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 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?

Minimal Sufficient Statistics Ancillary Statistics Location-scale Family Summary ...... Recap from last lecture

1. Is a sufficient statistic unique?

...... Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 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 θ?

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?

...... Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 2 / 23 4. Is a minimal sufficient statistic unique? 5. How can we show that a statistic is minimal sufficient for θ?

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?

...... Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 2 / 23 5. How can we show that a statistic is minimal sufficient for θ?

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?

...... Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 2 / 23 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 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 ′ ′ .other 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 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 . Solution . ∑ ∏n − − exp (− n (x − θ)) | exp( (xi θ)) ∏ i=1 i fX(x θ) = 2 = n 2 (1 + exp(−(xi − θ))) (1 + exp(−(xi − θ))) i=1 ∑ i=1 exp (− n x ) exp(nθ) = ∏ i=1 i n − − 2 . i=1(1 + exp( (xi θ)))

Minimal Sufficient Statistics Ancillary Statistics Location-scale Family Summary ...... Exercise from the textbook . Problem . X1, ··· , Xn are iid samples from

e−(x−θ) f (x|θ) = , −∞ < x < ∞, −∞ < θ < ∞ X (1 + e−(x−θ))2

.Find a minimal sufficient statistic for θ.

...... Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 5 / 23 ∑ − n − ∏ exp ( i=1(xi θ)) = n 2 (1 + exp(−(xi − θ))) ∑ i=1 exp (− n x ) exp(nθ) = ∏ i=1 i n − − 2 i=1(1 + exp( (xi θ)))

Minimal Sufficient Statistics Ancillary Statistics Location-scale Family Summary ...... Exercise from the textbook . Problem . X1, ··· , Xn are iid samples from

e−(x−θ) f (x|θ) = , −∞ < x < ∞, −∞ < θ < ∞ X (1 + e−(x−θ))2

.Find a minimal sufficient statistic for θ. . Solution .

∏n exp(−(x − θ)) f (x|θ) = i X (1 + exp(−(x − θ)))2 i=1 i

...... Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 5 / 23 ∑ exp (− n x ) exp(nθ) = ∏ i=1 i n − − 2 i=1(1 + exp( (xi θ)))

Minimal Sufficient Statistics Ancillary Statistics Location-scale Family Summary ...... Exercise from the textbook . Problem . X1, ··· , Xn are iid samples from

e−(x−θ) f (x|θ) = , −∞ < x < ∞, −∞ < θ < ∞ X (1 + e−(x−θ))2

.Find a minimal sufficient statistic for θ. . Solution . ∑ ∏n exp(−(x − θ)) exp (− n (x − θ)) f (x|θ) = i = ∏ i=1 i X (1 + exp(−(x − θ)))2 n (1 + exp(−(x − θ)))2 i=1 i i=1 i

...... Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 5 / 23 Minimal Sufficient Statistics Ancillary Statistics Location-scale Family Summary ...... Exercise from the textbook . Problem . X1, ··· , Xn are iid samples from

e−(x−θ) f (x|θ) = , −∞ < x < ∞, −∞ < θ < ∞ X (1 + e−(x−θ))2

.Find a minimal sufficient statistic for θ. . Solution . ∑ ∏n − − exp (− n (x − θ)) | exp( (xi θ)) ∏ i=1 i fX(x θ) = 2 = n 2 (1 + exp(−(xi − θ))) (1 + exp(−(xi − θ))) i=1 ∑ i=1 exp (− n x ) exp(nθ) = ∏ i=1 i n − − 2 . i=1(1 + exp( (xi θ))) ...... Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 5 / 23 ∑ ∏ exp (− n x ) n (1 + exp(−(y − θ)))2 = ∑i=1 i ∏i=1 i − n n − − 2 exp ( i=1 yi) i=1(1 + exp( (xi θ)))

The ratio above is constant to θ if and only if x1, ··· , xn are permutations ··· ··· of y1, , yn. So the order statistic T(X) = (X(1), , X(n)) is a minimal sufficient statistic.

