Scale space consistency of piecewise constant least squares estimators -- another look at the regressogram

We study the asymptotic behavior of piecewise constant least squares regression estimates, when the number of partitions of the estimate is penalized. We show that the estimator is consistent in the relevant metric if the signal is in $L^2([0,1])$, the space of c\`{a}dl\`{a}g functions equipped with the Skorokhod metric or $C([0,1])$ equipped with the supremum metric. Moreover, we consider the family of estimates under a varying smoothing parameter, also called scale space. We prove convergence of the empirical scale space towards its deterministic target.Comment: Published at http://dx.doi.org/10.1214/074921707000000274 in the IMS Lecture Notes Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org

A local maximal inequality under uniform entropy

Abstract: We derive an upper bound for the mean of the supremum of the empirical process indexed by a class of functions that are known to have variance bounded by a small constant δ. The bound is expressed in the uniform entropy integral of the class at δ. The bound yields a rate of convergence of minimum contrast estimators when applied to the modulus of continuity of the contrast functions

Optimal Concentration of Information Content For Log-Concave Densities

An elementary proof is provided of sharp bounds for the varentropy of random vectors with log-concave densities, as well as for deviations of the information content from its mean. These bounds significantly improve on the bounds obtained by Bobkov and Madiman ({\it Ann. Probab.}, 39(4):1528--1543, 2011).Comment: 15 pages. Changes in v2: Remark 2.5 (due to C. Saroglou) added with more general sufficient conditions for equality in Theorem 2.3. Also some minor corrections and added reference

D-optimal designs via a cocktail algorithm

A fast new algorithm is proposed for numerical computation of (approximate) D-optimal designs. This "cocktail algorithm" extends the well-known vertex direction method (VDM; Fedorov 1972) and the multiplicative algorithm (Silvey, Titterington and Torsney, 1978), and shares their simplicity and monotonic convergence properties. Numerical examples show that the cocktail algorithm can lead to dramatically improved speed, sometimes by orders of magnitude, relative to either the multiplicative algorithm or the vertex exchange method (a variant of VDM). Key to the improved speed is a new nearest neighbor exchange strategy, which acts locally and complements the global effect of the multiplicative algorithm. Possible extensions to related problems such as nonparametric maximum likelihood estimation are mentioned.Comment: A number of changes after accounting for the referees' comments including new examples in Section 4 and more detailed explanations throughou

Sedimentary Signatures of Persistent Subglacial Meltwater Drainage From Thwaites Glacier, Antarctica

Subglacial meltwater drainage can enhance localized melting along grounding zones and beneath the ice shelves of marine-terminating glaciers. Efforts to constrain the evolution of subglacial hydrology and the resulting influence on ice stability in space and on decadal to millennial timescales are lacking. Here, we apply sedimentological, geochemical, and statistical methods to analyze sediment cores recovered offshore Thwaites Glacier, West Antarctica to reconstruct meltwater drainage activity through the pre-satellite era. We find evidence for a long-lived subglacial hydrologic system beneath Thwaites Glacier and indications that meltwater plumes are the primary mechanism of sedimentation seaward of the glacier today. Detailed core stratigraphy revealed through computed tomography scanning captures variability in drainage styles and suggests greater magnitudes of sediment-laden meltwater have been delivered to the ocean in recent centuries compared to the past several thousand years. Fundamental similarities between meltwater plume deposits offshore Thwaites Glacier and those described in association with other Antarctic glacial systems imply widespread and similar subglacial hydrologic processes that occur independently of subglacial geology. In the context of Holocene changes to the Thwaites Glacier margin, it is likely that subglacial drainage enhanced submarine melt along the grounding zone and amplified ice-shelf melt driven by oceanic processes, consistent with observations of other West Antarctic glaciers today. This study highlights the necessity of accounting for the influence of subglacial hydrology on grounding-zone and ice-shelf melt in projections of future behavior of the Thwaites Glacier ice margin and marine-based glaciers around the Antarctic continent