(untitled) (Statistics, Optimization & Information Computing, 2023)
The weighted median plays a central role in the least absolute deviations (LAD). We propose a nonlinear regression using (LAD). Our objective function f(a, l, s) is non-convex with respect to the parameters a, l, s, and is such that for each fixed l, s the minimizer of a→ f (a, l, s) is the weighted median med(x(l, s), w(l, s)) of a sequence x(l, s) endowed with the weights w(l, s) (all depend on l, s). We analyse and compare theoretically the minimizers of the function (a, l, s)→ f (a, l, s) and the surface (l, s) → f (med(x(l, s), w(l, s)), l, s). As a numerical application we propose to fit the daily infections of COVID 19 in China using Gaussian model. We derive confident interval for the daily infections from each local minimum.