  Adaptive confidence intervals for regression functions are constructed under
shape constraints of monotonicity and convexity. A natural benchmark is
established for the minimum expected length of confidence intervals at a given
function in terms of an analytic quantity, the local modulus of continuity.
This bound depends not only on the function but also the assumed function
class. These benchmarks show that the constructed confidence intervals have
near minimum expected length for each individual function, while maintaining a
given coverage probability for functions within the class. Such adaptivity is
much stronger than adaptive minimaxity over a collection of large parameter
spaces.
