Is there an efficient estimator for $\theta$?

Question

Given a simple random sampling with density function:

$$f_{\theta}(x) = \frac{\ln(\theta)\theta^x}{\theta - 1} I_{(0,1)}(x)\,, \quad\theta \gt1$$

Is there an efficient estimator for $\theta$?

I know that the efficiency of an estimator is the ratio

$$ef_T(\theta) = \frac{(d'(\theta))^2}{I(\theta)Var[T]}$$

And the estimator is efficient if $ef_T(\theta) = 1$

I have tried to calculate Cramér–Rao bound since we have a simple random sample of size n the fisher information can be written as:

$I(\theta)= n I^*(\theta) = nE\left[(\frac{\partial }{\partial \theta} \ln f_\theta(x_i))^2\right] \Rightarrow \ln f_\theta(x) = \ln \frac{\ln(\theta)\theta^x}{\theta - 1} = \ln^2(\theta) + x\ln(\theta)-\ln(\theta - 1)$

And, $$\frac{\partial }{\partial \theta} \ln f_\theta(x) = \frac{2\ln(\theta)+x}{\theta} - \frac{1}{\theta - 1}$$

Hence $$I^*(\theta) = E\left[\left(\frac{2\ln(\theta)+x}{\theta} - \frac{1}{\theta - 1}\right)\right]$$

But at this point I get stuck as I get a complicated integral.

If I have not made mistakes, am I taking a good path or is there a better one? I would appreciate so much any hints or suggestions

Answer 1

Joint density of $X_1,X_2,\ldots,X_n$ at $\boldsymbol x=(x_1,x_2,\ldots,x_n)$ is

$$f_{\theta}(\boldsymbol x)=\left(\frac{\ln\theta}{\theta-1}\right)^n \theta^{\sum_{i=1}^n x_i}\,\mathbf1_{(0,1)^n}(\boldsymbol x)\quad,\,\theta>1$$

Therefore,

$$\ln f_{\theta}(\boldsymbol x)=n\ln\left(\frac{\ln\theta}{\theta-1}\right)+\ln\theta\sum_{i=1}^n x_i+\ln(\mathbf1_{(0,1)^n}(\boldsymbol x))$$

And

\begin{align} \frac{\partial}{\partial\theta}\ln f_{\theta}(\boldsymbol x)&=\frac{n}{\ln\theta}\left(\frac1{\theta}-\frac{\ln\theta}{\theta-1}\right)+\frac1{\theta}\sum_{i=1}^n x_i \\&=n\left(\frac1{\theta\ln\theta}-\frac1{\theta-1}\right)+\frac1{\theta}\sum_{i=1}^n x_i \\&=\frac{n}{\theta}\left[\frac1n\sum_{i=1}^n x_i-\left(\frac{\theta}{\theta-1}-\frac1{\ln\theta}\right)\right] \end{align}

So the score function is of the form

$$\frac{\partial}{\partial\theta}\ln f_{\theta}(\boldsymbol x)=k(\theta)(T(\boldsymbol x)-g(\theta))$$

The above is the equality condition of Cramér-Rao inequality.

This suggests that Cramér-Rao lower bound is attained only for functions like $g(\theta)=\frac{\theta}{\theta-1}-\frac1{\ln\theta}$ or constant multiples of $g(\theta)$. So by your definition of efficiency, efficient estimators exist only for these functions of $\theta$.