# Give an unbiased estimators for the third central moment of the underlying distribution.

Let $$X_{i}, i=1,\ldots, n$$ denote a random simple of size $$n$$ from a population with mean $$3$$. Assume that $$\hat{\theta}_{2}$$ is an unbiased estimator of $$\mathbb{E}[X^{2}]$$ and that $$\hat{\theta}_{3}$$ is an unbiased estimator for $$\mathbb{E}[X^3]$$. Give an unbiased estimator for the third central moment of the underlying distribution.

My approach:

I know that $$\mu_{3}=\mathbb{E}[(X-3)^{3}]=\mathbb{E}[X^{3}]-9\mathbb{E}[X^{2}]+54$$ so, we can see that $$\hat{\mu_{3}}=\hat{\theta}_{3}-9\hat{\theta}_{2}+54$$ is an unbiased estimator for $$\mu_{3}$$ of the underlying distribution.

Is it correct?

Yes it is unbiased and so correct. To check:

$$\mathbb E[\hat{\mu_{3}}]\\=\mathbb E[\hat{\theta}_{3}]-9\mathbb E[\hat{\theta}_{2}]+54 \\ = \mathbb E[X^3]-3\cdot3\cdot\mathbb E[X^2]+3 \cdot3^2\cdot\mathbb E[X] - 3^3 \\ = \mathbb E[(X-3)^3]$$

It is not the only unbiased estimator.

• Thank you so much for your verification. Nov 19 '20 at 11:17