Finding confidence bound using two-stage data

Multistage sampling schemes for statistical inference are widespread. In particular, computing one or two sided confidence limits from two-stage samples has many applications. Typically, in a two stage sampling scheme, at the end of the first stage, the investigator carries out a statistical testing procedure to decide whether to move on to the second stage or to stop sampling. If the investigator decides to go to the second stage, then more samples are taken and a final decision is reached. Since the cost of sampling is sometimes very high, it is desired to use all the samples from both the stages to make a terminal decision. Bias considerations usually make it improper to simply combine the samples and treat them as one sample. In this paper, we consider computing confidence bounds for a mean using data from two stages. Let n₁ sample points be taken in the first stage and a standard level-α t-test performed for testing the hypothesis (Formula presented.) If H₀ is rejected, then a second sample of size n₂ is taken. Now the goal is to compute a lower confidence bound for the true mean μ using all (n₁ + n₂) sample points. Here we propose three different methods to compute the lower confidence bound using bootstrap methodology. The procedures differ in their demands of computational power which is reflected in their accuracy of coverage probability. Performance of the procedures is evaluated by simulation. It is noteworthy that in almost all the research mentioned above, the values of n₁ and n₂ considered are relatively small. Hence we focus on small sample behavior of two stage confidence bound.