Estimation in the multipath change point problem for correlated data

In many experiments, several measurements on the same variable are taken over time, a geographic region, or some other index set. It is often of interest to know if there has been a change over the index set in the parameters of the distribution of the variable. Frequently, the data consist of a sequence of correlated random variables, and there may also be several experimental units under observation, each providing a sequence of data. A problem in ascertaining the boundaries between the layers in geological sedimentary beds is used to introduce the model and then to illustrate the proposed methodology. It is assumed that, conditional on the change point, the data from each sequence arise from an autoregressive process that undergoes a change in one or more of its parameters. Unconditionally, the model then becomes a mixture of nonstationary autoregressive processes. Maximum-likelihood methods are used, and results of simulations to evaluate the performance of these estimators under practical conditions are given.     

A New Method for Statistical Multidimensional Unfolding

ABSTRACT

Consider the problem of estimating the positions of a set of targets in a multidimensional Euclidean space from distances reported by a number of observers when the observers do not know their own positions in the space. Each observer reports the distance from the observer to each target plus a random error. This statistical problem is the basic model for the various forms of what is called multidimensional unfolding in the psychometric literature. Multidimensional unfolding methodology as developed in the field of cognitive psychology is basically a statistical estimation problem where the data structure is a set of measures that are monotonic functions of Euclidean distances between a number of observers and targets in a multidimensional space. The new method presented in this article deals with estimating the target locations and the observer positions when the observations are functions of the squared distances between observers and targets observed with an additive random error in a two-dimensional space. The method provides robust estimates of the target locations in a multidimensional space for the parametric structure of the data generating model presented in the article. The method also yields estimates of the orientation of the coordinate system and the mean and variances of the observer locations. The mean and the variances are not estimated by standard unfolding methods which yield targets maps that are invariant to a rotation of the coordinate system. The data is transformed so that the nonlinearity due to the squared observer locations is removed. The sampling properties of the estimates are derived from the asymptotic variances of the additive errors of a maximum likelihood factor analysis of the sample covariance matrix of the transformed data augmented with bootstrapping. The robustness of the new method is tested using artificial data. The method is applied to a 2001 survey data set from Turkey to provide a real data example.     

Expansion estimation by Bayes rules

In the problem of estimating a location parameter in any symmetric unimodal location parameter model, we demonstrate that Bayes rules with respect to squared error loss can be expanders for some priors that belong to the family of all symmetric priors. That generalizes the results obtained by DasGupta and Rubin for the one dimensional case. We also consider symmetric priors which either have an appropriate point mass at 0 or are unimodal, and prove that under the latter condition all Bayes rules are shrinkers. Results of such nature are important, for example, in wavelet based function estimation and data denoising, where shrinkage of wavelet coefficients is associated with smoothing the data. We illustrate the results using FIAT stock market data.     

Proportional hazards model for competing risks data with missing cause of failure

We consider the semiparametric proportional hazards model for the cause-specific hazard function in analysis of competing risks data with missing cause of failure. The inverse probability weighted equation and augmented inverse probability weighted equation are proposed for estimating the regression parameters in the model, and their theoretical properties are established for inference. Simulation studies demonstrate that the augmented inverse probability weighted estimator is doubly robust and the proposed method is appropriate for practical use. The simulations also compare the proposed estimators with the multiple imputation estimator of Lu and Tsiatis (2001). The application of the proposed method is illustrated using data from a bone marrow transplant study.     

Long memory and changepoint models: a spectral classification procedure

Time series within fields such as finance and economics are often modelled using long memory processes. Alternative studies on the same data can suggest that series may actually contain a ‘changepoint’ (a point within the time series where the data generating process has changed). These models have been shown to have elements of similarity, such as within their spectrum. Without prior knowledge this leads to an ambiguity between these two models, meaning it is difficult to assess which model is most appropriate. We demonstrate that considering this problem in a time-varying environment using the time-varying spectrum removes this ambiguity. Using the wavelet spectrum, we then use a classification approach to determine the most appropriate model (long memory or changepoint). Simulation results are presented across a number of models followed by an application to stock cross-correlations and US inflation. The results indicate that the proposed classification outperforms an existing hypothesis testing approach on a number of models and performs comparatively across others.     

