Summary: We study the problem of estimating a real valued function in \(\mathbb{R}^d\) under qualitative assumptions like monotonicity, symmetry, or modality. We consider density estimation in an i.i.d. setup. Assumptions on the underlying density \(f\) are formulated as shape restrictions. We assume that all density contour clusters \(\Gamma (\lambda) = \Gamma_f(\lambda) = \{x \in \mathbb{R}^d : f(x) \geq \lambda\}\), \(\lambda > 0\), lie in a given class \(\mathbb{C}\) of measurable subsets of \(\mathbb{R}^d\). Under this assumption the sets \(\Gamma(\lambda)\) are estimated by using the so-called excess mass approach. This leads to certain minumum volume sets (at random levels) as estimators for \(\Gamma (\lambda)\). A minimum volume set in \(\mathbb{C}\) at level \(\alpha\), by definition, has minimal Lebesgue measure among all sets in \(\mathbb{C}\) containing empirical mass not less than \(\alpha\). Using these estimators of \(\Gamma (\lambda)\) a density estimator called silhouette is constructed. The goal of this paper is to study the (asymptotic) behaviour of the silhouette.
The paper is organized as follows: In Section 2 estimators of the density contour clusters are given and the silhouette is defined. The connections of the silhouette to the Grenander estimator [U. Grenander, Skand. Aktuarietidskr. 1956, 125-153 (1957; Zbl 0077.33715)] and to the estimator of T. W. Sager [Lect. Notes Stat., Springer-Verlag 37, 69-90 (1986; Zbl 0609.62055)] are given in Section 2 also. Section 3 contains asymptotic results for the silhouette. Consistency results and rates of convergence in terms of \(L_1\)-distance are derived by means of empirical process theory. Section 4 contains some concluding remarks. Among others it is indicated there that the presented approach to density estimation can in principle also be applied to other (non-i.i.d.) situations such as nonparametric regression or spectral density estimation in time series analysis. All the proofs are given in Section 5.