Universitätsbibliothek

Prediction intervals are needed in many industrial applications. Frequently in mass production, small subgroups of unknown size with a lifetime behavior differing from the remainder of the population exist. A risk assessment for such a subgroup consists of two steps: i) the estimation of the subgroup size, and ii) the estimation of the lifetime behavior of this subgroup. This thesis covers both steps. An efficient practical method to estimate the size of a subgroup is presented and benchmarked against other methods. A prediction interval procedure which includes prior information in form of a Beta distribution is provided. This scheme is applied to the prediction of binomial and negative binomial counts. The effect of the population size on the prediction of the future number of failures is considered for a Weibull lifetime distribution, whose parameters are estimated from censored field data. Methods to obtain a prediction interval for the future number of failures with unknown sample size are presented. In many applications, failures are reported with a delay. The effects of such a reporting delay on the coverage properties of prediction intervals for the future number of failures are studied. The total failure probability of the two steps can be decomposed as a product probability. One-sided confidence intervals for such a product probability are presented.

The starting point of the thesis is the {\it universality} property of the Riemann Zeta-function $\zeta(s)$ which was proved by Voronin in 1975: {\it Given a positive number $\varepsilon>0$ and an analytic non-vanishing function $f$ defined on a compact subset $\mathcal{K}$ of the strip $\left\{s\in\mathbb{C}:1/2 < \Re s< 1\right\}$ with connected complement, there exists a real number $\tau$ such that \begin{align}\label{continuous} \max\limits_{s\in \mathcal{K}}|\zeta(s+i\tau)-f(s)|<\varepsilon. \end{align} } In 1980, Reich proved a discrete analogue of Voronin’s theorem, also known as {\it discrete universality theorem} for $\zeta(s)$: {\it If $\mathcal{K}$, $f$ and $\varepsilon$ are as before, then \begin{align}\label{discretee} \liminf\limits_{N\to\infty}\dfrac{1}{N}\sharp\left\{1\leq n\leq N:\max\limits_{s\in \mathcal{K}}|\zeta(s+i\Delta n)-f(s)|<\varepsilon\right\}>0, \end{align} where $\Delta$ is an arbitrary but fixed positive number. } We aim at developing a theory which can be applied to prove the majority of all so far existing discrete universality theorems in the case of Dirichlet $L$-functions $L(s,\chi)$ and Hurwitz zeta-functions $\zeta(s;\alpha)$, where $\chi$ is a Dirichlet character and $\alpha\in(0,1]$, respectively. Both of the aforementioned classes of functions are generalizations of $\zeta(s)$, since $\zeta(s)=L(s,\chi_0)=\zeta(s;1)$, where $\chi_0$ is the principal Dirichlet character mod 1. Amongst others, we prove statement (2) where instead of $\zeta(s)$ we have $L(s,\chi)$ for some Dirichlet character $\chi$ or $\zeta(s;\alpha)$ for some transcendental or rational number $\alpha\in(0,1]$, and instead of $(\Delta n)_{n\in\mathbb{N}}$ we can have: \begin{enumerate} \item \textit{Beatty sequences,} \item \textit{sequences of ordinates of $c$-points of zeta-functions from the Selberg class,} \item \textit{sequences which are generated by polynomials.} \end{enumerate} In all the preceding cases, the notion of {\it uniformly distributed sequences} plays an important role and we draw attention to it wherever we can. Moreover, for the case of polynomials, we employ more advanced techniques from Analytic Number Theory such as bounds of exponential sums and zero-density estimates for Dirichlet $L$-functions. This will allow us to prove the existence of discrete second moments of $L(s,\chi)$ and $\zeta(s;\alpha)$ on the left of the vertical line $1+i\mathbb{R}$, with respect to polynomials. In the case of the Hurwitz Zeta-function $\zeta(s;\alpha)$, where $\alpha$ is transcendental or rational but not equal to $1/2$ or 1, the target function $f$ in (1) or (2), where $\zeta(\cdot)$ is replaced by $\zeta(\cdot;\alpha)$, is also allowed to have zeros. Until recently there was no result regarding the universality of $\zeta(s;\alpha)$ in the literature whenever $\alpha$ is an algebraic irrational. In the second half of the thesis, we prove that a weak version of statement \eqref{continuous} for $\zeta(s;\alpha)$ holds for all but finitely many algebraic irrational $\alpha$ in $[A,1]$, where $A\in(0,1]$ is an arbitrary but fixed real number. Lastly, we prove that the ordinary Dirichlet series $\zeta(s;f)=\sum_{n\geq1}f(n)n^{-s}$ and $\zeta_\alpha(s)=\sum_{n\geq1}\lfloor P(\alpha n+\beta)\rfloor^{-s}$ are hypertranscendental, where $f:\mathbb{N}\to\mathbb{C}$ is a {\it Besicovitch almost periodic arithmetical function}, $\alpha,\beta>0$ are such that $\lfloor\alpha+\beta\rfloor>1$ and $P\in\mathbb{Z}[X]$ is such that $P(\mathbb{N})\subseteq\mathbb{N}$.

