SEQUENTIAL TOLERANCE REGIONS
A discussion of past results. Let X be a random variable (r.v.) on an induced probability space (X, 91, P) and let V = (X1, X2, * **, Xn) be a vector of n independent replications of X; denote the induced probability space on which V is a r.v. by (X*, W*, P*). If D is a function mapping I* into Xf, then the r.v. D(V) has been called a distribution-free tolerance region whenever the distribution of the random coverage Q, defined Q = P[D(V)], does not depend upon the measure P, under the condition that P belongs to some class of probability measures. Such tolerance regions as outlined above were first introduced by S. S. Wilks [1] in the following special case: If X is the real line, X the Borel subsets of X and L(V), U(V) are two statistics from X* into X such that U(V) _ L(V) almost surely (a.s.), then U(V) and L(V) are, respectively, upper and lower fl-tolerance limits of probability level a for a, i E (0, 1), if the coverage, letting D(V) be the open random interval (L(V), U(V)), Q = P[D(V)] is such that