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Poisson RP: MLE of k

  1. Jun 7, 2009 #1
    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?
     
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  3. Jun 7, 2009 #2

    EnumaElish

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    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
  4. Jun 8, 2009 #3
    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) = [tex]\frac{(k\tau)}{n!}[/tex] [tex]^{n}[/tex] e[tex]^{-k\tau}[/tex]

    • 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 [tex]N[/tex], 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...

    Thanks in advance!
     
  5. Jun 8, 2009 #4

    EnumaElish

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    Re: Poisson RP: MLE of "k"

    Is k a R.V., or is it a deterministic (although unknown) parameter (i.e. constant)?
     
  6. Jun 8, 2009 #5
    Re: Poisson RP: MLE of "k"

    k is a is an unknown nonrandom variable.

    Based on this....I would say that it is deterministic....
     
  7. Jun 8, 2009 #6

    EnumaElish

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    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?
     
  8. Jun 8, 2009 #7
    Re: Poisson RP: MLE of "k"

    YES - you are correct. Unfortunatly, the write up I have is written very poorly.
     
  9. Jun 8, 2009 #8

    EnumaElish

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    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
     
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