I'm trying to solve these two question below:
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 $10$.
I have the following questions:
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.
for the item a, using WLLN:
$$\bar X_n\xrightarrow{P}\lambda$$
and $\bar X_n$ is unbiased because $E[\bar X_n]=\frac{X_1+\ldots+X_n}{n}=\frac{n\lambda}{n}=\lambda$
- 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 $\lambda\le 10$.