Biostatistics 602 - Statistical Inference Lecture 04 Ancillary Statistics

84 Pages·2013·0.33 MB·English

by  Hyun Min Kang

MinimalSufficientStatistics AncillaryStatistics Location-scaleFamily Summary
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Biostatistics 602 - Statistical Inference
Lecture 04
Ancillary Statistics
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Hyun Min Kang
January 22th, 2013
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HyunMinKang Biostatistics602-Lecture04 January22th,2013 1/23
2.. What are examples obvious sufficient statistics for any distribution?
3.. What is a minimal sufficient statistic?
4.. Is a minimal sufficient statistic unique?
5.. How can we show that a statistic is minimal sufficient for (cid:18)?
MinimalSufficientStatistics AncillaryStatistics Location-scaleFamily Summary
..... ......... ....... .
Recap from last lecture
1.. Is a sufficient statistic unique?
. . . . . .
HyunMinKang Biostatistics602-Lecture04 January22th,2013 2/23
3.. What is a minimal sufficient statistic?
4.. Is a minimal sufficient statistic unique?
5.. How can we show that a statistic is minimal sufficient for (cid:18)?
MinimalSufficientStatistics AncillaryStatistics Location-scaleFamily Summary
..... ......... ....... .
Recap from last lecture
1.. Is a sufficient statistic unique?
2.. What are examples obvious sufficient statistics for any distribution?
. . . . . .
HyunMinKang Biostatistics602-Lecture04 January22th,2013 2/23
4.. Is a minimal sufficient statistic unique?
5.. How can we show that a statistic is minimal sufficient for (cid:18)?
MinimalSufficientStatistics AncillaryStatistics Location-scaleFamily Summary
..... ......... ....... .
Recap from last lecture
1.. Is a sufficient statistic unique?
2.. What are examples obvious sufficient statistics for any distribution?
3.. What is a minimal sufficient statistic?
. . . . . .
HyunMinKang Biostatistics602-Lecture04 January22th,2013 2/23
5.. How can we show that a statistic is minimal sufficient for (cid:18)?
MinimalSufficientStatistics AncillaryStatistics Location-scaleFamily Summary
..... ......... ....... .
Recap from last lecture
1.. Is a sufficient statistic unique?
2.. What are examples obvious sufficient statistics for any distribution?
3.. What is a minimal sufficient statistic?
4.. Is a minimal sufficient statistic unique?
. . . . . .
HyunMinKang Biostatistics602-Lecture04 January22th,2013 2/23
MinimalSufficientStatistics AncillaryStatistics Location-scaleFamily Summary
..... ......... ....... .
Recap from last lecture
1.. Is a sufficient statistic unique?
2.. What are examples obvious sufficient statistics for any distribution?
3.. What is a minimal sufficient statistic?
4.. Is a minimal sufficient statistic unique?
5.. How can we show that a statistic is minimal sufficient for (cid:18)?
. . . . . .
HyunMinKang Biostatistics602-Lecture04 January22th,2013 2/23
MinimalSufficientStatistics AncillaryStatistics Location-scaleFamily Summary
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Minimal Sufficient Statistic
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Definition 6.2.11
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A sufficient statistic T(X) is called a minimal sufficient statistic if, for any
′ ′
other sufficient statistic T(X), T(X) is a function of T(X).
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Why is this called ”minimal” sufficient statistic?
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• The sample space X consists of every possible sample - finest partition
• Given T(X), X can be partitioned into A where
t
t 2 T = ft : t = T(X) for some x 2 Xg
• Maximum data reduction is achieved when jTj is minimal.
• If size of T′ = t : t = T′(x) for some x 2 X is not less than jTj, then
jTj can be called as a minimal sufficient statistic.
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HyunMinKang Biostatistics602-Lecture04 January22th,2013 3/23
MinimalSufficientStatistics AncillaryStatistics Location-scaleFamily Summary
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Theorem for Minimal Sufficient Statistics
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Theorem 6.2.13
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• f (x) be pmf or pdf of a sample X.
X
• Suppose that there exists a function T(x) such that,
• For every two sample points x and y,
• The ratio f (xj(cid:18))/f (yj(cid:18)) is constant as a function of (cid:18) if and only if
X X
T(x) = T(y).
• Then T(X) is a minimal sufficient statistic for (cid:18).
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In other words..
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• f (xj(cid:18))/f (yj(cid:18)) is constant as a function of (cid:18) =) T(x) = T(y).
X X
• T(x) = T(y) =) f (xj(cid:18))/f (yj(cid:18)) is constant as a function of (cid:18)
X X
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HyunMinKang Biostatistics602-Lecture04 January22th,2013 4/23
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Solution
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∑
∏n exp((cid:0)(x (cid:0)(cid:18))) exp((cid:0) n (x (cid:0)(cid:18)))
f (xj(cid:18)) = i = ∏ i=1 i
X (1+exp((cid:0)(x (cid:0)(cid:18))))2 n (1+exp((cid:0)(x (cid:0)(cid:18))))2
i=1 ∑ i i=1 i
exp((cid:0) n x )exp(n(cid:18))
= ∏ i=1 i
n (1+exp((cid:0)(x (cid:0)(cid:18))))2
...... i=1 i
MinimalSufficientStatistics AncillaryStatistics Location-scaleFamily Summary
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Exercise from the textbook
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Problem
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X ;(cid:1)(cid:1)(cid:1) ;X are iid samples from
1 n
e(cid:0)(x(cid:0)(cid:18))
f (xj(cid:18)) = ;(cid:0)1 < x < 1;(cid:0)1 < (cid:18) < 1
X (1+e(cid:0)(x(cid:0)(cid:18)))2
F...... ind a minimal sufficient statistic for (cid:18).
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HyunMinKang Biostatistics602-Lecture04 January22th,2013 5/23
∑
exp((cid:0) n (x (cid:0)(cid:18)))
= ∏ i=1 i
n (1+exp((cid:0)(x (cid:0)(cid:18))))2
i=1 i
∑
exp((cid:0) n x )exp(n(cid:18))
= ∏ i=1 i
n (1+exp((cid:0)(x (cid:0)(cid:18))))2
i=1 i
MinimalSufficientStatistics AncillaryStatistics Location-scaleFamily Summary
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Exercise from the textbook
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Problem
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X ;(cid:1)(cid:1)(cid:1) ;X are iid samples from
1 n
e(cid:0)(x(cid:0)(cid:18))
f (xj(cid:18)) = ;(cid:0)1 < x < 1;(cid:0)1 < (cid:18) < 1
X (1+e(cid:0)(x(cid:0)(cid:18)))2
F...... ind a minimal sufficient statistic for (cid:18).
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Solution
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∏n exp((cid:0)(x (cid:0)(cid:18)))
f (xj(cid:18)) = i
X (1+exp((cid:0)(x (cid:0)(cid:18))))2
i
i=1
......
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HyunMinKang Biostatistics602-Lecture04 January22th,2013 5/23