Maximum Likelihood Estimation for Stochastic Differential Equations with Random Effects

Abstract. We consider N independent stochastic processes (X _i (t), t ∈ [0,T _i ]), i=1,…, N, defined by a stochastic differential equation with drift term depending on a random variable φ _i . The distribution of the random effect φ _i depends on unknown parameters which are to be estimated from the continuous observation of the processes X_i. We give the expression of the exact likelihood. When the drift term depends linearly on the random effect φ _i and φ _i has Gaussian distribution, an explicit formula for the likelihood is obtained. We prove that the maximum likelihood estimator is consistent and asymptotically Gaussian, when T _i =T for all i and N tends to infinity. We discuss the case of discrete observations. Estimators are computed on simulated data for several models and show good performances even when the length time interval of observations is not very large.     

Parameter Estimation for a Bidimensional Partially Observed Ornstein–Uhlenbeck Process with Biological Application

Abstract. We consider a bidimensional Ornstein–Uhlenbeck process to describe the tissue microvascularization in anti‐cancer therapy. Data are discrete, partial and noisy observations of this stochastic differential equation (SDE). Our aim is to estimate the SDE parameters. We use the main advantage of a one‐dimensional observation to obtain an easy way to compute the exact likelihood using the Kalman filter recursion, which allows to implement an easy numerical maximization of the likelihood. Furthermore, we establish the link between the observations and an ARMA process and we deduce the asymptotic properties of the maximum likelihood estimator. We show that this ARMA property can be generalized to a higher dimensional underlying Ornstein–Uhlenbeck diffusion. We compare this estimator with the one obtained by the well‐known expectation maximization algorithm on simulated data. Our estimation methods can be directly applied to other biological contexts such as drug pharmacokinetics or hormone secretions.     

Reasons for Starting and Continuing Gambling in a Mixed Ethnic Community Sample of Pathological and Non-problem Gamblers

Very few studies have investigated motivational differences between pathological gamblers (PG) and non-problem gamblers (NPG), or between men and women. Motives for starting gambling have not been distinguished from motives for continuing gambling. From a community survey questionnaire listing reasons generated from the population studied, the motives of 103 current PG met the DSM-IV-TR criteria of five or more symptoms within the 12 months to October 2004. NPG assented to less than three symptoms. Generally, PG had significantly stronger motives than NPG and preferred continuous forms of gambling. There were no overall gender differences in motives, but electronic gaming machines and bingo were the primary choices for female PG. Gambling to escape from stress and troubles increased for PG but not for NPG, while gambling for social reasons decreased for both groups. Because of disproportionate numbers of PG and NPG in the Caucasian, Maori, Pacific Island and Asian groups, ethnic differences were not examined. The findings supported some aspects of theories of gambling motivation. Lowering stress for PG, raising community awareness of the risk of gambling to socialize and undertaking longitudinal research in community samples were suggested.     

Spatial Pattern and Determinants of Age at Marriage in Nigeria Using a Geo-Additive Survival Model

Age at first marriage, after controlling for socio-economic and demographic variables, varies across regions and districts. A geo-additive hazard model allows for measuring spatial effects. The nonlinear and baseline effects are modeled by Bayesian penalized splines; spatial components are treated as correlated random effects following a Markov random field. Application is based on data from 1999, 2003, and 2008 Nigeria Demographic and Health Surveys. Age at first marriage is positively associated with education and urbanization and depends on religion. It presents a north-south divide.