For the sake of this question I want to focus on an unfair coin. Assuming we have a number of $i.i.d.$ samples $X_1, ..., X_n$ and a precision level requirement $\tau$. I search for the optimal estimator $\hat{P_n}$ minimizig the expression $P(|\hat{P_n} - {P(X_1 = 1)} |\geq \tau)$. Also I would like to know the convergence rate in this case.
Intiutively the law of large numbers should be the best choice, wich gives exponential convergence speed wich might not be enough for small $\tau$ and little $n$. So I just wanted to check if in this case we could possibly do any better. I guess the answer should not be due to the CLT, but I can't quite prove it.

For the sake of this question I want to focus on an unfair coin. Assuming we have a number of $i.i.d.$ samples $X_1, ..., X_n$ and a precision level requirement $\tau$. I search for the optimal estimator $\hat{P_n}$ minimizing the expression $P(|\hat{P_n} - {P(X_1 = 1)} |\geq \tau)$. Also I would like to know the convergence rate in this case.
Intiutively the law of large numbers should be the best choice, which gives exponential convergence speed. But that might not be enough for small $\tau$ and little $n$. So I just wanted to check if in this case we could possibly do any better. I guess the answer should not be due to the CLT, but I can't quite prove it.

For the sake of this question I want to focus on an unfair coin. Assuming we have a number of $iid.$ samples $X_1, ..., X_n$ and a precision level requirement $\tau$. I search for optimal the estimator $\hat{P_n}$ minimizig the expression $P(|\hat{P_n} - {P(X_1 = 1)} |\geq \tau)$. Also I would like to know the convergence rate in this case.
Intiutively the law of large numbers should be the best choice, with gives exponential convergence speed with might not be enough for small $\tau$ and little $n$. So I just wanted to check if in this case we could possibly do any better. I guess the answer should be no due to the CLT, but I cant quite proof it.

For the sake of this question I want to focus on an unfair coin. Assuming we have a number of $i.i.d.$ samples $X_1, ..., X_n$ and a precision level requirement $\tau$. I search for the optimal estimator $\hat{P_n}$ minimizig the expression $P(|\hat{P_n} - {P(X_1 = 1)} |\geq \tau)$. Also I would like to know the convergence rate in this case.
Intiutively the law of large numbers should be the best choice, wich gives exponential convergence speed wich might not be enough for small $\tau$ and little $n$. So I just wanted to check if in this case we could possibly do any better. I guess the answer should not be due to the CLT, but I can't quite prove it.