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 $\lambda\le 10$.
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
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