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. Summary January 22th, 2013 Biostatistics 602 - Lecture 04 Hyun Min Kang January 22th, 2013 Hyun Min Kang Ancillary Statistics Lecture 04 Biostatistics 602 - Statistical Inference . . . . Location-scale Family Ancillary Statistics

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. . . . . .

. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

. .

Biostatistics 602 - Statistical Inference Lecture 04 Ancillary Statistics

Hyun Min Kang January 22th, 2013

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 1 / 23

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. . . . . .

. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . 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

?

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 2 / 23

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. . . . . .

. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . 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

?

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 2 / 23

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. . . . . .

. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . 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

?

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 2 / 23

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. . . . . .

. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . 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

?

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 2 / 23

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. . . . . .

. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . 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 θ?

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 2 / 23

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. . . . . .

. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

Minimal Sufficient Statistic

.

Definition 6.2.11

. . A sufficient statistic T(X) is called a minimal sufficient statistic if, for any

  • ther sufficient statistic T′(X), T(X) is a function of T′(X).

.

Why is this called ”minimal” sufficient statistic?

. .

  • The sample space X consists of every possible sample - finest partition
  • Given T(X), X can be partitioned into At where

t ∈ T = {t : t = T(X) for some x ∈ X}

  • Maximum data reduction is achieved when |T | is minimal.
  • If size of T ′ = t : t = T′(x) for some x ∈ X is not less than |T |, then

|T | can be called as a minimal sufficient statistic.

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 3 / 23

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. . . . . .

. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

Theorem for Minimal Sufficient Statistics

.

Theorem 6.2.13

. .

  • fX(x) be pmf or pdf of a sample X.
  • Suppose that there exists a function T(x) such that,
  • For every two sample points x and y,
  • The ratio fX(x|θ)/fX(y|θ) is constant as a function of θ if and only if

T(x) = T(y).

  • Then T(X) is a minimal sufficient statistic for θ.

.

In other words..

. .

  • fX(x|θ)/fX(y|θ) is constant as a function of θ =

⇒ T(x) = T(y).

  • T(x) = T(y) =

⇒ fX(x|θ)/fX(y|θ) is constant as a function of θ

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 4 / 23

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. . . . . .

. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

Exercise from the textbook

.

Problem

. . X1, · · · , Xn are iid samples from fX(x|θ) = e−(x−θ) (1 + e−(x−θ))2 , −∞ < x < ∞, −∞ < θ < ∞ Find a minimal sufficient statistic for θ. .

Solution

. . . . . . . . fX x

n i

exp xi exp xi exp

n i

xi

n i

exp xi exp

n i

xi exp n

n i

exp xi

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 5 / 23

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. . . . . .

. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

Exercise from the textbook

.

Problem

. . X1, · · · , Xn are iid samples from fX(x|θ) = e−(x−θ) (1 + e−(x−θ))2 , −∞ < x < ∞, −∞ < θ < ∞ Find a minimal sufficient statistic for θ. .

Solution

. . fX(x|θ) =

n

∏

i=1

exp(−(xi − θ)) (1 + exp(−(xi − θ)))2 exp

n i

xi

n i

exp xi exp

n i

xi exp n

n i

exp xi

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 5 / 23

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. . . . . .

. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

Exercise from the textbook

.

Problem

. . X1, · · · , Xn are iid samples from fX(x|θ) = e−(x−θ) (1 + e−(x−θ))2 , −∞ < x < ∞, −∞ < θ < ∞ Find a minimal sufficient statistic for θ. .

Solution

. . fX(x|θ) =

n

∏

i=1

exp(−(xi − θ)) (1 + exp(−(xi − θ)))2 = exp (− ∑n

i=1(xi − θ))

∏n

i=1(1 + exp(−(xi − θ)))2

exp

n i

xi exp n

n i

exp xi

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 5 / 23

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. . . . . .

. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

Exercise from the textbook

.

Problem

. . X1, · · · , Xn are iid samples from fX(x|θ) = e−(x−θ) (1 + e−(x−θ))2 , −∞ < x < ∞, −∞ < θ < ∞ Find a minimal sufficient statistic for θ. .

Solution

. . fX(x|θ) =

n

∏

i=1

exp(−(xi − θ)) (1 + exp(−(xi − θ)))2 = exp (− ∑n

i=1(xi − θ))

∏n

i=1(1 + exp(−(xi − θ)))2

= exp (− ∑n

i=1 xi) exp(nθ)

∏n

i=1(1 + exp(−(xi − θ)))2

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 5 / 23

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. . . . . .

. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

Solution (cont’d)

.

Applying Theorem 6.2.13

. . fX(x|θ) fX(y|θ) = exp (− ∑n

i=1 xi) exp(nθ) ∏n i=1(1 + exp(−(yi − θ)))2

exp (− ∑n

i=1 yi) exp(nθ) ∏n i=1(1 + exp(−(xi − θ)))2

exp

n i

xi

n i

exp yi exp

n i

yi

n i

exp xi The ratio above is constant to if and only if x xn are permutations

  • f y
  • yn. So the order statistic T X

X X n is a minimal sufficient statistic.

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 6 / 23

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. . . . . .

. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

Solution (cont’d)

.

Applying Theorem 6.2.13

. . fX(x|θ) fX(y|θ) = exp (− ∑n

i=1 xi) exp(nθ) ∏n i=1(1 + exp(−(yi − θ)))2

exp (− ∑n

i=1 yi) exp(nθ) ∏n i=1(1 + exp(−(xi − θ)))2

= exp (− ∑n

i=1 xi) ∏n i=1(1 + exp(−(yi − θ)))2

exp (− ∑n

i=1 yi) ∏n i=1(1 + exp(−(xi − θ)))2

The ratio above is constant to if and only if x xn are permutations

  • f y
  • yn. So the order statistic T X

X X n is a minimal sufficient statistic.

