Complete Statistics Lecture 05 Biostatistics 602 - Statistical - - PowerPoint PPT Presentation

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

. .

Biostatistics 602 - Statistical Inference Lecture 05 Complete Statistics

Hyun Min Kang January 24th, 2013

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 1 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

. . 1 What is an ancillary statistic for θ? . . 2 Can an ancillary statistic be a sufficient statistic? . . 3 What are the location parameter and the scale parameter? . 4 In which case ancillary statistics would be helpful?

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 2 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

. . 1 What is an ancillary statistic for θ? . . 2 Can an ancillary statistic be a sufficient statistic? . . 3 What are the location parameter and the scale parameter? . 4 In which case ancillary statistics would be helpful?

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 2 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

. . 1 What is an ancillary statistic for θ? . . 2 Can an ancillary statistic be a sufficient statistic? . . 3 What are the location parameter and the scale parameter? . . 4 In which case ancillary statistics would be helpful?

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 2 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

. . 1 What is an ancillary statistic for θ? . . 2 Can an ancillary statistic be a sufficient statistic? . . 3 What are the location parameter and the scale parameter? . . 4 In which case ancillary statistics would be helpful?

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 2 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Last Lecture : 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

X 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 05 January 24th, 2013 3 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Last Lecture : 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(X1 − 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 05 January 24th, 2013 3 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Example : 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

. . . . . . . .

• Method 1 : Obtain the distribution of R and show that it is

independent of .

• Method 2 : Represent R as a function of ancillary statistics, which is

independent of .

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 4 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Example : 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

. .

• Method 1 : Obtain the distribution of R and show that it is

independent of θ.

• Method 2 : Represent R as a function of ancillary statistics, which is

independent of .

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 4 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Example : 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

. .

• Method 1 : Obtain the distribution of R and show that it is

independent of θ.

• Method 2 : Represent R as a function of ancillary statistics, which is

independent of θ.

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 4 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Method 2 - 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 05 January 24th, 2013 5 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Method 2 - 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 05 January 24th, 2013 5 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Method 2 - 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 05 January 24th, 2013 5 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . 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 05 January 24th, 2013 6 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . 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 05 January 24th, 2013 6 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Complete Statistics

.

Definition

. .

• Let T = {fT(t|θ), θ ∈ Ω} be a family of pdfs or pmfs for a statistic

T(X).

• The family of probability distributions is called complete if
• E g T

for all implies Pr g T for all .

• In other words, g T

almost surely.

• Equivalently, T X is called a complete statistic

.

Example

. . . . . . . .

• T X
• g

T X = Pr g T X .

• g

T X I T X = Pr g T X Pr T X . In this case, g T X is almost surely true.

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 7 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Complete Statistics

.

Definition

. .

• Let T = {fT(t|θ), θ ∈ Ω} be a family of pdfs or pmfs for a statistic

T(X).

• The family of probability distributions is called complete if
• E g T

for all implies Pr g T for all .

• In other words, g T

almost surely.

• Equivalently, T X is called a complete statistic

.

Example

. . . . . . . .

• T X
• g

T X = Pr g T X .

• g

T X I T X = Pr g T X Pr T X . In this case, g T X is almost surely true.

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 7 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Complete Statistics

.

Definition

. .

• Let T = {fT(t|θ), θ ∈ Ω} be a family of pdfs or pmfs for a statistic

T(X).

• The family of probability distributions is called complete if
• E[g(T)|θ] = 0 for all θ implies Pr[g(T) = 0|θ] = 1 for all θ.
• In other words, g T

almost surely.

• Equivalently, T X is called a complete statistic

.

Example

. . . . . . . .

• T X
• g

T X = Pr g T X .

• g

T X I T X = Pr g T X Pr T X . In this case, g T X is almost surely true.

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 7 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Complete Statistics

.

Definition

. .

• Let T = {fT(t|θ), θ ∈ Ω} be a family of pdfs or pmfs for a statistic

T(X).

• The family of probability distributions is called complete if
• E[g(T)|θ] = 0 for all θ implies Pr[g(T) = 0|θ] = 1 for all θ.
• In other words, g(T) = 0 almost surely.
• Equivalently, T X is called a complete statistic

.

Example

. . . . . . . .

• T X
• g

T X = Pr g T X .

• g

T X I T X = Pr g T X Pr T X . In this case, g T X is almost surely true.

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 7 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Complete Statistics

.

Definition

. .

• Let T = {fT(t|θ), θ ∈ Ω} be a family of pdfs or pmfs for a statistic

T(X).

• The family of probability distributions is called complete if
• E[g(T)|θ] = 0 for all θ implies Pr[g(T) = 0|θ] = 1 for all θ.
• In other words, g(T) = 0 almost surely.
• Equivalently, T(X) is called a complete statistic

.

Example

. . . . . . . .

• T X
• g

T X = Pr g T X .

• g

T X I T X = Pr g T X Pr T X . In this case, g T X is almost surely true.

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 7 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Complete Statistics

.

Definition

. .

• Let T = {fT(t|θ), θ ∈ Ω} be a family of pdfs or pmfs for a statistic

T(X).

• The family of probability distributions is called complete if
• E[g(T)|θ] = 0 for all θ implies Pr[g(T) = 0|θ] = 1 for all θ.
• In other words, g(T) = 0 almost surely.
• Equivalently, T(X) is called a complete statistic

.

Example

. .

• T(X) ∼ N(0, 1)
• g1(T(X)) = 0 =

⇒ Pr[g1(T(X)) = 0] = 1.

• g

T X I T X = Pr g T X Pr T X . In this case, g T X is almost surely true.

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 7 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Complete Statistics

.

Definition

. .

• Let T = {fT(t|θ), θ ∈ Ω} be a family of pdfs or pmfs for a statistic

T(X).

• The family of probability distributions is called complete if
• E[g(T)|θ] = 0 for all θ implies Pr[g(T) = 0|θ] = 1 for all θ.
• In other words, g(T) = 0 almost surely.
• Equivalently, T(X) is called a complete statistic

.

Example

. .

• T(X) ∼ N(0, 1)
• g1(T(X)) = 0 =

⇒ Pr[g1(T(X)) = 0] = 1.

