ABSTRACT
We introduced sequential tests in a variety of parametric problems earlier in Chapters 2, 3 and 4. These were heavily dependent on specific distributional features. Nonparametric methods, however, may not require as many specifications regarding a probability distribution as their parametric counterparts. Hence, we present some nonparametric methods of sequential tests in this chapter. But, sequential nonparametrics cover a vast area. Also, the mathemat-
ical mastery needed to appreciate many intricate details is beyond the scope of this book. Hence, we touch upon a small number of selected nonparametric sequential tests. We present this material mainly to instill some appreciation. One may obtain a broader perspective of this field from Sen (1981,1985), Govindarajulu (1981,2004) and other sources. In many statistical problems, continuous monitoring of ongoing produc-
tion processes takes the center stage. This involves some appropriate sampling inspection scheme. Under normal circumstances, an ongoing process produces items whose overall quality conforms to strict specifications and protocols. A null hypothesis H0 postulates that a production process is in control, that is, the overall quality of products meets specifications. On the other hand, an alternative hypothesis H1 postulates that a production process is out of control. Under routine monitoring, a quality control engineer may not tinker
with a process that continues to produce within specifications. If one does, then that will lead to significant loss in production of good items. So, we may think of observations X1,X2, . . . arriving in a sequence and sampling inspections being performed sequentially. One necessarily stops at some stage (that is, N <∞) if and only if a production line seems to be out of control. Inspection sampling or monitoring continues (that is, N = ∞) if and only if a production line seems to be within specifications. That is,
N <∞ ⇔ Accept H1 N =∞ ⇔ Accept H0
Our goal is two-fold: We would like
(i) type I error probability to be small, namely P(N <∞ | H0 is true) ≤ α or ≈ α where 0 < α < 1 is a small preassigned number;
1 Darling and Robbins (1968) developed ingenious methods to design sequential tests of power one with a small preassigned level or size α by exploiting the law of the iterated logarithm. Robbins (1970) included follow-up thoughts. Some details may be found
in Sen (1981, Chapter 9, pp. 233-243) and Govindarajulu (1981, Sections 3.13-3.14; 2004, Section 3.9). We first discuss some one-sample problems. Suppose that we have avail-
able a sequence of independent observations X1, . . . ,Xn, . . . from a population with a distribution function F. That is, F (x) ≡ P(X ≤ x), x ∈ IR, the real line. We denote the mean and the variance as follows:
Mean: µ ≡ µ(F ) = EF [X] = ∫∞ −∞ xdF (x);
Variance: σ2 ≡ σ2(F ) = EF [(X − µ)2] = ∫∞ −∞(x− µ)2dF (x).