Minimal Sufficient Statistics Ancillary Statistics Location-scale Family Summary ...... Solution (cont’d)

. Applying Theorem 6.2.13 . ∑ ∏ f (x|θ) exp (− n x ) exp(nθ) n (1 + exp(−(y − θ)))2 X = ∑i=1 i ∏i=1 i | − n n − − 2 fX(y θ) exp ( i=1 yi) exp(nθ) i=1(1 + exp( (xi θ)))

.

...... Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 6 / 23 The ratio above is constant to θ if and only if x1, ··· , xn are permutations ··· ··· of y1, , yn. So the order statistic T(X) = (X(1), , X(n)) is a minimal sufficient statistic.

Minimal Sufficient Statistics Ancillary Statistics Location-scale Family Summary ...... Solution (cont’d)

. Applying Theorem 6.2.13 . ∑ ∏ f (x|θ) exp (− n x ) exp(nθ) n (1 + exp(−(y − θ)))2 X = ∑i=1 i ∏i=1 i f (y|θ) exp (− n y ) exp(nθ) n (1 + exp(−(x − θ)))2 X ∑i=1 i ∏ i=1 i exp (− n x ) n (1 + exp(−(y − θ)))2 = ∑i=1 i ∏i=1 i − n n − − 2 exp ( i=1 yi) i=1(1 + exp( (xi θ)))

.

...... Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 6 / 23 Minimal Sufficient Statistics Ancillary Statistics Location-scale Family Summary ...... Solution (cont’d)

. Applying Theorem 6.2.13 . ∑ ∏ f (x|θ) exp (− n x ) exp(nθ) n (1 + exp(−(y − θ)))2 X = ∑i=1 i ∏i=1 i f (y|θ) exp (− n y ) exp(nθ) n (1 + exp(−(x − θ)))2 X ∑i=1 i ∏ i=1 i exp (− n x ) n (1 + exp(−(y − θ)))2 = ∑i=1 i ∏i=1 i − n n − − 2 exp ( i=1 yi) i=1(1 + exp( (xi θ)))

The ratio above is constant to θ if and only if x1, ··· , xn are permutations ··· ··· of 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 . Examples of Ancillary Statistics . X , ··· , X ∼i.i.d. N (µ, σ2) where σ2 is known. 1 n ∑ 2 1 n − 2 • sX = n−1 i=1(Xi X) is an ancillary statistic 2 • X1 − X2 ∼ N (0, 2σ ) is ancillary. 2 • (X1 + X2)/2 − X3 ∼ N (0, 1.5σ ) is ancillary. (n−1)s2 • X ∼ χ2 is ancillary. . σ2 n−1

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 .on θ.

...... Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 7 / 23 ∑ 2 1 n − 2 • sX = n−1 i=1(Xi X) is an ancillary statistic 2 • X1 − X2 ∼ N (0, 2σ ) is ancillary. 2 • (X1 + X2)/2 − X3 ∼ N (0, 1.5σ ) is ancillary. (n−1)s2 • X ∼ 2 σ2 χn−1 is ancillary.

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 .on θ. . Examples of Ancillary Statistics . i.i.d. 2 2 X1, ··· , Xn ∼ N (µ, σ ) where σ is known.

.

...... Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 7 / 23 2 • X1 − X2 ∼ N (0, 2σ ) is ancillary. 2 • (X1 + X2)/2 − X3 ∼ N (0, 1.5σ ) is ancillary. (n−1)s2 • X ∼ 2 σ2 χn−1 is ancillary.

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 .on θ. . Examples of Ancillary Statistics . X , ··· , X ∼i.i.d. N (µ, σ2) where σ2 is known. 1 n ∑ 2 1 n − 2 • sX = n−1 i=1(Xi X) is an ancillary statistic

.

...... Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 7 / 23 2 • (X1 + X2)/2 − X3 ∼ N (0, 1.5σ ) is ancillary. (n−1)s2 • X ∼ 2 σ2 χn−1 is ancillary.

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 .on θ. . Examples of Ancillary Statistics . X , ··· , X ∼i.i.d. N (µ, σ2) where σ2 is known. 1 n ∑ 2 1 n − 2 • sX = n−1 i=1(Xi X) is an ancillary statistic 2 • X1 − X2 ∼ N (0, 2σ ) is ancillary.

