# Uniformly Most Powerful Tests

```Uniformly Most Powerful Tests
Definition: Let f (x|θ), θ ∈ Θ ⊂ Rp , be the joint pdf/pmf of X = (X1 , . . . , Xn )
˜
˜
(the parameter θ can be vector-valued) and let Θ0 be a nonempty proper subset of
Θ. Then, a test rule ϕ(x) for testing H0 : θ ∈ Θ0 vs H1 : θ 6∈ Θ0 is called a
˜
uniformly most powerful (UMP) test of size α if
1. maxθ∈Θ0 Eθ ϕ(X ) = α
˜
2. it holds that Eθ ϕ(X ) ≥ Eθ ϕ̃(X ) for all θ 6∈ Θ0 , given any other test rule ϕ̃(x)
˜
˜
˜
with maxθ∈Θ0 Eθ ϕ̃(X ) ≤ α.
˜
Methods of finding UMP Tests
1. Method I: Based on Neyman-Pearson Lemma
2. Method II: Using the Monotone Likelihood Ratio (MLR) property
Method I (Neyman-Pearson Lemma-based)
To find a UMP size α test for H0 : θ ∈ Θ0 vs H1 : θ 6∈ Θ0 ,
1. first fix θ0 ∈ Θ0 (suitably) and also θ1 6∈ Θ0
2. then use the Neyman-Pearson lemma to find a MP size α test ϕ(x) for H0 :
˜
θ = θ0 vs H1 : θ = θ1 , where
(a) ϕ(x) does not depend on θ1 6∈ Θ0 and
˜
(b) maxθ∈Θ0 Eθ ϕ(X ) = α
˜
Then ϕ(x) is a UMP size α test for H0 : θ ∈ Θ0 vs H0 : θ 6∈ Θ0 .
˜
1
Example: Let X1 , . . . , Xn be a random sample from N(&micro;, 1), &micro; ∈ R. Find a UMP
test of size α for H0 : &micro; ≥ &micro;0 vs H1 : &micro; &lt; &micro;0 (where &micro;0 is a fixed real number).
2
Method II: Based on Monotone Likelihood Ratio, a property defined below
Definition: Let f (x|θ), θ ∈ Θ ⊂ R, be the joint pdf/pmf of X = (X1 , . . . , Xn )
˜
˜
(note: the parameter θ is real-valued). Then, {f (x|θ) : θ ∈ Θ} is said to have
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monotone likelihood ratio (MLR) in a real-valued statistic T = t(X ) if: for any
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θ1 &lt; θ2 , there exists a nondecreasing function gθ1 ,θ2 (&middot;) : R → [0, ∞] such that
f (x|θ2 )
&sup3;
&acute;
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= gθ1 ,θ2 t(x) ,
f (x|θ1 )
˜
˜
Here a/b = +∞ if a &gt; 0, b = 0
∀x ∈ {y : f (y |θ1 ) + f (y |θ2 ) &gt; 0}
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˜
˜
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Example: Let f (x|θ) = c(θ)h(x) exp[q1 (θ)t1 (x)], θ ∈ Θ ⊂ R be the joint pdf/pmf
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˜
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of X1 , . . . , Xn (i.e., in the exponential family) and q1 (θ) is nondecreasing. Show
that {f (x|θ) : θ ∈ Θ} has MLR in t1 (x).
˜
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Theorem: (Method II) Let f (x|θ), θ ∈ Θ ⊂ R, be the joint pdf/pmf of
˜
X1 , . . . , Xn . Assume that {f (x|θ) : θ ∈ Θ} has MLR in T = t(x). Then,
˜
˜
1. a size α UMP test for H0 : θ ≤ θ0 vs H1 : θ &gt; θ0 (θ0 ∈ Θ fixed) is given by


1
if t(x) &gt; k



˜

where γ ∈ [0, 1] and −∞ ≤ k ≤ ∞ are
γ
if t(x) = k
ϕ(x) =
constants satisfying Eθ0 ϕ(X ) = α.
˜

˜


˜

if t(x) &lt; k
0
˜
3
2. a size α UMP test for H0 : θ ≥ θ0 vs H1 : θ &lt; θ0 (θ0 ∈ Θ fixed) is given by


1
if t(x) &lt; k



˜

where γ ∈ [0, 1] and −∞ ≤ k ≤ ∞ are
γ
if t(x) = k
ϕ(x) =
constants satisfying Eθ0 ϕ(X ) = α.
˜

˜


˜

if t(x) &gt; k
0
˜
Caution: Method II of finding a UMP test applies only when Θ ⊂ R (i.e., parameter θ is real-valued) and the testing problem is of the form “H0 : θ ≤ θ0 vs
H1 : θ &gt; θ0 ” or “H0 : θ ≥ θ0 vs H1 : θ &lt; θ0 ”
Example: Let X1 , . . . , Xn be a random sample from uniform(0,θ), θ &gt; 0. Find a
UMP test of size α for H0 : θ ≤ θ0 vs H1 : θ &gt; θ0 (where θ0 &gt; 0 is fixed).
4
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