the earliest and latest dates has a uniform distribution. We show how such priors are derived from a simple physical model of the deposition and observation process. We illustrate this in a case-study, examining the effect that various priors have on the reconstructed dates. Bayes factors are used to help to decide model choice problems.     

Stochastic Volatility Models Based on OU-Gamma Time Change: Theory and Estimation

We consider stochastic volatility models that are defined by an Ornstein–Uhlenbeck (OU)-Gamma time change. These models are most suitable for modeling financial time series and follow the general framework of the popular non-Gaussian OU models of Barndorff-Nielsen and Shephard. One current problem of these otherwise attractive nontrivial models is, in general, the unavailability of a tractable likelihood-based statistical analysis for the returns of financial assets, which requires the ability to sample from a nontrivial joint distribution. We show that an OU process driven by an infinite activity Gamma process, which is an OU-Gamma process, exhibits unique features, which allows one to explicitly describe and exactly sample from relevant joint distributions. This is a consequence of the OU structure and the calculus of Gamma and Dirichlet processes. We develop a particle marginal Metropolis–Hastings algorithm for this type of continuous-time stochastic volatility models and check its performance using simulated data. For illustration we finally fit the model to S&P500 index data.     

DUAL‐OBJECTIVE OPTIMAL MIXTURE DESIGNS

Mixture experiments are widely used in many industries and particularly in the manufacture of consumer products. Almost all work to date assumes a single study objective, which is unrealistic. Researchers may want to estimate model parameters and make predictions or extrapolations at the same time. We discuss design issues for determining the optimal proportions of the mixture components when there are two or more objectives in the study and there is a large sample size. We present a general methodology for constructing two types of dual‐objective optimal design for mixture experiments and discuss the general applicability of the design strategy to more complicated types of mixture design problems, including mixture experiments.     

Explicit formulas for the cumulants and the vector-valued odd moments of the multivariate linearly skewed elliptical distributions

In this letter explicit expressions are derived for the cumulants and the vector-valued odd moments of the multivariate linearly skewed elliptical family of distributions. The general calculations of such moments are described by multivariate integrals which complicate the calculations. We show how such multivariate computations can be projected into a univariate framework, which extremely simplifies the computations.     

Adaptive Bayesian Procedures Using Random Series Priors

We consider a general class of prior distributions for nonparametric Bayesian estimation which uses finite random series with a random number of terms. A prior is constructed through distributions on the number of basis functions and the associated coefficients. We derive a general result on adaptive posterior contraction rates for all smoothness levels of the target function in the true model by constructing an appropriate ‘sieve’ and applying the general theory of posterior contraction rates. We apply this general result on several statistical problems such as density estimation, various nonparametric regressions, classification, spectral density estimation and functional regression. The prior can be viewed as an alternative to the commonly used Gaussian process prior, but properties of the posterior distribution can be analysed by relatively simpler techniques. An interesting approximation property of B‐spline basis expansion established in this paper allows a canonical choice of prior on coefficients in a random series and allows a simple computational approach without using Markov chain Monte Carlo methods. A simulation study is conducted to show that the accuracy of the Bayesian estimators based on the random series prior and the Gaussian process prior are comparable. We apply the method on Tecator data using functional regression models.     

Parametric inference in a perturbed gamma degradation process

Abstract

We consider a degradation model which is the sum of two independent processes: an homogeneous gamma process and a Brownian motion. This model is called perturbed gamma process. Based on independent copies of the perturbed gamma process observed at irregular instants we propose to estimate the unknown parameters of the model using the moment method. Some general conditions allow to derive the asymptotic behavior of the estimators. We also show that these general conditions are fulfilled for some specific observation schemes. Finally, we illustrate our method by a numerical study and an application to a real data set.     

Asymptotical improvement of maximum likelihood estimators on Kullback–Leibler loss

We discuss the general form of a first-order correction to the maximum likelihood estimator which is expressed in terms of the gradient of a function, which could for example be the logarithm of a prior density function. In terms of Kullback–Leibler divergence, the correction gives an asymptotic improvement over maximum likelihood under rather general conditions. The theory is illustrated for Bayes estimators with conjugate priors. The optimal choice of hyper-parameter to improve the maximum likelihood estimator is discussed. The results based on Kullback–Leibler risk are extended to a wide class of risk functions.     

