| This dissertation discusses two applied problems solved by saddlepoint approximation methods. The first part of this thesis concerns continuity corrected saddlepoint approximations for testing and confidence intervals for the difference of two independent binomial proportions. We propose two new continuity corrections, and compare them with the other four continuity corrections. Continuity corrections may give a more accurate approximations to the tail area of discrete random variables. To get a confidence interval for the difference of two independent binomial proportions, we proposed continuity corrections 1/2LCM( m, n) with the least common multiple (LCM) of the two binomial distributions sample sizes m and n considered by Xu and Kolassa[31], and R/2 LCM(m, n), an adjusted version of 1/2LCM (m, n) with R by Xu and Kolassa[32], a heuristic factor calculator from the standard error of the marginal null binomial distribution. We compare exact coverage probabilities for intervals calculated by using saddlepoint approximation with different corrections. The primary criterion for evaluating the corrections is agreement of actual with nominal coverage probabilities. Because of the ordering of the continuity corrections, the coverage probabilities will be ordered similarly. For all the cases considered with minimum expected cell size of at least 1, numerical results indicate that R/2LCM(m, n) has coverage probabilities very close to the nominal 95% and 90% even for the minimum of sample sizes as small as 5--9, and R /2LCM(m, n) improved uncorrected saddlepoint approximation methods moderately for the nominal 95% and 90% intervals; however, that the Yates continuity correction (2m) --1 +(2n)--1 is unnecessarily conservative for 95% and 90% but reasonable for 99% intervals.;The second part of this thesis concerns the sequential likelihood ratio test using in computerized adaptive testing. We consider sequential testing techniques, including the truncated sequential probability ratio test and the Haybittle-Peto test. Both of these tests in their original forms rely on approximate normality of the signed roots of the log likelihood ratio tests, and approximate boundary crossing probabilities for discrete normal-theory random walks. Bartroff, Finkelman and Lai[5] modify these techniques by using Monte Carlo approximations to calibrate the truncation boundary. We propose a hybrid Monte Carlo-Asymptotic approach, in which we substitute an easy Monte Carlo approximation in place of boundary crossing probabilities for Brownian motions, and use asymptotic approximations for the distribution of the signed root of the likelihood ratio test statistic. We found that after selecting stopping boundaries using normal-based Monte Carlo calculations, reliance on asymptotic normality of the signed root of the log likelihood ratio statistics provided adequate control of Type I error, without recourse to more complicated Monte Carlo operations. We also observe markable improvement using Barndorff-Nielsen's r* formula (Barndorff-Nielsen, 1991). |