This dissertation deals with composite-based methods for structural equation models with latent variables and their enhancement. It comprises five chapters. Besides a brief introduction in the first chapter, the remaining chapters consisting of four essays cover the results of my PhD studies.Two of the essays have already been published in an international journal. The first essay considers an alternative way of construct modeling in structural equation modeling.While in social and behavioral sciences theoretical constructs are typically modeled as common factors, in other sciences the common factor model is an inadequate way construct modeling due to its assumptions. This essay introduces the confirmatory composite analysis (CCA) analogous to confirmatory factor analysis (CFA). In contrast to CFA, CCA models theoretical constructs as composites instead of common factors. Besides the theoretical presentation of CCA and its assumptions, a Monte Carlo simulation is conducted which demonstrates that misspecifications of the composite model can be detected by the introduced test for overall model fit. The second essay rises the question of how parameter differences can be assessed in the framework of partial least squares path modeling. Since the standard errors of the estimated parameters have no analytical closed-form, the t- and F-test known from regression analysis cannot be directly used to test for parameter differences. However, bootstrapping provides a solution to this problem. It can be employed to construct confidence intervals for the estimated parameter differences, which can be used for making inferences about the parameter difference in the population. To guide practitioners, guidelines were developed and demonstrated by means of empirical examples. The third essay answers the question of how ordinal categorical indicators can be dealt with in partial least squares path modeling. A new consistent estimator is developed which combines the polychoric correlation and partial least squares path modeling to appropriately deal with the qualitative character of ordinal categorical indicators. The new estimator named ordinal consistent partial least squares combines consistent partial least squares with ordinal partial least squares. Besides its derivation, a Monte Carlo simulation is conducted which shows that the new estimator performs well in finite samples. Moreover, for illustration, an empirical example is estimated by ordinal consistent partial least squares. The last essay introduces a new consistent estimator for polynomial factor models. Similarly to consistent partial least squares, weights are determined to build stand-ins for the latent variables, however a non-iterative approach is used. A Monte Carlo simulation shows that the new estimator behaves well in finite samples.

