Find covariance of estimator and derivative of the log-likelihood function Problem:
Given and estimator $\hat k$. The estimation method is unknown (so, it can be max. likelihood, method of moments or another method), however, we know that $bias(\hat k) = 0$.
Let $L$ be the likelihood function and $\ell = ln L$.
Find $\Bbb Cov( \frac{d \ell}{d k},\hat k)$
My attempt:
$$\Bbb Cov( \frac{d \ell}{d k},\hat k) = E( \frac{d \ell}{d k} \cdot \hat k) - E( \frac{d \ell}{d k}) \cdot E( \hat k)$$
As unbiased $E( \hat k) = k$. Also,  $\frac{d \ell}{d k}$ is score function. We know that $E(s(k)) = 0$
Hence, we have
$$E( \frac{d \ell}{d k} \cdot \hat k) - E( \frac{d \ell}{d k}) \cdot E( \hat k) =E( \frac{d \ell}{d k} \cdot \hat k) $$
I decided to take $\hat k$ into the differential.
$$=E( \frac{d \ell  \cdot \hat k}{d k})$$
Since the expectation is linear, I can take the derivative out of it.
$$=  \frac{d E(\ell  \cdot \hat k)}{d k}$$
However, I am still stuck. Am I doing something wrong?
Can you give me some hint?
 A: Try things in the univariate case. I'll suppose $f(x; \theta)$ is the density and we're estimating $\theta$ with $\hat\theta$ which is unbiased but otherwise unknown. I'm changing to $\theta$ from $k$ just to be more consistent with the usual notation. I'll use $\nu$ as the dominating measure for these densities and $\mathcal X$ as the support.
We have$\newcommand{\E}{\operatorname{E}}$
$$
\E_X\left[\frac{\partial \log f(X; \theta)}{\partial \theta} \hat\theta(X)\right] = \int_{\mathcal X} \frac{1}{f(x; \theta)}\frac{\partial f(x; \theta)}{\partial \theta} \hat\theta(x) f(x;\theta)\,\text d\nu(x) \\
= \int_{\mathcal X}\frac{\partial f(x; \theta)}{\partial \theta} \hat\theta(x) \,\text d\nu(x) \\
= \frac{\partial }{\partial \theta}\int_{\mathcal X} \hat\theta(x)f(x; \theta) \,\text d\nu(x)
$$
assuming we're able to exchange integration and differentiation here (and this is an assumption; just because both are linear operators doesn't mean they commute). Can you see what this integral is?