• g2(T(X)) = I(T(X) = 0) =

⇒ Pr[g2(T(X)) = 0] = 1 − Pr[T(X) = 0)]. In this case, g2(T(X)) = 0 is almost surely true.

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 7 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Notes on Complete Statistics

• Notice that completeness is a property of a family of probability

distributions, not of a particular distribution.

• For example, X

and g x x makes E g X EX , but Pr g X instead of 1.

• The above example is only for a particular distribution, not a family
• f distributions.
• If X

, then no function of X except for g X satisfies E g X for all .

• Therefore, the family of

distributions, , is complete.

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 8 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Notes on Complete Statistics

• Notice that completeness is a property of a family of probability

distributions, not of a particular distribution.

• For example, X ∼ N(0, 1) and g(x) = x makes E[g(X)] = EX = 0,

but Pr(g(X) = 0) = 0 instead of 1.

• The above example is only for a particular distribution, not a family
• f distributions.
• If X

, then no function of X except for g X satisfies E g X for all .

• Therefore, the family of

distributions, , is complete.

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 8 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Notes on Complete Statistics

• Notice that completeness is a property of a family of probability

distributions, not of a particular distribution.

• For example, X ∼ N(0, 1) and g(x) = x makes E[g(X)] = EX = 0,

but Pr(g(X) = 0) = 0 instead of 1.

• The above example is only for a particular distribution, not a family
• f distributions.
• If X

, then no function of X except for g X satisfies E g X for all .

• Therefore, the family of

distributions, , is complete.

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 8 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Notes on Complete Statistics

• Notice that completeness is a property of a family of probability

distributions, not of a particular distribution.

• For example, X ∼ N(0, 1) and g(x) = x makes E[g(X)] = EX = 0,

but Pr(g(X) = 0) = 0 instead of 1.

• The above example is only for a particular distribution, not a family
• f distributions.
• If X ∼ N(θ, 1), −∞ < θ < ∞, then no function of X except for

g(X) = 0 satisfies E[g(X)|θ] for all θ.

• Therefore, the family of

distributions, , is complete.

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 8 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Notes on Complete Statistics

• Notice that completeness is a property of a family of probability

distributions, not of a particular distribution.

• For example, X ∼ N(0, 1) and g(x) = x makes E[g(X)] = EX = 0,

but Pr(g(X) = 0) = 0 instead of 1.

• The above example is only for a particular distribution, not a family
• f distributions.
• If X ∼ N(θ, 1), −∞ < θ < ∞, then no function of X except for

g(X) = 0 satisfies E[g(X)|θ] for all θ.

• Therefore, the family of N(θ, 1) distributions, −∞ < θ < ∞, is

complete.

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 8 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Why ”Complete” Statistics?

.

Stigler (1972) Am. Stat. 26(2):28-9

. .

• While a student encountering completeness for the first time is very

likely to appreciate its usefulness, he is just as likely to be puzzled by its name, and wonder what connection (if any) there is between the statistical use of the term ”complete”, and the dictionary definition: lacking none of the parts, whole, entire.

• Requiring g T to satisfy the definition puts a restriction on g. The

larger the family of pdfs/pmfs, the greater the restriction on g. When the family of pdfs/pmfs is augmented to the point that E g T for all , it rules out all g except for the trivial g T , then the family is said to be complete.

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 9 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Why ”Complete” Statistics?

.

Stigler (1972) Am. Stat. 26(2):28-9

. .

• While a student encountering completeness for the first time is very

likely to appreciate its usefulness, he is just as likely to be puzzled by its name, and wonder what connection (if any) there is between the statistical use of the term ”complete”, and the dictionary definition: lacking none of the parts, whole, entire.

• Requiring g T to satisfy the definition puts a restriction on g. The

larger the family of pdfs/pmfs, the greater the restriction on g. When the family of pdfs/pmfs is augmented to the point that E g T for all , it rules out all g except for the trivial g T , then the family is said to be complete.

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 9 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Why ”Complete” Statistics?

.

Stigler (1972) Am. Stat. 26(2):28-9

. .

• While a student encountering completeness for the first time is very

likely to appreciate its usefulness, he is just as likely to be puzzled by its name, and wonder what connection (if any) there is between the statistical use of the term ”complete”, and the dictionary definition: lacking none of the parts, whole, entire.

• Requiring g T to satisfy the definition puts a restriction on g. The

larger the family of pdfs/pmfs, the greater the restriction on g. When the family of pdfs/pmfs is augmented to the point that E g T for all , it rules out all g except for the trivial g T , then the family is said to be complete.

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 9 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Why ”Complete” Statistics?

.

Stigler (1972) Am. Stat. 26(2):28-9

. .

• While a student encountering completeness for the first time is very

likely to appreciate its usefulness, he is just as likely to be puzzled by its name, and wonder what connection (if any) there is between the statistical use of the term ”complete”, and the dictionary definition: lacking none of the parts, whole, entire.

• Requiring g T to satisfy the definition puts a restriction on g. The

larger the family of pdfs/pmfs, the greater the restriction on g. When the family of pdfs/pmfs is augmented to the point that E g T for all , it rules out all g except for the trivial g T , then the family is said to be complete.

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 9 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Why ”Complete” Statistics?

.

Stigler (1972) Am. Stat. 26(2):28-9

. .

• While a student encountering completeness for the first time is very

likely to appreciate its usefulness, he is just as likely to be puzzled by its name, and wonder what connection (if any) there is between the statistical use of the term ”complete”, and the dictionary definition: lacking none of the parts, whole, entire.

• Requiring g(T) to satisfy the definition puts a restriction on g.

The larger the family of pdfs/pmfs, the greater the restriction on g. When the family of pdfs/pmfs is augmented to the point that E g T for all , it rules out all g except for the trivial g T , then the family is said to be complete.

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 9 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Why ”Complete” Statistics?

.

Stigler (1972) Am. Stat. 26(2):28-9

. .

• While a student encountering completeness for the first time is very

likely to appreciate its usefulness, he is just as likely to be puzzled by its name, and wonder what connection (if any) there is between the statistical use of the term ”complete”, and the dictionary definition: lacking none of the parts, whole, entire.

• Requiring g(T) to satisfy the definition puts a restriction on g. The

larger the family of pdfs/pmfs, the greater the restriction on g. When the family of pdfs/pmfs is augmented to the point that E g T for all , it rules out all g except for the trivial g T , then the family is said to be complete.

