# What is the distribution of a chi-square minimizing function?

Suppose I have a set of $N$ experimental points of the form

\begin{equation} \{x_i, y_i, d_i\}, \end{equation}

where $i=1,...,N,$ and $d_i$ are errorbars for $y_i$. To fit the data, I minimize the reduced chi-square

\begin{equation} \chi^2(p) = \sum_{i=1}^N \frac{[y_i - f(x_i,p)]^2}{d_i^2}, \end{equation}

where $f(x,p)$ is a (generally non-linear) function parametrized by some parameter $p$ (there might be more than one parameter, but it doesn't really matter).

My question is: given the optimal parameter $p_0$, i.e. $\chi^2(p)$ is minimal at $p=p_0$, and assuming the $y_i$'s are independent and are Normally distributed, what can be said about the distribution of $f(x, p0)$?

Another term for your fitting procedure would be weighted non-linear least squares. The weights are a very minor complication. Fitting non-linear least squares is more tricky than ordinary least squares, but once the fitting is done the asymptotic ($N \to \infty$) distribution of the estimates is given by the same large-sample theory.