Estimation of Lower Bound in Poisson Distribution
Example 4.3: Let \( X_1, X_2, \dots, X_n \) be a random sample from the Poisson distribution \( P(\theta) \) with parameter \( \theta \). Obtain the CRLBs and examine whether there exist such estimators that their variances attain these lower bounds for estimating the parametric functions \( g(\theta) \): \( \theta \) \( e^{-\theta} \) \( \theta^2 \) Cramer Rao lower bound - statsclick Solution: The joint density of sample observations is \[ f(X; \theta) = \frac{\exp(-n\theta) \theta^{\sum x_i}}{\prod_{i=1}^{n} (x_i!)} \] Taking the log-likelihood: \[ \log f(X; \theta) = -n\theta + \left(\sum x_i\right) \log \theta - \sum_{i=1}^{n} \log (x_i!) \] \[ \frac{\partial}{\partial \theta} \log f(X; \theta) = -n + \frac{\sum x_i}{\theta} \] \[ I_x(\theta) = E \left[ \left( \frac{\partial}{\partial \theta} \log f(X; \theta) \right)^2 \right] = E \left[ \frac{n}{\theta} (\bar{X} - \th...
