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PART 4 - MINIMAL SUFFICIENT STATISTICS Intuitively, a minimal sufficient…
PART 4 - MINIMAL SUFFICIENT STATISTICS
Intuitively, a minimal sufficient statistic most efficiently captures all possible information about the parameter θ.
Definition
: A set of jointly sufficient statistics is defined to be minimal sufficient iff it is a function of every other set of sufficient statistics.
:black_flag:
Theorem 6.1.4 Casella & Berger.
Let f (x |θ) be the pmf/pdf of a sample X.Suppose that there exists a function T(x) such that for sample points x & y, the ratio
iff T(x)=T(y)
. Then
T(x) is a minimal sufficient statistics
for θ.
:pen:
Worked Example
[
Normal minimal sufficient statistic with 2 parameters
]
:pen:
Worked Example
[
Uniform minimal sufficient statistic
]
:<3:..
EXPONENTIAL FAMILY
...
........
ONE - PARAMETER
.......
Definition 1
: A one parameter family of densities that can be expressed as :star:
f (x;θ) = a(θ) b(x) exp [c(θ)d(x)]
is defined to be belong to the exponential family.
:pen:
Worked Example
[
Check for Exponential Family
]
........
K - PARAMETER
.......
Definition 2
: A family of densities . f( .; θ1,..., θk) that can be expressed as :star:
f (x;θ1,..., θk) = a(θ1,..., θk) b(x) exp [ summation c(θ1,..., θk)d(x)]
is defined to be belong to the exponential family.
:pen:
Worked Example
[
2-parameter Exponential Family
]
ANCILLARY STATISTICS
A complementary purpose to sufficient statistics
A statistics S(x) whose distribution
does not depend on the parameter θ
is called an ancillary statistics.
:pen:
Worked Example
[
Ancillary statistic Using Tranformation
]
:pen:
Worked Example
[
Ancillary statistic For Location Parameter Family
]
:pen:
Worked Example
[
Ancillary statistic For Scale Parameter Family
]
:warning:
Note
: :red_flag: Ancillary statistics contains no information about θ. :red_flag: It is an observation on a random variable whose distribution is fixed and known, unrelated to θ. :red_flag: when used in conjunction with other statistics, sometimes does contain valuable information for inferences about θ.
COMPLETE STATISTICS
:black_flag:
Definition 1
: The family of densities of T defined to be complete
iff Eθ[z(T)]=0
for all θ, implies that
Pθ[z(T)=0]=1
. Also, statistic T is said to be
complete iff family densities is complete.
:pen:
Worked Example
[
Complete Bernoulli Density
]
:pen:
Worked Example
[
Complete Uniform Density
]
TOOL FOR ARGUING COMPLETENESS
:check:
Theorem 1
: If
f (x;θ) = a(θ) b(x) exp [c(θ)d(x)]
is a member of one parameter
exponential family
, then
summation d(Xi)
is a complete minimal sufficient statistics
:check:
Theorem 2: (BASU's Theorem)
If T(X) is a complete & minimal sufficient statistics, then T(X) is independent of every ancillary statistics.
:pen:
Worked Example
[
Complete Statistics
]
:red_flag: Under random sampling, if a density
belongs to the one-parameter exponential family
, then there is a :check:
sufficient statistic
. In fact, it can be shown that it is also :check:
minimal sufficient statistic.
:<3: :<3: :<3: (this what makes Exp Method the best!)
:!: Basu's Theorem
Useful
- allow us to
deduce the independence of 2 statistics WITHOUT
ever finding the
joint distribution.
:star:
NOTE
: Please Click "Worked Example" to view
