Inferential Statistics 377
critical region ΔC. The probability of this statistic having a value in the critical
region is indicated by the shaded area.
Since, in general, the set Θ 0 belonging to the null hypothesis consists
of several values (e.g., Θ 0 ⊂ Θ) the probability of committing a type I error,
PI(δ), may vary for each parameter value θ in Θ 0. By convention, we set
the test size equal to the PI(δ) computed at that value θ in Θ 0 for which
this probability is maximal.^6 We illustrate this for some arbitrary test in
Figure C.2, where we depict the graph of the probability of rejection of the
null hypothesis depending on the parameter value θ. Over the set Θ 0 , as
indicated by the solid line in the figure, this is equal to the probability of a
type I error while, over Θ 1 , this is the probability of a correct decision (i.e.,
d 1 ). The latter is given by the dashed line.
Analogously to the probability PI(δ), we denote the probability of com-
mitting a type II error as PII(δ).
Deriving the wrong decision can lead to undesirable results. That is, the
errors related to a test may come at some cost. To handle the problem, the
hypotheses are generally chosen such that the type I error is more harmful
figURe C.2 Determining the Test Size α by Maximizing the Probability of a Type I
Error over the Set Θ 0 of Possible Parameter Values under the Null Hypothesis
P(δ(X) = d 1 )
Θ 0
θmax PI(δ)
Θ 1(^6) Theoretically, this may not be possible for any test.