We roll a non-symmetric die. Let $X_n$ be the reulst of $n$-th roll. $$P(X_n = 6)= \frac{1}{6} + \varepsilon, \ P(X_n = 1) = \frac{1}{6} - \varepsilon, \ P(X_n=2) = ... = P(X_n = 5) = \frac{1}{6} $$
How can I find a consistent and unbiased estimator for parameter $\varepsilon$?
I thought I could introduce to other variables: $A_n =$number of $1$s in the first $n$ rolls and $B_n =$number of $6$s in the first $n$ rolls.
Then for $\overline{\alpha_n} = \frac{A_n+B_n}{n}$ we have:
$\mathbb{E}(\overline{\alpha_n}) = \frac{1}{n}( \mathbb{E}(A_n) + \mathbb{E}(B_n)) = \frac{1}{n} (n \cdot (\frac{1}{6} - \varepsilon) + n \cdot (\frac{1}{6} - \varepsilon)) = \frac{1}{6} - \varepsilon +\frac{1}{6} - \varepsilon =\frac{2}{3} - 2 \varepsilon$ and I should get $\epsilon$.
So I suppose I could instead write $\overline{\alpha_n} = \frac{2}{3}- \frac{A_n+B_n}{2n}$
But the variable $\overline{\alpha_n}$ isn't convergent almost surely or in probability to $\varepsilon$.
Could you tell me how to solve such problems?
I haven't been able to find anything on the web that would help me come up with estimators given specific kinds of events.
Let $Y_n$ be $+1$ if $X_n$ is $6$, $-1$ if $X_n$ is $1$ and $0$ in all other cases. Then \begin{align*} E(Y_n)&=1\left(\frac{1}{6}+e\right)+(-1)\left(\frac{1}{6}-e\right)=2e,\\ E(Y_n^2)&=1^2\left(\frac{1}{6}+e\right)+(-1)^2\left(\frac{1}{6}-e\right)=\frac{1}{3}, \end{align*} and $V(Y_n)=\frac{1}{3}-4e^2$. Then, the estimator $\theta_n=\frac{1}{2n}\sum_{i=1}^n Y_n$ is unbiased and consistent for $e$. Checking for unbiasedness entails a straightforward application of linearity and because the variance is finite, you can use Chebyshev's inequality to prove consistency.