# What is the covariance between function and its argument?

For example, we have log-likelihood function $$\ell(\theta)$$ and maximum likelihood estimator $$\hat \theta_{ML}$$ obtained as $$\ell(\theta)' = 0$$. What would be $$Cov(\ell(\theta), \hat \theta_{ML}) ?$$

For simplicity let's assume that MLE is unbiased.

• should it be $\hat\theta_{ML}$ in the argument of $\ell$? – innisfree Mar 15 at 7:04
• Well, I think this is a function of the parameter $\theta$ – student Mar 15 at 7:06
• What are you viewing as random here? The data, the parameter, or both? – whuber Mar 15 at 14:39
• Data. Parameters are usually assumed to be fixed. – student Mar 15 at 14:46
• In a Bayesian setting parameters are treated as random. When the randomness occurs only through the data, I believe your question has no general answer: it must depend on the model as well as $\theta$ itself. In many cases this covariance must be tiny, but in some circumstances--such as when $\theta$ lies near the boundary of the parameter space--it cannot vanish. – whuber Mar 16 at 15:22