Determining Bias of MLE of k in Poisson RP

In summary, the conversation is about estimating the unknown nonrandom variable, k, in a Poisson random process. The maximum likelihood estimate of k is determined to be [1 / (n*tau) ] times the sum of the observed events, xi. The question is then raised about whether or not this estimate is biased, and it is determined that it is unbiased if E[k^ML] = k. The concept of a sufficient statistic is introduced as a way to model and think about the problem.
  • #1
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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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  • #2


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.
 
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  • #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!
 
  • #4


Is k a R.V., or is it a deterministic (although unknown) parameter (i.e. constant)?
 
  • #5


k is a is an unknown nonrandom variable.

Based on this...I would say that it is deterministic...
 
  • #6


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


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


"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
 

FAQ: Determining Bias of MLE of k in Poisson RP

1. What is MLE and how is it related to determining bias in Poisson RP?

MLE stands for Maximum Likelihood Estimation. It is a statistical method used to estimate the parameters of a probability distribution by maximizing the likelihood function. In the case of determining bias in Poisson RP, MLE is used to estimate the parameter k, which represents the rate of occurrence of events in a Poisson distribution. By analyzing the MLE of k, we can determine if there is any bias in the estimation of this parameter.

2. How is bias defined in the context of Poisson RP?

In the context of Poisson RP, bias refers to the tendency of the estimated value of k (the rate parameter) to consistently overestimate or underestimate the true value of k. This can lead to inaccurate conclusions and predictions based on the Poisson distribution.

3. What factors can contribute to bias in the MLE of k in Poisson RP?

There are several factors that can contribute to bias in the MLE of k in Poisson RP. These include small sample sizes, outliers in the data, and assumptions about the underlying distribution not being met. Additionally, the choice of estimation method and the quality of the data can also affect the bias in the MLE of k.

4. How can we determine if there is bias in the MLE of k in Poisson RP?

To determine if there is bias in the MLE of k, we can compare the estimated value of k to the true value of k. If the estimated value consistently deviates from the true value in a particular direction (e.g. always overestimating or always underestimating), then there is bias in the MLE of k. We can also use statistical tests, such as the Chi-Square test, to assess the bias in the estimation.

5. How can bias in the MLE of k in Poisson RP be reduced?

Bias in the MLE of k can be reduced by using larger sample sizes, removing outliers from the data, and ensuring that the assumptions of the Poisson distribution are met. It is also important to carefully select the appropriate estimation method and to assess the quality of the data before making conclusions based on the Poisson distribution. Regularly checking for bias and making adjustments as needed can also help reduce bias in the MLE of k.

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