Estimation for binary models generated by Gaussian autoregressive processes

Regression-type and partial likelihood models are presented for binary data obtained by clipping a Gaussian autoregressive process. Five methods for estimating parameters of the model are proposed and compared via a simulation study. A real data analysis is also presented.     

Bayesian Analysis of Ordinal Survey Data Using the Dirichlet Process to Account for Respondent Personality Traits

This article presents a Bayesian latent variable model used to analyze ordinal response survey data by taking into account the characteristics of respondents. The ordinal response data are viewed as multivariate responses arising from continuous latent variables with known cut-points. Each respondent is characterized by two parameters that have a Dirichlet process as their joint prior distribution. The proposed mechanism adjusts for classes of personalities. The model is applied to student survey data in course evaluations. Goodness-of-fit (GoF) procedures are developed for assessing the validity of the model. The proposed GoF procedures are simple, intuitive, and do not seem to be a part of current Bayesian practice.     

Test-Based Interval Estimation Under the Accelerated Failure Time Model

Some new results of a distance—based (DB) model for prediction with mixed variables are presented and discussed. This model can be thought of as a linear model where predictor variables for a response Y are obtained from the observed ones via classic multidimensional scaling. A coefficient is introduced in order to choose the most predictive dimensions, providing a solution to the problem of small variances and a very large number n of observations (the dimensionality increases as n). The problem of missing data is explored and a DB solution is proposed. It is shown that this approach can be regarded as a kind of ridge regression when the usual Euclidean distance is used.     

A Bayesian nonparametric estimator based on left censored data

Summary This paper introduces a Bayesian nonparametric estimator for an unknown distribution function based on left censored observations. Hjort (1990)/Lo (1993) introduced Bayesian nonparametric estimators derived from beta/beta-neutral processes which allow for right censoring. These processes are taken as priors from the class ofneutral to the right processes (Doksum, 1974). The Kaplan-Meier nonparametric product limit estimator can be obtained from these Bayesian nonparametric estimators in the limiting case of a vague prior. The present paper introduces what can be seen as the correspondingleft beta/beta-neutral process prior which allow for left censoring. The Bayesian nonparametyric estimator is obtained as in the corresponding product limit estimator based on left censored data.     

Curve prediction and clustering with mixtures of Gaussian process functional regression models

Shi, Wang, Murray-Smith and Titterington (Biometrics 63:714–723, 2007) proposed a Gaussian process functional regression (GPFR) model to model functional response curves with a set of functional covariates. Two main problems are addressed by their method: modelling nonlinear and nonparametric regression relationship and modelling covariance structure and mean structure simultaneously. The method gives very good results for curve fitting and prediction but side-steps the problem of heterogeneity. In this paper we present a new method for modelling functional data with ‘spatially’ indexed data, i.e., the heterogeneity is dependent on factors such as region and individual patient’s information. For data collected from different sources, we assume that the data corresponding to each curve (or batch) follows a Gaussian process functional regression model as a lower-level model, and introduce an allocation model for the latent indicator variables as a higher-level model. This higher-level model is dependent on the information related to each batch. This method takes advantage of both GPFR and mixture models and therefore improves the accuracy of predictions. The mixture model has also been used for curve clustering, but focusing on the problem of clustering functional relationships between response curve and covariates, i.e. the clustering is based on the surface shape of the functional response against the set of functional covariates. The model is examined on simulated data and real data.     

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.     

Nonparametric prediction of spatial multivariate data

This paper investigates a nonparametric spatial predictor of a stationary multidimensional spatial process observed over a rectangular domain. The proposed predictor depends on two kernels in order to control both the distance between observations and that between spatial locations. The uniform almost complete consistency and the asymptotic normality of the kernel predictor are obtained when the sample considered is an alpha-mixing sequence. Numerical studies were carried out in order to illustrate the behaviour of our methodology both for simulated data and for an environmental data set.     

