Cramer row inequality

Cramér-Rao Inequality
Subject:- PBD-2802
ADVANCE STATISTICAL METHODS
Presented by : Vashu Gupta 2020-PBD-012

Contents :-
Introduction
Basic conditions
Example
References

Introduction:-
The Cramer-Rao Inequality provides us with a lower bound on
the variance of an unbiased estimator for a parameter.
The Cramer-Rao Inequality Let X = (X1,X2,. . ., Xn) be a random
sample from a distribution with d.f. f(x|θ), where θ is a scalar
parameter. Under certain regularity conditions on f(x|θ), for any
unbiased estimator φˆ (X) of φ (θ)
Where
2
2
[ ( )]
ˆ
( )
( ( ))
d
d
Var
E S X
 

 

Continue…
 Most Efficient estimator:-
Note that 0 ≤ eff(φˆ) ≤ 1 for all θ ∈ Θ
ˆ
( )
ˆ
( )
CRLB
eff
Var




Continue…
The quantity I(θ) = E(S 2 (X)) is called the Fisher
information on the parameter θ contained in the
sample X.
The Fisher information is in general, a function of the
value of θ.
The higher the Fisher information i.e. the more
information there is in the sample, the smaller the
CRLB and consequently the smaller the variance of the
most efficient estimator.

Basic Conditions :-
We further make the following assumptions, which are known as
the Regularity conditions for Cramer-Roo Inequality.
The parameter space e is a non-degenerate open interval on the
real line
For almost all , and for all
exists,
the exceptional set, if any, is independent of
1
( , )
R  
1 2 3
( , , ............... )
n
x x x x x

, ( , )
L X
 






Continue…
The range of Integration is independent of the parameter
, so that is differentiable under integral sign.
The conditions of uniform convergence of integrals are
satisfied so that differentiation under the integral sign is
valid.
 exists and is positive for all .
( , )
f x 

2
( ) [{ log ( , )} ]
I E L x
 




 

Conditions for the Equality sign in
Cramer-Rao Inequality
The sign of equality will hold in C.R. Inequality if and only if the
sign of equality holds in
The sign o f equality will hold in by Cauchy Schwartz inequality,
if and only if the variables and are
proportional to each other
2 2. 2
[ ( )] [ ( )] ( log )
E t E L
   


  

[ ( )]
t  
 ( log )
L



( )
( )
log
t
L
 
  


 



Example of Cramer-Rao
inequality:-
If we have in Poisson distribution so
find the minimum variance unbiased estimator.
Solution:- firstly we have pdf of Poisson distribution is
And find likelihood function:
1 2
, .................... n
x x x
( )
x
e
p x
x




1
i
x
n
n
i
i
e
L
x








Continue…
Let log in both side,
Differentiating in both side-
=
1
log log log
n
i i
i
L n x n
  

   

log i
x nx
L n n
  

    


[ ]
n
x 



Continue…
Now we can say that,
S is a MUVE of .
={CRLB}
Cramer Rao lower bond
( )[ ( )]
A t
  
 
t x
 
( ) 1
( )
( )
Var t
n
A
n
 




 


References:-
https://web.williams.edu/Mathematics/sjmiller/public_html/Br
ownClasses/162/Handouts/CramerRaoHandout08.pdfhttps://w
eb.williams.edu/Mathematics/sjmiller/public_html/BrownClass
es/162/Handouts/CramerRaoHandout08.pdf
https://www.dcpehvpm.org/E-
Content/Stat/FUNDAMENTAL%20OF%20MATHEMATICAL%20ST
ATISTICS-S%20C%20GUPTA%20&%20V%20K%20KAPOOR.pdf

Cramer row inequality

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