Here we consider penalized regression methods, and extend on the results surrounding the l1 norm penalty. We address a more recent development that generalizes previous methods by penalizing a linear transformation of the coefficients of interest instead of penalizing just the coefficients themselves. We introduce an approximate algorithm to fit this generalization and a fully Bayesian hierarchical model that is a direct analogue of the frequentist version. A number of benefits are derived from the Bayesian persepective; most notably choice of the tuning parameter and natural means to estimate the variation of estimates – a notoriously difficult task for the frequentist formulation. We then introduce Bayesian trend filtering which exemplifies the benefits of our Bayesian version. Bayesian trend filtering is shown to be an empirically strong technique for fitting univariate, nonparametric regression. Through a simulation study, we show that Bayesian trend filtering reduces prediction error and attains more accurate coverage probabilities over the frequentist method. We then apply Bayesian trend filtering to real data sets, where our method is quite competitive against a number of other popular nonparametric methods.

We define a subclass of dynamic linear models with unknown hyperparameters called d-inverse-gamma models. We then approximate the marginal p.d.f.s of the hyperparameter and the state vector by the data augmentation algorithm of Tanner/Wong. We prove that the regularity conditions for convergence hold. A sampling based scheme for practical implementation is discussed. Finally, we illustrate how to obtain an iterative importance sampling estimate of the model likelihood. (author's abstract) / Series: Forschungsberichte / Institut für Statistik

This dissertation is comprised of a collection of five distinct research projects which apply, evaluate and extend common methods for land surface data assimilation. The introduction of novel diagnostics and extensions of existing algorithms is motivated by an example, related to estimating agricultural productivity, of failed application of current methods. We subsequently develop methods, based on Shannon's theory of communication, to quantify the contributions from all possible factors to the residual uncertainty in state estimates after data assimilation, and to measure the amount of information contained in observations which is lost due to erroneous assumptions in the assimilation algorithm. Additionally, we discuss an appropriate interpretation of Shannon information which allows us to measure the amount of information contained in a model, and use this interpretation to measure the amount of information introduced during data assimilation-based system identification. Finally, we propose a generalization of the ensemble Kalman filter designed to alleviate one of the primary assumptions - that the observation function is linear.

This dissertation develops Bayesian methods to analyze data from auctions and produce policy recommendations for auction design. The essay, "Auction Design Using Bayesian Methods," proposes a decision theoretic method to choose a reserve price in an auction using data from past auctions. Our method formally incorporates parameter uncertainty and the payoff structure into the decision procedure. When the sample size is modest, it produces higher expected revenue than the plug-in methods. Monte Carlo evidence for this is provided. The second essay, "Flexible Bayesian Analysis of First Price Auctions Using Simulated Likelihood," develops an empirical framework that fully exploits all the shape restrictions arising from economic theory: bidding monotonicity and density affiliation. We directly model the valuation density so that bidding monotonicity is automatically satisfied, and restrict the parameter space to rule out all the nonaffiliated densities. Our method uses a simulated likelihood to allow for a very exible specification, but the posterior analysis is exact for the chosen likelihood. Our method controls the smoothness and tail behavior of the valuation density and provides a decision theoretic framework for auction design. We reanalyze a dataset of auctions for drilling rights in the Outer Continental Shelf that has been widely used in past studies. Our approach gives significantly different policy prescriptions on the choice of reserve price than previous methods, suggesting the importance of the theoretical shape restrictions. Lastly, in the essay, "Simple Approximation Methods for Bayesian Auction Design," we propose simple approximation methods for Bayesian decision making in auction design problems. Asymptotic posterior distributions replace the true posteriors in the Bayesian decision framework, which are typically a Gaussian model (second price auction) or a shifted exponential model (first price auction). Our method first approximates the posterior payoff using the limiting models and then maximizes the approximate posterior payoff. Both the approximate and exact Bayes rules converge to the true revenue maximizing reserve price under certain conditions. Monte Carlo studies show that my method closely approximates the exact procedure even for fairly small samples.

This paper presents the fully Bayesian analysis of the latent class model using a new approach towards MCMC estimation in the context of mixture models. The approach starts with estimating unidentified models for various numbers of classes. Exact Bayes' factors are computed by the bridge sampling estimator to compare different models and select the number of classes. Estimation of the unidentified model is carried out using the random permutation sampler. From the unidentified model estimates for model parameters that are not class specific are derived. Then, the exploration of the MCMC output from the unconstrained model yields suitable identifiability constraints. Finally, the constrained version of the permutation sampler is used to estimate group specific parameters. Conjoint data from the Austrian mineral water market serve to illustrate the method. (author's abstract) / Series: Report Series SFB "Adaptive Information Systems and Modelling in Economics and Management Science"

