I'm trying to understand Epanechnikov's paper on nonparametric density estimation (https://search.proquest.com/docview/915574940?pq-origsite=gscholar&fromopenview=true) but I'm struggling with one of the equations he writes.
When he computes the "bias"-term $E[\Delta f_n(x_1,...,x_k)] = E[f_n(x_1,...,x_k)-f(x_1,...,x_k)]$ he obtains the asymptotic expression $$E[\Delta f_n(x_1,...,x_k)] \sim \frac12 \sum_{l = 1}^k \frac{\partial^2f(x_1,..,x_k)}{\partial x_l^2}h_l^2(n)$$ by taking the Taylor expansion of $f(x_1+h_1y_1,...,x_k+h_ky_k)$ around the point $(x_1,...,x_k)$.
But when I tried to verify the computation, I get the mixed partial derivatives $\frac12 \sum_{l = 1}^k\sum_{j = 1}^k \frac{\partial^2f(x_1,..,x_k)}{\partial x_l \partial x_j}h_l(n)h_j(n).$ Could anybody tell me what I am overlooking?
And also, I'm not quite sure what happens to the terms, where we take the first partial derivative (i.e. $- \sum_{l = 1}^k h_l(n)y_l \partial f(x_1,...,x_k)/\partial x_l$). Could anyone tell me what happens there?
Thank you. :-)