# Unbiased estimation for constrained input-output data?

I am trying to understand the following. I have a series of measured ground true data $$Y = (y_1,y_2,\ldots,y_m)$$ and a series of estimated data $$\hat Y = (\hat y_1, \hat y_2,\ldots,\hat y_m)$$. Then, it is not uncommon to measure the error using Least Squares, in specific: $$Err = \sum_{i=1}^m (y_i-\hat y_i)^2$$ and one usually tries to minimize this. This data can be matrix-valued, i.e., $$Y,\hat Y$$ can be a series/vector of matrices. Now, I want to understand how to show that this estimator is unbiased given that this estimator is subject to some polynomial constraints $$p(y_i),p(\hat y_i)$$ on the sets $$Y,\hat Y$$ (the constaints are equivalent for both sets).

Usually, in a linear regression model, one writes $$\hat y_i = b_0 + b_1x_i$$ and needs to prove that the following for the expectation values of the parameters $$b_0,b_1$$:

$$E[b_0]=\tilde b_0$$ and $$E[b_1]=\tilde b_1$$

where $$\tilde b_0$$ and $$\tilde b_1$$ are the real parameters $$(y_i = \tilde b_0 + \tilde b_1 x_i + \epsilon_i)$$.

I guess my question is, what other ways are there to show that given $$Y$$ and $$\hat Y$$ the LSE is an unbiased estimator? Are there other ways to define this unbiased notion? References would be greatly appreciated.

The definition of bias is that the expectation of the estimate minus its true value. By definition, a proof that some estimate $$\hat Y$$ is unbiased must show that $$E[\hat Y]$$ equals $$Y$$.

However, one can also prove that an estimator is asymptotically unbiased, or that the bias approaches 0 as N (or T, perhaps, for panel data) approaches infinity. A good example for reference is the demonstration of Nickell bias (dynamic panel bias).

• Thanks, but the problem is the fact there might be constraints, right? It is far by trivial to show the unbiasedness. Apr 29, 2021 at 13:41
• I totally overlooked that part of your question. Can you give an example of such a constraint? It strikes me as unusual there are the same constraints on both the true data and estimates, so I'm trying to understand what that would mean. Apr 29, 2021 at 19:36
• Well, for example, if you are doing quantum mechanics there are constraints that the density matrices must be Hermitian matrices and have unit trace. Imagine you record data of density matrices using quantum state tomography (of course you use some norm for defining the error function). Then you want to show your estimator is unbiased taking into account these constraints. Apr 30, 2021 at 9:36