.

...... Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 7 / 23 (n−1)s2 • X ∼ 2 σ2 χn−1 is ancillary.

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 .on θ. . Examples of Ancillary Statistics . X , ··· , X ∼i.i.d. N (µ, σ2) where σ2 is known. 1 n ∑ 2 1 n − 2 • sX = n−1 i=1(Xi X) is an ancillary statistic 2 • X1 − X2 ∼ N (0, 2σ ) is ancillary. 2 • (X1 + X2)/2 − X3 ∼ N (0, 1.5σ ) is ancillary.

.

...... Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 7 / 23 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 .on θ. . Examples of Ancillary Statistics . X , ··· , X ∼i.i.d. N (µ, σ2) where σ2 is known. 1 n ∑ 2 1 n − 2 • sX = n−1 i=1(Xi X) is an ancillary statistic 2 • X1 − X2 ∼ N (0, 2σ ) is ancillary. 2 • (X1 + X2)/2 − X3 ∼ N (0, 1.5σ ) is ancillary. (n−1)s2 • X ∼ χ2 is ancillary. . σ2 n−1

...... Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 7 / 23 • Y = X/σ is an ancillary statistic because Yi ∼ N (0, 1). • X1 = σY1 = Y1 also follows a cauchy distribution and is an ancillary X2 σY2 Y2 statistic. 2 • Any joint distribution of Y1, ··· , Yn does not depend on σ , and thus is an ancillary statistic. • For example, the following statistic is also ancillary. median(X ) σmedian(Y ) median(Y ) i = i = i X σY Y

Minimal Sufficient Statistics Ancillary Statistics Location-scale Family Summary ...... More Examples of Ancillary Statistics

. Examples with normal distribution at zero mean . i.i.d. 2 2 X1, ··· , Xn ∼ N (0, σ ) where σ is unknown

.

...... Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 8 / 23 • X1 = σY1 = Y1 also follows a cauchy distribution and is an ancillary X2 σY2 Y2 statistic. 2 • Any joint distribution of Y1, ··· , Yn does not depend on σ , and thus is an ancillary statistic. • For example, the following statistic is also ancillary. median(X ) σmedian(Y ) median(Y ) i = i = i X σY Y

Minimal Sufficient Statistics Ancillary Statistics Location-scale Family Summary ...... More Examples of Ancillary Statistics

. Examples with normal distribution at zero mean . i.i.d. 2 2 X1, ··· , Xn ∼ N (0, σ ) where σ is unknown

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

.

...... Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 8 / 23 2 • Any joint distribution of Y1, ··· , Yn does not depend on σ , and thus is an ancillary statistic. • For example, the following statistic is also ancillary. median(X ) σmedian(Y ) median(Y ) i = i = i X σY Y

Minimal Sufficient Statistics Ancillary Statistics Location-scale Family Summary ...... More Examples of Ancillary Statistics

. Examples with normal distribution at zero mean . i.i.d. 2 2 X1, ··· , Xn ∼ N (0, σ ) where σ is unknown

• Y = X/σ is an ancillary statistic because Yi ∼ N (0, 1). • X1 = σY1 = Y1 also follows a cauchy distribution and is an ancillary X2 σY2 Y2 statistic.

.

...... Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 8 / 23 • For example, the following statistic is also ancillary. median(X ) σmedian(Y ) median(Y ) i = i = i X σY Y

Minimal Sufficient Statistics Ancillary Statistics Location-scale Family Summary ...... More Examples of Ancillary Statistics

. Examples with normal distribution at zero mean . i.i.d. 2 2 X1, ··· , Xn ∼ N (0, σ ) where σ is unknown

• Y = X/σ is an ancillary statistic because Yi ∼ N (0, 1). • X1 = σY1 = Y1 also follows a cauchy distribution and is an ancillary X2 σY2 Y2 statistic. 2 • Any joint distribution of Y1, ··· , Yn does not depend on σ , and thus is an ancillary statistic.

.

...... Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 8 / 23 σmedian(Y ) median(Y ) = i = i σY Y

Minimal Sufficient Statistics Ancillary Statistics Location-scale Family Summary ...... More Examples of Ancillary Statistics

. Examples with normal distribution at zero mean . i.i.d. 2 2 X1, ··· , Xn ∼ N (0, σ ) where σ is unknown

• Y = X/σ is an ancillary statistic because Yi ∼ N (0, 1). • X1 = σY1 = Y1 also follows a cauchy distribution and is an ancillary X2 σY2 Y2 statistic. 2 • Any joint distribution of Y1, ··· , Yn does not depend on σ , and thus is an ancillary statistic. • For example, the following statistic is also ancillary. median(Xi) X .

...... Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 8 / 23 median(Y ) = i Y

Minimal Sufficient Statistics Ancillary Statistics Location-scale Family Summary ...... More Examples of Ancillary Statistics

. Examples with normal distribution at zero mean . i.i.d. 2 2 X1, ··· , Xn ∼ N (0, σ ) where σ is unknown

• Y = X/σ is an ancillary statistic because Yi ∼ N (0, 1). • X1 = σY1 = Y1 also follows a cauchy distribution and is an ancillary X2 σY2 Y2 statistic. 2 • Any joint distribution of Y1, ··· , Yn does not depend on σ , and thus is an ancillary statistic. • For example, the following statistic is also ancillary. median(X ) σmedian(Y ) i = i X σY .

...... Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 8 / 23 Minimal Sufficient Statistics Ancillary Statistics Location-scale Family Summary ...... More Examples of Ancillary Statistics

. Examples with normal distribution at zero mean . i.i.d. 2 2 X1, ··· , Xn ∼ N (0, σ ) where σ is unknown

• Y = X/σ is an ancillary statistic because Yi ∼ N (0, 1). • X1 = σY1 = Y1 also follows a cauchy distribution and is an ancillary X2 σY2 Y2 statistic. 2 • Any joint distribution of Y1, ··· , Yn does not depend on σ , and thus is an ancillary statistic. • For example, the following statistic is also ancillary. median(X ) σmedian(Y ) median(Y ) i = i = i X σY Y .

...... Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 8 / 23 . Solution . − • Let Zi = Xi θ. − dx • fZ(z) = fX(z + θ θ) dz = fX(z) ··· ∼i.i.d. • Z1, , Zn fX(z) does not depend on θ.

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

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

. Problem . ··· ∼i.i.d. − • X1, , Xn fX(x θ).

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

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

− dx • fZ(z) = fX(z + θ θ) dz = fX(z) ··· ∼i.i.d. • Z1, , Zn fX(z) does not depend on θ.

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

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

. Problem . ··· ∼i.i.d. − • X1, , Xn fX(x θ).

• Show that R = X(n) − X(1) is an ancillary statistic. . . Solution . • Let Zi = Xi − θ.

.

...... Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 9 / 23 ··· ∼i.i.d. • Z1, , Zn fX(z) does not depend on θ.

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

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

. Problem . ··· ∼i.i.d. − • X1, , Xn fX(x θ).

• Show that R = X(n) − X(1) is an ancillary statistic. . . Solution . − • Let Zi = Xi θ. − dx • fZ(z) = fX(z + θ θ) dz = fX(z)

.

...... Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 9 / 23 • R = X(n) − X(1) = Z(n) − Z(1) does not depend on θ.

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

. Problem . ··· ∼i.i.d. − • X1, , Xn fX(x θ).

• Show that R = X(n) − X(1) is an ancillary statistic. . . Solution . − • Let Zi = Xi θ. − dx • fZ(z) = fX(z + θ θ) dz = fX(z) ··· ∼i.i.d. • Z1, , Zn fX(z) does not depend on θ.

.

...... Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 9 / 23 Minimal Sufficient Statistics Ancillary Statistics Location-scale Family Summary ...... Range Statistics

. Problem . ··· ∼i.i.d. − • X1, , Xn fX(x θ).

• Show that R = X(n) − X(1) is an ancillary statistic. . . Solution . − • Let Zi = Xi θ. − dx • fZ(z) = fX(z + θ θ) dz = fX(z) ··· ∼i.i.d. • Z1, , Zn 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 • Obtain the distribution of R and show that it is independent of θ. • Represent R as a function of ancillary statistics, which is independent of θ.

