Parameter estimation from interval-valued data using the expectation-maximization algorithm

This paper investigates on the problem of parameter estimation in statistical model when observations are intervals assumed to be related to underlying crisp realizations of a random sample. The proposed approach relies on the extension of likelihood function in interval setting. A maximum likelihood estimate of the parameter of interest may then be defined as a crisp value maximizing the generalized likelihood function. Using the expectation-maximization (EM) to solve such maximizing problem therefore derives the so-called interval-valued EM algorithm (IEM), which makes it possible to solve a wide range of statistical problems involving interval-valued data. To show the performance of IEM, the following two classical problems are illustrated: univariate normal mean and variance estimation from interval-valued samples, and multiple linear/nonlinear regression with crisp inputs and interval output.     

Residual life estimation based on nonlinear-multivariate Wiener processes

For some operable products with critical reliability constraints, it is important to estimate accurately their residual lives so that maintenance actions can be arranged suitably and efficiently. In the literature, most publications have dealt with this issue by only considering one-dimensional degradation data. However, this may be not reasonable in situations wherein a product may have two or more performance characteristics (PCs). In such situations, multi-dimensional degradation data should be taken into account. Here, for the target product with multivariate PCs, methods of residual life (RL) estimation are developed. This is done with the assumption that the degradation of PCs over time is governed by a multivariate Wiener process with nonlinear drifts. Both the population-based degradation information and the degradation history of the target product up-to-date are combined to estimate the RL of the product. Specifically, the population-based degradation information is first used to obtain the estimates of the unknown parameters of the model through the EM algorithm. Then, the degradation history of the target product is adopted to update the degradation model, based on which the RL is estimated accordingly. To illustrate the validity and the usefulness of the proposed method, a numerical example about fatigue cracks is finally presented and analysed.     

Quantile regression based on a weighted approach under semi-competing risks data

In this article, we investigate the quantile regression analysis for semi-competing risks data in which a non-terminal event may be dependently censored by a terminal event. Due to the dependent censoring, the estimation of quantile regression coefficients on the non-terminal event becomes difficult. In order to handle this problem, we assume Archimedean Copula to specify the dependence of the non-terminal event and the terminal event. Portnoy [Censored regression quantiles. J Amer Statist Assoc. 2003;98:1001–1012] considered the quantile regression model under right-censoring data. We extend his approach to construct a weight function, and then impose the weight function to estimate the quantile regression parameter for the non-terminal event under semi-competing risks data. We also prove the consistency and asymptotic properties for the proposed estimator. According to the simulation studies, the performance of our proposed method is good. We also apply our suggested approach to analyse a real data.     

Mixture of linear mixed models using multivariate t distribution

Linear mixed models are widely used when multiple correlated measurements are made on each unit of interest. In many applications, the units may form several distinct clusters, and such heterogeneity can be more appropriately modelled by a finite mixture linear mixed model. The classical estimation approach, in which both the random effects and the error parts are assumed to follow normal distribution, is sensitive to outliers, and failure to accommodate outliers may greatly jeopardize the model estimation and inference. We propose a new mixture linear mixed model using multivariate t distribution. For each mixture component, we assume the response and the random effects jointly follow a multivariate t distribution, to conveniently robustify the estimation procedure. An efficient expectation conditional maximization algorithm is developed for conducting maximum likelihood estimation. The degrees of freedom parameters of the t distributions are chosen data adaptively, for achieving flexible trade-off between estimation robustness and efficiency. Simulation studies and an application on analysing lung growth longitudinal data showcase the efficacy of the proposed approach.     

Fitting asymmetric bimodal data with selected distributions

ABSTRACT

This article discusses two asymmetrization methods, Azzalini's representation and beta generation, to generate asymmetric bimodal models including two novel beta-generated models. The practical utility of these models is assessed with nine data sets from different fields of applied sciences. Besides this tutorial assessment, some methodological contributions are made: a random number generator for the asymmetric Rathie–Swamee model is developed (generators for the other models are already known and briefly described) and a new likelihood ratio test of unimodality is compared via simulations with other available tests. Several tools have been used to quantify and test for bimodality and assess goodness of fit including Bayesian information criterion, measures of agreement with the empirical distribution and the Kolmogorov–Smirnoff test. In the nine case studies, the results favoured models derived from Azzalini's asymmetrization, but no single model provided a best fit across the applications considered. In only two cases the normal mixture was selected as best model. Parameter estimation has been done by likelihood maximization. Numerical optimization must be performed with care since local optima are often present. We concluded that the models considered are flexible enough to fit different bimodal shapes and that the tools studied should be used with care and attention to detail.     

