2,893 research outputs found
Marshall's lemma for convex density estimation
No full textMarshall's [Nonparametric Techniques in Statistical Inference (1970)
174--176] lemma is an analytical result which implies --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 .Comment: Published at http://dx.doi.org/10.1214/074921707000000292 in the IMS
Lecture Notes Monograph Series
(http://www.imstat.org/publications/lecnotes.htm) by the Institute of
Mathematical Statistics (http://www.imstat.org
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A Conversation with Jon Wellner
No full textJon August Wellner was born in Portland, Oregon, in August 1945. He received his Bachelor’s degree from the University of Idaho in 1968 and his PhD degree from the University of Washington in 1975. From 1975 until 1983, he was an Assistant Professor and Associate Professor at the University of Rochester. In 1983, he returned to the University of Washington, and has remained at the UW as a faculty member since that time. Over the course of a long and distinguished career, Jon has made seminal contributions to a variety of areas including empirical processes, semiparametric theory and shape-constrained inference, and has co-authored a number of extremely influential books. He has been honored as the Le Cam lecturer by both the IMS (2015) and the French Statistical Society (2017). He is a Fellow of the IMS, the ASA and the AAAS, and an elected member of the International Statistical Institute. He has served as co-Editor of The Annals of Statistics (2001–2003) and Editor of Statistical Science (2010–2013), and President of IMS (2016–2017). In 2010, he was made a Knight of the Order of the Netherlands Lion. In his free time, Jon enjoys mountain climbing and backcountry skiing in the Cascades and British Columbia
Estimation of a k-monotone density: Limit distribution theory and the spline connection
No full texthttp://projecteuclid.org/euclid.aos/1201012971We study the asymptotic behavior of the Maximum Likelihood and Least Squares estimators of a monotone density at a fixed point when . In \mycite{balabwell:04a}, it was proved that both estimators exist and are splines of degree with simple knots. These knots, which are also the jump points of the st derivative of the estimators, cluster around a point under the assumption that has a continuous -th derivative in a neighborhood of and . If and are two successive knots, we prove that the random ``gap'' \ is for any if a conjecture about the upper bound on the error in a particular Hermite interpolation via odd-degree splines holds. Based on the order of the gap, the asymptotic distribution of the Maximum Likelihood and Least Squares estimators can be established. We find that the th derivative of the estimators at converges at the rate for . The limiting distribution depends on an almost surely uniquely defined stochastic process that stays above (below) the -fold integral of Brownian motion plus a deterministic drift, when is even (odd).ou
Limit distribution theory for maximum likelihood estimation of a log-concave density
No full textWe find limiting distributions of the nonparametric maximum likelihood
estimator (MLE) of a log-concave density, that is, a density of the form
where is a concave function on .
The pointwise limiting distributions depend on the second and third derivatives
at 0 of , the "lower invelope" of an integrated Brownian motion process
minus a drift term depending on the number of vanishing derivatives of
at the point of interest. We also establish the limiting
distribution of the resulting estimator of the mode and establish a
new local asymptotic minimax lower bound which shows the optimality of our mode
estimator in terms of both rate of convergence and dependence of constants on
population values.Comment: Published in at http://dx.doi.org/10.1214/08-AOS609 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Keynote: Jon Gertner
No full textThe symposium will start on the evening of April 16 with a keynote address by Jon Gertner. Jon is a journalist, historian, and feature writer for The New York Times Magazine
as well as the author of the NYTimes bestseller, The Idea Factory. His address will focus on the issue of intellectual property and the ethical questions around the huge amount of human-generated content that large language models use as they are developed
Jon Mirande eta ironia
No full textLa ironía es un elemento que ha ido siempre unido a la poesía, y especialmente a la poesía moderna.Tras un pequeño repaso a esta en diferentes épocas, se pasa a describir las tres diferentes ironías de Jon Mirande: la intelectual, la social y la filosófica. Todo ello acompañado de ejemplosIrony is an element that has always been united to poetry, and especially to modern poetry. After a small revision of irony in different eras, the author then describes the three different ironies of Jon Mirande: intellectual, social and philosophical irony. All this illustrated with example
A New Approach to Tests and Confidence Bands for Distribution Functions
No full textWe introduce new goodness-of-fit tests and corresponding confidence bands for
distribution functions. They are inspired by multi-scale methods of testing and
based on refined laws of the iterated logarithm for the normalized uniform
empirical process and its natural limiting
process, the normalized Brownian bridge process .
