Suppose $X_1,X_2,...,X_n$ be a random sample from a normal population with mean $\theta$ and variance $\theta^2$.Find an unbiased estimator of $\theta^3$ based on both sample mean and sample variance.
I tried by getting $\bar X \sim N(\theta,\frac{\theta^2}{n})$. Now, $E(X-\theta)^3=(\theta)^3+3 \theta^3 \implies \theta^3+E(3\bar X S^2)$ as $S, \bar X$ are independent. Therefore , $E[(\bar X -\theta)^3-3 \bar X S^2]=\theta^3$ So, will it be an unbiased estimator?