Analyses of periodic observations and time series data with applications to HIV prevention and state-space models

This thesis consists of two parts. The first part contains two studies, which are motivated from the prevention of mother-to-infant HIV transmission. The first study considers inferences about the distribution of time to HIV infection in infants. They are complicated because infection is a silent event and imperfect diagnostic tests are used to detect its occurrence, leading to false positive and false negative results. Non-parametric likelihood approaches are computationally hampered by large numbers of parameters and a possibly non-concave likelihood function. To overcome these difficulties, we develop one-sample and regression methods based on profile likelihood and Markov chain Monte Carlo techniques. The methods also provide a useful diagnostic for assessing the infection status of individual subjects, and are illustrated using results from a recent clinical trial for the prevention of mother-to-child HIV transmission. The second study considers the inference for a composite time-to-event composite when one component is silent and only observed periodically and with error, such as HIV-free survival in infants born to HIV-infected mothers. As in the first study, the imperfect n