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 6 / 23

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. . . . . .

. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

Solution (cont’d)

.

Applying Theorem 6.2.13

. . fX(x|θ) fX(y|θ) = exp (− ∑n

i=1 xi) exp(nθ) ∏n i=1(1 + exp(−(yi − θ)))2

exp (− ∑n

i=1 yi) exp(nθ) ∏n i=1(1 + exp(−(xi − θ)))2

= exp (− ∑n

i=1 xi) ∏n i=1(1 + exp(−(yi − θ)))2

exp (− ∑n

i=1 yi) ∏n i=1(1 + exp(−(xi − θ)))2

The ratio above is constant to θ if and only if x1, · · · , xn are permutations

  • f y1, · · · , yn. So the order statistic T(X) = (X(1), · · · , X(n)) is a minimal

sufficient statistic.

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 6 / 23

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. . . . . .

. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

Ancillary Statistics

.

Definition 6.2.16

. . A statistic S(X) is an ancillary statistic if its distribution does not depend

  • n θ.

.

Examples of Ancillary Statistics

. . . . . . . . X Xn

i.i.d.

where is known.

  • sX

n n i

Xi X is an ancillary statistic

  • X

X is ancillary.

  • X

X X is ancillary.

  • n

sX n

is ancillary.

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 7 / 23

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. . . . . .

. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

Ancillary Statistics

.

Definition 6.2.16

. . A statistic S(X) is an ancillary statistic if its distribution does not depend

  • n θ.

.

Examples of Ancillary Statistics

. . X1, · · · , Xn

i.i.d.

∼ N(µ, σ2) where σ2 is known.

  • sX

n n i

Xi X is an ancillary statistic

  • X

X is ancillary.

  • X

X X is ancillary.

  • n

sX n

is ancillary.

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 7 / 23

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. . . . . .

. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

Ancillary Statistics

.

Definition 6.2.16

. . A statistic S(X) is an ancillary statistic if its distribution does not depend

  • n θ.

.

Examples of Ancillary Statistics

. . X1, · · · , Xn

i.i.d.

∼ N(µ, σ2) where σ2 is known.

  • s2

X = 1 n−1

∑n

i=1(Xi − X)2 is an ancillary statistic

  • X

X is ancillary.

  • X

X X is ancillary.

  • n

sX n

is ancillary.

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 7 / 23

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. . . . . .

. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

Ancillary Statistics

.

Definition 6.2.16

. . A statistic S(X) is an ancillary statistic if its distribution does not depend

  • n θ.

.

Examples of Ancillary Statistics

. . X1, · · · , Xn

i.i.d.

∼ N(µ, σ2) where σ2 is known.

  • s2

X = 1 n−1

∑n

i=1(Xi − X)2 is an ancillary statistic

  • X1 − X2 ∼ N(0, 2σ2) is ancillary.
  • X

X X is ancillary.

  • n

sX n

is ancillary.

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 7 / 23

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. . . . . .

. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

Ancillary Statistics

.

Definition 6.2.16

. . A statistic S(X) is an ancillary statistic if its distribution does not depend

  • n θ.

.

Examples of Ancillary Statistics

. . X1, · · · , Xn

i.i.d.

∼ N(µ, σ2) where σ2 is known.

  • s2

X = 1 n−1

∑n

i=1(Xi − X)2 is an ancillary statistic

  • X1 − X2 ∼ N(0, 2σ2) is ancillary.
  • (X1 + X2)/2 − X3 ∼ N(0, 1.5σ2) is ancillary.
  • n

sX n

is ancillary.

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 7 / 23

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. . . . . .

. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

Ancillary Statistics

.

Definition 6.2.16

. . A statistic S(X) is an ancillary statistic if its distribution does not depend

  • n θ.

.

Examples of Ancillary Statistics

. . X1, · · · , Xn

i.i.d.

∼ N(µ, σ2) where σ2 is known.

  • s2

X = 1 n−1

∑n

i=1(Xi − X)2 is an ancillary statistic

  • X1 − X2 ∼ N(0, 2σ2) is ancillary.
  • (X1 + X2)/2 − X3 ∼ N(0, 1.5σ2) is ancillary.
  • (n−1)s2

X

σ2

∼ χ2

n−1 is ancillary.

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 7 / 23

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. . . . . .

. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

More Examples of Ancillary Statistics

.

Examples with normal distribution at zero mean

. . X1, · · · , Xn

i.i.d.

∼ N(0, σ2) where σ2 is unknown

  • Y

X is an ancillary statistic because Yi .

  • X

X Y Y Y Y

also follows a cauchy distribution and is an ancillary statistic.

  • Any joint distribution of Y

Yn does not depend on , and thus is an ancillary statistic.

  • For example, the following statistic is also ancillary.

median Xi X median Yi Y median Yi Y

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 8 / 23

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. . . . . .

. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

More Examples of Ancillary Statistics

.

Examples with normal distribution at zero mean

. . X1, · · · , Xn

i.i.d.

∼ N(0, σ2) where σ2 is unknown

  • Y = X/σ is an ancillary statistic because Yi ∼ N(0, 1).
  • X

X Y Y Y Y

also follows a cauchy distribution and is an ancillary statistic.

  • Any joint distribution of Y

Yn does not depend on , and thus is an ancillary statistic.

  • For example, the following statistic is also ancillary.

median Xi X median Yi Y median Yi Y

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 8 / 23

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. . . . . .

. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

More Examples of Ancillary Statistics

.

Examples with normal distribution at zero mean

. . X1, · · · , Xn

i.i.d.

