Confidence intervals from local minimums of objective function

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.