Anon

382 The Basics of financial economeTrics

Uniformly most powerful (UMP) test of size α. A test δ* of size α is

uniformly most powerful, if of all the tests δ of size α, this test δ*

has greatest power for any θ ∈ Θ 1.

Unbiased Test We know that when a test is of size α, the probability of it
causing a type I error is never greater than α. And when the design of the
test is reasonable, the power of the test should increase quickly once we
are considering parameter values in Θ 1. In both cases (i.e., when we compute
the probability of a type I error for θ ∈ Θ 0 , as well as when we compute
the power for θ ∈ Θ 1 ), we are dealing with the probability to reject the null
hypothesis (i.e., P(δ(X) = d 1 )). In case P(δ(X) = d 1 ) should be smaller than
α, then for certain parameter values θ ∈ Θ 1 it is more likely to accept the
null hypothesis when it is wrong than when it holds. This certainly does not
appear useful and we should try to avoid it when designing our test. This
concept is treated in the following definition.

Unbiased test. A test of size α is unbiased if the probability of a type II

error is never greater than 1 – α; formally,

figURe C.5 Decomposition 1(=δPPII )(+δθ 1 ()Xd= 1 ), over Θ 1

Θ 0 Θ 1 θ

1

PII(δ)

Power