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 9 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Why ”Complete” Statistics?

.

Stigler (1972) Am. Stat. 26(2):28-9

. .

• While a student encountering completeness for the first time is very

likely to appreciate its usefulness, he is just as likely to be puzzled by its name, and wonder what connection (if any) there is between the statistical use of the term ”complete”, and the dictionary definition: lacking none of the parts, whole, entire.

• Requiring g(T) to satisfy the definition puts a restriction on g. The

larger the family of pdfs/pmfs, the greater the restriction on g. When the family of pdfs/pmfs is augmented to the point that E[g(T)] = 0 for all θ, it rules out all g except for the trivial g T , then the family is said to be complete.

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 9 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Why ”Complete” Statistics?

.

Stigler (1972) Am. Stat. 26(2):28-9

. .

• While a student encountering completeness for the first time is very

likely to appreciate its usefulness, he is just as likely to be puzzled by its name, and wonder what connection (if any) there is between the statistical use of the term ”complete”, and the dictionary definition: lacking none of the parts, whole, entire.

• Requiring g(T) to satisfy the definition puts a restriction on g. The

larger the family of pdfs/pmfs, the greater the restriction on g. When the family of pdfs/pmfs is augmented to the point that E[g(T)] = 0 for all θ, it rules out all g except for the trivial g(T) = 0, then the family is said to be complete.

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 9 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Why ”Complete” Statistics?

.

Stigler (1972) Am. Stat. 26(2):28-9

. .

• While a student encountering completeness for the first time is very

likely to appreciate its usefulness, he is just as likely to be puzzled by its name, and wonder what connection (if any) there is between the statistical use of the term ”complete”, and the dictionary definition: lacking none of the parts, whole, entire.

• Requiring g(T) to satisfy the definition puts a restriction on g. The

larger the family of pdfs/pmfs, the greater the restriction on g. When the family of pdfs/pmfs is augmented to the point that E[g(T)] = 0 for all θ, it rules out all g except for the trivial g(T) = 0, then the family is said to be complete.

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 9 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Example - Poisson distribution

.

Problem

. .

• Suppose T =

{ fT : fT(t|λ) = λte−λ

t!

} for t ∈ {0, 1, 2, · · · }. Let λ ∈ Ω = {1, 2}. Show that this family is NOT complete .

Proof strategy

. . . . . . . .

• We need to find a counter example,
• which is a function g such that E g T

for but g T .

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 10 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Example - Poisson distribution

.

Problem

. .

• Suppose T =

{ fT : fT(t|λ) = λte−λ

t!

} for t ∈ {0, 1, 2, · · · }. Let λ ∈ Ω = {1, 2}. Show that this family is NOT complete .

Proof strategy

. .

• We need to find a counter example,
• which is a function g such that E g T

for but g T .

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 10 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Example - Poisson distribution

.

Problem

. .

• Suppose T =

{ fT : fT(t|λ) = λte−λ

t!

} for t ∈ {0, 1, 2, · · · }. Let λ ∈ Ω = {1, 2}. Show that this family is NOT complete .

Proof strategy

. .

• We need to find a counter example,
• which is a function g such that E[g(T)|λ] = 0 for λ = 1, 2 but

g(T) ̸= 0.

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 10 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Poisson distribution example : Proof

The function g must satisfy E[g(T)|λ] =

∞

∑

t=0

g(t)λte−λ t! = 0 for . Thus, E g T

t

g t

te

t

E g T

t

g t

te

t

The above equation can be rewritten as

t

g t t

t tg t

t

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 11 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Poisson distribution example : Proof

The function g must satisfy E[g(T)|λ] =

∞

∑

t=0

g(t)λte−λ t! = 0 for λ ∈ {1, 2}. Thus,      E[g(T)|λ = 1] = ∑∞

t=0 g(t) 1te−1 t!

= 0 E[g(T)|λ = 2] = ∑∞

t=0 g(t) 2te−2 t!

= 0 The above equation can be rewritten as

t

g t t

t tg t

t

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 11 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Poisson distribution example : Proof

The function g must satisfy E[g(T)|λ] =

∞

∑

t=0

g(t)λte−λ t! = 0 for λ ∈ {1, 2}. Thus,      E[g(T)|λ = 1] = ∑∞

t=0 g(t) 1te−1 t!

= 0 E[g(T)|λ = 2] = ∑∞

t=0 g(t) 2te−2 t!

= 0 The above equation can be rewritten as    ∑∞

t=0 g(t)/t!

= ∑∞

t=0 2tg(t)/t!

=

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 11 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Poisson distribution example : Proof (cont’d)

Define g(t) as g(t) =    2 t = 0 ∨ t = 2 −3 t = 1

• therwise

Then

∞

∑

t=0

g(t)/t! = g(0)/0! + g(1)/1! + g(2)/2! = 2 − 3 + 2/2 = 0

∞

∑

t=0

2tg(t)/t! = g(0)/0! + 2g(1)/1! + 22g(2)/2! = 2 − 6 + 8/2 = 0 There exists a non-zero function g that satisfies E[g(T)λ] = 0 for all λ ∈ Ω. Therefore this family is NOT complete.

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 12 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Another example with Poisson distribution

.

Problem

. .

• X1, · · · , Xn

i.i.d.

∼ Poisson(λ), λ > 0.

• Show that T(X) = ∑n

i=1 Xi is a complete statistic.

.

Proof strategy

. . . . . . . .

• Need to find the distribution of T X
• Show that there is no non-zero function g such that E g T

for all .

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 13 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Another example with Poisson distribution

.

Problem

. .

• X1, · · · , Xn

i.i.d.

∼ Poisson(λ), λ > 0.

• Show that T(X) = ∑n

i=1 Xi is a complete statistic.

.

Proof strategy

. .

• Need to find the distribution of T(X)
• Show that there is no non-zero function g such that E[g(T)|λ] = 0 for

all λ.

