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The problem asks to construct an asymptotic test of level \( \alpha \) for the null hypothesis \( H_0: \theta \leq 1 \) versus the alternative hypothesis \( H_1: \theta > 1 \), given a statistical model where \( n \) independent and identically distributed (i.i.d.) random variables follow a Poisson distribution with parameter \( \theta > 0 \).

### Solution Outline:

1. **Likelihood Ratio Test (LRT)**: 
   Since the problem involves an asymptotic test, a common approach is to use the likelihood ratio test. The likelihood function for the Poisson-distributed data \( X_1, X_2, ..., X_n \) is given by:

   \[
   L(\theta) = \prod_{i=1}^n \frac{e^{-\theta} \theta^{x_i}}{x_i!} = e^{-n\theta} \theta^{\sum_{i=1}^n x_i} \prod_{i=1}^n \frac{1}{x_i!}
   \]

   The log-likelihood function is:

   \[
   \ell(\theta) = -n\theta + \left( \sum_{i=1}^n x_i \right) \ln(\theta)
   \]

2. **Maximum Likelihood Estimator (MLE)**: 
   The MLE of \( \theta \) is found by solving \( \frac{\partial \ell(\theta)}{\partial \theta} = 0 \). This leads to the MLE:

   \[
   \hat{\theta} = \frac{\sum_{i=1}^n X_i}{n}
   \]

3. **Forming the Test**: 
   The likelihood ratio test statistic is given by:

   \[
   \Lambda = \frac{\sup_{\theta \leq 1} L(\theta)}{L(\hat{\theta})}
   \]

   In practice, this can be approximated using the log-likelihood function and asymptotic results (e.g., using a chi-squared distribution under \( H_0 \)).

4. **Critical Region**: 
   For large \( n \), under \( H_0 \), the LRT statistic converges in distribution to a chi-square distribution with one degree of freedom. We can reject \( H_0 \) if the test statistic exceeds the critical value from the chi-square distribution corresponding to the level \( \alpha \).

### Tip:
Remember that in hypothesis testing for large samples, the LRT statistic often follows a chi-square distribution asymptotically, which allows us to find critical values for the test.

Would you like a more detailed explanation of a particular part of this process?

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Here are 5 related questions to further your understanding:

1. What is the likelihood function in general for a Poisson-distributed random variable?
2. How do you derive the MLE of a parameter in a Poisson model?
3. What are the assumptions behind using a likelihood ratio test?
4. How is the asymptotic distribution of the likelihood ratio test statistic determined?
5. How does one compute critical values for a hypothesis test at a given significance level?



Si consideri il modello statistico corrispondente a n variabili aleatorie i.i.d. con distribuzione di Poisson di parametro θ > 0. Costruire un test asintotico di livello α per H0: θ ≤ 1 contro H1: θ > 1.

Construct an asymptotic hypothesis test for the Poisson distribution with parameter θ > 0, testing H0: θ ≤ 1 against H1: θ > 1 using the likelihood ratio test. Explore the MLE, log-likelihood function, and chi-square distribution.

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