This bouquet of essays contains four chapters.

In the first chapter I present a brief summary of the literature of misspecified models. I discuss what various estimators are actually estimating when the model is misspecified. Further I discuss corrections to standard errors and when they are useful. I briefly cover hypothesis testing in the presence of misspecified models. I cover both frequentist and Bayesian approached. I show that a misspecified model can indeed be useful and discuss some misconceptions with misspecified models.

The second chapter investigates the impact of misspecification in discrete choice models. I derive necessary and sufficient conditions for consistency of the maximum likelihood estimator from the misspecified model. A corollary is that the misspecified estimator is consistent for the correct sign, under certain conditions. It also follows that Huber-White standard errors can be used to obtain asymptotically conservative type one errors when testing the nullity of the coefficient.

The third chapter builds a Bayesian model for multiple-output quantiles using a commonly accepted definition for the quantile. The prior can be elicited as the ex-ante knowledge of Tukey depth, the first prior of its kind. I apply the model to the Tennessee Project STAR experiment and find there is a joint increase in \emph{all quantile subpopulations} for reading and mathematics scores given a decrease in the number of students per teacher. This result is consistent with, and much stronger than, the results from previous studies.

The four chapter I investigate if United States Supreme Court Justices recuse themselves strategically. I create a new structural model of recusals. Using this model I find causal evidence that justices recuse themselves strategically. I then calibrate and simulate the model to find the frequency of cases where at least one justice has a conflict of interest but does not recuse. I find at most 47\% of cases have at least one justice with a conflict of interest that did not recuse.

It was difficult to come up with an overarching theme for this bouquet of essays -- hence the title. The closest theme would be `model misspecification.' The first chapter provides an overview of misspecified models, the second chapter investigates a misspecified discrete choice models and the third chapter purposefully uses a misspecified model to get an interesting estimator. However, the closest the fourth chapter gets to `model misspecification' is the use of the Kullback-Leibler distance in a simulation. The Kullback-Leibler distance is often used in the misspecified literature but there is nothing about the distance that makes it inherently related to misspecified models.

Childhood trauma constitutes a major public health crisis in the United States, with an estimated two thirds of children experiencing at least one traumatic event by the age of 16. Despite the prevalence of childhood trauma, there has been very little sociological research on this subject. Using rich data from the National Longitudinal Study of Adolescent Health, the current dissertation project offers important new insights on the contours and consequences of childhood trauma. The first analytic chapter shows how neighborhood racial segregation is an important determinant for exposure to multiple types of violence. The findings suggest that adolescents living in more racially segregated communities are more likely to report exposure to violence. The second analytic chapter examines whether exposure to multiple traumas or “complex trauma” mediates the association between family structure and various “markers” of life chances in adulthood. The analysis shows that children born to single mother families, relative to children who are born to two biological or adoptive parent families, are more likely to experience complex trauma in childhood, which accounts for a significant portion of the association between family structure, adult incarceration, and college completion. Finally, the third analytic chapter examines the relationship between witnessing community violence and criminal legal system involvement in young adulthood. This analysis shows that adolescents who witness community violence are significantly more likely to become incarcerated in young adulthood. The analysis also finds significant racial disparities in witnessing violence, suggesting that exposure to community violence is a potentially important mechanism in the production of racial inequalities in criminal legal system involvement.

The focus of this dissertation is the development of a positron beam with significantly improved energy resolution over any beam resolution previously available. While positron interactions with matter are important in a variety of contexts, the range of experimental data available regarding fundamental positron-matter interactions is severely limited as compared to analogous electron-matter processes. This difference is due largely to the difficulties encountered in creating positron beams with narrow energy spreads. Described here is a detailed investigation into the physical processes operative during positron cooling and beam formation in state-of-the-art, trap-based beam systems. These beams rely on buffer gas traps (BGTs), in which positrons are trapped and cooled to the ambient temperature (300~K) through interactions with a molecular gas, and subsequently ejected as a high resolution pulsed beam.

