An estimator for the Poisson distribution with $\lambda\le 10$

Question

I'm trying to solve these two question below:

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For the following experiments, define a statistical model and check whether the parameter of interest is identified.

a. One observes $n$ i.i.d. Poisson random variables with unknown parameter $\lambda$.

b. One observes $n$ i.i.d. exponential random variables with parameter $\lambda$, which is unknown but a priori known to be no larger than $10$.

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I have the following questions:

1. What does he mean by "parameter of interest", an estimator with no bias or any estimator converging to the parameter is a valid one? in this case for any question asking an estimator of a parameter, that's enough to use WLLN and mapping theorem.

2. for the item a, using WLLN:

$$\bar X_n\xrightarrow{P}\lambda$$

and $\bar X_n$ is unbiased because $E[\bar X_n]=\frac{X_1+\ldots+X_n}{n}=\frac{n\lambda}{n}=\lambda$

3. The last part I found a little tricky.

Using the WLLN, mapping theorem and the fact $E[X_i]=1/\lambda$, I showed $$T_n=\frac{1}{\bar X_n}\xrightarrow{P}\lambda$$

But I'm having problems to find the bias of this estimator (I know this $E[\bar X_n]=\frac{1}{E[\bar X_n]}$is not always true by Jensen's inequality) and I don't know how to use the fact $\lambda\le 10$.

Answer 1

This seems to be from an MIT course (or at least I can find the same problem in the problem set 2 of Statistics for Applications Fall 2016). The question reads

Problem 2 Statistical models and identifiability

For the following experiments, define a statistical model and check whether the pa­rameter of interest is identified.

So the question relates to identifiability.

• Your approach is considering the question:

  Does the estimator converge to the parameter of interest (that is the parameter that you wish to estimate with the data, e.g. the parameter $\lambda$) when you increase the sample size?

  With that interpretation, the trick of the question about the item b. is probably that you are supposed to use the maximum likelihood estimate (which is the inverse of the sample mean) but cutoff by 10.

  $$\hat\lambda_{MLE} = \min \left( \frac{1}{ \bar{x}}, 10 \right) $$

  So you may have the situation that different observations give the same estimate $\hat\lambda_{MLE}$. Does that make it a non-identifiable model? (or is this alternative look at the problem only for finite samples and is it gone when we increase the sample size?)

• I believe that the problem is actually simpler and you are supposed to give the model (not the estimator). For instance in item 'a' the model is $$\left(\mathbb{N_0}, (\text{Pois}(\lambda))_{\lambda \in \mathbb{R_+}} \right)$$ which means

  • $\mathbb{N_0}$ the set of natural numbers (including zero) is the 'sample space'
  • $(\text{Pois}(\lambda))_{\lambda \in \mathbb{R_+}}$ the Poisson distribution is the 'probability measure'
  • $\lambda \in \mathbb{R_+}$ the positive real numbers is the 'parameter set'

  In this case, you consider "Can we have the same probability measure on this sample space for any two different parameters"?