Re: urn model

On Jun 10, 12:56 pm, isabelle...@xxxxxxxxxxx wrote:
On Jun 9, 11:22 pm, randerson1184 <randerson1...@xxxxxxxxx> wrote:



On Jun 9, 4:26 pm, randerson1184 <randerson1...@xxxxxxxxx> wrote:

On Jun 9, 3:08 pm, isabelle...@xxxxxxxxxxx wrote:

On Jun 9, 3:54 pm, randerson1184 <randerson1...@xxxxxxxxx> wrote:

On Jun 9, 2:51 pm, randerson1184 <randerson1...@xxxxxxxxx> wrote:

On Jun 9, 2:46 pm, Paul Rubin <ru...@xxxxxxx> wrote:

isabelle...@xxxxxxxxxxx wrote:
On Jun 9, 2:36 pm, Paul Rubin <ru...@xxxxxxx> wrote:
isabelle...@xxxxxxxxxxx wrote:
Hi everybody,
I am having a hard time writing the likelihood of the following urn
model. I would need to find the maximum likelihood estimator of p, so
I only need those terms of the likelihood that are functions of p. I
have not seen very many urn problems in my textbooks, if there is a
reference with urn problems, I will gladly read it.
We continue to draw balls with replacement from an urn containing both
black and white balls until two balls of the same color are drawn. Let
X be the number of draws (including the first draw) we take until we
stop. We are not told how many black and white balls that we see, but
only the value of X. Let p be the unknown proportion of white balls in
the urn.
I greatly appreciate your help.
Here is my work so far.
X can take only two values: 2 or 3.
The sample space is
BB, X=2
WW, X=2
BWB, X=3
BWW, X=3
WBB, X=3
WBW, X=3
When X=2, the likelihood can be proportional to p^2 or (1-p)^2.
When X=3, the likelihood can be proportional to p^1*(1-p)^2 or p^2*(1-
p)^1.
You need to rethink that last line a bit. If necessary, try assuming a
value such as 0.1 for p and see what you come up with for the
probabilities of X=2 and X=3.

/Paul

I am not sure how to do this. I do not have a method to write the
probabilities. Here is an attempt.

P[X=2]=P[BB or WW]=P[BB]+P[WW]=constant1*{p^2 + (1-p)^2}

P[X=3]=P[BWB]+P[BWW]+P[WBW]+...
=constant2*{2*p^1*(1-p)^2 + 2*p^2*(1- p)^1}

You are on the right track. Why do feel the need for constant1 and
constant2? You have six possible sequences listed above. Write the
probability of each in terms of p, and see where that takes you.

/Paul

I got that X~Ber(2p-2p^2-p^3)
I defined a success as X=3 for which the probability is the 2p-2p^2-
p^3
So from here, I just use the likelihood function to find the MLE of p
right?

I think I need to make the simple transformation Y=X-2 and then
declare it bernoulli...

I am finding
X-2~Bernoulli(2p-2p^2)

thanks a lot for your help

Yep, you're right, 2p - 2p^2. Let me know what you get for your MLE.

Alright, I re-read the thread, and I couldn't figure out why I thought
you were trying to find an MLE. Perhaps it's because you used the term
"likelihood." You're just trying to find an estimator for 'p' right?

I am a frequentist (go ahead, do your worst) so I would tend to go
with MLE or UMVUE (which I should have memorized for the bernoulli
case by now). If we were to find a bayes estimator, what prior would
we place on p? Beta?

If beta, how do you go about selecting values for your parameters for
your prior distribution? Hyperpriors?

R.G. Vickson said on another forum:

So P{X = 2} = p^2 + (1-p)^2 = 1 - 2*p + 2*p^2 and
P{X = 3} = 2*p*(1-p)^2 + 2*(1-p)*p^2 = 2*p*(1-p)*[(1-p) + p] = 2*p -
2*p^2. For observation {X = 2}, the probability is maximized at p = 0
or 1, since P{X=2} is a strictly convex function of p. For observation
{X = 3} the probability is maximized at 2 - 4*p = 0, giving p = 1/2.

I agree with Vickson. I was just kind of having fun and finding an
MLE. I came up with AN MLE, but it is almost certainly incorrect
because for several X=3 observations, it returns a probability of
either 1.366 or -.366, which makes no sense...

I think Vickson took the right approach.
.

Relevant Pages

  • Re: urn model
    ... I am having a hard time writing the likelihood of the following urn ... black and white balls until two balls of the same color are drawn. ... I just use the likelihood function to find the MLE of p ...
    (sci.stat.math)
  • Re: urn model
    ... I am having a hard time writing the likelihood of the following urn ... I would need to find the maximum likelihood estimator of p, ... black and white balls until two balls of the same color are drawn. ...
    (sci.stat.math)
  • Re: urn model
    ... I am having a hard time writing the likelihood of the following urn ... black and white balls until two balls of the same color are drawn. ... probabilities of X=2 and X=3. ...
    (sci.stat.math)
  • Re: urn model
    ... I am having a hard time writing the likelihood of the following urn ... black and white balls until two balls of the same color are drawn. ... probabilities of X=2 and X=3. ...
    (sci.stat.math)
  • Re: urn model
    ... I am having a hard time writing the likelihood of the following urn ... black and white balls until two balls of the same color are drawn. ... probabilities of X=2 and X=3. ...
    (sci.stat.math)