Let's say we want to do an estimation using iid samples $X_i, i=1,2,3,..., N$ the following formula, $$\hat{X}_1 = \frac{1}{N}(\sum_i\sqrt{X_i})^2$$ square sum of square roots.
This form also seems reasonable:
$$\hat{X}_2 = \frac{1}{N^2}(\sum_i\sqrt{X_i})^2$$
What are some nice properties that you could think of for this estimator, in addition to being a biased estimator of mean?
PS.1 What I am asking is a little different from what is conventionally being followed, I know!
PS.2 Assume that all variables can take only positive values.
PS.3 On the importance of this question: In image processing, square root of an image, because of its noise's distribution is of special importance. Sometimes it is interesting average the squared values instead of the values themselves. I was curious to see what other people (like many statisticians at this forum) can think of, about properties of estimation using squared values; though this might come up in many different ways.
To avoid all the square roots, define $Y=\sqrt X$. Then
$$E\hat{X}_1 = E\frac{1}{N}\left(\sum_i{Y_i}\right)^2 = E\frac{1}{N} \left(\sum_i{Y_i^2}+\sum_i{Y_i\sum_{j\neq i}{Y_j}}\right) $$
$$ = \frac{1}{N}\left(\sum_iE{X_i}\right) + E\frac{1}{N}\left(\sum_i{Y_i\sum_{j\neq i}{Y_j}}\right) $$
In the second term, the double sum translates into $N(N-1)$ pairwise products of rv's: $E(Y_iY_j)$. But that does not mean that you can actullay multiply the two r.v.'s - they are identically distributed, but they are not the same r.v.'s. So we get
$$E\hat{X}_1=EX+\frac{1}{N} N(N-1)E(Y_iY_j)= EX+(N-1)\left(E(\sqrt X)\right)^2$$
and
$$E\hat{X}_2= EX/N+(1-1/N)\left(E(\sqrt X)\right)^2$$
...meaning, that the expected value of your estimator increases with the size of the sample: not good, to put it politely. You should try divide your estimator by $N^2$, and see what happens.