# Kernel smoothing for Edgeworth expansion

Suppose I have an estimator which includes an indicator function in the objective function, then the objective function is not smooth. But if I want to approximate the behavior of this estimator in small samples I would need to have a certain number of higher order derivatives such that I can apply the usual approximation methods like Edgeworth, van Mises or Taylor expansions. I'm particularly interested in the Edgeworth expansion.

Would it be possible to smooth out somehow the objective function with a kernel such that I can apply the Edgeworth expansion? I thought of replacing the indicator with a symmetric bound kernel but I'm not sure of whether this would work. Any help would be appreciated!

Horowitz (1998) investigates whether the bootstrap can be used for asymptotic refinements of median regression. He faces the same problem as you given that the objective function has an embedded indicator $$\widehat{\beta} = min_{\beta} \frac{1}{n} \sum^{n}_{i=1} [q-1(u_i <0)]u_i$$ where the indicator is one for negative residuals. The problem is that when $u_i = 0$ we are at the kink of the check function $\rho (u_i) \equiv [q-1(u_i <0)]u_i$. Horowitz (1998) "smooths" this objective function by changing $\rho (u_i)$ to $$\rho^{S}(u_i) \equiv \left[2K\left(\frac{u_i}{h}\right) -1 \right]u_i$$ where $K(\cdot)$ is symmetric and bounded with $K \in [-1,1]$ with a differentiable function that satisfies $K(\nu)=0$ if $\nu \leq -1$ and $K(\nu) = 1$ if $\nu \geq 1$. In this context $K$ is similar to the integral of a kernel function but not a Kernel itself. Horowitz (1998) then applies the bootstrap for asymptotic refinements - which is the only difference to what you are planning to do but the reasoning is similar. For this purpose he needed the smooth objective function.