Annual Report 4 , Winter 2013

We study eective inference strategies for monotone trend functions in the presence of short range and long range dependence. Isotonic regression, which does not require specication of tuning parameters, is invoked to estimate the monotone trend. The problem of constructing reliable condence intervals for the trend function is addressed via the use of appropriate discrepancy statistics which arise from (hypothesis) testing problems for the value of the trend function at a time point. The superiority of these discrepancy statistics is underscored by the fact that they do not involve the estimation of tricky nuisance parameters which existing methods are forced to deal with. While the discrepancy statistics are motivated from earlier work on isotonic regression with i.i.d. data, the dependence of errors, especially long-range dependence, introduces formidable challenges in this problem, not least that of studying the behavior of convex minorants of drifted Fractional Brownian motion. We introduce the family of universal limit distributions (indexed by the Hurst parameter, H, of Fractional Brownian motion) that serves as the possible limits of a new statistic, n , introduced in this paper, and tabulate the quantiles for several values of H. Inversion of the n statistic using the quantiles of the appropriate limit distribution as critical values are shown to deliver condence intervals for the monotone function that achieve both the desired coverage and accuracy and are wonderfully robust to the behavior of the function around the point of interest.