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 13 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Proof : Finding the moment-generating function of X

MX(s) = E[esX] =

∞

∑

x=0

esx e−λλx x!

x

e es

x

x e

es ees x

e ees es

xe es

x e ees

x

fPoisson x es e

es

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 14 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Proof : Finding the moment-generating function of X

MX(s) = E[esX] =

∞

∑

x=0

esx e−λλx x! =

∞

∑

x=0

e−λ (esλ)x x! e−esλeesλ

x

e ees es

xe es

x e ees

x

fPoisson x es e

es

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 14 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Proof : Finding the moment-generating function of X

MX(s) = E[esX] =

∞

∑

x=0

esx e−λλx x! =

∞

∑

x=0

e−λ (esλ)x x! e−esλeesλ =

∞

∑

x=0

e−λeesλ (esλ)xe−esλ x! e ees

x

fPoisson x es e

es

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 14 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Proof : Finding the moment-generating function of X

MX(s) = E[esX] =

∞

∑

x=0

esx e−λλx x! =

∞

∑

x=0

e−λ (esλ)x x! e−esλeesλ =

∞

∑

x=0

e−λeesλ (esλ)xe−esλ x! = eλeesλ

∞

∑

x=0

fPoisson(x|esλ) e

es

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 14 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Proof : Finding the moment-generating function of X

MX(s) = E[esX] =

∞

∑

x=0

esx e−λλx x! =

∞

∑

x=0

e−λ (esλ)x x! e−esλeesλ =

∞

∑

x=0

e−λeesλ (esλ)xe−esλ x! = eλeesλ

∞

∑

x=0

fPoisson(x|esλ) = eλ(es−1)

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 14 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Proof : Finding the MGF of T(X) = ∑n

i=1 Xi

MT(s) = E(es ∑ Xi) = E

n i

esXi

n i

E esXi E esXi

n

e

es n

en

es

.

Theorem 2.3.11 (b)

. . . . . . . . Let FX x and FY y be two cdfs all of whose moments exists. If the moment generating functions exists and MX t MY t for all t in some neighborhood of 0, then FX u FY u for all u. By Theorem 2.3.11, T X Poisson n .

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 15 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Proof : Finding the MGF of T(X) = ∑n

i=1 Xi

MT(s) = E(es ∑ Xi) = E ( n ∏

i=1

esXi )

n i

E esXi E esXi

n

e

es n

en

es

.

Theorem 2.3.11 (b)

. . . . . . . . Let FX x and FY y be two cdfs all of whose moments exists. If the moment generating functions exists and MX t MY t for all t in some neighborhood of 0, then FX u FY u for all u. By Theorem 2.3.11, T X Poisson n .

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 15 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Proof : Finding the MGF of T(X) = ∑n

i=1 Xi

MT(s) = E(es ∑ Xi) = E ( n ∏

i=1

esXi ) =

n

∏

i=1

E ( esXi) = E esXi

n

e

es n

en

es

.

Theorem 2.3.11 (b)

. . . . . . . . Let FX x and FY y be two cdfs all of whose moments exists. If the moment generating functions exists and MX t MY t for all t in some neighborhood of 0, then FX u FY u for all u. By Theorem 2.3.11, T X Poisson n .

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 15 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Proof : Finding the MGF of T(X) = ∑n

i=1 Xi

MT(s) = E(es ∑ Xi) = E ( n ∏

i=1

esXi ) =

n

∏

i=1

E ( esXi) = [ E ( esXi)]n e

es n

en

es

.

Theorem 2.3.11 (b)

. . . . . . . . Let FX x and FY y be two cdfs all of whose moments exists. If the moment generating functions exists and MX t MY t for all t in some neighborhood of 0, then FX u FY u for all u. By Theorem 2.3.11, T X Poisson n .

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 15 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Proof : Finding the MGF of T(X) = ∑n

i=1 Xi

MT(s) = E(es ∑ Xi) = E ( n ∏

i=1

esXi ) =

n

∏

i=1

E ( esXi) = [ E ( esXi)]n = [ e−λ(es−1)]n = en

es

.

Theorem 2.3.11 (b)

. . . . . . . . Let FX x and FY y be two cdfs all of whose moments exists. If the moment generating functions exists and MX t MY t for all t in some neighborhood of 0, then FX u FY u for all u. By Theorem 2.3.11, T X Poisson n .

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 15 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Proof : Finding the MGF of T(X) = ∑n

i=1 Xi

MT(s) = E(es ∑ Xi) = E ( n ∏

i=1

esXi ) =

n

∏

i=1

E ( esXi) = [ E ( esXi)]n = [ e−λ(es−1)]n = enλ(es−1) .

Theorem 2.3.11 (b)

. . . . . . . . Let FX x and FY y be two cdfs all of whose moments exists. If the moment generating functions exists and MX t MY t for all t in some neighborhood of 0, then FX u FY u for all u. By Theorem 2.3.11, T X Poisson n .

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 15 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Proof : Finding the MGF of T(X) = ∑n

i=1 Xi

MT(s) = E(es ∑ Xi) = E ( n ∏

i=1

esXi ) =

n

∏

i=1

E ( esXi) = [ E ( esXi)]n = [ e−λ(es−1)]n = enλ(es−1) .

Theorem 2.3.11 (b)

. . Let FX(x) and FY(y) be two cdfs all of whose moments exists. If the moment generating functions exists and MX(t) = MY(t) for all t in some neighborhood of 0, then FX(u) = FY(u) for all u. By Theorem 2.3.11, T X Poisson n .

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 15 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Proof : Finding the MGF of T(X) = ∑n

i=1 Xi

MT(s) = E(es ∑ Xi) = E ( n ∏

i=1

esXi ) =

n

∏

i=1

E ( esXi) = [ E ( esXi)]n = [ e−λ(es−1)]n = enλ(es−1) .

Theorem 2.3.11 (b)

. . Let FX(x) and FY(y) be two cdfs all of whose moments exists. If the moment generating functions exists and MX(t) = MY(t) for all t in some neighborhood of 0, then FX(u) = FY(u) for all u. By Theorem 2.3.11, T(X) ∼ Poisson(nλ).

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 15 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Proof : Showing E[g(T)|λ] = 0 ⇐ ⇒ Pr[g(T) = 0] = 1

. Suppose that there exists a g(T) such that E[g(T)|λ] = 0 for all λ > 0. E g T

t

e

n

n

t

t e

n t

g t n

t

t Which is equivalent to

t

g t nt t

t

for all . Because the function above is a power series expansion of , g t nt t for all t. and g t for all t. Therefore T X

n i

Xi is a complete statistic.

