Inferential Statistics 373
Setting Up the Hypotheses Since in statistical inference we intend to gain
information about some unknown parameter θ, the possible results of the
test should refer to the parameter space Θ containing all possible values
that θ can assume. More precisely, to form the hypotheses, we divide the
parameter space into two disjoint sets Θ 0 and Θ 1 such that Θ = Θ 0 ∪ Θ 1.
We assume that the unknown parameter is either in Θ 0 or Θ 1 since it cannot
simultaneously be in both. Usually, the two alternative parameter sets either
divide the parameter space into two disjoint intervals or regions (depending
on the dimensionality of the parameter), or they contrast a single value with
any other value from the parameter space.
Now, with each of the two subsets Θ 0 and Θ 1 , we associate a hypothesis.
In the following two definitions, we present the most commonly applied
denominations for the hypotheses.
Null hypothesis. The null hypothesis, denoted by H 0 , states that the
parameter θ is in Θ 0.The null hypothesis may be interpreted as the assumption to be maintained
if we do not find ample evidence against it.
Alternative hypothesis. The alternative hypothesis, denoted by H 1 , is the
statement that the parameter θ is in Θ 1.We have to be aware that only one hypothesis can hold and, hence, the out-
come of our test should only support one.
When we test for a parameter or a single parameter component, we
usually encounter the following two constructions of hypothesis tests. In
the first construction, we split the parameter space Θ into a lower half up to
some boundary value θ and an upper half extending beyond this boundary
value. Then, we set the lower half either equal to Θ 0 or Θ 1. Consequently,
the upper half becomes the counterpart Θ 1 or Θ 0 , respectively. The bound-
ary value θ is usually added to Θ 0 ; that is, it is the set valid under the null
hypothesis. Such a test is referred to as a one-tailed test.
In the second construction, we test whether some parameter is equal
to a particular value or not. Accordingly, the parameter space is once
again divided into two sets Θ 0 and Θ 1. But this time, Θ 0 consists of only
one value (i.e., Θ= 0 θ ) while the set Θ 1 , corresponding to the alternative
hypothesis, is equal to the parameter space less the value belonging to the
null hypothesis (i.e., Θ= 1 Θθ\ ). This version of a hypothesis test is termed
a two-tailed test.