Testing of Hypotheses
Definition
1. A (statistical) hypothesis is a statement about a population parameter.
2. The two complementary hypotheses in a testing problem are called the null hypothesis (denoted
by H0 ) and the alternative hypothesis (denoted by H1 ).
Example :
θ = (θ1 , θ2 ) → average blood sugar level of a group of patients before and after taking
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a new drug
H0 : no effect ↔ H0 : θ1 = θ2
H1 : effective ↔ Ha : θ1 6= θ2 or θ1 > θ2
Definition: If a statistical hypothesis H completely specifies the distribution of (X1 , . . . , Xn ), then it
is simple; otherwise H is called composite.
Definition: Let X be the set of all possible values of X = (X1 , . . . , Xn ). Then, a test function or a test
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rule φ(X1 , . . . , Xn ) is a function from X into [0, 1] with the interpretation that if X = (X1 , . . . , Xn ) is
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observed then H0 is rejected with probability φ(X ).
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Definition: If φ(X ) ∈ {0, 1}, ∀X ∈ X , then
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1. φ(X ) is called a simple test function (rule).
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2. Rφ = {X : φ(X ) = 1} is called the rejection region of φ(X ).
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3. Aφ = {X : φ(X ) = 0} is called the acceptance region of φ(X ).
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Little Picture:
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action based on φ(·)
accept H0 reject H0
the
state
of
nature
H0 is true
H0 is false
Note: Suppose φ(X ) is a simple test rule for testing H0 : θ ∈ Θ0 vs. H1 : θ ∈ Θ0 . Then,
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1. for any θ ∈ Θ0 , the probability of type I error at θ
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Pθ (rejecting H0 ) = Pθ (X = (X1 , . . . , Xn ) ∈ Rφ ) = Eθ φ(X ),
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 1 if X ∈ Rφ
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since φ(X ) =
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0 if X 6∈ Rφ
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2. and for any θ 6∈ Θ0 , the probability of type II error at θ is
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Pθ (accepting H0 ) = Pθ (X ∈ Aφ ) = 1 − Pθ (X ∈ Rφ ) = 1 − Eθ φ(X )
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Remark: For any general test function φ(·)
1. Probability of a type I error at θ (θ ∈ Θ0 ) = Eθ φ(X ) AND
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2. Probability of a type II error at θ (θ 6∈ Θ0 ) = 1 − Eθ φ(X ).
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Definition: Let φ(X ) be a test rule for testing H0 : θ ∈ Θ0 vs H1 : θ 6∈ Θ0 ,
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1. maxθ∈Θ0 Eθ φ(X ) is called the size or the level of φ(X )
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2. Πφ (θ) = Eθ φ(X ) is called the power function of φ(X ).
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Note:
For θ ∈ Θ0 , Πφ (θ) = probability of type I error
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For θ 6∈ Θ0 , probability of type II error = 1 − Πφ (θ)
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