An additive–multiplicative mean model for panel count data with dependent observation and dropout processes

This paper discusses regression analysis of panel count data with dependent observation and dropout processes. For the problem, a general mean model is presented that can allow both additive and multiplicative effects of covariates on the underlying point process. In addition, the proportional rates model and the accelerated failure time model are employed to describe possible covariate effects on the observation process and the dropout or follow‐up process, respectively. For estimation of regression parameters, some estimating equation‐based procedures are developed and the asymptotic properties of the proposed estimators are established. In addition, a resampling approach is proposed for estimating a covariance matrix of the proposed estimator and a model checking procedure is also provided. Results from an extensive simulation study indicate that the proposed methodology works well for practical situations, and it is applied to a motivating set of real data.     

Sequential tuning of complex computer models

We propose a method that uses a sequential design instead of a space filling design for estimating tuning parameters of a complex computer model. The goal is to bring the computer model output closer to the real system output. The method fits separate Gaussian process (GP) models to the available data from the physical experiment and the computer experiment and minimizes the discrepancy between the predictions from the GP models to obtain estimates of the tuning parameters. A criterion based on the discrepancy between the predictions from the two GP models and the standard error of prediction for the computer experiment output is then used to obtain a design point for the next run of the computer experiment. The tuning parameters are re-estimated using the augmented data set. The steps are repeated until the budget for the computer experiment data is exhausted. Simulation studies show that the proposed method performs better in bringing a computer model closer to the real system than methods that use a space filling design.     

Accounting for Uncertainty in Heteroscedasticity in Nonlinear Regression

Toxicologists and pharmacologists often describe toxicity of a chemical using parameters of a nonlinear regression model. Thus estimation of parameters of a nonlinear regression model is an important problem. The estimates of the parameters and their uncertainty estimates depend upon the underlying error variance structure in the model. Typically, a priori the researcher would not know if the error variances are homoscedastic (i.e., constant across dose) or if they are heteroscedastic (i.e., the variance is a function of dose). Motivated by this concern, in this paper we introduce an estimation procedure based on preliminary test which selects an appropriate estimation procedure accounting for the underlying error variance structure. Since outliers and influential observations are common in toxicological data, the proposed methodology uses M-estimators. The asymptotic properties of the preliminary test estimator are investigated; in particular its asymptotic covariance matrix is derived. The performance of the proposed estimator is compared with several standard estimators using simulation studies. The proposed methodology is also illustrated using a data set obtained from the National Toxicology Program.     

Discriminating between the generalized Rayleigh and Weibull distributions

Generalized Rayleigh (GR) and Weibull (WE) distributions are used quite effectively for analysing skewed lifetime data. In this paper, we consider the problem of selecting either GR or WE distribution as a more appropriate fitting model for a given data set. We use the ratio of maximized likelihoods (RML) for discriminating between the two distributions. The asymptotic and simulated distributions of the logarithm of the RML are applied to determine the probability of correctly selecting between these two families of distributions. It is examined numerically that the asymptotic results work quite well even for small sample sizes. A real data set involving the annual rainfall recorded at Los Angeles Civic Center during 25 years is analysed to illustrate the procedures developed here.     

Augmented-limited regression models with an application to the study of the risk perceived using continuous scales

Studies of risk perceived using continuous scales of [0,100] were recently introduced in psychometrics, which can be transformed to the unit interval, but the presence of zeros or ones are commonly observed. Motivated by this, we introduce a full inferential set of tools that allows for augmented and limited data modeling. We considered parameter estimation, residual analysis, influence diagnostic and model selection for zero-and/or-one augmented beta rectangular (ZOABR) regression models and their particular nested models, which is based on a new parameterization of the beta rectangular distribution. Different from other alternatives, we performed maximum-likelihood estimation using a combination of the EM algorithm (for the continuous part) and Fisher scoring algorithm (for the discrete part). Also, we perform an additional step, by considering other link functions, besides the usual logistic link, for modeling the response mean. By considering randomized quantile residuals, (local) influence diagnostics and model selection tools, we identified that the ZOABR regression model is the best one. We also conducted extensive simulations studies, which indicate that all developed tools work properly. Finally, we discuss the use of this type of models to treat psychometric data. It is worthwhile to mention that applications of the developed methods go beyond to Psychometric data. Indeed, they can be useful when the response variable in bounded, including or not the respective limits.     