∼ N(0, σ2) where σ2 is unknown

  • Y = X/σ is an ancillary statistic because Yi ∼ N(0, 1).
  • X1

X2 = σY1 σY2 = Y1 Y2 also follows a cauchy distribution and is an ancillary

statistic.

  • Any joint distribution of Y

Yn does not depend on , and thus is an ancillary statistic.

  • For example, the following statistic is also ancillary.

median Xi X median Yi Y median Yi Y

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 8 / 23

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SLIDE 25

. . . . . .

. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

More Examples of Ancillary Statistics

.

Examples with normal distribution at zero mean

. . X1, · · · , Xn

i.i.d.

∼ N(0, σ2) where σ2 is unknown

  • Y = X/σ is an ancillary statistic because Yi ∼ N(0, 1).
  • X1

X2 = σY1 σY2 = Y1 Y2 also follows a cauchy distribution and is an ancillary

statistic.

  • Any joint distribution of Y1, · · · , Yn does not depend on σ2, and thus

is an ancillary statistic.

  • For example, the following statistic is also ancillary.

median Xi X median Yi Y median Yi Y

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 8 / 23

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SLIDE 26

. . . . . .

. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

More Examples of Ancillary Statistics

.

Examples with normal distribution at zero mean

. . X1, · · · , Xn

i.i.d.

∼ N(0, σ2) where σ2 is unknown

  • Y = X/σ is an ancillary statistic because Yi ∼ N(0, 1).
  • X1

X2 = σY1 σY2 = Y1 Y2 also follows a cauchy distribution and is an ancillary

statistic.

  • Any joint distribution of Y1, · · · , Yn does not depend on σ2, and thus

is an ancillary statistic.

  • For example, the following statistic is also ancillary.

median(Xi) X median Yi Y median Yi Y

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 8 / 23

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SLIDE 27

. . . . . .

. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

More Examples of Ancillary Statistics

.

Examples with normal distribution at zero mean

. . X1, · · · , Xn

i.i.d.

∼ N(0, σ2) where σ2 is unknown

  • Y = X/σ is an ancillary statistic because Yi ∼ N(0, 1).
  • X1

X2 = σY1 σY2 = Y1 Y2 also follows a cauchy distribution and is an ancillary

statistic.

  • Any joint distribution of Y1, · · · , Yn does not depend on σ2, and thus

is an ancillary statistic.

  • For example, the following statistic is also ancillary.

median(Xi) X = σmedian(Yi) σY median Yi Y

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 8 / 23

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SLIDE 28

. . . . . .

. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

More Examples of Ancillary Statistics

.

Examples with normal distribution at zero mean

. . X1, · · · , Xn

i.i.d.

∼ N(0, σ2) where σ2 is unknown

  • Y = X/σ is an ancillary statistic because Yi ∼ N(0, 1).
  • X1

X2 = σY1 σY2 = Y1 Y2 also follows a cauchy distribution and is an ancillary

statistic.

  • Any joint distribution of Y1, · · · , Yn does not depend on σ2, and thus

is an ancillary statistic.

  • For example, the following statistic is also ancillary.

median(Xi) X = σmedian(Yi) σY = median(Yi) Y

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 8 / 23

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. . . . . .

. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

Range Statistics

.

Problem

. .

  • X1, · · · , Xn

i.i.d.

∼ fX(x − θ).

  • Show that R = X(n) − X(1) is an ancillary statistic.

.

Solution

. . . . . . . .

  • Let Zi

Xi .

  • fZ z

fX z

dx dz

fX z

  • Z

Zn

i.i.d. fX z does not depend on

.

  • R

X n X Z n Z does not depend on .

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 9 / 23

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. . . . . .

. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

Range Statistics

.

Problem

. .

  • X1, · · · , Xn

i.i.d.

∼ fX(x − θ).

  • Show that R = X(n) − X(1) is an ancillary statistic.

.

Solution

. .

  • Let Zi = Xi − θ.
  • fZ z

fX z

dx dz

fX z

  • Z

Zn

i.i.d. fX z does not depend on

.

  • R

X n X Z n Z does not depend on .

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 9 / 23

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SLIDE 31

. . . . . .

. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

Range Statistics

.

Problem

. .

  • X1, · · · , Xn

i.i.d.

∼ fX(x − θ).

  • Show that R = X(n) − X(1) is an ancillary statistic.

.

Solution

. .

  • Let Zi = Xi − θ.
  • fZ(z) = fX(z + θ − θ)
  • dx

dz

  • = fX(z)
  • Z

Zn

i.i.d. fX z does not depend on

.

  • R

X n X Z n Z does not depend on .

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 9 / 23

slide-32
SLIDE 32

. . . . . .

. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

Range Statistics

.

Problem

. .

  • X1, · · · , Xn

i.i.d.

∼ fX(x − θ).

  • Show that R = X(n) − X(1) is an ancillary statistic.

.

Solution

. .

  • Let Zi = Xi − θ.
  • fZ(z) = fX(z + θ − θ)
  • dx

dz

  • = fX(z)
  • Z1, · · · , Zn

i.i.d.

∼ fX(z) does not depend on θ.

  • R

X n X Z n Z does not depend on .

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 9 / 23

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SLIDE 33

. . . . . .

. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

Range Statistics

.

Problem

. .

  • X1, · · · , Xn

i.i.d.

∼ fX(x − θ).

  • Show that R = X(n) − X(1) is an ancillary statistic.

.

Solution

. .

  • Let Zi = Xi − θ.
  • fZ(z) = fX(z + θ − θ)
  • dx

dz

  • = fX(z)
  • Z1, · · · , Zn

i.i.d.

