58 research outputs found
Estimation of a -monotone density: limit distribution theory and the spline connection
We study the asymptotic behavior of the Maximum Likelihood and Least Squares
Estimators of a -monotone density at a fixed point when .
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). Both the MLE and LSE are known to be splines of degree
with simple knots. Establishing the order of the random gap
, where denote two successive knots, is a key
ingredient of the proof of the main results. We show that this ``gap problem''
can be solved if a conjecture about the upper bound on the error in a
particular Hermite interpolation via odd-degree splines holds.Comment: Published in at http://dx.doi.org/10.1214/009053607000000262 the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Chernoff's density is log-concave
We show that the density of ,
sometimes known as Chernoff's density, is log-concave. We conjecture that
Chernoff's density is strongly log-concave or "super-Gaussian", and provide
evidence in support of the conjecture.Comment: Published in at http://dx.doi.org/10.3150/12-BEJ483 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
A Kiefer--Wolfowitz theorem for convex densities
Kiefer and Wolfowitz [Z. Wahrsch. Verw. Gebiete 34 (1976) 73--85] showed that
if is a strictly curved concave distribution function (corresponding to a
strictly monotone density ), then the Maximum Likelihood Estimator
, which is, in fact, the least concave majorant of the empirical
distribution function , differs from the empirical distribution
function in the uniform norm by no more than a constant times almost surely. We review their result and give an updated version of
their proof. We prove a comparable theorem for the class of distribution
functions with convex decreasing densities , but with the maximum
likelihood estimator of replaced by the least squares estimator
: if are sampled from a distribution function
with strictly convex density , then the least squares estimator
of and the empirical distribution function differ in the uniform norm by no more than a constant times almost surely. The proofs rely on bounds on the interpolation error
for complete spline interpolation due to Hall [J. Approximation Theory 1 (1968)
209--218], Hall and Meyer [J. Approximation Theory 16 (1976) 105--122],
building on earlier work by Birkhoff and de Boor [J. Math. Mech. 13 (1964)
827--835]. These results, which are crucial for the developments here, are all
nicely summarized and exposited in de Boor [A Practical Guide to Splines (2001)
Springer, New York].Comment: Published at http://dx.doi.org/10.1214/074921707000000256 in the IMS
Lecture Notes Monograph Series
(http://www.imstat.org/publications/lecnotes.htm) by the Institute of
Mathematical Statistics (http://www.imstat.org
Testing monotonicity via local least concave majorants
We propose a new testing procedure for detecting localized departures from
monotonicity of a signal embedded in white noise. In fact, we perform
simultaneously several tests that aim at detecting departures from concavity
for the integrated signal over various intervals of different sizes and
localizations. Each of these local tests relies on estimating the distance
between the restriction of the integrated signal to some interval and its least
concave majorant. Our test can be easily implemented and is proved to achieve
the optimal uniform separation rate simultaneously for a wide range of
H\"{o}lderian alternatives. Moreover, we show how this test can be extended to
a Gaussian regression framework with unknown variance. A simulation study
confirms the good performance of our procedure in practice.Comment: Published in at http://dx.doi.org/10.3150/12-BEJ496 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Least Squares estimation of two ordered monotone regression curves
In this paper, we consider the problem of finding the Least Squares
estimators of two isotonic regression curves and under
the additional constraint that they are ordered; e.g., . Given two sets of data points and observed at (the same) design points, the estimates of the true
curves are obtained by minimizing the weighted Least Squares criterion
over the class of pairs of vectors such that , , and
. The characterization of the estimators is
established. To compute these estimators, we use an iterative projected
subgradient algorithm, where the projection is performed with a "generalized"
pool-adjacent-violaters algorithm (PAVA), a byproduct of this work. Then, we
apply the estimation method to real data from mechanical engineering.Comment: 23 pages, 2 figures. Second revised version according to reviewer
comment
Limit distribution theory for maximum likelihood estimation of a log-concave density
We 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
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