Maximum Likelihood Estimator for Log Normal 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:

\[ \log f(X; \theta) = -n\theta + \left(\sum x_i\right) \log \theta - \sum_{i=1}^{n} \log (x_i!) \]

\[ \frac{\partial}{\partial \theta} \log f(X; \theta) = -n + \frac{\sum x_i}{\theta} \]

\[ I_x(\theta) = E \left[ \left( \frac{\partial}{\partial \theta} \log f(X; \theta) \right)^2 \right] = E \left[ \frac{n}{\theta} (\bar{X} - \theta) \right]^2 \]

\[ I_x(\theta) = \frac{n^2}{\theta^2} E(\bar{X} - \theta)^2 = \frac{n^2}{\theta^2} \theta = \frac{n\theta}{\theta^2} = \frac{n}{\theta} \]

\[ \text{Var}[s(X)] \geq \frac{1}{I_x(\theta)} = \frac{\theta}{n} \]

(ii) CRLB for estimating \( g(\theta) = e^{-\theta} \):

\[ \text{Var}[s(X)] \geq \frac{(g'(\theta))^2}{I_x(\theta)} = \frac{\theta e^{-2\theta}}{n} \]

\[ = \frac{\theta e^{2\theta}}{n} \]

Since this can't be expressed in the form \((*)\), there does not exist any unbiased estimator of \( e^{-\theta} \) whose variance attains the CRLB.

(iii) Here, \( \frac{\partial}{\partial \theta} \log f(X; \theta) \) cannot be expressed in the form of \( c(\theta)(s-\theta^2) \). Therefore, there does not exist any unbiased estimator of \( \theta^2 \). The CR lower bound for estimating \( \theta^2 \) is:

\[ \text{Var}[s(X)] \geq \frac{(2\theta)^2}{n/\theta} = \frac{4\theta^2}{n/\theta} = \frac{4\theta^3}{n} \]

\[ s(X) = \bar{X}^2 - \left(\frac{\bar{X}}{n}\right) \text{ is the UMVUE of } \theta^2. \]