On the theory of mortality measurement

By U. Grenander. 1956

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Convex-constrained Sparse Additive Modeling and Its Extensions (2017, background): “For instance, many functions that arise in econometrics (Varian, 1982, 1984), actuarial science (Grenander, 1956), statistics (Groeneboom & Jongbloed, 2014), and optimization (Hannah & Dunson, 2013; Hannah et al.”
Recent Progress in Log-Concave Density Estimation (2017, background): “Shape-constrained density estimation has a long history, dating back at least as far as Grenander (1956), who studied the maximum likelihood estimator of a decreasing density on the non-negative half-line.”
Inference and Modeling with Log-concave Distributions (2009, methodology): “In the context of density estimation, Grenander (1956) derived the nonparametric maximum likelihood Guenther Walther is Associate Professor, Department of Statistics, Stanford University, 390 Serra Mall, Stanford, California 94305, USA (e-mail: gwalther@stanford.edu). estimator of a density…”
Smoothed log-concave maximum likelihood estimation with applications (2011, background): “The general idea dates back to Grenander (1956), who derived the maximum likelihood estimator of a decreasing density on [0,∞).”
On risk bounds in isotonic and other shape restricted regression problems (2013, methodology): “mmonly used estimator for this problemis the monotone LSEdeﬁned as any nondecreasing function fˆ ls on Rfor which (fˆls(x1),...,fˆls(xn))=θˆ(Y;M). This estimator was proposed by [7] and [2]; also see [16] for the related problem of estimating a nonincreasing density. Note that θˆ(Y ;M) can be comput”
Risk bounds in isotonic regression (2002, methodology): “The LSE (1.2) and related methods for estimating a monotone regression or density function f (·) were proposed by Ayer, Brunk, Ewing, Reid and Silverman (1955), van Eeden (1956) and Grenander (1956).”
Divide and conquer in nonstandard problems and the super-efficiency phenomenon (2016, background): “Such estimators have been studied in a variety of interesting statistical contexts, for example, isotonic regression [8, 9], where a monotone regression function is estimated via least squares under that shape constraint, the current status model (and extensions thereof) [6, 17], where the distribut”
Nonparametric Least Squares Estimation of a Multivariate Convex Regression Function (2010, background): “Such problems have a long history in the statistical literature with seminal papers like Brunk (1955), Grenander (1956) and Hildreth (1954) written more than 50 years ago, albeit in simpler settings.”
A general asymptotic scheme for inference under order restrictions (2006, background): “etric maximum likelihood estimate (NPMLE) fˆ=argmax ( Yn i=1 z(ti):zincreasing,z≥0 and Z J z(u)du=1 ) (5) for independentdata can be written as fˆ=TJ(xn)′, where xn is the empirical distribution; see [20]. Asymptotic properties of fˆhave been obtained in [39] and [21]; see also [46]. In particular, ”
APPROXIMATION BY LOG-CONCAVE DISTRIBUTIONS, WITH APPLICATIONS TO REGRESSION (2010, background): “With these goals in mind, let us tell a short success story about Grenan-der’s estimator [Grenander (1956)], also a key example of Patilea (2001): let F mon be the class of all nonincreasing and left-continuous probability densities on X = (0 , ∞ ).”
Estimation of a convex function: characterizations and asymptotic theory. (2001, unknown): “In the related problem of estimating a monotone density, the least squares estimator and the MLE coincide: the least squares estimator is identical to the MLE found by Grenander (1956).”
Nonparametric Shape-Restricted Regression (2017, background): “The first result of this type was derived in [112] for the Grenander estimator — the maximum likelihood estimator of a nonincreasing density in [0,∞) (see [53]).”
Global risk bounds and adaptation in univariate convex regression (2013, background): “For example, in the related problem of monotone function estimation, which has an enormous literature (see e.g., Grenander (1956), Birgé (1989), Zhang (2002) and the references therein), the only result on adaptive global behavior of the LSE is found in Groeneboom and Pyke (1983); also see van de…”