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.     

Testing Monotonicity of Regression Functions – An Empirical Process Approach

We propose several new tests for monotonicity of regression functions based on different empirical processes of residuals and pseudo‐residuals. The residuals are obtained from an unconstrained kernel regression estimator whereas the pseudo‐residuals are obtained from an increasing regression estimator. Here, in particular, we consider a recently developed simple kernel‐based estimator for increasing regression functions based on increasing rearrangements of unconstrained non‐parametric estimators. The test statistics are estimated distance measures between the regression function and its increasing rearrangement. We discuss the asymptotic distributions, consistency and small sample performances of the tests.     

Robust location estimation with missing data

In a missing data setting, we have a sample in which a vector of explanatory variables ${\bf x}_i$ is observed for every subject i, while scalar responses $y_i$ are missing by happenstance on some individuals. In this work we propose robust estimators of the distribution of the responses assuming missing at random (MAR) data, under a semiparametric regression model. Our approach allows the consistent estimation of any weakly continuous functional of the response's distribution. In particular, strongly consistent estimators of any continuous location functional, such as the median, L‐functionals and M‐functionals, are proposed. A robust fit for the regression model combined with the robust properties of the location functional gives rise to a robust recipe for estimating the location parameter. Robustness is quantified through the breakdown point of the proposed procedure. The asymptotic distribution of the location estimators is also derived. The proofs of the theorems are presented in Supplementary Material available online. The Canadian Journal of Statistics 41: 111–132; 2013 © 2012 Statistical Society of Canada     

Characteristic Function‐based Semiparametric Inference for Skew‐symmetric Models

Skew‐symmetric models offer a very flexible class of distributions for modelling data. These distributions can also be viewed as selection models for the symmetric component of the specified skew‐symmetric distribution. The estimation of the location and scale parameters corresponding to the symmetric component is considered here, with the symmetric component known. Emphasis is placed on using the empirical characteristic function to estimate these parameters. This is made possible by an invariance property of the skew‐symmetric family of distributions, namely that even transformations of random variables that are skew‐symmetric have a distribution only depending on the symmetric density. A distance metric between the real components of the empirical and true characteristic functions is minimized to obtain the estimators. The method is semiparametric, in that the symmetric component is specified, but the skewing function is assumed unknown. Furthermore, the methodology is extended to hypothesis testing. Two tests for a null hypothesis of specific parameter values are considered, as well as a test for the hypothesis that the symmetric component has a specific parametric form. A resampling algorithm is described for practical implementation of these tests. The outcomes of various numerical experiments are presented.     

Generalized estimating equations for mixtures with varying concentrations

A finite mixture model is considered in which the mixing probabilities vary from observation to observation. A parametric model is assumed for one mixture component distribution, while the others are nonparametric nuisance parameters. Generalized estimating equations (GEE) are proposed for the semi‐parametric estimation. Asymptotic normality of the GEE estimates is demonstrated and the lower bound for their dispersion (asymptotic covariance) matrix is derived. An adaptive technique is developed to derive estimates with nearly optimal small dispersion. An application to the sociological analysis of voting results is discussed. The Canadian Journal of Statistics 41: 217–236; 2013 © 2013 Statistical Society of Canada     

Multivariate Kendall's tau for change‐point detection in copulas

Statistical procedures for the detection of a change in the dependence structure of a series of multivariate observations are studied in this work. The test statistics that are proposed are $L_1$ , $L_2$ , and $L_{\infty }$ distances computed from vectors of differences of Kendall's tau; two multivariate extensions of Kendall's measure of association are used. Since the distributions of these statistics under the null hypothesis of no change depend on the unknown underlying copula of the vectors, a procedure based on the multiplier central limit theorem is used for the computation of p‐values; the method is shown to be valid both asymptotically and for moderate sample sizes. Alternative versions of the tests that take into account possible breakpoints in the marginal distributions are also investigated. Monte Carlo simulations show that the tests are powerful under many scenarios of change‐point. In addition, two estimators of the time of change are proposed and their efficiency is carefully studied. The methodologies are illustrated on simulated series from the Canadian Regional Climate Model. The Canadian Journal of Statistics 41: 65–82; 2013 © 2012 Statistical Society of Canada     

Quasi-optimal Bayesian procedures of many hypotheses testing

Quasi-optimal procedures of testing many hypotheses are described in this paper. They significantly simplify the Bayesian algorithms of hypothesis testing and computation of the risk function. The relations allowing for obtaining the estimations for the values of average risks in optimum tasks are given. The obtained general solutions are reduced to concrete formulae for a multivariate normal distribution of probabilities. The methods of approximate computation of the risk functions in Bayesian tasks of testing many hypotheses are offered. The properties and interrelations of the developed methods and algorithms are investigated. On the basis of a simulation, the validity of the obtained results and conclusions drawn is presented.     

Patterns of individual change and program satisfaction in a positive parenting program for parents at psychosocial risk

Despite the evidence of effectiveness of positive parenting programs, little is known about the typology of changes that parents at psychosocial risk undergo after an intervention. We compared individual patterns of change in three parenting outcomes in 256 at risk parents with young children attending the group‐based Growing Up Happily in the Family program delivered in municipal social services. We identified four clusters of individual change: Cluster 1 (30.6%) had negative changes in parental child‐rearing attitudes and parenting stress, Cluster 2 (27.7%) had positive changes in child‐rearing attitudes and negative results in parental perceived competence, and Cluster 3 (24.1%) and Cluster 4 (17.6%) showed overall better results. Residential area, type of social support, and quality of implementation characterized cluster membership. Participants in clusters with better results were more satisfied with the program than those with worse results. Practical recommendations are provided for the successful implementation of group parenting programs in family preservation services.     

存在议价能力和副产品协同生产的制造商最优产量决策

考虑由环保型制造商和处理厂构成的两级供应链结构，其中上游制造商进行主产品的生产，而下游处理厂通过对废料加工进行副产品的生产。基于供应链成员之间存在不同议价能力的情形，构建了上游制造商将产生的废料给下游进行处理的协同生产模式，给出了主产品的均衡产量决策和废料的最优交易价格以及制造商关于废料的最优处理策略。研究结果表明，制造商的废料处理策略取决于消费者对副产品产量的敏感程度。当消费者对副产品产量较敏感（不敏感）时，制造商会选择自己处理废料（给处理厂进行加工）。此外，随着制造商议价能力的上升，制造商的均衡利润上升，而处理厂的均衡利润下降。最后发现本文的基本结论在随机产出环境下依然稳健。     

What you say,and how you say it: Preschoolers' growth in vocabulary and communication skills differentially predict kindergarten academic achievement and self‐regulation

The idea that language skills support school readiness, predicting later self‐regulation and academic success, is widely accepted. Although vocabulary is often emphasized in the developmental literature, the ability to use language appropriately in the classroom, or social communication skills, may also be critical. This article examined longitudinal contributions of children's vocabulary and social communication skills, from preschool to kindergarten, to kindergarten academic achievement (reading and math) and self‐regulation (executive functions and learning behaviors). Participants were 164 children (14% Latinx, 30% Black, 56% White; 57% girls) enrolled in Head Start programs. Results revealed that initial levels and growth in vocabulary and communication skills predicted better academic achievement. Social communication skills uniquely predicted self‐regulation, after accounting for vocabulary. We discuss potential mechanisms for these links and recommend that strategies to build social communication skills be incorporated in preschool interventions promoting school readiness.