Question

Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample from a Poisson distribution with \(\operatorname{mean} \theta>0\). (a) Statistician \(A\) observes the sample to be the values \(x_{1}, x_{2}, \ldots, x_{n}\) with sum \(y=\sum x_{i} .\) Find the mle of \(\theta\). (b) Statistician \(B\) loses the sample values \(x_{1}, x_{2}, \ldots, x_{n}\) but remembers the sum \(y_{1}\) and the fact that the sample arose from a Poisson distribution. Thus \(B\) decides to create some fake observations, which he calls \(z_{1}, z_{2}, \ldots, z_{n}\) (as

Short answer

The maximum likelihood estimates of \(\theta\) in the first case is \( \bar{x} = \frac{y}{n}\) and in the second case where the observations are lost is \( \bar{z} = \frac{y_{1}}{n}\). The same value is used because we simulate the lost observations as \( \frac{y_{1}}{n}\).

Step-by-step solution

Calculate sample mean for the original set

First calculate the sample mean by dividing the sum of the sample values by the number of samples. This gives \( \bar{x} = \frac{y}{n} \). This is the maximum likelihood estimate of \(\theta\) in the case of a Possion distribution when all sample values are known.

Simulate new sample set

Since the original sample values are lost and only sum \(y_{1}\) is known, we create some fake observations. These fake observations should have the sum \(y_{1}\). One simple way to accomplish this is to assign all the fake samples \(z_{1}, z_{2}, \ldots, z_{n}\) the same value of \( \frac{y_{1}}{n}\)

Calculate sample mean for the new set

We calculate the new sample mean as \( \bar{z} = \frac{y_{1}}{n}\). This is the maximum likelihood estimate of \(\theta\) in this case, when only sum of the sample values is known.