Improved maximum likelihood estimation in heteroscedastic nonlinear regression models.

Nonlinear heteroscedastic regression models are a widely used class of models in applied statistics, with applications especially in biology, medicine or chemistry. Nonlinearity and variance heterogeneity can make likelihood estimation for a scalar parameter of interest rather inaccurate for small or moderate samples. In this paper, we suggest a new approach to point estimation based on estimating equations obtained from higher-order pivots for the parameter of interest. In particular, we take as an estimating function the modified directed likelihood. This is a higher-order pivotal quantity that can be easily computed in practice for nonlinear heteroscedastic models with normally distributed errors , using a recently developed S-PLUS library (HOA, 2000) . The estimators obtained from this procedure are a refinement of the maximum likelihood estimators, improving their small sample properties and keeping equivariance under reparameterisation. Two applications to real data sets are discussed

A geometric characterization of c-optimal designs for heteroscedastic regression

We consider the common nonlinear regression model where the variance as well as the mean is a parametric function of the explanatory variables. The c-optimal design problem is investigated in the case when the parameters of both the mean and the variance function are of interest. A geometric characterization of c-optimal designs in this context is presented, which generalizes the classical result of Elfving (1952) for c-optimal designs. As in Elfving's famous characterization c-optimal designs can be described as representations of boundary points of a convex set. However, in the case where there appear parameters of interest in the variance, the structure of the Elfving set is different. Roughly speaking the Elfving set corresponding to a heteroscedastic regression model is the convex hull of a set of ellipsoids induced by the underlying model and indexed by the design space. The c-optimal designs are characterized as representations of the points where the line in direction of the vector c intersects the boundary of the new Elfving set. The theory is illustrated in several examples including pharmacokinetic models with random effects. --c-optimal design,heteroscedastic regression,Elfving's theorem,pharmacokinetic models,random effects,locally optimal design,geometric characterization

Bayesian calibration for multiple source regression model

In large variety of practical applications, using information from different sources or different kind of data is a reasonable demand. The problem of studying multiple source data can be represented as a multi-task learning problem, and then the information from one source can help to study the information from the other source by extracting a shared common structure. From the other hand, parameter evaluations obtained from various sources can be confused and conflicting. This paper proposes a Bayesian based approach to calibrate data obtained from different sources and to solve nonlinear regression problem in the presence of heteroscedastisity of the multiple-source model. An efficient algorithm is developed for implementation. Using analytical and simulation studies, it is shown that the proposed Bayesian calibration improves the convergence rate of the algorithm and precision of the model. The theoretical results are supported by a synthetic example, and a real-world problem, namely, modeling unsteady pitching moment coefficient of aircraft, for which a recurrent neural network is constructed

Mid-Epidemic Forecasts of COVID-19 Cases and Deaths: A Bivariate Model Applied to the UK.

Background: The evolution of the COVID-19 epidemic has been accompanied by efforts to provide comparable international data on new cases and deaths. There is also accumulating evidence on the epidemiological parameters underlying COVID-19. Hence, there is potential for epidemic models providing mid-term forecasts of the epidemic trajectory using such information. The effectiveness of lockdown or lockdown relaxation can also be assessed by modelling later epidemic stages, possibly using a multiphase epidemic model. Methods: Commonly applied methods to analyse epidemic trajectories or make forecasts include phenomenological growth models (e.g., the Richards family of densities) and variants of the susceptible-infected-recovered (SIR) compartment model. Here, we focus on a practical forecasting approach, applied to interim UK COVID data, using a bivariate Reynolds model (for cases and deaths), with implementation based on Bayesian inference. We show the utility of informative priors in developing and estimating the model and compare error densities (Poisson-gamma, Poisson-lognormal, and Poisson-log-Student) for overdispersed data on new cases and deaths. We use cross validation to assess medium-term forecasts. We also consider the longer-term postlockdown epidemic profile to assess epidemic containment, using a two-phase model. Results: Fit to interim mid-epidemic data show better fit to training data and better cross-validation performance for a Poisson-log-Student model. Estimation of longer-term epidemic data after lockdown relaxation, characterised by protracted slow downturn and then upturn in cases, casts doubt on effective containment. Conclusions: Many applications of phenomenological models have been to complete epidemics. However, evaluation of such models based simply on their fit to observed data may give only a partial picture, and cross validation against actual trends is also valuable. Similarly, it may be preferable to model incidence rather than cumulative data, although this raises questions about suitable error densities for modelling often erratic fluctuations. Hence, there may be utility in evaluating alternative error assumptions

Bayesian Beta Regression with the Bayesianbetareg R-Package

In this paper we summarize the main points of beta regression models under Bayesian perspective, including a presentation of the Bayesianbetareg R-package, used to fit the beta regression models under a Bayesian approach. Finally, beta regression models are fitted to a reading score database using, respectively, the Bayesianbetareg and betareg R-Packages for Bayesian and classic perspectives

Análise de redes policiais aplicadas ao controle da delincuência de rua em Bogotá

The community police models, together with the georeferencing of crime and work focused on solving specific problems, have allowed improving indicators of attention to crime and the perception of security in cities. However, little care has been placed in these models for cooperative networks between patrols organized by districts or urban sectors, to maintain an articulated action that compensates for the scarcity of police resources and avoids overtargeting. This article analyzes the interaction between the network of police reinforcements and homicides and personal thefts in Bogotá, capital of Colombia. Exponential random graph (ERG) models, network analysis, and georeferenced gamma estimates of criminal behavior are used. At the level of the police sectors known as Commands for Immediate Attention (CAI), it is found that patrol network organization schemes are conditioned by the city’s socioeconomic segregation patterns which limits the scope of the network in control of the crimes.Los modelos de policía de comunitaria, junto con la georreferenciación del delito y el trabajo enfocado a la resolución de problemas concretos, han permitido mejorar los indicadores de atención al delito y la percepción de seguridad en las ciudades. Sin embargo, poca atención se ha prestado en estos modelos a las redes de cooperación entre patrullas organizadas por distritos o sectores urbanos, para mantener una acción articulada que compense la escasez de recursos policiales y evite su sobrefocalización. Este artículo analiza la interacción entre la red de refuerzos policiales y los homicidios y hurtos a persona en Bogotá, capital de Colombia. Se utilizan modelos de gráficos aleatorios exponenciales (ERG), análisis de redes y estimaciones gamma georreferenciadas del comportamiento delictivo. A nivel de los sectores policiales conocidos como Comandos de Atención Inmediata (CAI), se encuentra que los esquemas de organización de las redes de patrullaje están condicionados por los patrones de segregación socioeconómica de la ciudad, lo que limita el alcance de la red en el control de los delitos.os modelos de polícia comunitária, conjuntamente com o georreferenciamento do delito e o policiamento focado em problemas concretos, têm permitido melhorar os indicadores de atenção ao delito e a percepção de segurança nas cidades. No entanto, nesses modelos, pouca atenção tem sido dada às redes de cooperação entre patrulhas organizadas por distritos ou setores urbanos, a fim de manter uma atuação articulada que compense a escassez de recursos policiais e evite sua focalização excessiva. Este artigo analisa a interação entre a rede de reforços policiais e os homicídios e roubos em Bogotá, capital da Colômbia. São usados modelos de grafos aleatórios exponenciais (ERG, da abreviatura em inglês), análise de redes, e estimativas gamma georreferenciadas de comportamento criminoso. No nível dos setores policiais conhecidos como Comandos de Atenção Imediata (CAI), se verifica que os esquemas de organização em rede das patrulhas são condicionados pelos padrões de segregação socioeconômica da cidade, o que limita o escopo da rede no controle do crime