In the present paper we discuss Bayesian estimation of a very general model class where the distribution of the observations is assumed to depend on a latent mixture or switching variable taking values in a discrete state space. This model class covers e.g. finite mixture modelling, Markov switching autoregressive modelling and dynamic linear models with switching. Joint Bayesian estimation of all latent variables, model parameters and parameters determining the probability law of the switching variable is carried out by a new Markov Chain Monte Carlo method called permutation sampling. Estimation of switching and mixture models is known to be faced with identifiability problems as switching and mixture are identifiable only up to permutations of the indices of the states. For a Bayesian analysis the posterior has to be constrained in such a way that identifiablity constraints are fulfilled. The permutation sampler is designed to sample efficiently from the constrained posterior, by first sampling from the unconstrained posterior - which often can be done in a convenient multimove manner - and then by applying a suitable permutation, if the identifiability constraint is violated. We present simple conditions on the prior which ensure that this method is a valid Markov Chain Monte Carlo method (that is invariance, irreducibility and aperiodicity hold). Three case studies are presented, including finite mixture modelling of fetal lamb data, Markov switching Autoregressive modelling of the U.S. quarterly real GDP data, and modelling the U .S./U.K. real exchange rate by a dynamic linear model with Markov switching heteroscedasticity. (author's abstract) / Series: Forschungsberichte / Institut für Statistik

This study examined how marital conflict and marital environment contribute to change in marital status over time; while controlling for gender and other demographic characteristics. The current study used all three waves, 1987-1988, 1992-1994, 2001-2002, of the nationally representative dataset National Survey of Families and Households (NSFH). Four longitudinal models were tested using path analysis and the Bayesian estimation technique. Findings indicate there is no effect of marital conflict on change in marital status when demographic and marital environment variables are in the model. Age has the strongest direct and indirect effects. An increase in number of times married consistently increases the chance of a change in marital status. Variables measuring the marital environment—with the exception of the effects of unfairness of chores and spending money in the male models—primarily, contribute direct and mediating effects on the two measures of marital conflict. Overall, when considering all models, the variable with the strongest direct and indirect effects, is age of the respondent. This finding indicates that the dominate influence on marital environment and marital conflict, and, ultimately, change in marital status, is that of age as a proxy for developmental change over the lifecycle.

This thesis develops techniques for adjusting for selection bias using Gaussian process models. Selection bias is a key issue both in sample surveys and in observational studies for causal inference. Despite recently emerged techniques for dealing with selection bias in high-dimensional or complex situations, use of Gaussian process models and Bayesian hierarchical models in general has not been explored. Three approaches are developed for using Gaussian process models to estimate the population mean of a response variable with binary selection mechanism. The first approach models only the response with the selection probability being ignored. The second approach incorporates the selection probability when modeling the response using dependent Gaussian process priors. The third approach uses the selection probability as an additional covariate when modeling the response. The third approach requires knowledge of the selection probability, while the second approach can be used even when the selection probability is not available. In addition to these Gaussian process approaches, a new version of the Horvitz-Thompson estimator is also developed, which follows the conditionality principle and relates to importance sampling for Monte Carlo simulations. Simulation studies and the analysis of an example due to Kang and Schafer show that the Gaussian process approaches that consider the selection probability are able to not only correct selection bias effectively, but also control the sampling errors well, and therefore can often provide more efficient estimates than the methods tested that are not based on Gaussian process models, in both simple and complex situations. Even the Gaussian process approach that ignores the selection probability often, though not always, performs well when some selection bias is present. These results demonstrate the strength of Gaussian process models in dealing with selection bias, especially in high-dimensional or complex situations. These results also demonstrate that Gaussian process models can be implemented rather effectively so that the benefits of using Gaussian process models can be realized in practice, contrary to the common belief that highly flexible models are too complex to use practically for dealing with selection bias.

In this paper we head for a fully Bayesian analysis of the latent class model with a priori unknown number of classes. Estimation is carried out by means of Markov Chain Monte Carlo (MCMC) methods. We deal explicitely with the consequences the unidentifiability of this type of model has on MCMC estimation. Joint Bayesian estimation of all latent variables, model parameters, and parameters determining the probability law of the latent process is carried out by a new MCMC method called permutation sampling. In a first run we use the random permutation sampler to sample from the unconstrained posterior. We will demonstrate that a lot of important information, such as e.g. estimates of the subject-specific regression coefficients, is available from such an unidentified model. The MCMC output of the random permutation sampler is explored in order to find suitable identifiability constraints. In a second run we use the permutation sampler to sample from the constrained posterior by imposing identifiablity constraints. The unknown number of classes is determined by formal Bayesian model comparison through exact model likelihoods. We apply a new method of computing model likelihoods for latent class models which is based on the method of bridge sampling. The approach is applied to simulated data and to data from a metric conjoint analysis in the Austrian mineral water market. (author's abstract) / Series: Working Papers SFB "Adaptive Information Systems and Modelling in Economics and Management Science"

We examine the effects of monetary policy on income inequality in Japan using a novel econometric approach that jointly estimates the Gini coefficient based on micro-level grouped data of households and the dynamics of macroeconomic quantities. Our results indicate different effects on income inequality for different types of households: A monetary tightening increases inequality when income data is based on households whose head is employed (workers' households), while the effect reverses over the medium term when considering a broader definition of households. Differences in the relative strength of the transmission channels can account for this finding. Finally we demonstrate that the proposed joint estimation strategy leads to more informative inference while results based on the frequently used two-step estimation approach yields inconclusive results. / Series: Working Papers in Regional Science