A two-stage procedure on comparing several experimental treatments and a control – the common and unknown variance case.

*(English)*Zbl 1094.62022Summary: This paper introduces a two-stage selection rule to compare several experimental treatments with a control when the variances are common and unknown. The selection rule integrates the indifference zone approach and the subset selection approach in multiple-decision theory. Two mutually exclusive subsets of the parameter space are defined, one is called the preference zone (PZ) and the other the indifference zone (IZ). The best experimental treatment is defined to be the experimental treatment with the largest population mean. The selection procedure opts to select only the experimental treatment which corresponds to the largest sample mean when the parameters are in the PZ, and selects a subset of the experimental treatments and the control when the parameters fall in the IZ.

The concept of a correct decision is defined differently in these two zones. A correct decision in the preference zone (CD\(_1\)) is defined to be the event that the best experimental treatment is selected. In the indifference zone, a selection is called correct (CD\(_2\)) if the selected subset contains the best experimental treatment. Theoretical results on the lower bounds for P(CD\(_1\)) in PZ and P(CD\(_2\)) in IZ are developed. A table is computed for the implementation of the selection procedure.

The concept of a correct decision is defined differently in these two zones. A correct decision in the preference zone (CD\(_1\)) is defined to be the event that the best experimental treatment is selected. In the indifference zone, a selection is called correct (CD\(_2\)) if the selected subset contains the best experimental treatment. Theoretical results on the lower bounds for P(CD\(_1\)) in PZ and P(CD\(_2\)) in IZ are developed. A table is computed for the implementation of the selection procedure.

##### MSC:

62F07 | Statistical ranking and selection procedures |