∼ fX(z) does not depend on θ.

  • R = X(n) − X(1) = Z(n) − Z(1) does not depend on θ.

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 9 / 23

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SLIDE 34

. . . . . .

. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

Uniform Ancillary Statistics

.

Problem

. .

  • X1, · · · , Xn

i.i.d.

∼ Uniform(θ, θ + 1).

  • Show that R = X(n) − X(1) is an ancillary statistic.

.

Possible Strategies

. .

  • Obtain the distribution of R and show that it is independent of

.

  • Represent R as a function of ancillary statistics, which is independent
  • f

.

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 10 / 23

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. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

Uniform Ancillary Statistics

.

Problem

. .

  • X1, · · · , Xn

i.i.d.

∼ Uniform(θ, θ + 1).

  • Show that R = X(n) − X(1) is an ancillary statistic.

.

Possible Strategies

. .

  • Obtain the distribution of R and show that it is independent of θ.
  • Represent R as a function of ancillary statistics, which is independent
  • f

.

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 10 / 23

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SLIDE 36

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. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

Uniform Ancillary Statistics

.

Problem

. .

  • X1, · · · , Xn

i.i.d.

∼ Uniform(θ, θ + 1).

  • Show that R = X(n) − X(1) is an ancillary statistic.

.

Possible Strategies

. .

  • Obtain the distribution of R and show that it is independent of θ.
  • Represent R as a function of ancillary statistics, which is independent
  • f θ.

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 10 / 23

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. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

Proof : Method I - 1/4

R is a function of (X(n), X(1)), so we need to derive the joint distribution

  • f (X(n), X(1)).

Define fX x I x If X X n , fX X X n n n X n X

n

and fX X X n

  • therwise.

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 11 / 23

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. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

Proof : Method I - 1/4

R is a function of (X(n), X(1)), so we need to derive the joint distribution

  • f (X(n), X(1)). Define

fX(x|θ) = I(θ < x < θ + 1) If X X n , fX X X n n n X n X

n

and fX X X n

  • therwise.

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 11 / 23

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. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

Proof : Method I - 1/4

R is a function of (X(n), X(1)), so we need to derive the joint distribution

  • f (X(n), X(1)). Define

fX(x|θ) = I(θ < x < θ + 1) If θ < X(1) ≤ X(n) < θ + 1, fX(X(1), X(n)|θ) = n n X n X

n

and fX X X n

  • therwise.

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 11 / 23

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. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

Proof : Method I - 1/4

R is a function of (X(n), X(1)), so we need to derive the joint distribution

  • f (X(n), X(1)). Define

fX(x|θ) = I(θ < x < θ + 1) If θ < X(1) ≤ X(n) < θ + 1, fX(X(1), X(n)|θ) = n! (n − 2)! ( X(n) − X(1) )(n−2) and fX X X n

  • therwise.

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 11 / 23

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. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

Proof : Method I - 1/4

R is a function of (X(n), X(1)), so we need to derive the joint distribution

  • f (X(n), X(1)). Define

fX(x|θ) = I(θ < x < θ + 1) If θ < X(1) ≤ X(n) < θ + 1, fX(X(1), X(n)|θ) = n! (n − 2)! ( X(n) − X(1) )(n−2) and fX(X(1), X(n)|θ) = 0 otherwise.

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 11 / 23

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. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

Proof : Method I - 2/4

Define R and M as follows { R = X(n) − X(1) M = (X(n) + X(1))/2 Then X M R X n M R The Jacobian is J X M X R X n M X n R

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 12 / 23

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. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

Proof : Method I - 2/4

Define R and M as follows { R = X(n) − X(1) M = (X(n) + X(1))/2 Then { X(1) = M − R/2 X(n) = M + R/2 The Jacobian is J X M X R X n M X n R

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 12 / 23

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. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

Proof : Method I - 2/4

Define R and M as follows { R = X(n) − X(1) M = (X(n) + X(1))/2 Then { X(1) = M − R/2 X(n) = M + R/2 The Jacobian is J =          ∂X(1) ∂M ∂X(1) ∂R ∂X(n) ∂M ∂X(n) ∂R          =     1 − 1

2

1

1 2

    = 1 2 − (−1 2) = 1

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 12 / 23

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. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

Proof : Method I - 3/4

The joint distribution of R and M is fR,M(r, m) = n(n − 1) (2m + r 2 − 2m − r 2 )(n−2) = n(n − 1)r(n−2) Because X X n , m r m r r r m r

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 13 / 23

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. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

Proof : Method I - 3/4

The joint distribution of R and M is fR,M(r, m) = n(n − 1) (2m + r 2 − 2m − r 2 )(n−2) = n(n − 1)r(n−2) Because θ < X(1) ≤ X(n) < θ + 1, θ < 2m − r 2 < 2m + r 2 < θ + 1 r r m r

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 13 / 23

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. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

Proof : Method I - 3/4

The joint distribution of R and M is fR,M(r, m) = n(n − 1) (2m + r 2 − 2m − r 2 )(n−2) = n(n − 1)r(n−2) Because θ < X(1) ≤ X(n) < θ + 1, θ < 2m − r 2 < 2m + r 2 < θ + 1 0 < r < 1 θ + r 2 < m < θ + 1 − r 2

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 13 / 23

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. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

Proof : Method I - 4/4

The distribution of R is fR(r|θ) = ∫ θ+1− r

2

θ+ r

2

n(n − 1)r(n−2)dm n n r n r r n n r n r r Therefore, fR r does not depend on , and R is an ancillary statistic.