Quantile regression in longitudinal studies with dropouts and measurement errors

ABSTRACT

Quantile regression models, as an important tool in practice, can describe effects of risk factors on the entire conditional distribution of the response variable with its estimates robust to outliers. However, there is few discussion on quantile regression for longitudinal data with both missing responses and measurement errors, which are commonly seen in practice. We develop a weighted and bias-corrected quantile loss function for the quantile regression with longitudinal data, which allows both missingness and measurement errors. Additionally, we establish the asymptotic properties of the proposed estimator. Simulation studies demonstrate the expected performance in correcting the bias resulted from missingness and measurement errors. Finally, we investigate the Lifestyle Education for Activity and Nutrition study and confirm the effective of intervention in producing weight loss after nine month at the high quantile.     

On testes for threshold–type nonlinearity in irregulaly spaced time series

Formulating the model first in continuous time, we have developed a state space approach to the problem of testing for threshold-type nonlinearity when the data are irregularly spaced.     

Understanding the epidemiology of foreign body injuries in children using a data-driven Bayesian network

Bayesian networks (BNs) are probabilistic expert systems which have emerged over the last few decades as a powerful data mining technique. Also, BNs have become especially popular in biomedical applications where they have been used for diagnosing diseases and studying complex cellular networks, among many other applications. In this study, we built a BN in a fully automated way in order to analyse data regarding injuries due to the inhalation, ingestion and aspiration of foreign bodies (FBs) in children. Then, a sensitivity analysis was carried out to characterize the uncertainty associated with the model. While other studies focused on characteristics such as shape, consistency and dimensions of the FBs which caused injuries, we propose an integrated environment which makes the relationships among the factors underlying the problem clear. The advantage of this approach is that it gives a picture of the influence of critical factors on the injury severity and allows for the comparison of the effect of different FB characteristics (volume, FB type, shape and consistency) and children's features (age and gender) on the risk of experiencing a hospitalization. The rates it consents to calculate provide a more rational basis for promoting care-givers’ education of the most influential risk factors regarding the adverse outcomes.     

An empirical comparison of Canonical Correspondence Analysis and STATICO in the identification of spatio-temporal ecological relationships

The wide-ranging and rapidly evolving nature of ecological studies mean that it is not possible to cover all existing and emerging techniques for analyzing multivariate data. However, two important methods enticed many followers: the Canonical Correspondence Analysis (CCA) and the STATICO analysis. Despite the particular characteristics of each, they have similarities and differences, which when analyzed properly, can, together, provide important complementary results to those that are usually exploited by researchers. If on one hand, the use of CCA is completely generalized and implemented, solving many problems formulated by ecologists, on the other hand, this method has some weaknesses mainly caused by the imposition of the number of variables that is required to be applied (much higher in comparison with samples). Also, the STATICO method has no such restrictions, but requires that the number of variables (species or environment) is the same in each time or space. Yet, the STATICO method presents information that can be more detailed since it allows visualizing the variability within groups (either in time or space). In this study, the data needed for implementing these methods are sketched, as well as the comparison is made showing the advantages and disadvantages of each method. The treated ecological data are a sequence of pairs of ecological tables, where species abundances and environmental variables are measured at different, specified locations, over the course of time.     

Robust Priors in Nonlinear Panel Data Models

Many approaches to estimation of panel models are based on an average or integrated likelihood that assigns weights to different values of the individual effects. Fixed effects, random effects, and Bayesian approaches all fall into this category. We provide a characterization of the class of weights (or priors) that produce estimators that are first‐order unbiased. We show that such bias‐reducing weights will depend on the data in general unless an orthogonal reparameterization or an essentially equivalent condition is available. Two intuitively appealing weighting schemes are discussed. We argue that asymptotically valid confidence intervals can be read from the posterior distribution of the common parameters when N and T grow at the same rate. Next, we show that random effects estimators are not bias reducing in general and we discuss important exceptions. Moreover, the bias depends on the Kullback–Leibler distance between the population distribution of the effects and its best approximation in the random effects family. Finally, we show that, in general, standard random effects estimation of marginal effects is inconsistent for large T, whereas the posterior mean of the marginal effect is large‐T consistent, and we provide conditions for bias reduction. Some examples and Monte Carlo experiments illustrate the results.