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 λα, of statistics
expressible as maxima. The upper bound approximation to λα
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.
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