This thesis deals with a new so-called sequential quadratic Hamiltonian (SQH) iterative scheme to solve optimal control problems with differential models and cost functionals ranging from smooth to discontinuous and non-convex. This scheme is based on the Pontryagin maximum principle (PMP) that provides necessary optimality conditions for an optimal solution. In this framework, a Hamiltonian function is defined that attains its minimum pointwise at the optimal solution of the corresponding optimal control problem. In the SQH scheme, this Hamiltonian function is augmented by a quadratic penalty term consisting of the current control function and the control function from the previous iteration. The heart of the SQH scheme is to minimize this augmented Hamiltonian function pointwise in order to determine a control update. Since the PMP does not require any differ- entiability with respect to the control argument, the SQH scheme can be used to solve optimal control problems with both smooth and non-convex or even discontinuous cost functionals. The main achievement of the thesis is the formulation of a robust and efficient SQH scheme and a framework in which the convergence analysis of the SQH scheme can be carried out. In this framework, convergence of the scheme means that the calculated solution fulfills the PMP condition. The governing differential models of the considered optimal control problems are ordinary differential equations (ODEs) and partial differential equations (PDEs). In the PDE case, elliptic and parabolic equations as well as the Fokker-Planck (FP) equation are considered. For both the ODE and the PDE cases, assumptions are formulated for which it can be proved that a solution to an optimal control problem has to fulfill the PMP. The obtained results are essential for the discussion of the convergence analysis of the SQH scheme. This analysis has two parts. The first one is the well-posedness of the scheme which means that all steps of the scheme can be carried out and provide a result in finite time. The second part part is the PMP consistency of the solution. This means that the solution of the SQH scheme fulfills the PMP conditions. In the ODE case, the following results are obtained that state well-posedness of the SQH scheme and the PMP consistency of the corresponding solution. Lemma 7 states the existence of a pointwise minimum of the augmented Hamiltonian. Lemma 11 proves the existence of a weight of the quadratic penalty term such that the minimization of the corresponding augmented Hamiltonian results in a control updated that reduces the value of the cost functional. Lemma 12 states that the SQH scheme stops if an iterate is PMP optimal. Theorem 13 proves the cost functional reducing properties of the SQH control updates. The main result is given in Theorem 14, which states the pointwise convergence of the SQH scheme towards a PMP consistent solution. In this ODE framework, the SQH method is applied to two optimal control problems. The first one is an optimal quantum control problem where it is shown that the SQH method converges much faster to an optimal solution than a globalized Newton method. The second optimal control problem is an optimal tumor treatment problem with a system of coupled highly non-linear state equations that describe the tumor growth. It is shown that the framework in which the convergence of the SQH scheme is proved is applicable for this highly non-linear case. Next, the case of PDE control problems is considered. First a general framework is discussed in which a solution to the corresponding optimal control problem fulfills the PMP conditions. In this case, many theoretical estimates are presented in Theorem 59 and Theorem 64 to prove in particular the essential boundedness of the state and adjoint variables. The steps for the convergence analysis of the SQH scheme are analogous to that of the ODE case and result in Theorem 27 that states the PMP consistency of the solution obtained with the SQH scheme. This framework is applied to different elliptic and parabolic optimal control problems, including linear and bilinear control mechanisms, as well as non-linear state equations. Moreover, the SQH method is discussed for solving a state-constrained optimal control problem in an augmented formulation. In this case, it is shown in Theorem 30 that for increasing the weight of the augmentation term, which penalizes the violation of the state constraint, the measure of this state constraint violation by the corresponding solution converges to zero. Furthermore, an optimal control problem with a non-smooth L\(^1\)-tracking term and a non-smooth state equation is investigated. For this purpose, an adjoint equation is defined and the SQH method is used to solve the corresponding optimal control problem. The final part of this thesis is devoted to a class of FP models related to specific stochastic processes. The discussion starts with a focus on random walks where also jumps are included. This framework allows a derivation of a discrete FP model corresponding to a continuous FP model with jumps and boundary conditions ranging from absorbing to totally reflecting. This discussion allows the consideration of the drift-control resulting from an anisotropic probability of the steps of the random walk. Thereafter, in the PMP framework, two drift-diffusion processes and the corresponding FP models with two different control strategies for an optimal control problem with an expectation functional are considered. In the first strategy, the controls depend on time and in the second one, the controls depend on space and time. In both cases a solution to the corresponding optimal control problem is characterized with the PMP conditions, stated in Theorem 48 and Theorem 49. The well-posedness of the SQH scheme is shown in both cases and further conditions are discussed that ensure the convergence of the SQH scheme to a PMP consistent solution. The case of a space and time dependent control strategy results in a special structure of the corresponding PMP conditions that is exploited in another solution method, the so-called direct Hamiltonian (DH) method.

The work at hand discusses various universality results for locally univalent and conformal metrics. In Chapter 2 several interesting approximation results are discussed. Runge-type Theorems for holomorphic and meromorphic locally univalent functions are shown. A well-known local approximation theorem for harmonic functions due to Keldysh is generalized to solutions of the curvature equation. In Chapter 3 and 4 these approximation theorems are used to establish universality results for locally univalent functions and conformal metrics. In particular locally univalent analogues for well-known universality results due Birkhoff, Seidel & Walsh and Heins are shown.