Dtjytyk

Why didn't Eitri join the fight?

Is it cost-effective to upgrade an old-ish Giant Escape R3 commuter bike with entry-level branded parts (wheels, drivetrain)?

How would a mousetrap for use in space work?

Compare a given version number in the form major.minor.build.patch and see if one is less than the other

old style "caution" boxes

What's the meaning of "fortified infraction restraint"?

Amount of permutations on an NxNxN Rubik's Cube

Is there any way for the UK Prime Minister to make a motion directly dependent on Government confidence?

Using et al. for a last / senior author rather than for a first author

Do I really need recursive chmod to restrict access to a folder?

What does this Jacques Hadamard quote mean?

Is there a kind of relay only consumes power when switching?

Circuit to "zoom in" on mV fluctuations of a DC signal?

What does the "x" in "x86" represent?

Extracting terms with certain heads in a function

Is this homebrew Lady of Pain warlock patron balanced?

Did MS DOS itself ever use blinking text?

What do you call the main part of a joke?

How to deal with a team lead who never gives me credit?

Is grep documentation wrong?

Crossing US/Canada Border for less than 24 hours

Significance of Cersei's obsession with elephants?

Irreducible of finite Krull dimension implies quasi-compact?

What does "lightly crushed" mean for cardamon pods?

Sufficiency of $|X|$ when $Xsim N(0,sigma^2)$ without using Factorization theorem

.everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty,.everyoneloves__bot-mid-leaderboard:empty{ margin-bottom:0;
}

1

$begingroup$

Question:

Given, $Xsim N(0,sigma^2)$. By means of conditional approach show that $|X|$ is a sufficient estimator for $sigma^2$.

My Attempt:

This problem is very easy if we use Fisher–Neyman Factorisation theorem. But in this problem we have to use the basic definition of Sufficient Statistics, i.e,

A statistic $t = T(X)$ is sufficient for underlying parameter $theta$ precisely if the conditional probability distribution of the data $X$, given the statistic $t = T(X)$, does not depend on the parameter $θ$.

$Xsim N(0,sigma^2)$, here parameter, $theta=sigma^2$.
So we have to show that $P_{sigma^2}(Xle xmid T(X)le t)$ is independent of $sigma^2$.

For $xin (-t,t]$,

begin{align}
P_{sigma^2}(Xle x mid T(X)le t)&= frac{P_{sigma^2}(Xle x,T(X)le t)}{P_{sigma^2}(T(X)le t)}
\&=frac{P_{sigma^2}(Xle x,|X|le t)}{P_{sigma^2}(|X|le t)}
\&=frac{P_{sigma^2}(-tle Xle x)}{P_{sigma^2}(-tle Xle t)}
end{align}

But how to show that this expression is free of $sigma^2$?

normal-distribution estimation

share|cite|improve this question

edited Mar 27 at 12:26

Ferdi

3,87952355

asked Mar 24 at 19:50

RATNODEEP BAINRATNODEEP BAIN

62

$endgroup$

migrated from math.stackexchange.com Mar 26 at 7:49

This question came from our site for people studying math at any level and professionals in related fields.

• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  Possible duplicate of Sufficiency of $|X|$ when $Xsim N(0,sigma^2)$ without using Factorization theorem
  $endgroup$
  – StubbornAtom
  Mar 26 at 10:05
  

  
  
  
  

add a comment | 

1

$begingroup$

Question:

Given, $Xsim N(0,sigma^2)$. By means of conditional approach show that $|X|$ is a sufficient estimator for $sigma^2$.

My Attempt:

This problem is very easy if we use Fisher–Neyman Factorisation theorem. But in this problem we have to use the basic definition of Sufficient Statistics, i.e,

A statistic $t = T(X)$ is sufficient for underlying parameter $theta$ precisely if the conditional probability distribution of the data $X$, given the statistic $t = T(X)$, does not depend on the parameter $θ$.

$Xsim N(0,sigma^2)$, here parameter, $theta=sigma^2$.
So we have to show that $P_{sigma^2}(Xle xmid T(X)le t)$ is independent of $sigma^2$.

For $xin (-t,t]$,

begin{align}
P_{sigma^2}(Xle x mid T(X)le t)&= frac{P_{sigma^2}(Xle x,T(X)le t)}{P_{sigma^2}(T(X)le t)}
\&=frac{P_{sigma^2}(Xle x,|X|le t)}{P_{sigma^2}(|X|le t)}
\&=frac{P_{sigma^2}(-tle Xle x)}{P_{sigma^2}(-tle Xle t)}
end{align}

But how to show that this expression is free of $sigma^2$?

normal-distribution estimation

share|cite|improve this question

edited Mar 27 at 12:26

Ferdi

3,87952355

asked Mar 24 at 19:50

RATNODEEP BAINRATNODEEP BAIN

62

$endgroup$

migrated from math.stackexchange.com Mar 26 at 7:49

This question came from our site for people studying math at any level and professionals in related fields.

• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  Possible duplicate of Sufficiency of $|X|$ when $Xsim N(0,sigma^2)$ without using Factorization theorem
  $endgroup$
  – StubbornAtom
  Mar 26 at 10:05
  

  
  
  
  

add a comment | 

$begingroup$

Question:

Given, $Xsim N(0,sigma^2)$. By means of conditional approach show that $|X|$ is a sufficient estimator for $sigma^2$.

My Attempt:

This problem is very easy if we use Fisher–Neyman Factorisation theorem. But in this problem we have to use the basic definition of Sufficient Statistics, i.e,

A statistic $t = T(X)$ is sufficient for underlying parameter $theta$ precisely if the conditional probability distribution of the data $X$, given the statistic $t = T(X)$, does not depend on the parameter $θ$.

$Xsim N(0,sigma^2)$, here parameter, $theta=sigma^2$.
So we have to show that $P_{sigma^2}(Xle xmid T(X)le t)$ is independent of $sigma^2$.

For $xin (-t,t]$,

begin{align}
P_{sigma^2}(Xle x mid T(X)le t)&= frac{P_{sigma^2}(Xle x,T(X)le t)}{P_{sigma^2}(T(X)le t)}
\&=frac{P_{sigma^2}(Xle x,|X|le t)}{P_{sigma^2}(|X|le t)}
\&=frac{P_{sigma^2}(-tle Xle x)}{P_{sigma^2}(-tle Xle t)}
end{align}

But how to show that this expression is free of $sigma^2$?

normal-distribution estimation

share|cite|improve this question

edited Mar 27 at 12:26

Ferdi

3,87952355

asked Mar 24 at 19:50

RATNODEEP BAINRATNODEEP BAIN

62

$endgroup$

share|cite|improve this question

edited Mar 27 at 12:26

Ferdi

3,87952355

asked Mar 24 at 19:50

RATNODEEP BAINRATNODEEP BAIN

62

share|cite|improve this question

edited Mar 27 at 12:26

Ferdi

3,87952355

asked Mar 24 at 19:50

RATNODEEP BAINRATNODEEP BAIN

62

edited Mar 27 at 12:26

Ferdi

3,87952355

edited Mar 27 at 12:26

Ferdi

3,87952355

asked Mar 24 at 19:50

RATNODEEP BAINRATNODEEP BAIN

62

asked Mar 24 at 19:50

RATNODEEP BAINRATNODEEP BAIN

62