Marshall’s [Nonparametric Techniques in Statistical Inference (1970) 174–176] lemma is an analytical result which implies √ n–consistency of the distribution function corresponding to the Grenander [Skand. Aktuarietidskr. 39 (1956) 125–153] estimator of a non-decreasing probability density. The present paper derives analogous results for the setting of convex densities on [0,∞).

There is no fundamental difference between lag-windowing a correlation sequence and multiple-windowing a data sequence when the objective is to reduce the mean-squared error of a spectrum estimator. By analyzing the approximate low-rank factorization of a bandlimiting Toeplitz operator, we find that lag-windowed (or spectrally smoothed) spectrum estimators have multiple-data-windowed implementa...

In this paper we will consider the estimation of a monotone regression (or density) function in a fixed point by the least squares (Grenander) estimator. We will show that this estimator is fully adaptive, in the sense that the attained rate is given by a functional relation using the underlying function f 0 , and not by some smoothness parameter, and that this rate is optimal when considering ...

In this paper we will consider the estimation of a monotone regression (or density) function in a fixed point by the least squares (Grenander) estimator. We will show that this estimator is fully adaptive, in the sense that the attained rate is given by a functional relation using the underlying function f0, and not by some smoothness parameter, and that this rate is optimal when considering th...

Under the assumption that the true density is decreasing, it is well known that the Grenander estimator converges at rate n−1/3 if the true density is curved (Prakasa Rao, 1969) and at rate n−1/2 if the density is flat (Groeneboom and Pyke, 1983; Carolan and Dykstra, 1999). In the case that the true density is misspecified, the results of Patilea (2001) tell us that the global convergence rate ...

In literature several estimates have been proposed for unimodal densities. They are typically derived from the Grenander estimate (see Grenander (1956)) for decreasing densities which can be easily extended to the case of unimodal densities with a known mode at . The resulting estimator is the nonparametric maximum likelihood estimator (NPMLE). If the mode is unknown, the NPMLE does not exist a...

We aim at estimating a function λ : [0,1]→ R, subject to the constraint that it is decreasing (or increasing). We provide a unified approach for studying the Lp-loss of an estimator defined as the slope of a concave (or convex) approximation of an estimator of a primitive of λ, based on n observations. Our main task is to prove that the Lp-loss is asymptotically Gaussian with explicit (though u...

In this note we prove the following law of the iterated logarithm for the Grenander estimator of a monotone decreasing density: If f(t0) > 0, f'(t0) < 0, and f' is continuous in a neighborhood of t0, then [Formula: see text]almost surely where [Formula: see text]here [Formula: see text] is the two-sided Strassen limit set on [Formula: see text]. The proof relies on laws of the iterated logarith...