This is actually an engineering problem. I have a linear regression of the following form: $$ Y = X(p(t),\dot p(t), q) \beta, $$ where the regressor $X$ is a nonlinear function of $p$, $\dot p(t) = \frac{dp}{dt}$ ($t$ is an independent variable) and $q$ and $Y$ denotes measurements. In fact, we design a rich set of $x(p, \dot p,q)$s and apply to our system and measure $y$s. $$ X = \begin{pmatrix} x_1(p_1,\dot p_1,q_1)\\ x_2(p_2,\dot p_2,q_2)\\ \vdots\\ x_N(p_N,\dot p_N,q_N)\end{pmatrix}, Y = \begin{pmatrix} y_1\\y_2\\ \vdots\\ y_N \end{pmatrix} $$ Given enough and rich measurement/observations the desired parameter set $\beta$ can be estimated using least squares method by $$ \beta = X^{*}(p, \dot p,q)Y, $$ with $X^{*}(p, \dot p,q)$ being the pseudo-inverse of $X$. Now imagine $p$ has some error which can not be quantified or measured.
What performance measure can show best the $\beta$ estimation divergence due to $p$ errors?
I will provide an example to clarify the final goal of this question:
If $p$ has $\%a$ error, the estimation error of $\beta$ will be $\%b$.
Please also note that since $x$ is a nonlinear function of $p$, one can not simply get error ratio $\delta p/p$ in $x$.
For this I came across sensitivity analysis and error/uncertainty propagation. The problem with sensitivity analysis is that I do not know how to use it for an optimization problem (least squares in my case). I would also like to carry out the analysis analytically and avoid numbers as much as possible. Furthermore, as far as I know error propagation is normally used for showing measurement errors effect on the system output. Therefore, since $p$ is not measured, I am not sure if I can use error propagation. Thank you in advance for comments and helps.
Update: After doing some research, I found out once you consider regressor error, the problem becomes total least squares.