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

. Problem . i.i.d. • X1, ··· , Xn ∼ Uniform(θ, θ + 1).

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

.

...... Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 10 / 23 • Represent R as a function of ancillary statistics, which is independent of θ.

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

. Problem . i.i.d. • X1, ··· , Xn ∼ 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.

.

...... Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 10 / 23 Minimal Sufficient Statistics Ancillary Statistics Location-scale Family Summary ...... Uniform Ancillary Statistics

. Problem . i.i.d. • X1, ··· , Xn ∼ 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 of θ. .

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

| fX(x θ) = I(θ < x < θ + 1)

If θ < X(1) ≤ X(n) < θ + 1,

n! ( ) − f (X , X |θ) = X − X (n 2) X (1) (n) (n − 2)! (n) (1) | and fX(X(1), X(n) θ) = 0 otherwise.

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 of (X(n), X(1)).

...... Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 11 / 23 If θ < X(1) ≤ X(n) < θ + 1,

n! ( ) − f (X , X |θ) = X − X (n 2) X (1) (n) (n − 2)! (n) (1) | and fX(X(1), X(n) θ) = 0 otherwise.

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 of (X(n), X(1)). Define | fX(x θ) = I(θ < x < θ + 1)

...... Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 11 / 23 n! ( ) − X − X (n 2) (n − 2)! (n) (1) | and fX(X(1), X(n) θ) = 0 otherwise.

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 of (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) =

...... Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 11 / 23 | and fX(X(1), X(n) θ) = 0 otherwise.

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 of (X(n), X(1)). Define | fX(x θ) = I(θ < x < θ + 1)

If θ < X(1) ≤ X(n) < θ + 1,

n! ( ) − f (X , X |θ) = X − X (n 2) X (1) (n) (n − 2)! (n) (1)

...... Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 11 / 23 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 of (X(n), X(1)). Define | fX(x θ) = I(θ < x < θ + 1)

If θ < X(1) ≤ X(n) < θ + 1,

n! ( ) − f (X , X |θ) = X − X (n 2) X (1) (n) (n − 2)! (n) (1) | and fX(X(1), X(n) θ) = 0 otherwise.

...... Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 11 / 23 Then { X(1) = M − R/2 X(n) = M + R/2 The Jacobian is      ∂X(1) ∂X(1)       ∂M ∂R   1 − 1  1 1 J =   =  2  = − (− ) = 1    1 1   ∂X ∂X  2 2 2  (n) (n)  ∂M ∂R

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

...... Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 12 / 23 The Jacobian is      ∂X(1) ∂X(1)       ∂M ∂R   1 − 1  1 1 J =   =  2  = − (− ) = 1    1 1   ∂X ∂X  2 2 2  (n) (n)  ∂M ∂R

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

...... Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 12 / 23 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      ∂X(1) ∂X(1)       ∂M ∂R   1 − 1  1 1 J =   =  2  = − (− ) = 1    1 1   ∂X ∂X  2 2 2  (n) (n)  ∂M ∂R ...... Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 12 / 23 Because θ < X(1) ≤ X(n) < θ + 1, 2m − r 2m + r θ < < < θ + 1 2 2

0 < r < 1 r r θ + < m < θ + 1 − 2 2

Minimal Sufficient Statistics Ancillary Statistics Location-scale Family Summary ...... Proof : Method I - 3/4

The joint distribution of R and M is

( ) − 2m + r 2m − r (n 2) f (r, m) = n(n − 1) − = n(n − 1)r(n−2) R,M 2 2

...... Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 13 / 23 0 < r < 1 r r θ + < m < θ + 1 − 2 2

Minimal Sufficient Statistics Ancillary Statistics Location-scale Family Summary ...... Proof : Method I - 3/4

The joint distribution of R and M is

( ) − 2m + r 2m − r (n 2) f (r, m) = n(n − 1) − = n(n − 1)r(n−2) R,M 2 2

Because θ < X(1) ≤ X(n) < θ + 1, 2m − r 2m + r θ < < < θ + 1 2 2

...... Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 13 / 23 Minimal Sufficient Statistics Ancillary Statistics Location-scale Family Summary ...... Proof : Method I - 3/4

The joint distribution of R and M is

( ) − 2m + r 2m − r (n 2) f (r, m) = n(n − 1) − = n(n − 1)r(n−2) R,M 2 2

Because θ < X(1) ≤ X(n) < θ + 1, 2m − r 2m + r θ < < < θ + 1 2 2

0 < r < 1 r r θ + < m < θ + 1 − 2 2

...... Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 13 / 23 ( ) r r = n(n − 1)r(n−2) θ + 1 − − θ − 2 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.