On the efficiency of some non-orthogonal split-plot×split-block designs with control treatments

Many split-plot×split-block (SPSB) type experiments used in agriculture, biochemistry or plant protection are designed to study new crop plant cultivars or chemical agents. In these experiments it is usually very important to compare test treatments with the so-called control treatments. It happens yet that experimental material is limited and it does not allow using a complete (orthogonal) SPSB design. In the paper we propose a non-orthogonal SPSB design for consideration. Two cases of the design are presented here, i.e. when its incompleteness is connected with a crossed treatment structure only or with a nested treatment structure only. It is assumed the factors' levels connected with the incompleteness of the design are split into two groups: a set of test treatments and a set of control treatments. The method of constructions involves applying augmented block designs for some factors' levels. In a modelling data obtained from such experiments the structure of experimental material and appropriate randomization scheme of the different kinds of units before they enter the experiment are taken into account. With respect to the analysis of the obtained randomization model the approach typical to the multistratum experiments with orthogonal block structure is adapted. The proposed statistical analysis of linear model obtained includes estimation of parameters, testing general and particular hypotheses defined by the (basic) treatment contrasts with special reference to the notion of general balance.     

The variability of rainfall acidity revisited

Six years of rainfall-event pH measurements from the nine-station MAP3S/PCN monitoring network in the eastern United States were analyzed. The initial objective was an attempted validation of the model developed by Eynon and Switzer (1983, Canad. J. Statist. 11, 11–24) on this independent data set. Because some features of the structure presumed in the model are not evident in this data set, the underlying structure of the data was then explored in some detail. Both aspects of the investigation confirmed that the identification of an appropriate statistical model for such data is a difficult undertaking; anticipated structure may not be evident, and the data corresponding to specific stations or years may exhibit anomalous behavior.     

Model selection for probabilistic clustering using cross-validated likelihood

Cross-validated likelihood is investigated as a tool for automatically determining the appropriate number of components (given the data) in finite mixture modeling, particularly in the context of model-based probabilistic clustering. The conceptual framework for the cross-validation approach to model selection is straightforward in the sense that models are judged directly on their estimated out-of-sample predictive performance. The cross-validation approach, as well as penalized likelihood and McLachlan's bootstrap method, are applied to two data sets and the results from all three methods are in close agreement. The second data set involves a well-known clustering problem from the atmospheric science literature using historical records of upper atmosphere geopotential height in the Northern hemisphere. Cross-validated likelihood provides an interpretable and objective solution to the atmospheric clustering problem. The clusters found are in agreement with prior analyses of the same data based on non-probabilistic clustering techniques.     

A Bayesian approach to inference about a change point model with application to DNA copy number experimental data

In this paper, we study the change-point inference problem motivated by the genomic data that were collected for the purpose of monitoring DNA copy number changes. DNA copy number changes or copy number variations (CNVs) correspond to chromosomal aberrations and signify abnormality of a cell. Cancer development or other related diseases are usually relevant to DNA copy number changes on the genome. There are inherited random noises in such data, therefore, there is a need to employ an appropriate statistical model for identifying statistically significant DNA copy number changes. This type of statistical inference is evidently crucial in cancer researches, clinical diagnostic applications, and other related genomic researches. For the high-throughput genomic data resulting from DNA copy number experiments, a mean and variance change point model (MVCM) for detecting the CNVs is appropriate. We propose to use a Bayesian approach to study the MVCM for the cases of one change and propose to use a sliding window to search for all CNVs on a given chromosome. We carry out simulation studies to evaluate the estimate of the locus of the DNA copy number change using the derived posterior probability. These simulation results show that the approach is suitable for identifying copy number changes. The approach is also illustrated on several chromosomes from nine fibroblast cancer cell line data (array-based comparative genomic hybridization data). All DNA copy number aberrations that have been identified and verified by karyotyping are detected by our approach on these cell lines.     

Empirical Analysis of Interval-Censored Failure Time Data with Linear Transformation Models

Interval-censored data naturally arise in many studies. For their regression analysis, many approaches have been proposed under various models and for most of them, the inference is carried out based on the asymptotic normality. In particular, Zhang et al. (2005) discussed the procedure under the linear transformation model. It is well-known that the symmetric property implied by the normal distribution may not be appropriate sometimes. Also the method could underestimate the variance of estimated parameters. This paper proposes an empirical likelihood-based procedure for the problem. Simulation and the analysis of a real data set are conducted to assess the performance of the procedure.     

