In this thesis two approaches of inferences on median failure times are considered and developed to compare the difference of median failure times between two groups of censored survival data.The first one is to generalize the Mood's median test - which is designed to deal with complete data - to censored survival data. To this end, two groups of censored survival data are pooled and then the estimated pooled median failure time is obtained from the product-limit method. A score is assigned for each observation to indicate the probability whether it survives after the pooled median failure time or not and for each group the scores are summed to summarize the number of observations whose survival time is larger than or equal to pooled median survival time, which results in a 2x2 contingency table with non-integer entries. Four 2x2 contingency tables with integer entries are then derived and a test statistic is defined as the weighted sum of the statistics from the four 2x2 contingency tables which is shown to be approximately distributed as chi-square distribution with 1 degree of freedom for large samples.The second approach is proposed to construct a 95% confidence interval for the difference of median failure times between two groups of censored survival distributions. Since the median failure time is approximately normally distributed for large samples, the estimated median failure times for each group are obtained by product-limit method and their standard errors are computed through bootstrap samples from the original data. Theory of construction for 95% confidence interval for the difference of median failure times is investigated for the standard normal distributions and it can be used for general normal distributions by translation and rescaling.Extensive numerical studies are carried out to test the appropriateness of the two approaches and the results show that the approaches developed in the thesis are easy to implement and the results are promising, compared to the results from published papers. The proposed methods will facilitate more accurate analysis of survival data under censoring, which are commonly collected from clinical studies that influence public health
