  When a parameter of interest is nondifferentiable in the probability, the
existing theory of semiparametric efficient estimation is not applicable, as it
does not have an influence function. Song (2014) recently developed a local
asymptotic minimax estimation theory for a parameter that is a
nondifferentiable transform of a regular parameter, where the nondifferentiable
transform is a composite map of a continuous piecewise linear map with a single
kink point and a translation-scale equivariant map. The contribution of this
paper is two fold. First, this paper extends the local asymptotic minimax
theory to nondifferentiable transforms that are a composite map of a Lipschitz
continuous map having a finite set of nondifferentiability points and a
translation-scale equivariant map. Second, this paper investigates the
discontinuity of the local asymptotic minimax risk in the true probability and
shows that the proposed estimator remains to be optimal even when the risk is
locally robustified not only over the scores at the true probability, but also
over the true probability itself. However, the local robustification does not
resolve the issue of discontinuity in the local asymptotic minimax risk.
