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Asymptotic normality of the Lkerror of the Grenander estimator
 Ann. Statist
, 2005
"... We investigate the limit behavior of the Lkdistance between a decreasing density f and its nonparametric maximum likelihood estimator ˆ fn for k ≥ 1. Due to the inconsistency of ˆ fn at zero, the case k = 2.5 turns out to be a kind of transition point. We extend asymptotic normality of the L1dista ..."
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Cited by 13 (3 self)
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We investigate the limit behavior of the Lkdistance between a decreasing density f and its nonparametric maximum likelihood estimator ˆ fn for k ≥ 1. Due to the inconsistency of ˆ fn at zero, the case k = 2.5 turns out to be a kind of transition point. We extend asymptotic normality of the L1
CONVERGENCE OF LINEAR FUNCTIONALS OF THE GRENANDER ESTIMATOR UNDER MISSPECIFICATION
"... Abstract. 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 t ..."
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Cited by 5 (1 self)
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Abstract. 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
Solving geometric problems with the rotating calipers
, 1983
"... Shamos [1] recently showed that the diameter of a convex nsided polygon could be computed in O(n) time using a very elegant and simple procedure which resembles rotating a set of calipers around the polygon once. In this paper we show that this simple idea can be generalized in two ways: several se ..."
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Cited by 147 (11 self)
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the critical support lines, in turn, leads to obtaining solutions to several additional problems concerned with visibility, collision, avoidance, range fitting, linear separability, and computing the Grenander distance between sets. 1.
Object Restoration Through Dynamic Polygons
 J. R. Statist. Soc. B
, 1995
"... this article we will try to recover the boundary of a single object starting from a noisy version of it, relying on the prior information that the boundary is a closed and nonintersecting curve in the plane, that is, fitting a nonintersecting polygon to the object. In doing this, we can use more i ..."
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Cited by 2 (0 self)
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information than that required by a simple edge detector (such as the gradient of greylevel intensity, whose nature is essentially local), because a polygon, besides this information, conveys the idea of connectivity between the pairs of its vertices (not depending on their distance), and the global idea
GoodnessofFit Test for Monotone Functions
"... ABSTRACT. In this article, we develop a test for the null hypothesis that a realvalued function belongs to a given parametric set against the nonparametric alternative that it is monotone, say decreasing. The method is described in a general model that covers the monotone density model, the monoto ..."
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, the monotone regression and the rightcensoring model with monotone hazard rate. The criterion for testing is an L pdistance between a Grenandertype nonparametric estimator and a parametric estimator computed under the null hypothesis. A normalized version of this distance is shown to have an asymptotic
i i
"... , and we can say the Campbell bandwidth is the minimum average bandwidth for encoding the process across all possible distortion levels. IX. CONCLUSION We have presented two new derivations of the coefficient rate introduced by Campbell. One derivation solidifies its interpretation as a coefficien ..."
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, and we can say the Campbell bandwidth is the minimum average bandwidth for encoding the process across all possible distortion levels. IX. CONCLUSION We have presented two new derivations of the coefficient rate introduced by Campbell. One derivation solidifies its interpretation as a coefficient rate, and shows that the spectral entropy of a random process is proportional to the logarithm of the equivalent bandwidth of the smallest frequency band that contains most of the energy. The second derivation implies that the number of samples of a particular component should be proportional to the variance of that component. We discussed the implications of the latter result for realizationadaptive source coding and provided a connection with the familiar reverse waterfilling result from rate distortion theory. From the coefficient rate, we defined a quantity called the Campbell bandwidth of a random process, and we contrasted Fourier bandwidth, Shannon bandwidth, and Campbell bandwidth. ACKNOWLEDGMENT The authors are indebted to the referees for their constructive comments and insights.