Poisson RP: MLE of k

1. Jun 7, 2009

ACM_acm

Poisson RP: MLE of "k"

P(n,tau) = [ [ (k*tau)^n ] / n! ] * exp(-k*tau)

Parameter k is the process of an unknown non random variable that I want to estimate.

I have determined that k^ML = [1 / (n*tau) ] sigma (xi)

I believe this is correct...

How do I determine if K^ML is biased?

2. Jun 7, 2009

EnumaElish

Re: Poisson RP: MLE of "k"

k^ML is unbiased if E[k^ML] = k, otherwise it is biased.

Hint: since each x is distributed Poisson with mean = k, ∑x is distributed Poisson with mean = Nk, where N is the number of x's.

Last edited: Jun 8, 2009
3. Jun 8, 2009

ACM_acm

Poisson Random Process, Sufficient Statistic

OK - I think I understand you...I would like to rewrite the problem using the LATEX symbology...This is my first time to this website and I would like to learn this program....

My problem is stated as follows...

• Stationary Poisson Random Process
• The probability of n events in an interval of time tau is

P(n,tau) = $$\frac{(k\tau)}{n!}$$ $$^{n}$$ e$$^{-k\tau}$$

• parameter k is an unknown RV that I want to estiamte
• I will observe x(t) over an interval (0,T)

My questions are as follows...

(1) is $$N$$, the number of events that occur in the interval (),T), a sufficient statistic, or is it necessary to record the actual event times?

I am not sure what this question is looking for....how can I model this? or think of it? Once I get this part, I will move on to the rest of the problem...

4. Jun 8, 2009

EnumaElish

Re: Poisson RP: MLE of "k"

Is k a R.V., or is it a deterministic (although unknown) parameter (i.e. constant)?

5. Jun 8, 2009

ACM_acm

Re: Poisson RP: MLE of "k"

k is a is an unknown nonrandom variable.

Based on this....I would say that it is deterministic....

6. Jun 8, 2009

EnumaElish

Re: Poisson RP: MLE of "k"

In your later post you wrote N is the number of events. I had used N as the sample size (number of x's). Did you mean to write n instead?

7. Jun 8, 2009

ACM_acm

Re: Poisson RP: MLE of "k"

YES - you are correct. Unfortunatly, the write up I have is written very poorly.

8. Jun 8, 2009

EnumaElish

Re: Poisson RP: MLE of "k"

"I am not sure what this question is looking for....how can I model this? or think of it? Once I get this part, I will move on to the rest of the problem..."

You can start with studying the concept of Sufficient Statistic. See, for example, http://en.wikipedia.org/wiki/Sufficient_statistic