# Visualization of unbiasedness of high dimensional paramter estimates

Assume a statistical model $$f_{\theta}(X)$$ that allows to estimate a parameter vector $$\hat{\theta}\in \mathbb{R}^p$$ from data $$X$$ and assume that $$p$$ is high dimensional (you may assume something like $$p=500$$).

Since the properties of the estimator cannot be evaluated analytically, I want to show some of them via simulations. Here I draw $$N$$ times from my data generating process and try to re-estimate the parameters: $$\hat{\theta}_{j,i}$$ for the $$j$$th paramter from simulation $$i$$. My major focus is thereby on unbiasedness, i.e. I want to demonstrate that

$$E[\hat{\theta}]=\theta,$$ This would be a simple task with a low-dimensional parameter vector, where I simply could use boxplots or kernel-density estimates, showing that the distributions of the parameters are centered around the true value. However, this is not possible (or at least extremely confusing) with my high-dimensional parameter vector.

Another issue that complicates matters is given by the fact that the estimation procedure is constructed such that $$\sum_{j=1}^p \theta_{j,i} = \sum_{j=1}^p \hat{\theta}_{j,i}$$ for $$i=1,...,N$$ and it is therefore not an option for me to use a measure like $$N^{-1}\sum_{i=1}^N\sum_{j=1}^p( \theta_{j,i} -\hat{\theta}_{j,i}),$$ since it is zero by construction.

Do you have some ideas of how to show unbiasedness in such a setting?