Let $X$, $Y$ be two independent normal random variable, say $N(0,1)$. And I want to estimate $\operatorname{cov}(X,Y)$. What is the distribution of the $\widehat{\operatorname{cov}}(X,Y)$?
- Of course, I can calculate the true value directly with $\operatorname{cov}(X,Y)=0$.
- Suppose I have data $X_1,\cdots,X_n \sim N(0,1)$ and $Y_1,\cdots,Y_n \sim \text{ (another independent) } N(0,1)$
- Consider the estimator $\widehat{\operatorname{cov}}(X,Y)=\frac{1}{n} \sum_{i=1}^n(x_i-\bar{x})(y_i-\bar{y})$
Then my question is what is the distribution of $\widehat{\operatorname{cov}}(X,Y)$?