Rates of consistency for nonparametric estimation of the mode in absence of smoothness assumptions

Abstract

Nonparametric estimation of the mode of a density or regression function via kernel methods is considered. It is shown that the rate of consistency of the mode estimator can be determined without the typical smoothness conditions. Only the uniform rate of the so-called stochastic part of the problem together with some mild conditions characterizing the shape or "acuteness" of the mode influence the rate of the mode estimator. In particular, outside the location of the mode, our assumptions do not even imply continuity. Overall, it turns out that the location of the mode can be estimated at a rate that is the better the "peakier" (and hence non-smooth) the mode is, while the contrary holds with estimation of the size of the mode.

Inference for the mode of a log-concave density

“…Second derivative type assumptions of this type are made in Parzen [1962], Has'minskii [1979], Eddy [1980], Donoho and Liu [1991], Romano [1988a,b], and Pfanzagl [2000]. Exceptions include Müller [1989], Ehm [1996], Herrmann and Ziegler [2004], Balabdaoui, Rufibach and Wellner [2009].…”

Minimax estimation of the mode of functional data

“…The properties of that kernel mode estimator (KME) ✓n of the actual mode ✓ have been thoroughly investigated and extended to a multivariate setting in the last decades. Some significant progress was achieved by the results of Vieu (1996), Abraham et al (2003), Herrmann and Ziegler (2004) and Shi et al (2008), who proved that the measurement error k ✓n ✓k (with respect to the Euclidean norm) attains an upper bound of the order ln(n) c 1 • n c 2 almost surely. The constants c 1 and c 2 typically depend on the dimension of the support of the density, the smoothness of the density at or around the mode and the steepness of it in a neighbourhood of the mode.…”

“…The constants c 1 and c 2 typically depend on the dimension of the support of the density, the smoothness of the density at or around the mode and the steepness of it in a neighbourhood of the mode. The paper of Herrmann and Ziegler (2004) examines the KME in the absence of any smoothness conditions. Eddy (1980) has shown for a univariate setting that by selecting a specific kernel and appropriately adjusting the bandwidth parameter one achieves E ✓n ✓ 2 2 O(n c ), where c is a smoothness parameter.…”

On inference for modes under long memory

“…Chan and Tong (2004) propose a test for the number of modes in one-dimensional marginal densities of stationary Markov processes. Selected references to mode estimation for iid data are, for instance, Parzen (1962), Chernoff (1964), Grenander (1965), Sager (1978), Eddy (1980), Romano (1988aRomano ( , 1988b, Tsybakov (1990), Kim (1994), Ziegler (2001), Walther (2002), Abraham, Biau, and Benoît (2003), Herrmann and Ziegler (2004), Klemelä (2005), Samworth and Wand (2010), Doss and Wellner (2019). References to the extended literature on kernel density estimation can be found in Silverman (1986), Wand and Jones (1995), Scott (2015), Chacón and Duong (2018), Gramacki (2018), and Ghosh (2018).…”