Medical cost has received increasing interest recently in Biostatistics and public health. Statistical analysis and inference of life time medical cost have been challenging by the fact that the survival times are censored on some study subjects and their subsequent cost are unknown. Huang (2002) proposed the calibration regression model which is a semiparametric regression tool to study the medical cost associated with covariates. In this thesis, an inference procedure is investigated using empirical likelihood ratio method. The unadjusted and adjusted empirical likelihood confidence regions are constructed for the regression parameters. We compare the proposed empirical likelihood methods with normal approximation based method. Simulation results show that the proposed empirical likelihood ratio method outperforms the normal approximation based method in terms of coverage probability. In particular, the adjusted empirical likelihood is the best one which overcomes the under coverage problem.

The ranking method used for testing the equivalence of two distributions has been studied for decades and is widely adopted for its simplicity. However, due to the complexity of calculations, the power of the test is either estimated by normal approximation or found when an appropriate alternative is given. Here, via a Finite Markov chain imbedding (FMCI) technique, we are able to establish the marginal and joint distributions of the rank statistics considering the shift and scale parameters, respectively and simultaneously, under two continuous distribution functions. Furthermore, the procedures of distribution equivalence tests and their power functions are discussed. Numerical results of a joint distribution of two rank statistics under the standard normal distribution and the powers for a sequence of alternative normal distributions with mean from -20 to 20 and standard deviation from 1 to 9 and their reciprocals are presented. In addition, we discuss the powers of the rank statistics under the Lehmann alternatives. Wallenstein et. al. (1993, 1994) discussed power via combinatorial calculations for the scan statistic against a pulse alternative; however, unless certain proper conditions are given, computational difficulties exist. Our work extends their results and provides an alternative way to obtain the distribution of a scan statistic under various alternative conditions. An efficient and intuitive expression for the distribution as well as the power of the scan statistic are introduced via the FMCI. The numerical results of the exact power for a discrete scan statistic against various conditions are presented. Powers through the finite Markov chain imbedding method and a combinatorial algorithm for a continuous scan statistic against a pulse alternative of a higher risk for a disease on a specified subinterval time are also discussed and compared.