Given a sample $X_1, \ldots, X_n$ that is geomterically distributed, i.e. $X_i$ ~ $G_{\theta}$, how does one have to choose the constants $a$ and $b$, such that the following estimating function $T$ is unbiased for $\tau = \frac{1}{\theta}$:
$$ T = b \sum_{i=1}^{n-1} X_i(aX_{i+1}-X_i) $$
Furthermore, is $T$ consistent for $\theta$?
To Show that $T$ is unbiased, I'd have to Show that $E[T] = \frac{1}{\theta}$ right?
Therefore I'd get:
$$ E[T] = b \sum_{i=1}^{n-1} E[X_i(aX_{i+1}-X_i)]$$
Is it allowed to assume Independence here or is there another way to determine $a$ and $b$?