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 nature of the diagnostic test leads to observations for which time of occurrence of the composite event is never observed with certainty. Interpretation of such data is further complicated by deaths or possibly informative dropouts that occur prior to all scheduled diagnostic tests.

With minimal assumptions, we first discuss the identifiable aspects of data arising from such a setting, and determine sharp upper and lower bounds for the distribution of time to the composite endpoint in the general case and when dropouts lead to non-informative censoring. We then discuss likelihood-based methods for estimating the identifiability bounds, and illustrate the results from a recently-completed clinical trial for the prevention of mother-to-child HIV transmission in Botswana. The final study, which is the second part of the thesis, considers analyzing state-space models using particle filter under the framework of sequential Monte Carlo. Consider the nature of importance sampling, a well chosen sampling distribution between two importance sampling steps is the key to a successful algorithm. Different candidates sampling distributions may perform differently in various settings. For instance, we would prefer independent particle filter to bootstrap particle filter when the observation variance is small. We usually need to run a SMC with a large number of particles, and use it to decide which method performs best. However, this is not possible for real-time problems. In addition, different sampling distributions may be more suitable than others at different time points. If we can wisely select a mixture of different sampling distributions where good proposals get larger weights, we can improve the performance of SMC. In this article, we propose the use of pilot particles to select such an optimal mixture of sampling distributions to propagate particles. The method will adaptively determine the importance of individual sampling distribution and mix them together at each time point giving us more robust and accurate estimates.