This thesis takes a step towards developing a vine-based regression method tailored to mixed-type (continuous & discrete) data. We visualize the difference between continuous conditional quantile functions of bivariate copulas and their discretized (discretized by binning the continuous variable into bins of equal probability) conditional quantile functions. This showed how discretization into a small number of bins loses dependence, especially in the tails. We show a different binning procedure can improve the preservation of dependence after discretization, motivating the development of methodologies which compute an ‘optimal’ (optimal in the sense of balancing tail dependence with dependence around the median) binning procedure given some data. For application in computing an ‘optimal’ binning procedure, we make an attempt at analytically quantifying the difference between discretized conditional quantiles and continuous conditional quantiles of bivariate copulas, but the analytical derivation is not possible for most settings (with exceptions for bivariate Clayton and bivariate Frank copulas, and the smallest conditioning value of discretized covariate).

From the aforementioned conclusion that discretization loses dependence, we infer a variable
selection measure tailored to mixed-type data should be biased against discretized covariates,
and that this bias should be monotone decreasing with the number of bins of the discretized
covariate. The bias for/against discretized covariates of various variable selection measures is
investigated, in a scenario with a single covariate and a scenario with two covariates. With a single covariate, Pearson’s/polyserial correlation and Kendall’s tau/tau-b were found unsuitable as variable selection measures for mixed-type data, due to a lack of bias against discretized covariates. Conditional log-likelihood and check-loss at the quantile level 0.05 seem nearly identically biased in the bivariate setting, although this should be said with the caveat that all simulation scenarios are homoskedastic. Finally, we show in a three-dimensional setting that correlation between covariates does not seem to affect the predictive performance when both covariates are continuous, but correlation between covariates has significant negative effects on the predictive performance when one of the covariates is discretized. This difference between the effect of discretization in two dimensions and three dimensions should be kept in mind when developing variable selection procedures for mixed-type data.

Statistical models often require more insights than the estimated values of a model. When these insights are not available by analytical means, one often resorts to resampling schemes. The most well known of which are bootstrapping and cross-validation. These techniques are very flexible and allow you to gain insight on whether you are overfitting your model, the distribution and bias of your estimators, the expected prediction error of regression models and many more. These then allow you to make statistical inferences such as hypothesis testing. The problem with these methods is the computational cost of repeatedly refitting models. When you are using Zestimators, the Swiss army infinitesimal jackknife is an approximation to what your parameters would be had they been estimated from a different weighting of the data. This approximation is very easy to compute and allows for very quick approximation of the reweighted parameter estimates. Giordano et al. 2020 provides a bound on the error of this approximation. This error bound relies on constants that are hard to compute, thus the result is not easily interpretable. In the case of applying the Swiss army infinitesimal jackknife to the case of estimating the parameter from a subset of the data, we build on the results by Giordano et al. 2020 by applying asymptotic theory. This leads to a much more interpretable result about the approximation error. By a simulation study we show that this asymptotic result still works well in a finite sample.

This thesis investigates the Ground Motion Model (GMM) for the Groningen Seismic Hazard & Risk Analysis. We look at various aspects of the model to see where improvements can be made. We start by looking at model calibration and validation, where we check to what extent the proposed model along with its parameter fits the data. Here we come to the conclusion that both the parameters and the model itself have room for further optimization to reflect the data set more accurately. In addition to this, a new model for correlations of site-response amplifications is presented. The topic of how the dependence of these quantities should be modeled is still unclear. The lack of coherent solution to this modeling problem makes the proposed model valuable, as it is simple in nature and reflects the data we have the best.
Lastly, the main improvements that come out of this research are in on the computational front, by finding a novel method for calculating average spectral accelerations, which is the main quantity used in risk assessment. This method generates the distribution of this quantity in one step using numerical integration rather than the previously used Monte Carlo method. The method speeds up computations by a factor of 550 times.