Limit Theory in Monotone Function Estimation

Abstract

We give an overview of the different concepts and methods that are commonly used when studying the asymptotic properties of isotonic estimators. After introducing the inverse process, we illustrate its use in establishing weak convergence of the estimators at a fixed point and also weak convergence of global distances, such as the L pdistance and supremum distance. Furthermore, we discuss the developments on smooth isotonic estimation.

Local continuity of log-concave projection, with applications to estimation under model misspecification

“…Entry points to the field include the book by Groeneboom and Jongbloed [22], as well as the 2018 special issue of the journal Statistical Science (Samworth and Sen [40]). Other important shape-constrained problems that could benefit from the perspective taken in this work include decreasing density estimation (Grenander [20], Prakasa Rao [38], Groeneboom [21], Birgé [4], Jankowski [29]), isotonic regression (Barlow et al [2], Zhang [56], Chatterjee, Guntuboyina and Sen [7], Durot and Lopuhaä [16], Bellec [3], Yang and Barber [55], Han et al [25]) and convex regression (Hildreth [28], Seijo and Sen [44], Cai and Low [5], Guntuboyina and Sen [23], Han and Wellner [26], Fang and Guntuboyina [17]), among many others. In these cases, the analysis is likely to be more straightforward, since the canonical least squares/maximum likelihood estimator can be characterised as an L 2 -projection onto a convex set.…”

Local continuity of log-concave projection, with applications to estimation under model misspecification

“…Entry points to the field include the book by Groeneboom and Jongbloed (2014), as well as the 2018 special issue of the journal Statistical Science (Samworth and Sen, 2018). Other important shape-constrained problems that could benefit from the perspective taken in this work include decreasing density estimation (Grenander, 1956;Rao, 1969;Groeneboom, 1985;Birgé, 1989;Jankowski, 2014), isotonic regression (Brunk et al, 1972;Zhang, 2002;Chatterjee et al, 2015;Durot and Lopuhaä, 2018;Bellec, 2018;Yang and Barber, 2019; and convex regression (Hildreth, 1954;Seijo and Sen, 2011;Cai and Low, 2015;Han and Wellner, 2016b;Fang and Guntuboyina, 2019), among many others. In these cases, the analysis is likely to be more straightforward, since the canonical least squares/maximum likelihood estimator can be characterised as an L 2 -projection onto a convex set.…”

Quantile regression approach to conditional mode estimation

“…The second result (6) follows from the √ n-uniform consistency of the quantile regression estimator and the delta method. To prove the first result (5), we begin with expanding the objective function s x (τ ) and showing that τ x is an approximate minimizer of the sample average of kernel functions with a uniform kernel; see (13) in the proof. Since those kernel functions depend on the sample size n via the bandwidth h = h n , the result (5) does not follow from the general theorem, Theorem 1.1, in [27], which is a pioneering work on cube root asymptotic theory.…”

“…To the best of our knowledge, however, much less is known about the rate of convergence and (especially) limiting distribution for the L ∞ -distance in nonstandard nonparametric estimation problems than standard nonparametric estimation problems with Gaussian limits, and we believe that the problem is challenging. One exception is the work of [12], which derives the uniform rate of convergence and the limiting distribution of the L ∞ -distance for the Grenander [18] estimator (precisely speaking [12] cover more general Grenander-type estimators); see also the recent review article by [13]. Their argument depends substantially on the specific construction of the Grenander estimator and can not be directly extended to our estimator.…”