Non parametric mixture priors based on an exponential random scheme

We propose a general procedure for constructing nonparametric priors for Bayesian inference. Under very general assumptions, the proposed prior selects absolutely continuous distribution functions, hence it can be useful with continuous data. We use the notion ofFeller-type approximation, with a random scheme based on the natural exponential family, in order to construct a large class of distribution functions. We show how one can assign a probability to such a class and discuss the main properties of the proposed prior, namedFeller prior. Feller priors are related to mixture models with unknown number of components or, more generally, to mixtures with unknown weight distribution. Two illustrations relative to the estimation of a density and of a mixing distribution are carried out with respect to well known data-set in order to evaluate the performance of our procedure. Computations are performed using a modified version of an MCMC algorithm which is briefly described.     

Improved estimation of the stable laws

Fitting general stable laws to data by maximum likelihood is important but difficult. This is why much research has considered alternative procedures based on empirical characteristic functions. Two problems then are how many values of the characteristic function to select, and how to position them. We provide recommendations for both of these topics. We propose an arithmetic spacing of transform variables, coupled with a recommendation for the location of the variables. It is shown that arithmetic spacing, which is far simpler to implement, closely approximates optimum spacing. The new methods that result are compared in simulation studies with existing methods, including maximum-likelihood. The main conclusion is that arithmetic spacing of the values of the characteristic function, coupled with appropriately limiting the range for these values, improves the overall performance of the regression-type method of Koutrouvelis, which is the standard procedure for estimating general stable law parameters.     

Estimation of noncentrality parameters

We propose some estimators of noncentrality parameters which improve upon usual unbiased estimators under quadratic loss. The distributions we consider are the noncentral chi-square and the noncentral F. However, we give more general results for the family of elliptically contoured distributions and propose a robust dominating estimator.     

Nonparametric estimation of the mean function of a stochastic process with missing observations

In an attempt to identify similarities between methods for estimating a mean function with different types of response or observation processes, we explore a general theoretical framework for nonparametric estimation of the mean function of a response process subject to incomplete observations. Special cases of the response process include quantitative responses and discrete state processes such as survival processes, counting processes and alternating binary processes. The incomplete data are assumed to arise from a general response-independent observation process, which includes right- censoring, interval censoring, periodic observation, and mixtures of these as special cases. We explore two criteria for defining nonparametric estimators, one based on the sample mean of available data and the other inspired by the construction of Kaplan-Meier (or product-limit) estimator [J. Am. Statist. Assoc. 53 (1958) 457] for right-censored survival data. We show that under regularity conditions the estimated mean functions resulting from both criteria are consistent and converge weakly to Gaussian processes, and provide consistent estimators of their covariance functions. We then evaluate these general criteria for specific responses and observation processes, and show how they lead to familiar estimators for some response and observation processes and new estimators for others. We illustrate the latter with data from an recently completed AIDS clinical trial.     

Second-order least squares estimation of censored regression models

This paper proposes the second-order least squares estimation, which is an extension of the ordinary least squares method, for censored regression models where the error term has a general parametric distribution (not necessarily normal). The strong consistency and asymptotic normality of the estimator are derived under fairly general regularity conditions. We also propose a computationally simpler estimator which is consistent and asymptotically normal under the same regularity conditions. Finite sample behavior of the proposed estimators under both correctly and misspecified models are investigated through Monte Carlo simulations. The simulation results show that the proposed estimator using optimal weighting matrix performs very similar to the maximum likelihood estimator, and the estimator with the identity weight is more robust against the misspecification.     

Representations of the covariance matrix for robustness in the gauss-markov model

This paper considers some extensions of the results of Rao and Rao and Mitra. They gave a table of general representations of the covariance matrix in terms of the given design matrix, under which various statistical procedures in the least squares theory based on the simple Gauss-Markov model with the spherical covariance matrix are also valid under the general Gauss-Markov model. We shall give extended tables adding some more results relating to robustness, especially in connection with the estimation and testing of hypotheses on linear parametric functions