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

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

2006-09-13

arxiv.org/abs/math/0609347v2

IMS Lecture Notes Monograph Series 2007, Vol. 55, 65-84

Mathematics

Statistics Theory

Published at http://dx.doi.org/10.1214/074921707000000274 in the IMS Lecture Notes Monograph Series (http://www.imstat.org/p

Scientific paper

10.1214/074921707000000274

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.

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