Some Studies On Multiple Comparisons

In this thesis, new simultaneous confidence intervals for parameters are built and some well-known simultaneous confidence intervals are compared in term of their length; the way to measure type I error is deeply discussed and two step-up multiple testing procedures are produced under the control of PFER; in large scale multiple comparisons, the asymptotic operating properties of generic step-up procedures are studied as well as asymptotic FDR and FNR; step-up procedures are compared in terms of classification risk.In Chapter 2 multiple comparisons are studied in terms of simultaneous confidence intervals instead of hypothesis testing. First Simplified Scheffe simultaneous confidence intervals are constructed under multivariate normal distributions. Secondly, we theoret-ically compared several well-known simultaneous confidence intervals including Bonfer-roni simultaneous confidence intervals and Scheffe simultaneous confidence intervals as well as simplified Scheffe simultaneous confidence intervals. Even though a number of researchers have contributed their efforts to examine the differences among these simulta-neous confidence intervals,their results are based either on numerical outcomes or under the framework of some special statistical models.In addition, all the previous outcomes involve only the univariate situation and no such results have yet been reported regarding the multivariate scenario. You will see that the theory for multivariate circumstance is not only trivial extension of the theory obtained from univariate analysis in this issue. The confidence bounds comparison results are presented when there are more than two multivariate normal populations which is not as easy as it looks. In addition,sharing the same form makes the confidence bounds comparable.In chapter 3 we have put forward an opinion in which the task of designing a procedure for a set of multiple comparisons should be considered as a decision-making procedure under uncertainty, in which the risk faced by the decision-maker is the false discovery:the number of falsely rejected hypotheses or the proportion of the falsely rejected hypotheses among all rejected hypotheses. Once the risk faced by the decision-maker is explicitly determined,a logical approach to control the risk should be broken into two steps. The first step is to find a functional of the random variable (or its distribution) to measure the risk characterized by the random variable and then in the second step one develops multiple comparison methodologies under which the measured risk, which is now a real number,can be controlled.The risk measure reflects the attitude of the decision maker towards the risk; different attitudes towards a risk surely lead to different choices of risk measurements. First,a naive single step procedure is suggested. Secondly, two step-up procedures are developed one of which is of a generic set of critical values and another uses BH type critical values. A simulation study is conducted to compare their powers in terms of some specified power definitions.In chapter 4 the asymptotic operating properties of generic step-up procedure which can be seen as the family of step-up procedures are studied when there are infinite number of hypotheses. The results are produced under the different fixed alternatives instead of the same fixed alternative. Meanwhile the asymptotic FDR and FNR are deduced for generic step-up procedures. In addition, a classification risk is defined as a power to compare different multiple comparison procedures in asymptotic way.