Automatic estimation procedure in partial linear model with functional data

Partial linear modelling ideas have recently been adapted to situations when functional data are observed. This paper aims to complete the study of such model by proposing a fully automatic estimation procedure. This is achieved by constructing a data-driven method to choose the smoothing parameters entered in the nonparametric components of the model. The asymptotic optimality of the method is stated and its practical interest is illustrated on finite size Monte Carlo simulated samples.     

Estimation using plug-in of the stationary distribution and Shannon entropy of continuous time Markov processes

A natural way to deal with the uncertainty of an ergodic finite state space Markov process is to investigate the entropy of its stationary distribution. When the process is observed, it becomes necessary to estimate this entropy.We estimate both the stationary distribution and its entropy by plug-in of the estimators of the infinitesimal generator. Three situations of observation are discussed: one long trajectory is observed, several independent short trajectories are observed, or the process is observed at discrete times. The good asymptotic behavior of the plug-in estimators is established. We also illustrate the behavior of the estimators through simulation.     

Mean estimation with data missing at random for functional covariables

In a missing-data setting, we want to estimate the mean of a scalar outcome, based on a sample in which an explanatory variable is observed for every subject while responses are missing by happenstance for some of them. We consider two kinds of estimates of the mean response when the explanatory variable is functional. One is based on the average of the predicted values and the second one is a functional adaptation of the Horvitz–Thompson estimator. We show that the infinite dimensionality of the problem does not affect the rates of convergence by stating that the estimates are root-n consistent, under missing at random (MAR) assumption. These asymptotic features are completed by simulated experiments illustrating the easiness of implementation and the good behaviour on finite sample sizes of the method. This is the first paper emphasizing that the insensitiveness of averaged estimates, well known in multivariate non-parametric statistics, remains true for an infinite-dimensional covariable. In this sense, this work opens the way for various other results of this kind in functional data analysis.     

Estimation of generalized linear latent variable models

Summary.  Generalized linear latent variable models (GLLVMs), as defined by Bartholomew and Knott, enable modelling of relationships between manifest and latent variables. They extend structural equation modelling techniques, which are powerful tools in the social sciences. However, because of the complexity of the log-likelihood function of a GLLVM, an approximation such as numerical integration must be used for inference. This can limit drastically the number of variables in the model and can lead to biased estimators. We propose a new estimator for the parameters of a GLLVM, based on a Laplace approximation to the likelihood function and which can be computed even for models with a large number of variables. The new estimator can be viewed as an M -estimator, leading to readily available asymptotic properties and correct inference. A simulation study shows its excellent finite sample properties, in particular when compared with a well-established approach such as LISREL. A real data example on the measurement of wealth for the computation of multidimensional inequality is analysed to highlight the importance of the methodology.     

Quantitative Microbial Risk Assessment of Cryptosporidiosis and Giardiasis from Very Small Private Water Supplies

This article reports a quantitative microbial risk assessment of the risk of Giardia and Cryptosporidium in very small private water supplies. Both pathogens have been implicated in causing outbreaks of waterborne disease associated with such supplies, though the risk of endemic disease is not known. For exposure assessments, we used existing data to derive regression equations describing the relationships between the concentration of these pathogens and Escherichia coli in private water supplies. Pathogen concentrations were then estimated using national surveillance data of E. coli in private water supplies in England and France. The estimated risk of infection was very high with the median annual risk being of the order of 25–28% for Cryptosporidium and 0.4% to 0.7% for Giardia, though, in the poorer quality supplies the risk could be much higher. These risks are substantially greater than for public water supplies and well above the risk considered tolerable. The observation that observed infection rates are generally much lower may indicate increased immunity in people regularly consuming water from private supplies. However, this increased immunity is presumed to derive from increased disease risk in young children, the group most at risk from severe disease.     

CIVIL SOCIETY AND THE STATE: THE INTERPLAY BETWEEN COOPERATION AND MINIMUM WAGE REGULATION

In a cross‐section of countries, state regulation of labor markets is negatively correlated with the quality of labor relations. In this paper, we argue that these facts reflect different ways of regulating labor markets, either through the state or through the civil society, depending on the degree of cooperation in the economy. We rationalize these facts with a model of learning of the quality of labor relations. Distrustful labor relations lead to low unionization and high demand for direct state regulation of wages. In turn, state regulation crowds out the possibility for workers to experiment negotiation and learn about the potential cooperative nature of labor relations. This crowding out effect can give rise to multiple equilibria: a “good” equilibrium characterized by cooperative labor relations and high union density, leading to low state regulation; and a “bad” equilibrium, characterized by distrustful labor relations, low union density, and strong state regulation of the minimum wage.