An application of the principle of inclusion and exclusion

Abstract

This thesis is concerned with an application of the principle of inclusion and exclusion and with related approximation techniques. These procedures are extensively employed for developing test criteria based on statistics expressible as maxima.

Upper percentage points of a number of such statistics have been tabulated by various special methods. However, the application of the principle of inclusion and exclusion, coupled with the Bonferroni inequalities, is often useful in providing good approximations. An extensive review of this method is presented in this report.

This procedure allows one to establish upper and lower limits to upper percentage points, say λ_α, of statistics expressible as maxima. The upper bound approximation to λ_α requires only the knowledge of the distribution(s) of the variates under consideration. The lower bound, however, requires also the joint distribution(s) of pairs of the variates. Since the joint distribution is often difficult to calculate, an approximation technique may be necessary. A detailed discussion of such an approximation with guidelines for its applicability to statistics other than those discussed is presented.

Two alternative methods for the determination of upper percentage points for statistics expressed as maxima are discussed: Whittle's lower bound approximation and the assumption of independence. It is pointed out that Whittle's lower bound is stronger than that of Bonferroni only under certain conditions. The assumption of independence leads to approximately the same result as Bonferroni.