Isotontic regression for interval censored survival data using an e-m algorithm

This paper develops a nonparametric model of the relationship between survival S and a dichotomous random variable X under the order constraint that P(X=1|S=s) is increasing (or decreasing) with s. The estimation procedure, called isotonic regression, has been studied in some depth for the case of uncensored data, but we give a methodology which is appropriate in the more general context of right, left, and interval censored data. An E-M Algorithm (Dempster et. al., 1977) is used for maximum likelihood estimation.     

Correcting for nonlinear measurement errors in the dependent variable in the general linear model

Linear models are considered in which measurement error is present in the dependent variable. Observed values are related to true values via nonlinear regression models with the parameters in the measurement error models being estimated with the use of independent, external data, collected using standards. Pseudo-maximum likelihood estimators and their asymptotic properties are developed under normality assumptions and the common approach of simply analyzing imputed values obtained from the nestimated calibration curves is assessed. A small simulation evaluates the procedures. An example is presented in which urinary neopterin (measured via radioimmunoassay) is nbeing compared between two groups of individuals.     

Estimation in a truncated bivariate poisson distribution using the EM algorithm

An EM algorithm (Dempster et al., 1977) is derived for the estimation of parameters of the truncated bivariate Poisson distribution with zeros rnissing from both margins. The observed inforrnation matrix is obtained and a numerical exarnple is given where the convergence of the EM algorithm is accelerated by the methods of Louis (1982) and conjugate gradients (Jamshidian antl Jennrich, 1993).     

An augmented data scoring algorithm for maximum likelihood

The expectation-maximization (EM) method facilitates computation of max¬imum likelihood (ML) and maximum penalized likelihood (MPL) solutions. The procedure requires specification of unobservabie complete data which augment the measured or incomplete data. This specification defines a conditional expectation of the complete data log-likelihood function which is computed in the E-stcp. The EM algorithm is most effective when maximizing the iunction Q{0) denned in the F-stnp is easier than maximizing the likelihood function.

The Monte Carlo EM (MCEM) algorithm of Wei & Tanner (1990) was introduced for problems where computation of Q is difficult or intractable. However Monte Carlo can he computationally expensive, e.g. in signal processing applications involving large numbers of parameters. We provide another approach: a modification of thc standard EM algorithm avoiding computation of conditional expectations.     

D-optimal designs for combined polynomial and trigonometric regression on a partial circle

Consider the D-optimal designs for a combined polynomial and trigonometric regression on a partial circle. It is shown that the optimal design is equally supported and the structure of the optimal design depends only on the length of the design interval and the support points are analytic functions of this parameter. Moreover, the Taylor expansion of the optimal support points can be determined efficiently by a recursive procedure. Examples are presented to illustrate the procedures for computing the optimal designs.     

Fast and Accurate Inference for the Smoothing Parameter in Semiparametric Models

A fast and accurate method of confidence interval construction for the smoothing parameter in penalised spline and partially linear models is proposed. The method is akin to a parametric percentile bootstrap where Monte Carlo simulation is replaced by saddlepoint approximation, and can therefore be viewed as an approximate bootstrap. It is applicable in a quite general setting, requiring only that the underlying estimator be the root of an estimating equation that is a quadratic form in normal random variables. This is the case under a variety of optimality criteria such as those commonly denoted by maximum likelihood (ML), restricted ML (REML), generalized cross validation (GCV) and Akaike's information criteria (AIC). Simulation studies reveal that under the ML and REML criteria, the method delivers a near‐exact performance with computational speeds that are an order of magnitude faster than existing exact methods, and two orders of magnitude faster than a classical bootstrap. Perhaps most importantly, the proposed method also offers a computationally feasible alternative when no known exact or asymptotic methods exist, e.g. GCV and AIC. An application is illustrated by applying the methodology to well‐known fossil data. Giving a range of plausible smoothed values in this instance can help answer questions about the statistical significance of apparent features in the data.     

Dirichlet Processes in Nonlinear Mixed Effects Models

In this article, we use two efficient approaches to deal with the difficulty in computing the intractable integrals when implementing Gibbs sampling in the nonlinear mixed effects model (NLMM) based on Dirichlet processes (DP). In the first approach, we compute the Laplace's approximation to the integral for its high accuracy, low cost, and ease of implementation. The second approach uses the no-gaps algorithm of MacEachern and Müller (1998 MacEachern , S. , Müller , P. ( 1998 ). Estimating mixtures of Dirichlet process models . Journal of Computational and Graphical Statistics 7 : 223 – 238 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) to perform Gibbs sampling without evaluating the difficult integral. We apply both approaches to real problems and simulations. Results show that both approaches perform well in density estimation and prediction and are superior to the parametric analysis in that they can detect important model features, such as skewness, long tails, and multimodality, whereas the parametric analysis cannot.     

An EM and a Stochastic Version of the EM Algorithm for Nonparametric Hidden Semi-Markov Models

The Hidden semi-Markov models (HSMMs) were introduced to overcome the constraint of a geometric sojourn time distribution for the different hidden states in the classical hidden Markov models. Several variations of HSMMs were proposed that model the sojourn times by a parametric or a nonparametric family of distributions. In this article, we concentrate our interest on the nonparametric case where the duration distributions are attached to transitions and not to states as in most of the published papers in HSMMs. Therefore, it is worth noticing that here we treat the underlying hidden semi-Markov chain in its general probabilistic structure. In that case, Barbu and Limnios (2008 Barbu , V. , Limnios , N. ( 2008 ). Semi-Markov Chains and Hidden Semi-Markov Models Toward Applications: Their Use in Reliability and DNA Analysis . New York : Springer . [Google Scholar]) proposed an Expectation–Maximization (EM) algorithm in order to estimate the semi-Markov kernel and the emission probabilities that characterize the dynamics of the model. In this article, we consider an improved version of Barbu and Limnios' EM algorithm which is faster than the original one. Moreover, we propose a stochastic version of the EM algorithm that achieves comparable estimates with the EM algorithm in less execution time. Some numerical examples are provided which illustrate the efficient performance of the proposed algorithms.     

An algorithm for the minimum sum of weighted absolute errors regression

We propose an algorithm to estimate the unknown constants in a multiple linear regression model under the minimum sum of weighted absolute errors (MSWAE). The proposed algorithm, a generalization of an earlier algorithm, is compared to a bounded variable algorithm. Some somputational experience is reported.