Models and estimation methods for clinical HIV-1 data

Clinical HIV-1 data include many individual factors, such as compliance to treatment, pharmacokinetics, variability in respect to viral dynamics, race, sex, income, etc., which might directly influence or be associated with clinical outcome. These factors need to be taken into account to achieve a better understanding of clinical outcome and mathematical models can provide a unifying framework to do so. The first objective of this paper is to demonstrate the development of comprehensive HIV-1 dynamics models that describe viral dynamics and also incorporate different factors influencing such dynamics. The second objective of this paper is to describe alternative estimation methods that can be applied to the analysis of data with such models. In particular, we consider: (i) simple but effective two-stage estimation methods, in which data from each patient are analyzed separately and summary statistics derived from the results, (ii) more complex nonlinear mixed effect models, used to pool all the patient data in a single analysis. Bayesian estimation methods are also considered, in particular: (iii) maximum posterior approximations, MAP, and (iv) Markov chain Monte Carlo, MCMC. Bayesian methods incorporate prior knowledge into the models, thus avoiding some of the model simplifications introduced when the data are analyzed using two-stage methods, or a nonlinear mixed effect framework. We demonstrate the development of the models and the different estimation methods using real AIDS clinical trial data involving patients receiving multiple drugs regimens.     

Parameter estimation in stochastic scenario generation systems

Scenario analysis offers an effective tool for addressing the stochastic elements in multi-period financial planning models. Critical to any scenario generation process is the estimation of the input parameters of the underlying stochastic model for economic factors. In this paper, we propose a new approach for estimation, known as the integrated parameter estimation (IPE). This approach combines the significant features of other well-known estimation techniques within a non-convex multiple objective optimization framework, with the objective weights controlling the relative importance of the features. We solve the non-convex optimization problem using adaptive memory programming – a variation of tabu search. Based on a short interest rate model using UK treasury rates from 1980 to 1995, the integrated approach compares favorably with maximum likelihood and the generalized method of moments. We also evaluate performance with Towers Perrin's CAP:Link scenario generation system.     

Learning curve parameter estimation beyond traditional statistics

Due to decreasing order quantities, increasing product variety and fluctuating production orders, manufacturing companies have been encountering an increased occurrence of repetitive learning-forgetting phenomenon. In this paper, deterministic methods for the learning curve parameter estimation from the limited production data available from the unstable production environment are studied. Two main learning curve models: cumulative average (Wright) and unit (Crawford) were considered and several different mathematically proven methods were proposed for the parameter estimation. The calculation results illustrated that learning curve parameters can be unequivocally estimated from the limited production data (single random sample) by using deterministic methods for both of the learning curve models, although more accurate estimation was provided by the cumulative average model based methods. Newly proposed methods enable sufficiently accurate parameter estimation from the limited production data where traditional statistical parameter estimation methods cannot be applied.     

Bayes estimation in linear models: A coordinate-free approach

The unified theory of Bayes estimation in linear models is presented, using a coordinate-free approach. The results are applied to the problem of linear and quadratic estimation in linear regression model.     

Maximum likelihood estimation in exponential orthogeodesic models

An orthogeodesic statistical model is defined in terms of five conditions of differential geometric nature. These conditions are reviewed together with a characterization theorem for exponential orthogeodesic models. Orthogonal projections, relevant for maximum likelihood estimation in exponential orthogeodesic models, are described in a simple way in terms of some of the quantities in the characterization theorem. A unified procedure for performing maximum likelihood estimation in exponential orthogenodesic models is given and the use of this procedure is illustrated for some of the most important models of this kind such as -parallel models, -parallel models and certain transformation models.     

Improved probabilistic point estimation schemes for uncertainty analysis

The probabilistic point estimation (PPE) methods replace the probability distribution of the random parameters of a model with a finite number of discrete points in sample space selected in such a way to preserve limit probabilistic information of involved random parameters. Most PPE methods developed thus far match the distribution of random parameters up to the third statistical moment and, in general, could provide reasonable accurate estimation only for the first two statistical moments of model output. This study proposes two optimization-based point selection schemes for the PPE methods to enhance the accuracy of higher-order statistical moments estimation for model output. Several test models of varying degrees of complexity and nonlinearity are used to examine the performance of the proposed point selection schemes. The results indicate that the proposed point selection schemes provide significantly more accurate estimation of model output uncertainty features than the existing schemes.     

