Most Likely Transformations

We propose and study properties of maximum likelihood estimators in the class of conditional transformation models. Based on a suitable explicit parameterisation of the unconditional or conditional transformation function, we establish a cascade of increasingly complex transformation models that can be estimated, compared and analysed in the maximum likelihood framework. Models for the unconditional or conditional distribution function of any univariate response variable can be set-up and estimated in the same theoretical and computational framework simply by choosing an appropriate transformation function and parameterisation thereof. The ability to evaluate the distribution function directly allows us to estimate models based on the exact likelihood, especially in the presence of random censoring or truncation. For discrete and continuous responses, we establish the asymptotic normality of the proposed estimators. A reference software implementation of maximum likelihood-based estimation for conditional transformation models allowing the same flexibility as the theory developed here was employed to illustrate the wide range of possible applications.Comment: Accepted for publication by the Scandinavian Journal of Statistics 2017-06-1

Bayesian semiparametric multi-state models

Multi-state models provide a unified framework for the description of the evolution of discrete phenomena in continuous time. One particular example is Markov processes which can be characterised by a set of time-constant transition intensities between the states. In this paper, we will extend such parametric approaches to semiparametric models with flexible transition intensities based on Bayesian versions of penalised splines. The transition intensities will be modelled as smooth functions of time and can further be related to parametric as well as nonparametric covariate effects. Covariates with time-varying effects and frailty terms can be included in addition. Inference will be conducted either fully Bayesian (using Markov chain Monte Carlo simulation techniques) or empirically Bayesian (based on a mixed model representation). A counting process representation of semiparametric multi-state models provides the likelihood formula and also forms the basis for model validation via martingale residual processes. As an application, we will consider human sleep data with a discrete set of sleep states such as REM and non-REM phases. In this case, simple parametric approaches are inappropriate since the dynamics underlying human sleep are strongly varying throughout the night and individual specific variation has to be accounted for using covariate information and frailty terms

Gaussian processes and Bayesian moment estimation

Given a set of moment restrictions (MRs) that overidentify a parameter $\theta$, we investigate a semiparametric Bayesian approach for inference on $\theta$ that does not restrict the data distribution $F$ apart from the MRs. As main contribution, we construct a degenerate Gaussian process prior that, conditionally on $\theta$, restricts the $F$ generated by this prior to satisfy the MRs with probability one. Our prior works even in the more involved case where the number of MRs is larger than the dimension of $\theta$. We demonstrate that the corresponding posterior for $\theta$ is computationally convenient. Moreover, we show that there exists a link between our procedure, the Generalized Empirical Likelihood with quadratic criterion and the limited information likelihood-based procedures. We provide a frequentist validation of our procedure by showing consistency and asymptotic normality of the posterior distribution of $\theta$. The finite sample properties of our method are illustrated through Monte Carlo experiments and we provide an application to demand estimation in the airline market

Bayesian Semiparametric Multi-State Models

Multi-state models provide a unified framework for the description of the evolution of discrete phenomena in continuous time. One particular example are Markov processes which can be characterised by a set of time-constant transition intensities between the states. In this paper, we will extend such parametric approaches to semiparametric models with flexible transition intensities based on Bayesian versions of penalised splines. The transition intensities will be modelled as smooth functions of time and can further be related to parametric as well as nonparametric covariate effects. Covariates with time-varying effects and frailty terms can be included in addition. Inference will be conducted either fully Bayesian using Markov chain Monte Carlo simulation techniques or empirically Bayesian based on a mixed model representation. A counting process representation of semiparametric multi-state models provides the likelihood formula and also forms the basis for model validation via martingale residual processes. As an application, we will consider human sleep data with a discrete set of sleep states such as REM and Non-REM phases. In this case, simple parametric approaches are inappropriate since the dynamics underlying human sleep are strongly varying throughout the night and individual-specific variation has to be accounted for using covariate information and frailty terms

Bayesian Semiparametric Multi-State Models

Multi-state models provide a unified framework for the description of the evolution of discrete phenomena in continuous time. One particular example are Markov processes which can be characterised by a set of time-constant transition intensities between the states. In this paper, we will extend such parametric approaches to semiparametric models with flexible transition intensities based on Bayesian versions of penalised splines. The transition intensities will be modelled as smooth functions of time and can further be related to parametric as well as nonparametric covariate effects. Covariates with time-varying effects and frailty terms can be included in addition. Inference will be conducted either fully Bayesian using Markov chain Monte Carlo simulation techniques or empirically Bayesian based on a mixed model representation. A counting process representation of semiparametric multi-state models provides the likelihood formula and also forms the basis for model validation via martingale residual processes. As an application, we will consider human sleep data with a discrete set of sleep states such as REM and Non-REM phases. In this case, simple parametric approaches are inappropriate since the dynamics underlying human sleep are strongly varying throughout the night and individual-specific variation has to be accounted for using covariate information and frailty terms

Multiple Imputation Using Gaussian Copulas

Missing observations are pervasive throughout empirical research, especially in the social sciences. Despite multiple approaches to dealing adequately with missing data, many scholars still fail to address this vital issue. In this paper, we present a simple-to-use method for generating multiple imputations using a Gaussian copula. The Gaussian copula for multiple imputation (Hoff, 2007) allows scholars to attain estimation results that have good coverage and small bias. The use of copulas to model the dependence among variables will enable researchers to construct valid joint distributions of the data, even without knowledge of the actual underlying marginal distributions. Multiple imputations are then generated by drawing observations from the resulting posterior joint distribution and replacing the missing values. Using simulated and observational data from published social science research, we compare imputation via Gaussian copulas with two other widely used imputation methods: MICE and Amelia II. Our results suggest that the Gaussian copula approach has a slightly smaller bias, higher coverage rates, and narrower confidence intervals compared to the other methods. This is especially true when the variables with missing data are not normally distributed. These results, combined with theoretical guarantees and ease-of-use suggest that the approach examined provides an attractive alternative for applied researchers undertaking multiple imputations