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 14 / 23

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. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

Proof : Method I - 4/4

The distribution of R is fR(r|θ) = ∫ θ+1− r

2

θ+ r

2

n(n − 1)r(n−2)dm = n(n − 1)r(n−2) ( θ + 1 − r 2 − θ − r 2 ) n n r n r r Therefore, fR r does not depend on , and R is an ancillary statistic.

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 14 / 23

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. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

Proof : Method I - 4/4

The distribution of R is fR(r|θ) = ∫ θ+1− r

2

θ+ r

2

n(n − 1)r(n−2)dm = n(n − 1)r(n−2) ( θ + 1 − r 2 − θ − r 2 ) = n(n − 1)r(n−2)(1 − r) , 0 < r < 1 Therefore, fR r does not depend on , and R is an ancillary statistic.

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 14 / 23

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. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

Proof : Method I - 4/4

The distribution of R is fR(r|θ) = ∫ θ+1− r

2

θ+ r

2

n(n − 1)r(n−2)dm = n(n − 1)r(n−2) ( θ + 1 − r 2 − θ − r 2 ) = n(n − 1)r(n−2)(1 − r) , 0 < r < 1 Therefore, fR(r|θ) does not depend on θ, and R is an ancillary statistic.

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 14 / 23

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. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

Method II : Probably A Simpler Proof

fX(x|θ) = I(θ < x < θ + 1) = I(0 < x − θ < 1) Let Yi Xi Uniform . Then Xi Yi ,

dx dy

holds. fY y I y

dx dy

I y Then, the range statistic R can be rewritten as follows. R X n X Y n Y Y n Y As Y n Y is a function of Y

  • Yn. Any joint distribution of

Y Yn does not depend on . Therefore, R is an ancillary statistic.

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 15 / 23

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. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

Method II : Probably A Simpler Proof

fX(x|θ) = I(θ < x < θ + 1) = I(0 < x − θ < 1) Let Yi = Xi − θ ∼ Uniform(0, 1). Then Xi = Yi + θ, | dx

dy| = 1 holds.

fY y I y

dx dy

I y Then, the range statistic R can be rewritten as follows. R X n X Y n Y Y n Y As Y n Y is a function of Y

  • Yn. Any joint distribution of

Y Yn does not depend on . Therefore, R is an ancillary statistic.

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 15 / 23

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. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

Method II : Probably A Simpler Proof

fX(x|θ) = I(θ < x < θ + 1) = I(0 < x − θ < 1) Let Yi = Xi − θ ∼ Uniform(0, 1). Then Xi = Yi + θ, | dx

dy| = 1 holds.

fY(y) = I(0 < y + θ − θ < 1)| dx

dy| = I(0 < y < 1)

Then, the range statistic R can be rewritten as follows. R X n X Y n Y Y n Y As Y n Y is a function of Y

  • Yn. Any joint distribution of

Y Yn does not depend on . Therefore, R is an ancillary statistic.

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 15 / 23

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. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

Method II : Probably A Simpler Proof

fX(x|θ) = I(θ < x < θ + 1) = I(0 < x − θ < 1) Let Yi = Xi − θ ∼ Uniform(0, 1). Then Xi = Yi + θ, | dx

dy| = 1 holds.

fY(y) = I(0 < y + θ − θ < 1)| dx

dy| = I(0 < y < 1)

Then, the range statistic R can be rewritten as follows. R = X(n) − X(1) = (Y(n) + θ) − (Y(1) + θ) = Y(n) − Y(1) As Y n Y is a function of Y

  • Yn. Any joint distribution of

Y Yn does not depend on . Therefore, R is an ancillary statistic.

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 15 / 23

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. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

Method II : Probably A Simpler Proof

fX(x|θ) = I(θ < x < θ + 1) = I(0 < x − θ < 1) Let Yi = Xi − θ ∼ Uniform(0, 1). Then Xi = Yi + θ, | dx

dy| = 1 holds.

fY(y) = I(0 < y + θ − θ < 1)| dx

dy| = I(0 < y < 1)

Then, the range statistic R can be rewritten as follows. R = X(n) − X(1) = (Y(n) + θ) − (Y(1) + θ) = Y(n) − Y(1) As Y(n) − Y(1) is a function of Y1, · · · , Yn. Any joint distribution of Y1, · · · , Yn does not depend on θ. Therefore, R is an ancillary statistic.

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 15 / 23

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. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

A brief review on location and scale family

.

Theorem 3.5.1

. . Let f(x) be any pdf and let µ and σ > 0 be any given constant, then, g(x|µ, σ) = 1 σf (x − µ σ ) is a pdf. .

Proof

. . . . . . . . Because f x is a pdf, then f x , and g x for all x. Let y x , then x y , and dx dy . f x dx f y dy f y dy Therefore, g x is also a pdf.

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 16 / 23

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SLIDE 58

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. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

A brief review on location and scale family

.

Theorem 3.5.1

. . Let f(x) be any pdf and let µ and σ > 0 be any given constant, then, g(x|µ, σ) = 1 σf (x − µ σ ) is a pdf. .

Proof

. . Because f(x) is a pdf, then f(x) ≥ 0, and g(x|µ, σ) ≥ 0 for all x. Let y x , then x y , and dx dy . f x dx f y dy f y dy Therefore, g x is also a pdf.

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 16 / 23

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SLIDE 59

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. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

A brief review on location and scale family

.

Theorem 3.5.1

. . Let f(x) be any pdf and let µ and σ > 0 be any given constant, then, g(x|µ, σ) = 1 σf (x − µ σ ) is a pdf. .

Proof

. . Because f(x) is a pdf, then f(x) ≥ 0, and g(x|µ, σ) ≥ 0 for all x. Let y = (x − µ)/σ, then x = σy + µ, and dx/dy = σ. f x dx f y dy f y dy Therefore, g x is also a pdf.