Optimal Maintenance Time for Repairable Systems Under Two Types of Failures

Determination of preventive maintenance is an important issue for systems under degradation. A typical maintenance policy calls for complete preventive repair actions at pre-scheduled times and minimal repair actions whenever a failure occurs. Under minimal repair, failures are modeled according to a non homogeneous Poisson process. A perfect preventive maintenance restores the system to the as good as new condition. The motivation for this article was a maintenance data set related to power switch disconnectors. Two different types of failures could be observed for these systems according to their causes. The major difference between these types of failures is their costs. Assuming that the system will be in operation for an infinite time, we find the expected cost per unit of time for each preventive maintenance policy and hence obtain the optimal strategy as a function of the processes intensities. Assuming a parametrical form for the intensity function, large sample estimates for the optimal maintenance check points are obtained and discussed.     

ESTIMATING A LOGIT MODEL WITH RANDOMIZED DATA: THE CASE OF COCAINE USE

In his influential book, Maddala (1983) suggests combining randomized response survey data with other personal information to estimate logit models predicting immoral, unpopular, or unlawful behaviour. This study is one of the first to implement this technique using real data. Models of college students' recent cocaine use are estimated with academic performance and socio-economic characteristics as determinants. Parameter estimates obtained from randomized response surveys are compared to those obtained using conventional, direct question surveys. The results indicate that randomized response estimates provide useful information on the degree to which inferences regarding the determinants of cocaine use are sensitive to survey type.     

Parameter inference for time series with regular and seasonal unit roots

It is common to have both regular and seasonal roots present in many time series data. It may occur that one or both of the roots are just close but not equal to unity. Parameter inference for this situation is considered both when the time series has a finite or an infinite variance. Asymptotic char-acterizations of the test statistics were obtained via functionals of Ornstein-Uhlenbeck processes and Lévy processes. Tabulations for the large sample distributions are obtained. The results will be useful in applications deciding whether both regular and seasonal differencing are needed in fitting a time series model.     

Estimation of percentile residual life function with left-truncated and right-censored data

This article focuses on the estimation of percentile residual life function with left-truncated and right-censored data. Asymptotic normality and a pointwise confidence interval that does not require estimating the unknown underlying distribution function of the proposed empirical estimator are obtained. Some simulation studies and a real data example are used to illustrate our results.     

Regression model for surrogate data in high dimensional statistics

Abstract

This paper deals with the problem of estimating the regression of a surrogated scalar response variable given a functional random one. We construct an estimator of the regression operator by using, in addition to the available (true) response data, a surrogate data. We then establish some asymptotic properties of the constructed estimator in terms of the almost-complete and the quadratic mean convergences. Notice that the obtained results generalize a part of the results obtained in the finite dimensional framework. Finally, an illustration on the applicability of our results on both simulated data and a real dataset was realized. We have thus shown the superiority of our estimator on classical estimators when we are lacking complete data.     

Statistical Modelling and Deconvolution of Yield Meter Data

Abstract.  This paper considers the problem of mapping spatial variation of yield in a field using data from a yield monitoring system on a combine harvester. The unobserved yield is assumed to be a Gaussian random field and the yield monitoring system data is modelled as a convolution of the yield and an impulse response function. This results in an unusual spatial covariance structure (depending on the driving pattern of the combine harvester) for the yield monitoring system data. Parameters of the impulse response function and the spatial covariance function of the yield are estimated using maximum likelihood methods. The fitted model is assessed using certain empirical directional covariograms and the yield is finally predicted using the inferred statistical model.     

The ARHD model

We introduce and study a new model for functional data. The ARHD is an autoregressive model in which the first order derivative of the random curves appears explicitly. Convergent estimates are obtained through an original double penalization method. The prediction method is applied to a real set of data already studied in the literature.     