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 16 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Proof : Showing E[g(T)|λ] = 0 ⇐ ⇒ Pr[g(T) = 0] = 1

. Suppose that there exists a g(T) such that E[g(T)|λ] = 0 for all λ > 0. E[g(T)|λ] =

∞

∑

t=0

e−nλ(nλ)t t! e

n t

g t n

t

t Which is equivalent to

t

g t nt t

t

for all . Because the function above is a power series expansion of , g t nt t for all t. and g t for all t. Therefore T X

n i

Xi is a complete statistic.

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 16 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Proof : Showing E[g(T)|λ] = 0 ⇐ ⇒ Pr[g(T) = 0] = 1

. Suppose that there exists a g(T) such that E[g(T)|λ] = 0 for all λ > 0. E[g(T)|λ] =

∞

∑

t=0

e−nλ(nλ)t t! = e−nλ

∞

∑

t=0

g(t)(nλ)t t! = 0 Which is equivalent to

t

g t nt t

t

for all . Because the function above is a power series expansion of , g t nt t for all t. and g t for all t. Therefore T X

n i

Xi is a complete statistic.

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 16 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Proof : Showing E[g(T)|λ] = 0 ⇐ ⇒ Pr[g(T) = 0] = 1

. Suppose that there exists a g(T) such that E[g(T)|λ] = 0 for all λ > 0. E[g(T)|λ] =

∞

∑

t=0

e−nλ(nλ)t t! = e−nλ

∞

∑

t=0

g(t)(nλ)t t! = 0 Which is equivalent to

∞

∑

t=0

g(t)nt t! λt = for all . Because the function above is a power series expansion of , g t nt t for all t. and g t for all t. Therefore T X

n i

Xi is a complete statistic.

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 16 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Proof : Showing E[g(T)|λ] = 0 ⇐ ⇒ Pr[g(T) = 0] = 1

. Suppose that there exists a g(T) such that E[g(T)|λ] = 0 for all λ > 0. E[g(T)|λ] =

∞

∑

t=0

e−nλ(nλ)t t! = e−nλ

∞

∑

t=0

g(t)(nλ)t t! = 0 Which is equivalent to

∞

∑

t=0

g(t)nt t! λt = for all λ > 0. Because the function above is a power series expansion of , g t nt t for all t. and g t for all t. Therefore T X

n i

Xi is a complete statistic.

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 16 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Proof : Showing E[g(T)|λ] = 0 ⇐ ⇒ Pr[g(T) = 0] = 1

. Suppose that there exists a g(T) such that E[g(T)|λ] = 0 for all λ > 0. E[g(T)|λ] =

∞

∑

t=0

e−nλ(nλ)t t! = e−nλ

∞

∑

t=0

g(t)(nλ)t t! = 0 Which is equivalent to

∞

∑

t=0

g(t)nt t! λt = for all λ > 0. Because the function above is a power series expansion of λ, g(t)nt/t! = 0 for all t. and g(t) = 0 for all t. Therefore T X

n i

Xi is a complete statistic.

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 16 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Proof : Showing E[g(T)|λ] = 0 ⇐ ⇒ Pr[g(T) = 0] = 1

. Suppose that there exists a g(T) such that E[g(T)|λ] = 0 for all λ > 0. E[g(T)|λ] =

∞

∑

t=0

e−nλ(nλ)t t! = e−nλ

∞

∑

t=0

g(t)(nλ)t t! = 0 Which is equivalent to

∞

∑

t=0

g(t)nt t! λt = for all λ > 0. Because the function above is a power series expansion of λ, g(t)nt/t! = 0 for all t. and g(t) = 0 for all t. Therefore T(X) = ∑n

i=1 Xi is

a complete statistic.

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 16 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Example : Uniform Distribution

.

Problem

. . Let X1, · · · , Xn

i.i.d.

∼ Uniform(0, θ), θ > 0, Ω = (0, ∞).

Show that X(n) is complete. .

Proof

. . . . . . . . We need to obtain the distribution of T X X n . Let fX x I x , then its cdf is FX x

xI

x I x . fT t n n fX t FX t n n t

n

I t n

ntn

I t

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 17 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Example : Uniform Distribution

.

Problem

. . Let X1, · · · , Xn

i.i.d.

∼ Uniform(0, θ), θ > 0, Ω = (0, ∞).

Show that X(n) is complete. .

Proof

. . We need to obtain the distribution of T(X) = X(n). Let fX x I x , then its cdf is FX x

xI

x I x . fT t n n fX t FX t n n t

n

I t n

ntn

I t

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 17 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Example : Uniform Distribution

.

Problem

. . Let X1, · · · , Xn

i.i.d.

∼ Uniform(0, θ), θ > 0, Ω = (0, ∞).

Show that X(n) is complete. .

Proof

. . We need to obtain the distribution of T(X) = X(n). Let fX(x) = 1

θI(0 < x < θ), then its cdf is

FX x

xI

x I x . fT t n n fX t FX t n n t

n

I t n

ntn

I t

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 17 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Example : Uniform Distribution

.

Problem

. . Let X1, · · · , Xn

i.i.d.

∼ Uniform(0, θ), θ > 0, Ω = (0, ∞).

Show that X(n) is complete. .

Proof

. . We need to obtain the distribution of T(X) = X(n). Let fX(x) = 1

θI(0 < x < θ), then its cdf is FX(x) = x θI(0 < x < θ) + I(x ≥ θ).

fT t n n fX t FX t n n t

n

I t n

ntn

I t

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 17 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Example : Uniform Distribution

.

Problem

. . Let X1, · · · , Xn

i.i.d.

∼ Uniform(0, θ), θ > 0, Ω = (0, ∞).

Show that X(n) is complete. .

Proof

. . We need to obtain the distribution of T(X) = X(n). Let fX(x) = 1

θI(0 < x < θ), then its cdf is FX(x) = x θI(0 < x < θ) + I(x ≥ θ).

fT(t|θ) = n! (n − 1)!fX(t)FX(t)n−1 n t

n

I t n

ntn

I t

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 17 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Example : Uniform Distribution

.

Problem

. . Let X1, · · · , Xn

i.i.d.

∼ Uniform(0, θ), θ > 0, Ω = (0, ∞).

Show that X(n) is complete. .

