I am trying to calibrate a sensor and I have a data set consisting of several observations of a 3-dimensional vector $X_i$, with
$X_i=w_i + \epsilon_i$
where $w_i$ is the value that the sensor should measure (if uncalibrated but unaffected by noise) which is fixed for each $i$ and unobservable, and $\epsilon_i$ is gaussian error vector with zero mean and a known diagonal covariance matrix, which is the same for all $i$. The error vectors $\epsilon_i$ are uncorrelated for different $i$.
The response variable is
$Y_i=\|C(w_i-B)\|=1$ for all $i$.
$C$ is a 3x3 matrix that can be eigendecomposed by $C=R(\psi,\theta,\phi)\,E\,R(\psi,\theta,\phi)^T$, where $R$ is a rotation matrix parametrized by the three angles $\psi,\theta,\phi$. $E$ is a diagonal matrix where each entry in the diagonal is positive and bounded from above by a known value. $B$ is a 3x1 vector.
This is a typical nonlinear regression with additive measurement error, but without additional additive error in $Y$ (as opposed to the traditional setting of this problem that I have seen so far). Is it possible to obtain an unbiased estimate of the model parameters $p$ (the three attitude angles that parametrize $R$, the three diagonal entries of $E$ and the vector $B$) from the observations $X_i$?
Alternatively, is there a way to find out if solving the nonlinear least squares problem $\hat{p}= \min_p \sum_{i=1}^n (\|C(X_i-B)\|-1)^2$ yields an unbiased estimate of $p$?