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 by the minimization
algorithm 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 derivative 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.
