This dissertation concerns several statistical problems arising in Naval Engineering which relate to statistical uncertainty and characterizing rare events, and naturally involve a stochastic component to be accounted for through statistical methods. Chapter 1 has been adapted from Glotzer and Pipiras [43] and provides an overview of the problems to be discussed in later chapters.

In Chapter 2, statistical inference of a probability of exceeding a large critical value is studied in the peaks-over-threshold (POT) approach. The focus is on assessing the performance of various confidence intervals for the exceed anceprobability,and several approaches to uncertainty reduction are considered. This chapter has been published as Glotzer et al. [44].

Chapter 3 concerns the study of a single-degree-of-freedom random oscillator with a piecewise linear restoring force (experiencing softening after a certain point value of the response, called a "knuckle" point), with the goal of understanding the structure of the distribution tail of its response or (local) maximum. The random oscillator considered here serves as a prototypical model for ship roll motion in beam seas. A theoretical analysis is carried out first by focusing on the maximum and response after crossing the knuckle point, where explicit calculations can be performed assuming standard distributions for the derivative at the crossing; and second by considering the white noise random external excitation, where the stationary distribution of the response is readily available from the literature. Both approaches reveal the structure of the distribution tails where a Gaussian core is followed by a heavier tail, possibly having a power-law form, ultimately resulting in a tail having a finite upper bound. The extent of the light tail region is shown to be the iii result of conditioning on the system not reaching the unstable equilibrium (associated with "capsizing").

In Chapter 4, the distributions of certain response rates of the above random oscillator are investigated analytically and numerically. These include the minimum response rate leading to capsizing, referred to as the critical response rate, and the split-time metric, which measures the closeness of an observed response rate to the critical response rate and is used to assess the probability of capsizing. Three nonlinear restoring forces are considered: piecewise linear (experiencing linearly softening stiffness above a knuckle point), doubly piecewise linear (experiencing piecewise linearly softening stiffness above a knuckle point), and the cubic restoring force of the Duffing oscillator. Numerical simulation of the critical response rate and split-time metric is proposed from a derived distribution in the first two cases; in the latter case, the density of the critical response rate is approximated assuming white noise excitation. The distribution of the split-time metric is found to have a "light" tail, motivating the use of an exponential distribution or a Weibull distribution tail rather than the generalized Pareto distribution for exceedances above threshold in the POT approach.

Finally, Chapter 5 considers inference for the mean and variance of a stationary continuous-time stochastic process, for which the so-called long-run variances of the process and its square play a central role. The well-known problem of estimating the non-zero long-run variances is revisited here in the context of random oscillatory processes such as random (non-)linear oscillators and related models. The less studied case of the zero long-run variance is also considered. The approaches are extended to the situation where multiple independent records of the stochastic process are available, by introducing an estimator of the long-run variance which improves on other natural candidate estimators. A simulation study is provided to assess the performance of the proposed methods in estimating the long-run variances and constructing confidence intervals.