Maximum Likelihood Estimator for Log Normal Distribution

Lower bound and UMVUE estimation 

Let $X_1, X_2, \dots, X_n$ be a random sample from $U(0, \theta)$. We examine the application of the Cramér-Rao Lower Bound (CRLB) theorem for the estimation of $\theta$.

Solution:

The support of $U(0, \theta)$, which is $S(\theta) = \{ x \mid 0 < x < \theta \}$, depends on the parameter $\theta$. This violates the regularity conditions (ii) and (iv), meaning the CRLB theorem does not provide meaningful results.

Step 1: Computing Fisher Information

The probability density function is:

`f(x; \theta) = \frac{1}{\theta}`

Taking the logarithm:

`\log f(x; \theta) = -\log \theta`

Computing the derivative:

`\frac{\partial}{\partial \theta} \log f(x; \theta) = -\frac{1}{\theta}`

Thus, Fisher information is:

`E \left( \frac{\partial}{\partial \theta} \log f(x; \theta) \right)^2 = \frac{n}{\theta^2} = n I_X(\theta) = I_X(\theta)`

Step 2: CRLB Calculation

Let $T$ be an unbiased estimator of $\theta$. The CRLB is given by:

`V_{\theta}(T) \geq \frac{1}{I_X(\theta)} = \frac{\theta^2}{n}`

We now consider an unbiased estimator of $\theta$:

`2\bar{X} = \frac{2}{n} \sum X_i`

Its variance is:

`V(2\bar{X}) = \frac{\theta^2}{3n} < V_{\theta}(T) = \frac{\theta^2}{n}`

This contradicts the claim that CRLB($\theta$) is the lower bound on the variance of unbiased estimators of $\theta$.

Step 3: Another Unbiased Estimator

We also consider:

`T = \frac{n+1}{n} X_{(m)}`

where $X_{(m)} = \max(X_1, X_2, \dots, X_n)$ is the complete-sufficient statistic for $\theta$.

Its expected value is:

`E(T) = \theta`

and its variance is:

`V_{\theta}(T) = \frac{\theta^2}{n(n+2)} < \frac{\theta^2}{n}`

Since we have found unbiased estimators with variance smaller than the CRLB, the Cramér-Rao inequality does not hold in this case.