Statistical estimation with model selection

The purpose of this paper is to explain the interest and importance of (approximate) models and model selection in Statistics. Starting from the very elementary example of histograms we present a general notion of finite dimensional model for statistical estimation and we explain what type of risk bounds can be expected from the use of one such model. We then give the performance of snitable model selection procedures from a family of such models. We illustrate our point of view by two main examples: the choice of a partition for designing a histogram from an n-sample and the problem of variable selection in the context of Gaussian regression.     

The covariance structure and likelihood function for multivariate dyadic data

This paper extends the results in Li and Loken [A unified theory of statistical analysis and inference for variance component models for dyadic data, Statist. Sinica 12 (2002) 519-535] on the statistical analysis of measurements taken on dyads to the situations in which more than one attribute are measured on each dyad. Starting from the covariance structure for the univariate case obtained in Li and Loken (2002), the covariance structure for the multivariate case is derived based on the group symmetry induced by the assumed exchangeability in the units. Our primary objective is to document the Gaussian likelihood and the sufficient statistics for multivariate dyadic data in closed form, so that they can be referenced by researchers as they analyze those data. The derivation carried out can also serve as an example of multivariate extension of univariate models based on exchangeability.     

Asymptotics for estimation and testing procedures under loss of identifiability

Statistical analyses commonly make use of models that suffer from loss of identifiability. In this paper, we address important issues related to the parameter estimation and hypothesis testing in models with loss of identifiability. That is, there are multiple parameter points corresponding to the same true model. We refer the set of these parameter points to as the set of true parameter values. We consider the case where the set of true parameter values is allowed to be very large or even infinite, some parameter values may lie on the boundary of the parameter space, and the data are not necessarily independently and identically distributed. Our results are applicable to a large class of estimators and their related testing statistics derived from optimizing an objective function such as a likelihood. We examine three specific examples: (i) a finite mixture logistic regression model; (ii) stationary ARMA processes; (iii) general quadratic approximation using Hellinger distance. The applications to these examples demonstrate the applicability of our results in a broad range of difficult statistical problems.     

On optimal parameter estimation in credibility

The present paper is concerned with optimal estimation of the 1st and 2nd order structural moments appearing in credibility formulas. In a recent paper De Vylder has treated the problem in the case of multinormal conditional distributions under quite restrictive assumptions. He minimizes, within a certain restricted class of unbiased estimators, the variance (or the sum of variances if the estimand is a matrix) and next replaces all structural moments (up to fourth order) in the solution by estimates based on the data. This paper is an attempt to simplify the method and extend it so as to make it applicable in more general situations. By suitable choice of a (sufficient) set of statistics and a suitable parametrization, the powerful theory of estimation in linear models can be employed, which makes cumbersome minimization procedures superfluous. The theory is applied to the cases with binomial. Poisson, compound Poisson, and multinormal conditional distributions. Some simulation studies have been performed to assess the performance of the estimators.     

A nonlinear time series model and estimation of missing observations

This paper formulates a nonlinear time series model which encompasses several standard nonlinear models for time series as special cases. It also offers two methods for estimating missing observations, one using prediction and fixed point smoothing algorithms and the other using optimal estimating equation theory. Recursive estimation of missing observations in an autoregressive conditionally heteroscedastic (ARCH) model and the estimation of missing observations in a linear time series model are shown to be special cases. Construction of optimal estimates of missing observations using estimating equation theory is discussed and applied to some nonlinear models.Authors were supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada.     

Semiparametric estimation of mean and variance functions for non-Gaussian data

Flexible modelling of the response variance in regression is interesting for understanding the causes of variability in the responses, and is crucial for efficient estimation and correct inference for mean parameters. In this paper we describe methods for mean and variance estimation where the responses are modelled using the double exponential family of distributions and mean and dispersion parameters are described as an additive function of predictors. The additive terms in the model are represented by penalized splines. A simple and unified computational methodology is presented for carrying out the calculations required for Bayesian inference in this class of models based on an adaptive Metropolis algorithm. Application of the adaptive Metropolis algorithm is fully automatic and does not require any kind of pretuning runs. The methodology presented provides flexible methods for modelling heterogeneous Gaussian data, as well as overdispersed and underdispersed count data. Performance is considered in a variety of examples involving real and simulated data sets.     