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 16 / 23

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SLIDE 60

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. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

A brief review on location and scale family

.

Theorem 3.5.1

. . Let f(x) be any pdf and let µ and σ > 0 be any given constant, then, g(x|µ, σ) = 1 σf (x − µ σ ) is a pdf. .

Proof

. . Because f(x) is a pdf, then f(x) ≥ 0, and g(x|µ, σ) ≥ 0 for all x. Let y = (x − µ)/σ, then x = σy + µ, and dx/dy = σ. ∫ ∞

−∞

1 σf (x − µ σ ) dx = ∫ ∞

−∞

1 σf(y)σdy = ∫ ∞

−∞

f(y)dy = 1 Therefore, g x is also a pdf.

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 16 / 23

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SLIDE 61

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. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

A brief review on location and scale family

.

Theorem 3.5.1

. . Let f(x) be any pdf and let µ and σ > 0 be any given constant, then, g(x|µ, σ) = 1 σf (x − µ σ ) is a pdf. .

Proof

. . Because f(x) is a pdf, then f(x) ≥ 0, and g(x|µ, σ) ≥ 0 for all x. Let y = (x − µ)/σ, then x = σy + µ, and dx/dy = σ. ∫ ∞

−∞

1 σf (x − µ σ ) dx = ∫ ∞

−∞

1 σf(y)σdy = ∫ ∞

−∞

f(y)dy = 1 Therefore, g(x|µ, σ) is also a pdf.

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 16 / 23

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. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

Location Family and Parameter

.

Definition 3.5.2

. . Let f(x) be any pdf. Then the family of pdfs f(x − µ), indexed by the parameter −∞ < µ < ∞, is called the location family with standard pdf f(x), and µ is called the location parameter for the family. .

Example

. . . . . . . .

  • f x

e

x

  • f x

e

x

  • f x

I x Uniform

  • f x

I x Uniform

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 17 / 23

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. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

Location Family and Parameter

.

Definition 3.5.2

. . Let f(x) be any pdf. Then the family of pdfs f(x − µ), indexed by the parameter −∞ < µ < ∞, is called the location family with standard pdf f(x), and µ is called the location parameter for the family. .

Example

. .

  • f(x) =

1 √ 2πe−x2/2 ∼ N(0, 1)

  • f(x − µ) =

1 √ 2πe−(x−µ)2/2 ∼ N(µ, 1)

  • f x

I x Uniform

  • f x

I x Uniform

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 17 / 23

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. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

Location Family and Parameter

.

Definition 3.5.2

. . Let f(x) be any pdf. Then the family of pdfs f(x − µ), indexed by the parameter −∞ < µ < ∞, is called the location family with standard pdf f(x), and µ is called the location parameter for the family. .

Example

. .

  • f(x) =

1 √ 2πe−x2/2 ∼ N(0, 1)

  • f(x − µ) =

1 √ 2πe−(x−µ)2/2 ∼ N(µ, 1)

  • f(x) = I(0 < x < 1) ∼ Uniform(0, 1)
  • f(x − θ) = I(θ < x < θ + 1) ∼ Uniform(θ, θ + 1)

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 17 / 23

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. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

Scale Family and Parameter

.

Definition 3.5.4

. . Let f(x) be any pdf. Then for any σ > 0 the family of pdfs f(x/σ)/σ, indexed by the parameter σ is called the scale family with standard pdf f(x), and σ is called the scale parameter for the family. .

Example

. . . . . . . .

  • f x

e

x

  • f x

e

x

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 18 / 23

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. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

Scale Family and Parameter

.

Definition 3.5.4

. . Let f(x) be any pdf. Then for any σ > 0 the family of pdfs f(x/σ)/σ, indexed by the parameter σ is called the scale family with standard pdf f(x), and σ is called the scale parameter for the family. .

Example

. .

  • f(x) =

1 √ 2πe−x2/2 ∼ N(0, 1)

  • f(x/σ)/σ =

1 √ 2πσ2 e−x2/2σ2 ∼ N(0, σ2)

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 18 / 23

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. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

Location-Scale Family and Parameters

.

Definition 3.5.5

. . Let f(x) be any pdf. Then for any µ, −∞ < µ < ∞, and any σ > 0 the family of pdfs f((x − µ)/σ)/σ, indexed by the parameter (µ, σ) is called the location-scale family with standard pdf f(x), and µ is called the location parameter and σ is called the scale parameter for the family. .

Example

. . . . . . . .

  • f x

e

x

  • f x

e

x

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 19 / 23

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. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

Location-Scale Family and Parameters

.

Definition 3.5.5

. . Let f(x) be any pdf. Then for any µ, −∞ < µ < ∞, and any σ > 0 the family of pdfs f((x − µ)/σ)/σ, indexed by the parameter (µ, σ) is called the location-scale family with standard pdf f(x), and µ is called the location parameter and σ is called the scale parameter for the family. .

Example

. .

  • f(x) =

1 √ 2πe−x2/2 ∼ N(0, 1)

  • f((x − µ)/σ)/σ =

1 √ 2πσ2 e−(x−µ)2/2σ2 ∼ N(µ, σ2)

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 19 / 23

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. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

Theorem for location and scale family

.

Theorem 3.5.6

. .

  • Let f(·) be any pdf.
  • Let µ be any real number.
  • Let σ be any positive real number.
  • Then X is a random variable with pdf 1

σf

(x−µ

σ

)

  • if and only if there exists a random variable Z with pdf f(z) and

X = σZ + µ.

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 20 / 23

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. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

Ancillary Statistics for Location Family

.

Problem

. . Let X1, · · · , Xn be iid from a location family with pdf f(x − µ) where −∞ < µ < ∞. Show that the range R = X(n) − X(1) is an ancillary statistic. .

