• Multiple comparison, simultaneous inference, and ranking and selections
• Sequential methods

Some of my contributions (a number in [] indicates the paper in my refereed journal publications)

• In many real problems, one has to make several inferences (e.g. to test several hypotheses) at the same time and it is required that the inferences are simultaneously correct `with a good chance' in certain sense. The classical problem is to make inferences about several normal means. One can construct suitable simultaneous confidence intervals for specific comparisons. The frequently used simutaneous confidence intervals are Tukey's (1953) intervals for pairwise comparisons ([1,2,5,13,21]), Dunnett's (1955) intervals for treatments-control comparisons ([4,27,32]), Hsu's (1981, 1984) intervals for comparisons with the best ([6,15]), Hochberg and Marcus (1978) and Lee and Spurrier's (1995) intervals for successive comparisons ([37]), and Hayter's (1990) intervals for one-sided-pariwise comparisons ([19,35]). One can also use type I familywise error rate controlled tests which are often stepwise. For pairwise comparisons, my work includes ([18,26]). For treatmants-control comparisons, my work includes ([16,24,30,32,40]). One may also consider the control of type II error probability (power of a test) ([1,2,4,21,27]). Type III (directional) error probability can also be considered even though it is probably less well known ([25,28,49]). Control of FDR (false discovery rate) or its variations has become very popular when the number of hypotheses is very large (many thousands) following the celebrated work of Benjamini and Hochberg (1995) ([33]). Another thread of my research in multiple comparison is to use recursive method to compute exactly multivariate t or normal probabilities ([19,22,35,37,38])
• Many multiple comparison and simultaneous inference procedures can be generalized to situations that observations can be taken sequentially ([11,17]). My prime research interest in sequential methods is in the so called sequential estimation problems. The classical problem is the construction of a fixed-wdith 2d and level 1-a confidence interval for the mean of a normal population, assuming the variance is unknown either. Since the required total sample size depends on the variance and the variance is assumed to be unknown, it is necessary to collect observations in at least two batches; the unknown variance and the required total sample can be estimated after the first batch of observations. The most famous sequential estimation procedures are Stein's (1945) two-stage procedure, Anscombe (1953) and Chow and Robbins's (1965) one-observation-at-a-time procedure, and Hall's (1981) three-stage procedure. My contributions include a new procedure ([23]), a demonstration that a five-stage procedure can be as good as the one-observation-at-a-time procedure ([29]), and application of sequential estimation procedure to some substantial new problems ([13], [31], [66]) in addition to some related but more general theoretical work ([36,46,47,51,56]).
• Using a simultaneous confidence band to assess a linear regression model is part of simultaneous inference (see e.g. Miller, 1981). Much of the published work is on approximate or conservative methods when there are more than one explanatory variable. Simulation-based methods ([58,59,61,72]) allow the critical constant be calculated as accurate as requird by using a sufficiently large number of simulations, in addition to exact methods for some special situations ([63,73,74]). While the average width is the standard criterion for comparison of different confidence bands, [65,78] propose use of the area of confidence set for the regression coefficients as a criterion. Traditional comparison of means can be generalized to comparison of regresion models following Spurrier (1999) ([55,59,67,68,69,70,71,75]). Using simultaneous confidence bands to assess regression models is a fertile research area; see my book Simultaneous Inference in Regression.

My book Simultaneous Inference in Regression, Chapman and Hall, 2010.