Experimental measurements, analytic models, and simulation results are used to understand the creation and characterization of these beams, with a focus on the mechanisms responsible for setting beam energy resolution. The information gained from these experimental and theoretical studies was then used to design, construct, and operate a next-generation high-energy-resolution beam system. In this new system, the pulsed beam from the BGT is magnetically guided into a new apparatus which re-traps the positrons, cools them to 50~K, and re-emits them as a pulsed beam with superior beam characteristics.

Using these techniques, positron beams with total energy spreads as low as 6.9~meV FWHM are produced. This represents a factor of $\sim 5$ improvement over the previous state-of-the-art, making it the largest increase in positron beam energy resolution since the development of advanced moderator techniques in the early 1980's. These beams also have temporal spreads of $0.9~\mu$s FWHM and radial spreads of 1~mm FWHM. This represents improvements by factors of $\sim 2$ and $10$, respectively, over those of the previous beam resolutions. Future experimental applications of this new technology are also discussed.

What does a molecule do after it absorbs a photon? This dissertation contains a compilation of my published work on this very question on dye molecules and alternative nucleobases, as well as notes on laboratory operation. All experimental work presented is done with some variation of Resonance Enhanced Multi-Photon Ionization (REMPI) on neutral gas phase molecules around 10 K prepared by laser desorption and entrained into a supersonic molecular beam. Detection is achieved with a time of flight mass spectrometer. When a molecule absorbs a photon, it must find a way to deal with that energy. Pigments in paintings must absorb light as part of their function, otherwise they would have no color. DNA bases absorb in the UV region because they are aromatic, which provides them the rigidity needed for a pairing structure. In both cases, some of these compounds deal with radiation better than others, displaying superior photostability. The ones that do well use some pathway that safely and rapidly converts the energy to heat, which can then be safely dissipated to the environment. Those pathways are a function of the potential energy surface, which is acutely dependent on the molecular structure. As a result, isomers, and even tautomers of a molecule can display very different dynamics. The experiment performed allows for a separation of tautomers and isomers, and as such can provide a benchmark to state-of-the-art calculations. In the first two chapters, examples of visible absorbing pigments and alternative DNA bases will be presented. In the first chapter, two papers on visible absorbing dyes are presented, with examination of the excited state intramolecular proton transfer mechanism in each system. In the first paper, we find that the indigo dye molecule is quite extraordinary in that both a hydrogen transfer and proton transfer geometry are populated following excitation. The dominant geometry varies depending on the vibronic mode excited, with each geometry displaying different characteristic lifetimes. The different lifetimes for each of these geometries allows some estimates of their yields as a function of vibronic band excitation. At high enough energy excitation, there is an interconversion between the two geometries, allowing indigo to use the fastest path available and safely dissipate the energy following excitation. The second paper analyzes a series of hydroxyanthraquinone molecules, which are red dyes or pigments. By studying a series of these dyes and comparing with high levels of calculations, the reason for the lack of photostability for hydroxyanthraquinones substituted with a 1,4- motif becomes clear; the proton transfer geometry is destabilized relative to the Frank-Condon region for molecules bearing this motif, and as such they cannot use this efficient relaxation pathway. The next chapter contains a pair of articles on a pair of alternative nucleobases, isocytosine and isoguanine. These articles seek to identify if there was a photochemical reason for cytosine and guanine to be chosen over isocytosine and isoguanine. Interestingly, we find that isocytosine bears some similarity in dynamics to guanine, while isoguanine bears some similarities to cytosine. The lower photostability of the biologically relevant tautomeric form of isoguanine is suggested to be a reason for the exclusion of isocytosine and isoguanine from biology. In the last chapter, the focus shifts from studying photophysics to analyzing mass spectra of Poly-Aromatic Hydrocarbons (PAHs) prepared in a plasma jet. These molecules are found in a wide variety of settings, from soots and combustion products to the interstellar media. In this paper, the early stages and mechanism of carbon growth are explored in a simulation of space conditions.

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