ABSTRACT
Consider a sequence of i.i.d. random variables X1,X2, . . ., having the common probability density function (p.d.f.) f(x|θ) for x ∈ X ⊆ IR and θ ∈ Θ ⊆ IR. The methodology and the results would be laid out pretending that X’s are continuous variables. However these are general enough to include discrete distributions as well. In the situations where X’s are discrete, f(x|θ) would stand for the probability mass function (p.m.f.), and in the derivations, one should replace the integrals with the appropriate sums for a smooth transition. The problem is one of testing a simple null hypothesis H0 : θ = θ0 versus
a simple alternative hypothesis H1 : θ = θ1 where θ0 6= θ1 are two specified numbers in the parameter space Θ. We also have two preassigned numbers 0 < α, β < 1 corresponding to the type I and type II error probabilities. From a practical point of view, let us assume that α+ β < 1. Let us write Xj = (X1, . . . ,Xj), j = 1, 2, . . .. Now, when θ is the true
parameter value, the likelihood function of Xj is given by
L (θ; j,Xj) = j∏ l=1
f(Xl|θ). (3.1.1)
We denote L0(j) = L(θ0; j,Xj) and L1(j) = L(θ1; j,Xj) respectively under H0 and H1. Having observed Xj , the most powerful (MP) level α test has the form
Reject H0 if and only if L1(j) L0(j) ≥ k (3.1.2)
where k(> 0) is to be determined appropriately. One should keep in mind that in a discrete case, appropriate randomiza-
tion would be mandatory to make sure that the size of this test is indeed exactly α. The test given in (3.1.2) is the best among all fixed-sample-size (= j)
tests at level α. But its type II error probability can be quite a bit larger than β, the preassigned target. In order to meet both (α, β) requirements, in the early 1940’s, Abraham Wald, among others, looked at this problem from a different angle. Having observed Xj let us consider the sequence
Λj = L1(j) L0(j) , j = 1, 2, . . . .
