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We use p.values as a discrepancy criterion for identifying the threshold
level at which a regression function takes off from its baseline value . a
problem motivated by applications in toxicological and pharmacological
dose-response studies and environmental statistics. In this paper, we
study the problem in two different sampling settings: one where multiple
responses can be obtained at a number of different covariate-levels and
the other being the standard regression setting (limited number of
response values at each covariate). Our procedure involves testing the
hypothesis that the regression function is at its baseline at each
covariate value and then computing the (potentially approximate) p.value
of the test. An estimate of the threshold is obtained by fitting a
piecewise constant function with a single jump discontinuity (stump) to
these observed p.values (or their surrogates), as they behave in markedly
different ways on the two sides of the threshold. The estimate is shown
to be consistent and its finite sample properties are studied through a
simulation study. Our approach is computationally simple and extends to
the estimation of the baseline value of the regression function,
heteroscedastic errors and to time.series models. It is further
illustrated on some motivating real data applications. Some asymptotic
properties are also discussed.
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