Confidence Bands for Convex Median Curves using Sign-Tests

Suppose that one observes pairs (x1,Y1), (x2,Y2), ..., (xn,Yn), where x1 < x2 < ... < xn are fixed numbers while Y1, Y2, ..., Yn are independent random variables with unknown distributions. The only assumption is that Median(Yi) = f(xi) for some unknown convex or concave function f. We present a confidence band for this regression function f using suitable multiscale sign tests. While the exact computation of this band seems to require O(n4) steps, good approximations can be obtained in O(n2) steps. In addition the confidence band is shown to have desirable asymptotic properties as the sample size n tends to infinity.

Optimal Confidence Bands for Shape-Restricted Curves

Let Y be a stochastic process on [0,1] satisfying dY(t)=n 1/2 f(t)dt+dW(t) , where n≥1 is a given scale parameter (`sample size'), W is standard Brownian motion and f is an unknown function. Utilizing suitable multiscale tests, we construct confidence bands for f with guaranteed given coverage probability, assuming that f is isotonic or convex. These confidence bands are computationally feasible and shown to be asymptotically sharp optimal in an appropriate sense.

Confidence Bands for Isotonic Median Curves using Sign-Tests

Suppose that one observes independent random variables (X1, Y1), (X2, Y2), …, (Xn, Yn) in R2 with unknown distributions, except that Median(Yi | Xi = M(x) for some unknown isotonic function M. We describe an explicit algorithm for the computation of confidence bands for the median function M whose running time is of order O(n2). The bands rely on multiscale sign tests and are shown to have desirable asymptotic properties.