Solution

. . . . . . . . Assume that cdf is F x . Using Theorem 3.5.6, Z X Zn Xn are iid observations from pdf f x and cdf F x . Then the cdf of the range statistic R becomes FR r Pr R r Pr X n X r Pr Z n Z r Pr Z n Z r which does not depend on because Z Zn does not depend on . Therefore, R is an ancillary statistic.

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 21 / 23

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. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

Ancillary Statistics for Location Family

.

Problem

. . Let X1, · · · , Xn be iid from a location family with pdf f(x − µ) where −∞ < µ < ∞. Show that the range R = X(n) − X(1) is an ancillary statistic. .

Solution

. . Assume that cdf is F(x − µ). Using Theorem 3.5.6, Z1 = X1 − µ, · · · , Zn = Xn − µ are iid observations from pdf f(x) and cdf F(x). Then the cdf of the range statistic R becomes FR r Pr R r Pr X n X r Pr Z n Z r Pr Z n Z r which does not depend on because Z Zn does not depend on . Therefore, R is an ancillary statistic.

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 21 / 23

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. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

Ancillary Statistics for Location Family

.

Problem

. . Let X1, · · · , Xn be iid from a location family with pdf f(x − µ) where −∞ < µ < ∞. Show that the range R = X(n) − X(1) is an ancillary statistic. .

Solution

. . Assume that cdf is F(x − µ). Using Theorem 3.5.6, Z1 = X1 − µ, · · · , Zn = Xn − µ are iid observations from pdf f(x) and cdf F(x). Then the cdf of the range statistic R becomes FR r Pr R r Pr X n X r Pr Z n Z r Pr Z n Z r which does not depend on because Z Zn does not depend on . Therefore, R is an ancillary statistic.

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 21 / 23

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SLIDE 73

. . . . . .

. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

Ancillary Statistics for Location Family

.

Problem

. . Let X1, · · · , Xn be iid from a location family with pdf f(x − µ) where −∞ < µ < ∞. Show that the range R = X(n) − X(1) is an ancillary statistic. .

Solution

. . Assume that cdf is F(x − µ). Using Theorem 3.5.6, Z1 = X1 − µ, · · · , Zn = Xn − µ are iid observations from pdf f(x) and cdf F(x). Then the cdf of the range statistic R becomes FR(r|µ) = Pr(R ≤ r|µ) = Pr(X(n) − X(1) ≤ r|µ) Pr Z n Z r Pr Z n Z r which does not depend on because Z Zn does not depend on . Therefore, R is an ancillary statistic.

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 21 / 23

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SLIDE 74

. . . . . .

. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

Ancillary Statistics for Location Family

.

Problem

. . Let X1, · · · , Xn be iid from a location family with pdf f(x − µ) where −∞ < µ < ∞. Show that the range R = X(n) − X(1) is an ancillary statistic. .

Solution

. . Assume that cdf is F(x − µ). Using Theorem 3.5.6, Z1 = X1 − µ, · · · , Zn = Xn − µ are iid observations from pdf f(x) and cdf F(x). Then the cdf of the range statistic R becomes FR(r|µ) = Pr(R ≤ r|µ) = Pr(X(n) − X(1) ≤ r|µ) = Pr(Z(n) + µ − Z(1) − µ ≤ r|µ) = Pr(Z(n) − Z(1) ≤ r|µ) which does not depend on because Z Zn does not depend on . Therefore, R is an ancillary statistic.

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 21 / 23

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SLIDE 75

. . . . . .

. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

Ancillary Statistics for Location Family

.

Problem

. . Let X1, · · · , Xn be iid from a location family with pdf f(x − µ) where −∞ < µ < ∞. Show that the range R = X(n) − X(1) is an ancillary statistic. .

Solution

. . Assume that cdf is F(x − µ). Using Theorem 3.5.6, Z1 = X1 − µ, · · · , Zn = Xn − µ are iid observations from pdf f(x) and cdf F(x). Then the cdf of the range statistic R becomes FR(r|µ) = Pr(R ≤ r|µ) = Pr(X(n) − X(1) ≤ r|µ) = Pr(Z(n) + µ − Z(1) − µ ≤ r|µ) = Pr(Z(n) − Z(1) ≤ r|µ) which does not depend on µ because Z1, · · · , Zn does not depend on µ. Therefore, R is an ancillary statistic.

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 21 / 23

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SLIDE 76

. . . . . .

. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

Ancillary Statistics for Scale Family

.

Problem

. . Let X1, · · · , Xn be iid from a location family with pdf f(x/σ)/σ where σ > 0. Show that the following statistic T(X) is ancillary. T(X) = (X1/Xn, · · · , Xn−1/Xn) .

Solution

. . . . . . . . Assume that cdf is F x , and let Z X Zn Xn be iid

  • bservations from pdf f x and cdf F x . Then the joint cdf of the T X is

FT t tn Pr X Xn t Xn Xn tn Pr Z Zn t Zn Zn tn Pr Z Zn t Zn Zn tn Because Z Zn does not depend on , T X is an ancillary statistic.

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 22 / 23

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SLIDE 77

. . . . . .

. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

Ancillary Statistics for Scale Family

.

Problem

. . Let X1, · · · , Xn be iid from a location family with pdf f(x/σ)/σ where σ > 0. Show that the following statistic T(X) is ancillary. T(X) = (X1/Xn, · · · , Xn−1/Xn) .

Solution

. . Assume that cdf is F(x/σ), and let Z1 = X1/σ, · · · , Zn = Xn/σ be iid

  • bservations from pdf f(x) and cdf F(x).