Proof

. . We need to obtain the distribution of T(X) = X(n). Let fX(x) = 1

θI(0 < x < θ), then its cdf is FX(x) = x θI(0 < x < θ) + I(x ≥ θ).

fT(t|θ) = n! (n − 1)!fX(t)FX(t)n−1 = n θ ( t θ )n−1 I(0 < t < θ) n

ntn

I t

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 17 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Example : Uniform Distribution

.

Problem

. . Let X1, · · · , Xn

i.i.d.

∼ Uniform(0, θ), θ > 0, Ω = (0, ∞).

Show that X(n) is complete. .

Proof

. . We need to obtain the distribution of T(X) = X(n). Let fX(x) = 1

θI(0 < x < θ), then its cdf is FX(x) = x θI(0 < x < θ) + I(x ≥ θ).

fT(t|θ) = n! (n − 1)!fX(t)FX(t)n−1 = n θ ( t θ )n−1 I(0 < t < θ) = nθ−ntn−1I(0 < t < θ)

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 17 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Proof : Uniform Distribution (cont’d)

Consider a function g(T) such that E[g(T)|θ] = 0 for all θ > 0 E g T g t n

ntn

I t dt n

n

g t tn dt Taking derivative of both sides, n

n g n

n

n

g t tn dt ng n n

n

g t tn dt nE g T Because g T holds for all , T X X n is a complete statistic.

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 18 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Proof : Uniform Distribution (cont’d)

Consider a function g(T) such that E[g(T)|θ] = 0 for all θ > 0 E[g(T)|θ] = ∫ θ g(t)nθ−ntn−1I(0 < t < θ)dt n

n

g t tn dt Taking derivative of both sides, n

n g n

n

n

g t tn dt ng n n

n

g t tn dt nE g T Because g T holds for all , T X X n is a complete statistic.

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 18 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Proof : Uniform Distribution (cont’d)

Consider a function g(T) such that E[g(T)|θ] = 0 for all θ > 0 E[g(T)|θ] = ∫ θ g(t)nθ−ntn−1I(0 < t < θ)dt = n θn ∫ θ g(t)tn−1dt = 0 Taking derivative of both sides, n

n g n

n

n

g t tn dt ng n n

n

g t tn dt nE g T Because g T holds for all , T X X n is a complete statistic.

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 18 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Proof : Uniform Distribution (cont’d)

Consider a function g(T) such that E[g(T)|θ] = 0 for all θ > 0 E[g(T)|θ] = ∫ θ g(t)nθ−ntn−1I(0 < t < θ)dt = n θn ∫ θ g(t)tn−1dt = 0 Taking derivative of both sides, n θn g(θ)θn−1 − n2 θn+1 ∫ θ g(t)tn−1dt = 0 ng n n

n

g t tn dt nE g T Because g T holds for all , T X X n is a complete statistic.

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 18 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Proof : Uniform Distribution (cont’d)

Consider a function g(T) such that E[g(T)|θ] = 0 for all θ > 0 E[g(T)|θ] = ∫ θ g(t)nθ−ntn−1I(0 < t < θ)dt = n θn ∫ θ g(t)tn−1dt = 0 Taking derivative of both sides, n θn g(θ)θn−1 − n2 θn+1 ∫ θ g(t)tn−1dt = 0 ng(θ) θ = n θ n θn ∫ θ g(t)tn−1dt = n θ E[g(T)|θ] = 0 Because g T holds for all , T X X n is a complete statistic.

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 18 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Proof : Uniform Distribution (cont’d)

Consider a function g(T) such that E[g(T)|θ] = 0 for all θ > 0 E[g(T)|θ] = ∫ θ g(t)nθ−ntn−1I(0 < t < θ)dt = n θn ∫ θ g(t)tn−1dt = 0 Taking derivative of both sides, n θn g(θ)θn−1 − n2 θn+1 ∫ θ g(t)tn−1dt = 0 ng(θ) θ = n θ n θn ∫ θ g(t)tn−1dt = n θ E[g(T)|θ] = 0 Because g T holds for all , T X X n is a complete statistic.

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 18 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Proof : Uniform Distribution (cont’d)

Consider a function g(T) such that E[g(T)|θ] = 0 for all θ > 0 E[g(T)|θ] = ∫ θ g(t)nθ−ntn−1I(0 < t < θ)dt = n θn ∫ θ g(t)tn−1dt = 0 Taking derivative of both sides, n θn g(θ)θn−1 − n2 θn+1 ∫ θ g(t)tn−1dt = 0 ng(θ) θ = n θ n θn ∫ θ g(t)tn−1dt = n θ E[g(T)|θ] = 0 Because g(T) = 0 holds for all θ > 0, T(X) = X(n) is a complete statistic.

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 18 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

A simpler proof (how it was solved in the class)

Consider a function g(T) such that E[g(T)|θ] = 0 for all θ > 0 E[g(T)|θ] = ∫ θ g(t)nθ−ntn−1I(0 < t < θ)dt n

n

g t tn dt g t tn dt Taking derivative of both sides, g

n

g for all . Because g T holds for all , T X X n is a complete statistic.

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 19 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

A simpler proof (how it was solved in the class)

Consider a function g(T) such that E[g(T)|θ] = 0 for all θ > 0 E[g(T)|θ] = ∫ θ g(t)nθ−ntn−1I(0 < t < θ)dt = n θn ∫ θ g(t)tn−1dt = 0 g t tn dt Taking derivative of both sides, g

n

g for all . Because g T holds for all , T X X n is a complete statistic.

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 19 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

A simpler proof (how it was solved in the class)

Consider a function g(T) such that E[g(T)|θ] = 0 for all θ > 0 E[g(T)|θ] = ∫ θ g(t)nθ−ntn−1I(0 < t < θ)dt = n θn ∫ θ g(t)tn−1dt = 0 ∫ θ g(t)tn−1dt = Taking derivative of both sides, g

n

g for all . Because g T holds for all , T X X n is a complete statistic.

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 19 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

A simpler proof (how it was solved in the class)

Consider a function g(T) such that E[g(T)|θ] = 0 for all θ > 0 E[g(T)|θ] = ∫ θ g(t)nθ−ntn−1I(0 < t < θ)dt = n θn ∫ θ g(t)tn−1dt = 0 ∫ θ g(t)tn−1dt = Taking derivative of both sides, g(θ)θn−1 = g(θ) = for all . Because g T holds for all , T X X n is a complete statistic.

