  This study investigates two-stage plans based on nonparametric procedures for
estimating an inverse regression function at a given point. Specifically,
isotonic regression is used at stage one to obtain an initial estimate followed
by another round of isotonic regression in the vicinity of this estimate at
stage two. It is shown that such two stage plans accelerate the convergence
rate of one-stage procedures and are superior to existing two-stage procedures
that use local parametric approximations at stage two when the available budget
is moderate and/or the regression function is 'ill-behaved'. Both Wald and
Likelihood Ratio type confidence intervals for the threshold value of interest
are investigated and the latter are recommended in applications due to their
simplicity and robustness. The developed plans are illustrated through a
comprehensive simulation study and an application to car fuel efficiency data.
