How to estimate the axis of symmetry for an even function with error? I have a situation here, where, for an unknown $t$, and an unknown but nice* real function $f$, for which $x\rightarrow f(x-t)$ is even, I measure $f(x) + \epsilon_x$, where $\epsilon_x$ is some kind of more or less random error, hopefully small.
Now, I am looking for a sensible estimator for $t$.
I note that my situation is not well defined, but is there some theory I might look into that adresses this or similar situations?
*: Having all properties you desire  to get a meaningful result.

Since it looks like you are going to measure some physical quantity, it seems fair to assume that $x=t$ is a local maximum or minimum for $f$. This would not be the case with a function like $$f(x)=x^3{\sin{\frac{1}{x}}}$$ that is even, differentiable but doesn't exhibit a maximum or a minimum at $x=0$.
If you have a rough idea of where the simmetry axis $x=t$ is and how does $f$ behave in the neighborood of $t$, you could try to look for the zero of the first derivative, by measuring the ratios: $$\Delta y /\Delta x.$$
More precisely, you can choose some points $x_1,x_2,x_3,\dots$ around $t$ at some fixed distance $D$ and measure $$r_i=\dfrac{y(x_i+\frac{\delta}{2})-y(x_i-\frac{\delta}{2})}{\delta},$$
with $\delta \ll D$. If you expect only one zero for the derivative in the range of the $x_i$'s, then the $x_i$ with the smallest $r_i$ would give an extimation of $t$ ($r_i$ will be an increasing function of $x_i$). The uncertainty associated to this $t$ would be $\Delta t \sim D$, the details depending on how you do the measure and the analysis.

Maybe a more practical way would be to take several points around the expected $t$ and to perform a quadratic (or quartic) fit of the form: $$y_i = ax_i^2+bx_i+c,$$
obtaining $$t=-\frac{b}{2a}.$$
You can do this with the aid of a software like QtiPlot or, if you are very patient, also by hand. This method is applicable if $$|\dfrac{\text d y}{\text d x}|_{x=x_i}\sigma _{x_i}\ll \sigma _{y_i}.$$
If you can give a good extimation of the uncertainties $\sigma _{y_i}$, the fit will also give the uncertainties  for the parameters $a,b,c$ and you can test the goodness of the quadratic approximation with a $\chi ^2$ test.