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 19 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

A simpler proof (how it was solved in the class)

Consider a function g(T) such that E[g(T)|θ] = 0 for all θ > 0 E[g(T)|θ] = ∫ θ g(t)nθ−ntn−1I(0 < t < θ)dt = n θn ∫ θ g(t)tn−1dt = 0 ∫ θ g(t)tn−1dt = Taking derivative of both sides, g(θ)θn−1 = g(θ) = for all θ > 0. Because g(T) = 0 holds for all θ > 0, T(X) = X(n) is a complete statistic.

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 19 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Another Example of Uniform Distribution

.

Problem

. .

• Let X1, · · · , Xn

i.i.d.

∼ Uniform(θ, θ + 1), θ ∈ R.

• We have previously shown that T X

X X n is a minimal sufficient statistic for .

• Show that T X is not a complete statistic.

.

Proof - Using a range statistic

. . . . . . . . Define R X n X . We have previously shown that fR r n n r n r r Then R Beta n , and E R

n n

.

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 20 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Another Example of Uniform Distribution

.

Problem

. .

• Let X1, · · · , Xn

i.i.d.

∼ Uniform(θ, θ + 1), θ ∈ R.

• We have previously shown that T(X) = (X(1), X(n)) is a minimal

sufficient statistic for θ.

• Show that T X is not a complete statistic.

.

Proof - Using a range statistic

. . . . . . . . Define R X n X . We have previously shown that fR r n n r n r r Then R Beta n , and E R

n n

.

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 20 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Another Example of Uniform Distribution

.

Problem

. .

• Let X1, · · · , Xn

i.i.d.

∼ Uniform(θ, θ + 1), θ ∈ R.

• We have previously shown that T(X) = (X(1), X(n)) is a minimal

sufficient statistic for θ.

• Show that T(X) is not a complete statistic.

.

Proof - Using a range statistic

. . . . . . . . Define R X n X . We have previously shown that fR r n n r n r r Then R Beta n , and E R

n n

.

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 20 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Another Example of Uniform Distribution

.

Problem

. .

• Let X1, · · · , Xn

i.i.d.

∼ Uniform(θ, θ + 1), θ ∈ R.

• We have previously shown that T(X) = (X(1), X(n)) is a minimal

sufficient statistic for θ.

• Show that T(X) is not a complete statistic.

.

Proof - Using a range statistic

. . Define R = X(n) − X(1). We have previously shown that fR r n n r n r r Then R Beta n , and E R

n n

.

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 20 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Another Example of Uniform Distribution

.

Problem

. .

• Let X1, · · · , Xn

i.i.d.

∼ Uniform(θ, θ + 1), θ ∈ R.

• We have previously shown that T(X) = (X(1), X(n)) is a minimal

sufficient statistic for θ.

• Show that T(X) is not a complete statistic.

.

Proof - Using a range statistic

. . Define R = X(n) − X(1). We have previously shown that fR(r|θ) = n(n − 1)r(n−2)(1 − r) , 0 < r < 1 Then R Beta n , and E R

n n

.

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 20 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Another Example of Uniform Distribution

.

Problem

. .

• Let X1, · · · , Xn

i.i.d.

∼ Uniform(θ, θ + 1), θ ∈ R.

• We have previously shown that T(X) = (X(1), X(n)) is a minimal

sufficient statistic for θ.

• Show that T(X) is not a complete statistic.

.

Proof - Using a range statistic

. . Define R = X(n) − X(1). We have previously shown that fR(r|θ) = n(n − 1)r(n−2)(1 − r) , 0 < r < 1 Then R ∼ Beta(n − 1, 2), and E[R|θ] = n−1

n+1.

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 20 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Proof

Define g(T(X)) = X(n) − X(1) − n−1

n+1

E g T E X n X n n n n n n Therefore, there exist a g T such that Pr g T for all , so T X X X n is not a complete statistic.

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 21 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Proof

Define g(T(X)) = X(n) − X(1) − n−1

n+1

E[g(T)|θ] = E[X(n) − X(1)|θ] − n − 1 n + 1 n n n n Therefore, there exist a g T such that Pr g T for all , so T X X X n is not a complete statistic.

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 21 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Proof

Define g(T(X)) = X(n) − X(1) − n−1

n+1

E[g(T)|θ] = E[X(n) − X(1)|θ] − n − 1 n + 1 = n − 1 n + 1 − n − 1 n + 1 = 0 Therefore, there exist a g(T) such that Pr[g(T)|θ] < 1 for all θ, so T(X) = (X(1), X(n)) is not a complete statistic.

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 21 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Even Simpler Proof

• We know that R = X(n) − X(1) is an ancillary statistic, which do not

depend on θ.

• Define g T

X n X E R . Note that E R is constant to .

• Then E g T

E R E R , so T is not a complete statistic.

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 22 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Even Simpler Proof

• We know that R = X(n) − X(1) is an ancillary statistic, which do not

depend on θ.

• Define g(T) = X(n) − X(1) − E(R). Note that E(R) is constant to θ.
• Then E g T

E R E R , so T is not a complete statistic.

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 22 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Even Simpler Proof

• We know that R = X(n) − X(1) is an ancillary statistic, which do not

depend on θ.

• Define g(T) = X(n) − X(1) − E(R). Note that E(R) is constant to θ.
• Then E[g(T)|θ] = E(R) − E(R) = 0, so T is not a complete statistic.

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 22 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Example from Stigler (1972) Am. Stat.

.

Problem

. . Let X is a uniform random sample from {1, · · · , θ} where θ ∈ Ω = N. Is T X X a complete statistic? .

Solution

. . . . . . . . Consider a function g T such that E g T for all . Note that fX x I x I x . E g T E g X

x

g x

x

g x

x

g x

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 23 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Example from Stigler (1972) Am. Stat.

.

Problem

. . Let X is a uniform random sample from {1, · · · , θ} where θ ∈ Ω = N. Is T(X) = X a complete statistic? .

Solution

. . . . . . . . Consider a function g T such that E g T for all . Note that fX x I x I x . E g T E g X

x

g x

x

g x

x

g x

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 23 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Example from Stigler (1972) Am. Stat.

.