Then the joint cdf of the T X is FT t tn Pr X Xn t Xn Xn tn Pr Z Zn t Zn Zn tn Pr Z Zn t Zn Zn tn Because Z Zn does not depend on , T X is an ancillary statistic.

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 22 / 23

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SLIDE 78

. . . . . .

. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

Ancillary Statistics for Scale Family

.

Problem

. . Let X1, · · · , Xn be iid from a location family with pdf f(x/σ)/σ where σ > 0. Show that the following statistic T(X) is ancillary. T(X) = (X1/Xn, · · · , Xn−1/Xn) .

Solution

. . Assume that cdf is F(x/σ), and let Z1 = X1/σ, · · · , Zn = Xn/σ be iid

  • bservations from pdf f(x) and cdf F(x). Then the joint cdf of the T(X) is

FT t tn Pr X Xn t Xn Xn tn Pr Z Zn t Zn Zn tn Pr Z Zn t Zn Zn tn Because Z Zn does not depend on , T X is an ancillary statistic.

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 22 / 23

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SLIDE 79

. . . . . .

. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

Ancillary Statistics for Scale Family

.

Problem

. . Let X1, · · · , Xn be iid from a location family with pdf f(x/σ)/σ where σ > 0. Show that the following statistic T(X) is ancillary. T(X) = (X1/Xn, · · · , Xn−1/Xn) .

Solution

. . Assume that cdf is F(x/σ), and let Z1 = X1/σ, · · · , Zn = Xn/σ be iid

  • bservations from pdf f(x) and cdf F(x). Then the joint cdf of the T(X) is

FT(t1, · · · , tn−1|σ) = Pr(X1/Xn ≤ t1, · · · , Xn−1/Xn ≤ tn−1|σ) Pr Z Zn t Zn Zn tn Pr Z Zn t Zn Zn tn Because Z Zn does not depend on , T X is an ancillary statistic.

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 22 / 23

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SLIDE 80

. . . . . .

. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

Ancillary Statistics for Scale Family

.

Problem

. . Let X1, · · · , Xn be iid from a location family with pdf f(x/σ)/σ where σ > 0. Show that the following statistic T(X) is ancillary. T(X) = (X1/Xn, · · · , Xn−1/Xn) .

Solution

. . Assume that cdf is F(x/σ), and let Z1 = X1/σ, · · · , Zn = Xn/σ be iid

  • bservations from pdf f(x) and cdf F(x). Then the joint cdf of the T(X) is

FT(t1, · · · , tn−1|σ) = Pr(X1/Xn ≤ t1, · · · , Xn−1/Xn ≤ tn−1|σ) = Pr(σZ1/σZn ≤ t1, · · · , σZn−1/σZn ≤ tn−1|σ) Pr Z Zn t Zn Zn tn Because Z Zn does not depend on , T X is an ancillary statistic.

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 22 / 23

slide-81
SLIDE 81

. . . . . .

. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

Ancillary Statistics for Scale Family

.

Problem

. . Let X1, · · · , Xn be iid from a location family with pdf f(x/σ)/σ where σ > 0. Show that the following statistic T(X) is ancillary. T(X) = (X1/Xn, · · · , Xn−1/Xn) .

Solution

. . Assume that cdf is F(x/σ), and let Z1 = X1/σ, · · · , Zn = Xn/σ be iid

  • bservations from pdf f(x) and cdf F(x). Then the joint cdf of the T(X) is

FT(t1, · · · , tn−1|σ) = Pr(X1/Xn ≤ t1, · · · , Xn−1/Xn ≤ tn−1|σ) = Pr(σZ1/σZn ≤ t1, · · · , σZn−1/σZn ≤ tn−1|σ) = Pr(Z1/Zn ≤ t1, · · · , Zn−1/Zn ≤ tn−1|σ) Because Z Zn does not depend on , T X is an ancillary statistic.

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 22 / 23

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SLIDE 82

. . . . . .

. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

Ancillary Statistics for Scale Family

.

Problem

. . Let X1, · · · , Xn be iid from a location family with pdf f(x/σ)/σ where σ > 0. Show that the following statistic T(X) is ancillary. T(X) = (X1/Xn, · · · , Xn−1/Xn) .

Solution

. . Assume that cdf is F(x/σ), and let Z1 = X1/σ, · · · , Zn = Xn/σ be iid

  • bservations from pdf f(x) and cdf F(x). Then the joint cdf of the T(X) is

FT(t1, · · · , tn−1|σ) = Pr(X1/Xn ≤ t1, · · · , Xn−1/Xn ≤ tn−1|σ) = Pr(σZ1/σZn ≤ t1, · · · , σZn−1/σZn ≤ tn−1|σ) = Pr(Z1/Zn ≤ t1, · · · , Zn−1/Zn ≤ tn−1|σ) Because Z1, · · · , Zn does not depend on σ, T(X) is an ancillary statistic.

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 22 / 23

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SLIDE 83

. . . . . .

. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

Summary

.

Today

. .

  • Minimal Sufficient Statistics
  • Recap from last lecture
  • Example from the textbook
  • Ancillary Statistics
  • Definition
  • Examples
  • Location-scale family and parameters

.

Next Lecture

. . . . . . . .

  • Complete Statistics

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 23 / 23

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SLIDE 84

. . . . . .

. . . . . Minimal Sufficient Statistics . . . . . . . . . Ancillary Statistics . . . . . . . Location-scale Family . Summary

Summary

.

Today

. .

  • Minimal Sufficient Statistics
  • Recap from last lecture
  • Example from the textbook
  • Ancillary Statistics
  • Definition
  • Examples
  • Location-scale family and parameters

.

Next Lecture

. .

  • Complete Statistics

Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 23 / 23