Hypothesis Testing
Hypothesis Testing Basics
Tests for Normal/Binomial distributions
Most Powerful Tests (MP)
UMP (Uniformly Most Powerful) Tests
MLR (Monotone Likelihood Ratio)
Likelihood Ratio Tests
Sample size/power/Chi-square Tests
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Notes on Hypothesis Testing - Inference Theory | MATH 7654, Study notes of Mathematics
Material Type: Notes; Class: Inference Theory; Subject: MATH Mathematics; University: University of Memphis; Term: Unknown 1989;
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Hypothesis Testing^ ^ Hypothesis Testing Basics^ ^ Tests for Normal/Binomial distributions^ ^ Most Powerful Tests (MP)^ ^ UMP (Uniformly Most Powerful) Tests^ ^ MLR (Monotone Likelihood Ratio)^ ^ Likelihood Ratio Tests^ ^ Sample size/power/Chi-square Tests
Hypothesis Testing Basics^ ^ Formulation of H
vs H^0 ^ (Ch 8.1, page 373-374) Type I vs. type II error calculation ^ (Ch 8.3, page 382) Power function calculation ^ (Def. 8.3.1, page 383-385) size^ α^ test vs. level
α^ test ^ (Def. 8.3.5 and Def. 8.3.6, page 385)
Types of Hypothesis Testing^ X^ , X, …, X^1
i.i.d. ~p.d.f. f(x;n
θ).
^ H^ :^ θ^0
=^ θvs. H^0
:^ θ^ =^ θ 1
1 (simple vs. simple) ^ H^ :^ θ^0
=^ θvs. H^0
:^ θ ≠ θ 1
0
(simple vs. composite, two-sided) H :^ θ ≤ θ 0
vs. H 0 :^ θ^ >^ θ 1
0
(composite vs. composite, one-sided)
Test about Normal Distribution^ ^ One sample:^ ^ Mean μ, (
(^2) σknown/unknown) ^ Variance
(^2) σ. ^ Two samples:^ ^ Difference μ
(^2) - μ(σ 12 (^2) = σ, 1 2 (^22) σ≠ σ^1
) 2
^ Ratio or equality of variances
(^2) σand^1 (^2) σ.^2
Two samples tests^ ^ Equality of μ
and μ 12 (^2) (σ=^1 (^2) σ) : use pooled-variance.^2 (^2) (σ≠ σ^1 2 ) : use Welch approximation.^2 ^ Equality of variances
(^2) σand^1
(^2) σ.^2
^ F-test can be used. ^ Important to known the relation betweenupper/lower percentiles of F-table.
- Test about BinomialDistributions One sample X ~ B(n,p) H: p = p
- vs H: p
- ≠ p
- Test statistic/critical region? Two samples: X~ B(n
- ,p) independent of X
- ~ B(n
- ,p )
- H: p
- = pvs. H
- : p≠ p
- Test statistic/critical region? Two samples: X~ B(n
Most Powerful Tests (MP)^ Neyman-Pearson Lemma^ X^ , X, …, X^1
i.i.d. ~p.d.f. f(x;n
θ).
^ H^ :^ θ^0
=^ θvs. H^0
:^ θ^ =^ θ 1
1 (simple vs. simple) ^ f( x ;θ)=
Π^ f(x^ ;θi^
) = L(θ
| x ).
^ MP Test: reject H
:^ θ^ =^0
θif^0 f( x ;^ θ)/f(^0
x ;^ θ)≤^1
k.
Uniformly Most Powerful Tests^ X, X, …, X^12
i.i.d. ~p.d.f. f(x;n
θ). H:^ θ ≤ θ^0
vs. H 0 1 :^ θ^ >^ θ^0
^ If the rejection region of the MP Test^ H
:^ θ^ =^ θ 0
vs. H 0 1 :^ θ^ =^ θ 1
^ θ. 0
is the same for any
θ, then MP is UMP.^1 ^ UMP Test does not exist for two sided
H:^ θ^ =^0
θvs. H^0
:^ θ ≠ θ 1 0
^ In that case, we find unbiased test. (UMPU)
Likelihood Ratio Tests
H:^ θ ε Θ^0
vs. H 0 1 :^ θ ε Θ 0 c Θ^ =^ ΘU^0
c Θ 0
^ λ( x )= sup
L( x θ ε Θ 0 |θ)/ supθ ε
L( x |θ)Θ
^ LRT: (Def. 8.2.1, page 375)
Reject H
if^ λ( x ) 0
≤^ k.
^ Techniques needed:^ ^ Find MLEs under the condition
θ ε Θand^0
θ ε Θ.
^ Get a specific critical region.
Likelihood Ratio Tests^ X^ , X, …, X^1
i.i.d. ~p.d.f. f(x;n
θ).
H^ :^ θ ≤ θ^0
vs. H 0 :^ θ^ >^1
θ^0
^ supθ ≤ θ
L( x |θ) is a constrained 0 maximization problem. Solution may beat its boundary (
θ).^0 ^ supθ^
L( x |θ) is a regular MLE problem.
MLR and Exponential family^ ^ Exponential family
f(x;^ θ)=a(x) b(
θ) exp(c(
θ) d(x))
^ If X, X^1
, …, Xi.i.d. ~ f(x; 2 n^
θ),
^ T( X )=^ ∑
d(X) is SS fori
θ. ^ If c(θ) is monotone, then f(x;
θ) has MLR and
critical region of UMP test for H
:^ θ ≤ θ 0
vs. 0
H:^ θ^ >^1
θis of the form^0 T( X ) > k.