Problem

. . Let X is a uniform random sample from {1, · · · , θ} where θ ∈ Ω = N. Is T(X) = X a complete statistic? .

Solution

. . Consider a function g(T) such that E[g(T)|θ] = 0 for all θ ∈ N. Note that fX(x) = 1

θI(x ∈ {1, · · · , θ}) = 1 θINθ(x).

E g T E g X

x

g x

x

g x

x

g x

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 23 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Example from Stigler (1972) Am. Stat.

.

Problem

. . Let X is a uniform random sample from {1, · · · , θ} where θ ∈ Ω = N. Is T(X) = X a complete statistic? .

Solution

. . Consider a function g(T) such that E[g(T)|θ] = 0 for all θ ∈ N. Note that fX(x) = 1

θI(x ∈ {1, · · · , θ}) = 1 θINθ(x).

E[g(T)|θ] = E[g(X)|θ] =

θ

∑

x=1

1 θg(x) = 1 θ

θ

∑

x=1

g(x) = 0

x

g x

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 23 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Example from Stigler (1972) Am. Stat.

.

Problem

. . Let X is a uniform random sample from {1, · · · , θ} where θ ∈ Ω = N. Is T(X) = X a complete statistic? .

Solution

. . Consider a function g(T) such that E[g(T)|θ] = 0 for all θ ∈ N. Note that fX(x) = 1

θI(x ∈ {1, · · · , θ}) = 1 θINθ(x).

E[g(T)|θ] = E[g(X)|θ] =

θ

∑

x=1

1 θg(x) = 1 θ

θ

∑

x=1

g(x) = 0

θ

∑

x=1

g(x) =

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 23 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Solution (cont’d)

for all θ ∈ N, which implies

• if θ = 1, ∑θ

x=1 g(x) = g(1) = 0

• if

,

x

g x g g g .

• .

. .

• if

k,

x

g x g g k g g k . Therefore, g x for all x , and T X X is a complete statistic for .

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 24 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Solution (cont’d)

for all θ ∈ N, which implies

• if θ = 1, ∑θ

x=1 g(x) = g(1) = 0

• if θ = 2, ∑θ

x=1 g(x) = g(1) + g(2) = g(2) = 0.

• .

. .

• if

k,

x

g x g g k g g k . Therefore, g x for all x , and T X X is a complete statistic for .

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 24 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Solution (cont’d)

for all θ ∈ N, which implies

• if θ = 1, ∑θ

x=1 g(x) = g(1) = 0

• if θ = 2, ∑θ

x=1 g(x) = g(1) + g(2) = g(2) = 0.

• .

. .

• if θ = k, ∑θ

x=1 g(x) = g(1) + · · · + g(k − 1) + g(2) = g(k) = 0.

Therefore, g x for all x , and T X X is a complete statistic for .

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 24 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Solution (cont’d)

for all θ ∈ N, which implies

• if θ = 1, ∑θ

x=1 g(x) = g(1) = 0

• if θ = 2, ∑θ

x=1 g(x) = g(1) + g(2) = g(2) = 0.

• .

. .

• if θ = k, ∑θ

x=1 g(x) = g(1) + · · · + g(k − 1) + g(2) = g(k) = 0.

Therefore, g(x) = 0 for all x ∈ N, and T(X) = X is a complete statistic for θ ∈ Ω = N.

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 24 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Is the previous example barely complete?

.

Modified Problem

. . Let X is a uniform random sample from {1, · · · , θ} where θ ∈ Ω = N − {n}. Is T X X a complete statistic? .

Solution

. . . . . . . . Define a nonzero g x as follows g x x n x n

• therwise

E g T

x

g x n n Because does not include n, g x for all n , and T X X is not a complete statistic.

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 25 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Is the previous example barely complete?

.

Modified Problem

. . Let X is a uniform random sample from {1, · · · , θ} where θ ∈ Ω = N − {n}. Is T(X) = X a complete statistic? .

Solution

. . . . . . . . Define a nonzero g x as follows g x x n x n

• therwise

E g T

x

g x n n Because does not include n, g x for all n , and T X X is not a complete statistic.

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 25 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Is the previous example barely complete?

.

Modified Problem

. . Let X is a uniform random sample from {1, · · · , θ} where θ ∈ Ω = N − {n}. Is T(X) = X a complete statistic? .

Solution

. . Define a nonzero g(x) as follows g(x) =    1 x = n −1 x = n + 1

• therwise

E g T

x

g x n n Because does not include n, g x for all n , and T X X is not a complete statistic.

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 25 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Is the previous example barely complete?

.

Modified Problem

. . Let X is a uniform random sample from {1, · · · , θ} where θ ∈ Ω = N − {n}. Is T(X) = X a complete statistic? .

Solution

. . Define a nonzero g(x) as follows g(x) =    1 x = n −1 x = n + 1

• therwise

E[g(T)|θ] = 1 θ

θ

∑

x=1

g(x) = { 0 θ ̸= n

1 θ

θ = n Because does not include n, g x for all n , and T X X is not a complete statistic.

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 25 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Is the previous example barely complete?

.

Modified Problem

. . Let X is a uniform random sample from {1, · · · , θ} where θ ∈ Ω = N − {n}. Is T(X) = X a complete statistic? .

Solution

. . Define a nonzero g(x) as follows g(x) =    1 x = n −1 x = n + 1

• therwise

E[g(T)|θ] = 1 θ

θ

∑

x=1

g(x) = { 0 θ ̸= n

1 θ

θ = n Because Ω does not include n, g(x) = 0 for all θ ∈ Ω = N − {n}, and T(X) = X is not a complete statistic.

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 25 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Summary

.

Today - Complete Statistics

. .

• Examples of complete statistics
• Two Poisson distribution examples
• Two Uniform distribution examples
• Example of barely complete statistics.

.

Next Lecture

. . . . . . . . • Basu’s Theorem

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 26 / 26

. . . . . .

. . . . . Ancillary Statistics . . . . . . . . . . . . . . . . . . . Complete Statistics . Summary

Summary

.

Today - Complete Statistics

. .

• Examples of complete statistics
• Two Poisson distribution examples
• Two Uniform distribution examples
• Example of barely complete statistics.

.

Next Lecture

. . • Basu’s Theorem

Hyun Min Kang Biostatistics 602 - Lecture 05 January 24th, 2013 26 / 26