Estimation $\mu^2$ under certain conditions.

Question

Let $X_1,X_2,....,X_n$ be a random sample of size $n$ from a population with cdf $F()$. Let $E(X)=\mu$ exist. Then estimate $\mu^2$ unbiasedly for the following three cases:-

(i) $Var(X)=\sigma^2$ exists and is known.

(ii) $Var(X)=\sigma^2$ exists and is unknown.

(iii) $Var(X)$ does not exist.

Now for (i) & (ii) I have obtained the following solutions:-

(i) We define $$\bar X=\frac1n\sum_{i=1}^n X_i$$ The UE of $\mu^2$ is given as follows $$\bar X^2-\frac{\sigma^2}{n}$$ (ii) We define $$S^2=\frac{1}{n-1}\sum_{i=1}^n (X_i-\bar X)^2$$ The UE of $\mu^2$ is given as follows $$\bar X^2-\frac{S^2}{n}$$

For (iii) I need help. And please let me know if my answers thus far are correct. Thanks.

Answer 1

Even if the variance does not exist, you still have $E(X_i X_j) = \mu^2$ if $i \ne j$. Thus you could take $$ \frac{2}{n(n-1)} \sum_{i=2}^n \sum_{j = 1}^{i-1} X_i X_j = \frac{1}{n(n-1)} \left( \left(\sum_i X_i\right)^2 - \sum_i X_i^2 \right) $$