Page 133 - Introduction to Statistical Pattern Recognition

3 Hypothesis Testing 115

where s, is used instead of s to indicate the number of observations, and a and
b are thresholds to determine wl and 02, respectively. This decision rule is
called the Wald sequential test [20].
The error of the Wald sequential test is controlled by a and b; that is, as
the absolute values of a and h increase, the error decreases, while the number
of observations required to reach the decision increases. The relation between
the threshold values and the error can be expressed by

02)dsj . (3.186)

Theoretically, we should be able to find u and b from (3.185) and (3.186) for
any given and ~2.
A simpler way to find the threshold values was developed by Wald. The
procedure is as follows: At the mth observation, the likelihood ratio is tested as

IB +X‘s E y . (3.187)

Therefore,

p1(X,, . . .,X,)dXI. . . dX,
m=l
(3.188)