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Capture-recapture (CRC) models use two or more samples, or lists, to estimate the size of a population. In the canonical example, a researcher captures, marks, and releases several samples of fish in a lake. When the fish that are captured more than once are few compared to the total number that are captured, one suspects that the lake contains many more uncaptured fish. This basic intuition motivates CRC models in fields as diverse as epidemiology, entomology, and computer science. We use simulations to study the performance of conventional log-linear models for CRC. Specifically we evaluate model selection criteria, model averaging, an asymptotic variance formula, and several small-sample data adjustments. Next, we argue that interpretable models are essential for credible inference, since sets of models that fit the data equally well can imply vastly different estimates of the population size. A secondary analysis of data on survivors of the World Trade Center attacks illustrates this issue. Our main chapter develops local log-linear models. Heterogeneous populations tend to bias conventional log-linear models. Post-stratification can reduce the effects of heterogeneity by using covariates, such as the age or size of each observed unit, to partition the data into relatively homogeneous post-strata. One can fit a model to each post-stratum and aggregate the resulting estimates across post-strata. We extend post-stratification to its logical extreme by selecting a local log-linear model for each observed point in the covariate space, while smoothing to achieve stability. Local log-linear models serve a dual purpose. Besides estimating the population size, they estimate the rate of missingness as a function of covariates. Simulations demonstrate the superiority of local log-linear models for estimating local rates of missingness for special cases in which the generating model varies over the covariate space. We apply the method to estimate bird species richness in continental North America and to estimate the prevalence of multiple sclerosis in a region of France.

We consider a time series model with autoregressive conditional heteroskedasticity that is subject to changes in regime. The regimes evolve according to a multistate latent Markov switching process with unknown transition probabilities, and it is the constant in the variance process of the innovations that is subject to regime shifts. The joint estimation of the latent process and all model parameters is performed within a Bayesian framework using the method of Markov Chain Monte Carlo simulation. We perform model selection with respect to the number of states and the number of autoregressive parameters in the variance process using Bayes factors and model likelihoods. To this aim, the model likelihood is estimated by combining the candidate's formula with importance sampling. The usefulness of the sampler is demonstrated by applying it to the dataset previously used by Hamilton and Susmel who investigated models with switching autoregressive conditional heteroskedasticity using maximum likelihood methods. The paper concludes with some issues related to maximum likelihood methods, to classical model select ion, and to potential straightforward extensions of the model presented here. (author's abstract) / Series: Forschungsberichte / Institut für Statistik

abstract: The primary objective in time series analysis is forecasting. Raw data often exhibits nonstationary behavior: trends, seasonal cycles, and heteroskedasticity. After data is transformed to a weakly stationary process, autoregressive moving average (ARMA) models may capture the remaining temporal dynamics to improve forecasting. Estimation of ARMA can be performed through regressing current values on previous realizations and proxy innovations. The classic paradigm fails when dynamics are nonlinear; in this case, parametric, regime-switching specifications model changes in level, ARMA dynamics, and volatility, using a finite number of latent states. If the states can be identified using past endogenous or exogenous information, a threshold autoregressive (TAR) or logistic smooth transition autoregressive (LSTAR) model may simplify complex nonlinear associations to conditional weakly stationary processes. For ARMA, TAR, and STAR, order parameters quantify the extent past information is associated with the future. Unfortunately, even if model orders are known a priori, the possibility of over-fitting can lead to sub-optimal forecasting performance. By intentionally overestimating these orders, a linear representation of the full model is exploited and Bayesian regularization can be used to achieve sparsity. Global-local shrinkage priors for AR, MA, and exogenous coefficients are adopted to pull posterior means toward 0 without over-shrinking relevant effects. This dissertation introduces, evaluates, and compares Bayesian techniques that automatically perform model selection and coefficient estimation of ARMA, TAR, and STAR models. Multiple Monte Carlo experiments illustrate the accuracy of these methods in finding the "true" data generating process. Practical applications demonstrate their efficacy in forecasting. / Dissertation/Thesis / Doctoral Dissertation Statistics 2018

Reliable forecasting of wind power generation is crucial to optimal control of costs in generation of electricity with respect to the electricity demand. Here, we propose and analyze stochastic wind power forecast models described by parametrized stochastic differential equations, which introduce appropriate fluctuations in numerical forecast outputs. We use an approximate maximum likelihood method to infer the model parameters taking into account the time correlated sets of data. Furthermore, we study the validity and sensitivity of the parameters for each model. We applied our models to Uruguayan wind power production as determined by historical data and corresponding numerical forecasts for the period of March 1 to May 31, 2016.

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