Inference for Monotone Functions Under Short- and Long-Range Dependence: Confidence Intervals and New Universal Limits

Abstract

We introduce new point-wise confidence interval estimates for monotone functions observed with additive, dependent noise. Our methodology applies to both short-and long-range dependence regimes for the errors. The interval estimates are obtained via the method of inversion of certain discrepancy statistics. This approach avoids the estimation of nuisance parameters such as the derivative of the unknown function, which previous methods are forced to deal with. The resulting estimates are therefore more accurate, stable, and widely applicable in practice under minimal assumptions on the trend and error structure. The dependence of the errors especially long-range dependence leads to new phenomena, where new universal limits based on convex minorant functionals of drifted fractional Brownian motion emerge. Some extensions to uniform confidence bands are also developed.

Nonparametric Shape-Restricted Regression

“…This methodology has been applied successfully in several monotone function estimation problems; see e.g., [15,13,60]. The method, and extensions thereof, also applies to both short-and long-range dependence regimes for the errors; see [7]. Also see [16,7] for illustrations and examples of the superior performance of the LR based method over plug-in methods for constructing confidence intervals for f (t), especially when the estimation of the derivative f (t) is difficult.…”

Bootstrap‐Based Inference for Cube Root Asymptotics

“…Seo and Otsu () gave conditions under which results of the form () can be obtained also when the data exhibit weak dependence; see also Bagchi, Banerjee, and Stoev (), and references therein. It seems plausible that a version of our procedure, implemented with a resampling procedure suitable for dependent data, can be shown to be consistent under similar conditions, but it is beyond the scope of this paper to substantiate that conjecture.…”

Limit Theory in Monotone Function Estimation

“…A direct approach similar to the one in [84] is used in [3], who consider Grenander-type estimators obtained as the greatest convex minorants (and derivatives thereof) of partial sum and empirical processes for independent, weakly dependent and long range dependent data. For other extensions of the direct approach to dependent data, see [4,21,5].…”

“…The size of the small blocks should be chosen large enough such that the integrals over the big blocks B i become independent, but small enough in comparison with the size of the big blocks so that the summation over the small blocks S i is negligible. One possibility, see [25], is to take big blocks of length n −1/3 (log n) 5 and small blocks of length n −1/3 (log n) 2 . When the contribution of the small blocks is negligible, then n 1/6 (J n − E[J n ]) is asymptotically equivalent to the contribution of the big blocks, which is a sum of independent centered variables.…”

“…Although [25] allows for some form of dependence, in the present paper we have mainly considered setups of independent data. Results about the pointwise limiting distribution for isotonic estimators in the regression setting with short and long range dependent errors can be found in [21,98], whereas [5] considers the construction of confidence intervals in a similar setting. Estimation of a monotone spectral density is considered in [4], for short memory linear processes and long memory Gaussian processes.…”