Sufficient Statistics: Examples

[Pages:31]Sufficient Statistics: Examples

Mathematics 47: Lecture 8

Dan Sloughter

Furman University

March 16, 2006

Dan Sloughter (Furman University)

Sufficient Statistics: Examples

March 16, 2006 1 / 12

Example

Dan Sloughter (Furman University)

Sufficient Statistics: Examples

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Example

Let X1, X2, . . . , Xn be a random sample from a Bernoulli distribution with probability of success p.

Dan Sloughter (Furman University)

Sufficient Statistics: Examples

March 16, 2006 2 / 12

Example

Let X1, X2, . . . , Xn be a random sample from a Bernoulli distribution with probability of success p.

Let T = X1 + X2 + ? ? ? + Xn and let f be the joint density of X1, X2, . . . , Xn.

Dan Sloughter (Furman University)

Sufficient Statistics: Examples

March 16, 2006 2 / 12

Example (cont'd)

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Sufficient Statistics: Examples

March 16, 2006 3 / 12

Example (cont'd)

Then

f (x1, x2, . . . ,xn | p)

=

n i =1

pxi

(1

-

p)1-xi

,

if xi = 0, 1, i = 1, 2, . . . , n,

0,

otherwise,

=

Pn

p i=1

xi

(1

-

p )n-Pni =1

xi

,

if xi = 0, 1, i = 1, 2, . . . , n,

0,

otherwise,

= pt (1 - p)n-t h(x1, x2, . . . , xn)

= g (t, p)h(x1, x2, . . . , xn),

where g (t, p) = pt (1 - p)n-t and

h(x1, x2, . . . , xn) =

1, 0,

if xi = 0, 1, i =, 2, . . . , n, otherwise.

Dan Sloughter (Furman University)

Sufficient Statistics: Examples

March 16, 2006 3 / 12

Example (cont'd)

Then

f (x1, x2, . . . ,xn | p)

=

n i =1

pxi

(1

-

p)1-xi

,

if xi = 0, 1, i = 1, 2, . . . , n,

0,

otherwise,

=

Pn

p i=1

xi

(1

-

p )n-Pni =1

xi

,

if xi = 0, 1, i = 1, 2, . . . , n,

0,

otherwise,

= pt (1 - p)n-t h(x1, x2, . . . , xn)

= g (t, p)h(x1, x2, . . . , xn),

where g (t, p) = pt (1 - p)n-t and

h(x1, x2, . . . , xn) =

1, 0,

if xi = 0, 1, i =, 2, . . . , n, otherwise.

Hence T is a sufficient statistic for p.

Dan Sloughter (Furman University)

Sufficient Statistics: Examples

March 16, 2006 3 / 12

Example

Dan Sloughter (Furman University)

Sufficient Statistics: Examples

March 16, 2006 4 / 12

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