A mixture model-based approach to the clustering of exponential repeated data

The analysis of finite mixture models for exponential repeated data is considered. The mixture components correspond to different unknown groups of the statistical units. Dependency and variability of repeated data are taken into account through random effects. For each component, an exponential mixed model is thus defined. When considering parameter estimation in this mixture of exponential mixed models, the EM-algorithm cannot be directly used since the marginal distribution of each mixture component cannot be analytically derived. In this paper, we propose two parameter estimation methods. The first one uses a linearisation specific to the exponential distribution hypothesis within each component. The second approach uses a Metropolis–Hastings algorithm as a building block of a general MCEM-algorithm.     

Bayesian estimation of generalized gamma shared frailty model

Multivariate survival analysis comprises of event times that are generally grouped together in clusters. Observations in each of these clusters relate to data belonging to the same individual or individuals with a common factor. Frailty models can be used when there is unaccounted association between survival times of a cluster. The frailty variable describes the heterogeneity in the data caused by unknown covariates or randomness in the data. In this article, we use the generalized gamma distribution to describe the frailty variable and discuss the Bayesian method of estimation for the parameters of the model. The baseline hazard function is assumed to follow the two parameter Weibull distribution. Data is simulated from the given model and the Metropolis–Hastings MCMC algorithm is used to obtain parameter estimates. It is shown that increasing the size of the dataset improves estimates. It is also shown that high heterogeneity within clusters does not affect the estimates of treatment effects significantly. The model is also applied to a real life dataset.     

Nearest neighbor conditional estimation for Harris recurrent Markov chains

This paper is concerned with consistent nearest neighbor time series estimation for data generated by a Harris recurrent Markov chain on a general state space. It is shown that nearest neighbor estimation is consistent in this general time series context, using simple and weak conditions. The results proved here, establish consistency, in a unified manner, for a large variety of problems, e.g. autoregression function estimation, and, more generally, extremum estimators as well as sequential forecasting. Finally, under additional conditions, it is also shown that the estimators are asymptotically normal.     

An implicit least squares algorithm for nonlinear rational model parameter estimation

In this study a new insight into least squares regression is identified and immediately applied to estimating the parameters of nonlinear rational models. From the beginning the ordinary explicit expression for linear in the parameters model is expanded into an implicit expression. Then a generic algorithm in terms of least squares error is developed for the model parameter estimation. It has been proved that a nonlinear rational model can be expressed as an implicit linear in the parameters model, therefore, the developed algorithm can be comfortably revised for estimating the parameters of the rational models. The major advancement of the generic algorithm is its conciseness and efficiency in dealing with the parameter estimation problems associated with nonlinear in the parameters models. Further, the algorithm can be used to deal with those regression terms which are subject to noise. The algorithm is reduced to an ordinary least square algorithm in the case of linear or linear in the parameters models. Three simulated examples plus a realistic case study are used to test and illustrate the performance of the algorithm.     

Specifying measurement errors for required lifetime estimation performance

Lifetime estimation based on the measured health monitoring data has long been investigated and applied in reliability and operational management communities and practices, such as planning maintenance schedules, logistic supports, and production planning. It is known that measurement error (ME) is a source of uncertainty in the measured data considerably affecting the performance of data driven lifetime estimation. While the effect of ME on the performance of data driven lifetime estimation models has been studied recently, a reversed problem—“the specification of the ME range to achieve a desirable lifetime estimation performance” has not been addressed. This problem is related to the usability of the measured health monitoring data for estimating the lifetime. In this paper, we deal with this problem and develop guidelines regarding the formulation of specification limits to the distribution-related ME characteristics. By referring to one widely applied Wiener process-based degradation model, permissible values for the ME bias and standard deviation can be given under a specified lifetime estimation requirement. If the performance of ME does not satisfy the permissible values, the desirable performance for lifetime estimation cannot be ensured by the measured health monitoring data. We further analyze the effect of ME on an age based replacement decision, which is one of the most common and popular maintenance policies in maintenance scheduling. Numerical examples and a case study are provided to illustrate the implementation procedure and usefulness of theoretical results.     

Asymptotic properties of a particular nonlinear regression quantile estimation

In this paper we shall be concerned with the asymptotic properties of the regression quantile estimation in the nonlinear regression time series models. For these, first we prove the strong consistency and derive the asymptotic normality of the regression quantile estimators for a particular sinusoidal regression model with a simple harmonic component. Next, we extend the results to more complicated sinusoidal models of several harmonic components.     

Weibull分布恒定应力加速寿命试验的Bayes估计

对定时和定数截尾样本情形CE模型下Weibull分布场合恒定应力加速寿命试验进行了Bayes统计分析,利用Laplace方法给出了该模型的近似Bayes估计.最后通过模拟实例表明该Bayes估计是有效的.