An estimator for the Exponential distribution with $\lambda\le 10$ 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 $\lambda\le 10$.
 A: They simply ask you to define the Statistical Model for the two experiments.
The statisical model is definded through the following phases

*

*Definition of the Base Model


*Bernullian Sampling (with replacement)
1. Base Model
In your cases, the base model is the following:
$$\Big(X, p(x|\theta), \theta \in \Theta\Big)$$
2. Statistical Model
The given Sampling rule induces the following Stat Model
$$\Big(\mathcal{X}^{(n)}, \prod _{i=1}^{n}p(x_i|\theta), \theta \in \Theta\Big)$$
If the question is what you posted

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

no calculations are needed
For the first experiment you have
$$\Big(\mathcal{X}^{(n)}, \frac{e^{-n\lambda}\lambda^{\Sigma_i X_i}}{\Pi_iX_i!}, \lambda>0 \Big)$$
Where
a. $\mathcal{X}^{(n)}$ is the set of all possible n-tuples where any element can takes the values $\{0,1,2,3,...\}$
b. The function $\frac{e^{-n\lambda}\lambda^{\Sigma_i X_i}}{\Pi_iX_i!}$, given the knowledge of $\lambda$, assigns the probability to any possible n-tuple.
c. $\lambda$ is the parameter of interest

Similar reasoning for the 2nd experiment