Minimal Sufficient Statistics Ancillary Statistics Location-scale Family Summary ...... Proof : Method I - 4/4

The distribution of R is

∫ θ+1− r | 2 − (n−2) fR(r θ) = n(n 1)r dm r θ+ 2

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

Minimal Sufficient Statistics Ancillary Statistics Location-scale Family Summary ...... Proof : Method I - 4/4

The distribution of R is

∫ θ+1− r | 2 − (n−2) fR(r θ) = n(n 1)r dm θ+ r 2 ( ) r r = n(n − 1)r(n−2) θ + 1 − − θ − 2 2

...... Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 14 / 23 | Therefore, fR(r θ) does not depend on θ, and R is an ancillary statistic.

Minimal Sufficient Statistics Ancillary Statistics Location-scale Family Summary ...... Proof : Method I - 4/4

The distribution of R is

∫ θ+1− r | 2 − (n−2) fR(r θ) = n(n 1)r dm θ+ r 2 ( ) r r = n(n − 1)r(n−2) θ + 1 − − θ − 2 2 = n(n − 1)r(n−2)(1 − r) , 0 < r < 1

...... Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 14 / 23 Minimal Sufficient Statistics Ancillary Statistics Location-scale Family Summary ...... Proof : Method I - 4/4

The distribution of R is

∫ θ+1− r | 2 − (n−2) fR(r θ) = n(n 1)r dm θ+ r 2 ( ) r r = n(n − 1)r(n−2) θ + 1 − − θ − 2 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 − ∼ | dx | Let Yi = Xi θ Uniform(0, 1). Then Xi = Yi + θ, dy = 1 holds.

− | dx | fY(y) = I(0 < y + θ θ < 1) 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.

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)

...... Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 15 / 23 − | dx | fY(y) = I(0 < y + θ θ < 1) 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.

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)

− ∼ | dx | Let Yi = Xi θ Uniform(0, 1). Then Xi = Yi + θ, dy = 1 holds.

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

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)

− ∼ | dx | Let Yi = Xi θ Uniform(0, 1). Then Xi = Yi + θ, dy = 1 holds.

− | dx | fY(y) = I(0 < y + θ θ < 1) dy = I(0 < y < 1)

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

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)

− ∼ | dx | Let Yi = Xi θ Uniform(0, 1). Then Xi = Yi + θ, dy = 1 holds.

− | dx | fY(y) = I(0 < y + θ θ < 1) 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)

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

− ∼ | dx | Let Yi = Xi θ Uniform(0, 1). Then Xi = Yi + θ, dy = 1 holds.

− | dx | fY(y) = I(0 < y + θ θ < 1) 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 . 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 x − µ ∞ 1 ∞ f dx = f(y)σdy = f(y)dy = 1 −∞ σ σ −∞ σ −∞

.Therefore, g(x|µ, σ) is also a pdf.

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, ( ) 1 x − µ g(x|µ, σ) = f σ σ

.is a pdf.

...... Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 16 / 23 Let y = (x − µ)/σ, then x = σy + µ, and dx/dy = σ. ∫ ( ) ∫ ∫ ∞ 1 x − µ ∞ 1 ∞ f dx = f(y)σdy = f(y)dy = 1 −∞ σ σ −∞ σ −∞

Therefore, g(x|µ, σ) is also a pdf.

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, ( ) 1 x − µ g(x|µ, σ) = f σ σ

.is a pdf. . Proof . Because f(x) is a pdf, then f(x) ≥ 0, and g(x|µ, σ) ≥ 0 for all x.

...... Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 16 / 23 ∫ ( ) ∫ ∫ ∞ 1 x − µ ∞ 1 ∞ f dx = f(y)σdy = f(y)dy = 1 −∞ σ σ −∞ σ −∞

Therefore, g(x|µ, σ) is also a pdf.

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, ( ) 1 x − µ g(x|µ, σ) = f σ 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.

...... Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 16 / 23 Therefore, g(x|µ, σ) is also a pdf.

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, ( ) 1 x − µ g(x|µ, σ) = f σ 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 x − µ ∞ 1 ∞ f dx = f(y)σdy = f(y)dy = 1 −∞ σ σ −∞ σ −∞

...... Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 16 / 23 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, ( ) 1 x − µ g(x|µ, σ) = f σ 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 x − µ ∞ 1 ∞ f dx = f(y)σdy = f(y)dy = 1 −∞ σ σ −∞ σ −∞

Therefore, g(x|µ, σ) is also a pdf...... Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 16 / 23 . Example . 2 • f(x) = √1 e−x /2 ∼ N (0, 1) 2π 2 • f(x − µ) = √1 e−(x−µ) /2 ∼ N (µ, 1) 2π • f(x) = I(0 < x < 1) ∼ Uniform(0, 1) • f(x − θ) = I(θ < x < θ + 1) ∼ Uniform(θ, θ + 1) .

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.

...... Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 17 / 23 • f(x) = I(0 < x < 1) ∼ Uniform(0, 1) • f(x − θ) = I(θ < x < θ + 1) ∼ Uniform(θ, θ + 1)

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 . 2 • f(x) = √1 e−x /2 ∼ N (0, 1) 2π 2 • f(x − µ) = √1 e−(x−µ) /2 ∼ N (µ, 1) 2π

.

...... Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 17 / 23 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 . 2 • f(x) = √1 e−x /2 ∼ N (0, 1) 2π 2 • f(x − µ) = √1 e−(x−µ) /2 ∼ N (µ, 1) 2π • 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 . Example . 2 • f(x) = √1 e−x /2 ∼ N (0, 1) 2π 2 2 • f(x/σ)/σ = √ 1 e−x /2σ ∼ N (0, σ2) 2πσ2 .

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.

...... Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 18 / 23 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 . 2 • f(x) = √1 e−x /2 ∼ N (0, 1) 2π 2 2 • f(x/σ)/σ = √ 1 e−x /2σ ∼ N (0, σ2) 2πσ2 .

...... Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 18 / 23 . Example . 2 • f(x) = √1 e−x /2 ∼ N (0, 1) 2π 2 2 • f((x − µ)/σ)/σ = √ 1 e−(x−µ) /2σ ∼ N (µ, σ2) 2πσ2 .

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.

...... Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 19 / 23 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 . 2 • f(x) = √1 e−x /2 ∼ N (0, 1) 2π 2 2 • f((x − µ)/σ)/σ = √ 1 e−(x−µ) /2σ ∼ N (µ, σ2) 2πσ2 .

...... Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 19 / 23 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. ( ) 1 x−µ • Then X is a random variable with pdf σ f 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 . 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.

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.

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

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).

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

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

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

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)

...... Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 21 / 23 which does not depend on µ because Z1, ··· , Zn does not depend on µ. Therefore, R is an ancillary statistic.

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)

...... Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 21 / 23 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 . Solution . Assume that cdf is F(x/σ), and let Z1 = X1/σ, ··· , Zn = Xn/σ be iid observations 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.

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)

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

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 observations from pdf f(x) and cdf F(x).

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

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

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

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 observations 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)

...... Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 22 / 23 = Pr(Z1/Zn ≤ t1, ··· , Zn−1/Zn ≤ tn−1|σ)

Because Z1, ··· , Zn does not depend on σ, T(X) is an ancillary statistic.

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 observations 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)

...... Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 22 / 23 Because Z1, ··· , Zn does not depend on σ, T(X) is an ancillary statistic.

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 observations 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)

...... Hyun Min Kang Biostatistics 602 - Lecture 04 January 22th, 2013 22 / 23 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 observations 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 . Next Lecture . • Complete Statistics .

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 .

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