Bernoulli estimator

1. Jun 1, 2012

chicory

A Bernoulli random sample :

X=1 , Pr(X=1)=p;
X=0 , Pr(X=0)=1-p;

taken X1, ..., Xn

and if it is known that 0=< p =<0.5

find a method of moments estimator of p

If I just take estimator of P =(sum Xi )/n than the estimate p may be bigger than 0.5 as in extreme case all Xi =1 .
What should I do , take estimator of p = Max (0.5*n , Sum Xi) /n ?

2. Jun 1, 2012

viraltux

Just simply state that fact, that is

P= sum Xi /n if sum Xi /n < 0.5
P= 0.5 otherwise.

3. Jun 1, 2012

chicory

But... is it too artificial?
Pr( estimator (p) = 0.5 ) is slightly bigger ...?

4. Jun 1, 2012

viraltux

Artificial? You like better min( sum Xi/n, 0.5) ?

No, p will not be generally bigger and you will rarely get estimations of 0.7 or 0.8, remember that they already assure you that the real p is no higher than 0.5

Adjusting the estimation makes the value p=0.5 more common but that is irrelevant and only makes the estimation more accurate.

Last edited: Jun 1, 2012
5. Jun 1, 2012

chicory

Thank you so much !

6. Jun 2, 2012

viraltux

You're welcome