  Multi-stage (designed) procedures, obtained by splitting the sampling budget
suitably across stages, and designing the sampling at a particular stage based
on information about the parameter obtained from previous stages, are often
advantageous from the perspective of precise inference. We develop a generic
framework for M-estimation in a multistage setting and apply empirical process
techniques to develop limit theorems that describe the large sample behavior of
the resulting M-estimates. Applications to change-point estimation, inverse
isotonic regression, classification and mode estimation are provided: it is
typically seen that the multistage procedure accentuates the efficiency of the
M-estimates by accelerating the rate of convergence, relative to one-stage
procedures. The step-by-step process induces dependence across stages and
complicates the analysis in such problems, which we address through careful
conditioning arguments.