Semiparametric inference for a general class of models for recurrent events

Procedures for estimating the parameters of the general class of semiparametric models for recurrent events proposed by Peña and Hollander [(2004). Models for recurrent events in reliability and survival analysis. In: Soyer R., Mazzuchi T., Singpurwalla N. (Eds.), Mathematical Reliability: An Expository Perspective. Kluwer Academic Publishers, Dordrecht, pp. 105–123 (Chapter 6)] are developed. This class of models incorporates an effective age function encoding the effect of changes after each event occurrence such as the impact of an intervention, it models the impact of accumulating event occurrences on the unit, it admits a link function in which the effect of possibly time-dependent covariates are incorporated, and it allows the incorporation of unobservable frailty components which induce dependencies among the inter-event times for each unit. The estimation procedures are semiparametric in that a baseline hazard function is nonparametrically specified. The sampling distribution properties of the estimators are examined through a simulation study, and the consequences of mis-specifying the model are analyzed. The results indicate that the flexibility of this general class of models provides a safeguard for analyzing recurrent event data, even data possibly arising from a frailty-less mechanism. The estimation procedures are applied to real data sets arising in the biomedical and public health settings, as well as from reliability and engineering situations. In particular, the procedures are applied to a data set pertaining to times to recurrence of bladder cancer and the results of the analysis are compared to those obtained using three methods of analyzing recurrent event data.     

A new geometric first-order integer-valued autoregressive (NGINAR(1)) process

A new stationary first-order integer-valued autoregressive process with geometric marginal distributions is introduced based on negative binomial thinning. Some properties of the process are established. Estimators of the parameters of the process are obtained using the methods of conditional least squares, Yule–Walker and maximum likelihood. Also, the asymptotic properties of the estimators are derived involving their distributions. Some numerical results of the estimators are presented with a discussion to the obtained results. Real data are used and a possible application is discussed.     

POISSON REGRESSION WITH A PERIODIC FUNCTION

ABSTRACT

Let {y_t } be a Poisson-like process with the mean μ _t which is a periodic function of time t. We discuss how to fit this type of data set using quasi-likelihood method. Our method provides a new avenue to fit a time series data when the usual assumption of stationarity and homogeneous residual variances are invalid. We show that the estimators obtained are strongly consistent and also asymptotically normal.     

Adaptive modelling of dose–response relationships using smoothing splines

We consider the use of smoothing splines for the adaptive modelling of dose–response relationships. A smoothing spline is a nonparametric estimator of a function that is a compromise between the fit to the data and the degree of smoothness and thus provides a flexible way of modelling dose–response data. In conjunction with decision rules for which doses to continue with after an interim analysis, it can be used to give an adaptive way of modelling the relationship between dose and response. We fit smoothing splines using the generalized cross‐validation criterion for deciding on the degree of smoothness and we use estimated bootstrap percentiles of the predicted values for each dose to decide upon which doses to continue with after an interim analysis. We compare this approach with a corresponding adaptive analysis of variance approach based upon new simulations of the scenarios previously used by the PhRMA Working Group on Adaptive Dose‐Ranging Studies. The results obtained for the adaptive modelling of dose–response data using smoothing splines are mostly comparable with those previously obtained by the PhRMA Working Group for the Bayesian Normal Dynamic Linear model (GADA) procedure. These methods may be useful for carrying out adaptations, detecting dose–response relationships and identifying clinically relevant doses. Copyright © 2009 John Wiley & Sons, Ltd.     

Optimal Selection of the Most Reliable Design Based on Gamma Degradation Processes

Manufacturers are often faced with the problem of how to select the most reliable design among several competing designs in the stage of development. It becomes complicated if products are highly reliable. Under the circumstances, recent work has focused on the study with degradation data by assuming that degradation paths follow Wiener processes or random-effect models. However, it is more appropriate to use gamma processes to model degradation data with monotone-increasing pattern. This article deals with the selection problem for such processes. With a minimum probability of correct decision, optimal test plans can be obtained by minimizing the total cost.     

On semi-markov compartmental models with branching particles under the influence of disasters

A semi-Markov multi-compart mental system in which particles reproduce similar particles as a Markov branching process and being subjected to disasters is studied. Expressions for the mean number of particles alive at time t in each compartment are obtained. The results concerning irreversible, mammillarian and catenary compartmental systems have been discussed.