The new tests and confidence bands refine the procedures of Berk and Jones
(1979) and Owen (1995). Roughly speaking, the high power and accuracy of the
latter methods in the tail regions of distributions are essentially preserved
while gaining considerably in the central region. The goodness-of-fit tests
perform well in signal detection problems involving sparsity, as in Ingster
(1997), Donoho and Jin (2004) and Jager and Wellner (2007), but also under
contiguous alternatives. Our analysis of the confidence bands sheds new light
on the influence of the underlying -divergences
Jon Pineda, 32nd Annual ODU Literary Festival
No full textJon Pineda is the author of The Translator\u27s Diary, winner of the Green Rose Prize for Poetry, and BIrthmark, winner of the Crab Orchard Award Series in Poetry Open Competition. His memoir, Sleep in Me, is forthcoming in 2010 from the University of Nebraska Press. He teaches in the low-residency MFA program at Queens University of Charlotte
Likelihood Ratio, Score, and Wald Statistics in Models with Monotone Functions: Some Comparisons
No full textBanerjee and Wellner (2001) introduced and studied the likelihood ratio statistic for testing the hypothesis that a monotone function takes on a xed value at a xed point in the context of estimating the distribution function of the survival time in the interval censoring model. In this paper we continue to use the interval censoring model as a simple \test problem". We introduce three natural \score statistics" for the same testing problem studied in Banerjee and Wellner (2001) which have natural intepretations in terms of certain (weighted) L 2 distances. We compare these new test statistics with an analogue of the classical Wald statistic and the likelihood ratio statistic introduced in Banerjee and Wellner (2001). We rst establish limiting distribution theory of the statistics under the null hypothesis, and discuss calculation of the relevant critical points for the test statistics. We then establish the limiting behavior of all ve statistics under both xed and local alternatives. Although the asymptotic theory does allow some qualitative conclusions, it unfortunately does not (yet) lead to explicit quantitative comparison of the power behavior of the ve dierent tests. We therefore also compare the power of ve dierent statistics via a limited Monte-Carlo study. Our preliminary conclusion is that one of the three score tests seem to slightly dominate all the other test statistics, including the likelihood ratio and the Wald statistics
Estimation of a k-monotone density, part 2: algorithms for computation and numerical results
No full textThe iterative (2k − 1)−spline algorithm is an extension of the iterative cubic spline algorithm developed and used by Groeneboom, Jongbloed, and Wellner (2001b) to compute the Least Squares Estimator (LSE) of a nonincreasing and convex density on (0, ∞), and to find an approximation of the “invelope ” of the integrated two-sided Brownian motion+t 4 that is involved in the limiting distribution of both the Maximum Likelihood Estimator (MLE) and the LSE (Groeneboom, Jongbloed, and Wellner (2001a)). The iterative (2k − 1) − spline algorithm was developed to compute the LSE of a k-monotone density on (0, ∞) for any integer k>2, and also to calculate an approximation of the envelopes ( “ invelopes”) of the (k − 1)-fold integral of two-sided Brownian motion + (k!/(2k)!) t 2k when k is odd (even) on a finite interval [−c, c] for some fixed c>0. Existence and uniqueness of the latter processes are the subject of Balabdaoui and Wellner (2004c). To compute the MLE of a k-monotone density, another variation of the algorithm involving quadratic approximation is described. This algorithm involves the computation of a spline of degree k − 1 instead of a spline of degree 2k − 1. The principles of both algorithms are explained in detail. We also give several applications to real and artificial data
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