Simultaneous confidence intervals for the successive ratios of scale parameters

Consider k(k⩾3) independent populations π1,…,πk such that the observations from populations πi follow distribution with cumulative distribution function (cdf) Fi(x)=F[(x-μi)/θi], where F(.) is any absolutely continuous cdf, i=1,…,k. It is assumed that scale parameters θ's satisfy a simple ordering, e.g., populations may be the outcome of successive runs of production process in quality control where it is the endeavor of the management to reduce the variation or spread in the quality of the product by adjusting the machine. In such situations, the interest of the experimenter may be in the pairwise comparisons. In this paper, one-sided and two-sided simultaneous confidence intervals for the ratios θ2/θ1,…,θk/θk-1 are proposed. A recursive method to compute the required critical points for the simultaneous confidence intervals is demonstrated by taking k two-parameter exponential distributions. The required points are tabulated and the uses of these points for variance ratios for Normal probability model and Pareto and Gamma probability model are discussed. Finally, the application of the proposed procedure is illustrated with an example.