Bayesian estimation and case influence diagnostics for the zero-inflated negative binomial regression model

In recent years, there has been considerable interest in regression models based on zero-inflated distributions. These models are commonly encountered in many disciplines, such as medicine, public health, and environmental sciences, among others. The zero-inflated Poisson (ZIP) model has been typically considered for these types of problems. However, the ZIP model can fail if the non-zero counts are overdispersed in relation to the Poisson distribution, hence the zero-inflated negative binomial (ZINB) model may be more appropriate. In this paper, we present a Bayesian approach for fitting the ZINB regression model. This model considers that an observed zero may come from a point mass distribution at zero or from the negative binomial model. The likelihood function is utilized to compute not only some Bayesian model selection measures, but also to develop Bayesian case-deletion influence diagnostics based on q-divergence measures. The approach can be easily implemented using standard Bayesian software, such as WinBUGS. The performance of the proposed method is evaluated with a simulation study. Further, a real data set is analyzed, where we show that ZINB regression models seems to fit the data better than the Poisson counterpart.     

Modelling spatial intensity for replicated inhomogeneous point patterns in brain imaging

Summary.  Pharmacological experiments in brain microscopy study patterns of cellular activ- ation in response to psychotropic drugs for connected neuroanatomic regions. A typical ex- perimental design produces replicated point patterns having highly complex spatial variability. Modelling this variability hierarchically can enhance the inference for comparing treatments. We propose a semiparametric formulation that combines the robustness of a nonparametric kernel method with the efficiency of likelihood-based parameter estimation. In the convenient framework of a generalized linear mixed model, we decompose pattern variation by kriging the intensities of a hierarchically heterogeneous spatial point process. This approximation entails discretizing the inhomogeneous Poisson likelihood by Voronoi tiling of augmented point patterns. The resulting intensity-weighted log-linear model accommodates spatial smoothing through a reduced rank penalized linear spline. To correct for anatomic distortion between subjects, we interpolate point locations via an isomorphic mapping so that smoothing occurs relative to common neuroanatomical atlas co-ordinates. We propose a criterion for choosing the degree and spatial locale of smoothing based on truncating the ordered set of smoothing covariates to minimize residual extra-dispersion. Additional spatial covariates, experimental design factors, hierarchical random effects and intensity functions are readily accommodated in the linear predictor, enabling comprehensive analyses of the salient properties underlying replicated point patterns. We illustrate our method through application to data from a novel study of drug effects on neuronal activation patterns in the brain of rats.     

Consistency of the structured total least squares estimator in a multivariate errors-in-variables model

The structured total least squares estimator, defined via a constrained optimization problem, is a generalization of the total least squares estimator when the data matrix and the applied correction satisfy given structural constraints. In the paper, an affine structure with additional assumptions is considered. In particular, Toeplitz and Hankel structured, noise free and unstructured blocks are allowed simultaneously in the augmented data matrix. An equivalent optimization problem is derived that has as decision variables only the estimated parameters. The cost function of the equivalent problem is used to prove consistency of the structured total least squares estimator. The results for the general affine structured multivariate model are illustrated by examples of special models. Modification of the results for block-Hankel/Toeplitz structures is also given. As a by-product of the analysis of the cost function, an iterative algorithm for the computation of the structured total least squares estimator is proposed.     

LOCATION OF THE OPTIMAL SAMPLING POINT FOR THE QUALITY ASSESSMENT OF CONTINUOUS STREAMS

The paper makes an appraisal of the most appropriate sampling point for situations where a single sample must be used to estimate the mean flow of a continuous stream during a set time interval. Taking ‘optimal’ to mean the point at which the estimation error variance is minimised, optimal sampling locations are obtained for constant, linear and exponential flow rates when the process variogram is assumed linear or exponential. Numerical results illustrate the significance of failing to sample at the optimal point.     

Adaptive Jonckheere-type tests for ordered alternatives

Testing against ordered alternatives in the c -sample location problem plays an important role in statistical practice. The parametric test proposed by Barlow et al .-in the following, called the 'B-test'-is an appropriate test under the model of normality. For non-normal data, however, there are rank tests which have higher power than the B-test, such as the Jonckheere test or so-called Jonckheere-type tests introduced and studied by Buning and Kossler. However, we usually have no information about the underlying distribution. Thus, an adaptive test should be applied which takes into account the given data set. Two versions of such an adaptive test are proposed, which are based on the concept introduced by Hogg in 1974. These adaptive tests are compared with each of the single Jonckheere-type tests in the adaptive scheme and also with the B-test. It is shown via Monte Carlo simulation that the adaptive tests behave well over a broad class of symmetric distributions with short, medium and long tails, as well as for asymmetric distributions.