### Repeated Confidence Intervals

In a group sequential test, repeated confidence intervals for a parameter are defined as a sequence of intervals , , for which a simultaneous coverage probability is maintained (Jennison and Turnbull, 2000, p. 189). That is, a sequence of repeated confidence intervals has These confidence limits and can be created from observed statistic and boundary values at each stage.

#### Two-Sided Repeated Confidence Intervals

Two sequences of repeated confidence intervals can be derived for a two-sided test. One is a rejection repeated confidence intervals , , and the other is a acceptance repeated confidence intervals , , where and are the lower and upper Type I error probabilities for the test and and are the lower and upper Type II error probabilities for the test (Jennison and Turnbull, 2000, p. 196).

The rejection lower and upper repeated confidence limits at stage k are The hypothesis is rejected for upper alternative if the lower limit and is rejected for lower alternative if the upper limit . That is, the hypothesis is rejected if both and are not in a rejection repeated confidence interval .

The acceptance lower and upper repeated confidence limits at stage k are The hypothesis is accepted if the lower limit and the upper limit . That is, a repeated confidence interval is contained in the interval .

#### One-Sided Repeated Confidence Intervals

Like the two-sided repeated confidence intervals, two sequences of repeated confidence intervals can be derived for a one-sided test. Suppose the one-sided test has an upper alternative . Then one sequence of repeated confidence intervals is a rejection repeated confidence intervals , , and the other is a acceptance repeated confidence intervals , , where and are the upper Type I and Type II error probabilities for the test. Thus, a sequence of repeated confidence intervals with confidence level greater than or equal to is given by .

The rejection lower repeated confidence limit and the acceptance upper repeated confidence limit at stage k are The hypothesis is rejected if the lower limit . and it is accepted if the upper limit .