Given a set of data {U(s_{i}) = u(s_{i
}; p*)} corresponding to the delay differential equation
 u'(t ; p) = f(t, u(t ; p), u(a(t ; p)
; p) ; p) for t >= t_{0}(p),
 u(t ; p) = U(t ; p) for
t <= t_{0}(p),
the basic problem addressed here is that of calculating
the vector p*. (We also consider neutral differential equations.)
Most approaches to parameter estimation calculate p* by minimizing
a suitable objective function that is assumed to be sufficiently
smooth. In this paper, we use derivative discontinuity tracking theory
for delay differential equations to analyze how jumps can arise in the
derivatives of a natural objective function. These jumps can occur when
estimating parameters in lag functions and estimating the position of the
initial point, and as such are not expected to occur in parameter estimation
problems for ordinary differential equations.
