59 research outputs found
Nonparametric confidence intervals for monotone functions
We study nonparametric isotonic confidence intervals for monotone functions.
In Banerjee and Wellner (2001) pointwise confidence intervals, based on
likelihood ratio tests for the restricted and unrestricted MLE in the current
status model, are introduced. We extend the method to the treatment of other
models with monotone functions, and demonstrate our method by a new proof of
the results in Banerjee and Wellner (2001) and also by constructing confidence
intervals for monotone densities, for which still theory had to be developed.
For the latter model we prove that the limit distribution of the LR test under
the null hypothesis is the same as in the current status model. We compare the
confidence intervals, so obtained, with confidence intervals using the smoothed
maximum likelihood estimator (SMLE), using bootstrap methods. The
`Lagrange-modified' cusum diagrams, developed here, are an essential tool both
for the computation of the restricted MLEs and for the development of the
theory for the confidence intervals, based on the LR tests.Comment: 31 pages, 13 figure
Smooth plug-in inverse estimators in the current status continuous mark model
We consider the problem of estimating the joint distribution function of the
event time and a continuous mark variable when the event time is subject to
interval censoring case 1 and the continuous mark variable is only observed in
case the event occurred before the time of inspection. The nonparametric
maximum likelihood estimator in this model is known to be inconsistent. We
study two alternative smooth estimators, based on the explicit (inverse)
expression of the distribution function of interest in terms of the density of
the observable vector. We derive the pointwise asymptotic distribution of both
estimators.Comment: 29 pages, 12 figure
Maximum smoothed likelihood estimation and smoothed maximum likelihood estimation in the current status model
We consider the problem of estimating the distribution function, the density
and the hazard rate of the (unobservable) event time in the current status
model. A well studied and natural nonparametric estimator for the distribution
function in this model is the nonparametric maximum likelihood estimator (MLE).
We study two alternative methods for the estimation of the distribution
function, assuming some smoothness of the event time distribution. The first
estimator is based on a maximum smoothed likelihood approach. The second method
is based on smoothing the (discrete) MLE of the distribution function. These
estimators can be used to estimate the density and hazard rate of the event
time distribution based on the plug-in principle.Comment: Published in at http://dx.doi.org/10.1214/09-AOS721 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Testing monotonicity of a hazard: asymptotic distribution theory
Two new test statistics are introduced to test the null hypotheses that the
sampling distribution has an increasing hazard rate on a specified interval
[0,a]. These statistics are empirical L_1-type distances between the isotonic
estimates, which use the monotonicity constraint, and either the empirical
distribution function or the empirical cumulative hazard. They measure the
excursions of the empirical estimates with respect to the isotonic estimates,
due to local non-monotonicity. Asymptotic normality of the test statistics, if
the hazard is strictly increasing on [0,a], is established under mild
conditions. This is done by first approximating the global empirical distance
by an distance with respect to the underlying distribution function. The
resulting integral is treated as sum of increasingly many local integrals to
which a CLT can be applied. The behavior of the local integrals is determined
by a canonical process: the difference between the stochastic process x ->
W(x)+x^2 where W is standard two-sided Brownian Motion, and its greatest convex
minorant.Comment: 28 pages, 1 figur
Confidence intervals in monotone regression
We construct bootstrap confidence intervals for a monotone regression
function. It has been shown that the ordinary nonparametric bootstrap, based on
the nonparametric least squares estimator (LSE) is inconsistent in
this situation. We show, however, that a consistent bootstrap can be based on
the smoothed , to be called the SLSE (Smoothed Least Squares
Estimator).
The asymptotic pointwise distribution of the SLSE is derived. The confidence
intervals, based on the smoothed bootstrap, are compared to intervals based on
the (not necessarily monotone) Nadaraya Watson estimator and the effect of
Studentization is investigated. We also give a method for automatic bandwidth
choice, correcting work in Sen and Xu (2015). The procedure is illustrated
using a well known dataset related to climate change.Comment: 22 pages, 8 figure
A maximum smoothed likelihood estimator in the current status continuous mark model
We consider the problem of estimating the joint distribution function of the
event time and a continuous mark variable based on censored data. More
specifically, the event time is subject to current status censoring and the
continuous mark is only observed in case inspection takes place after the event
time. The nonparametric maximum likelihood estimator (MLE) in this model is
known to be inconsistent. We propose and study an alternative likelihood based
estimator, maximizing a smoothed log-likelihood, hence called a maximum
smoothed likelihood estimator (MSLE). This estimator is shown to be well
defined and consistent, and a simple algorithm is described that can be used to
compute it. The MSLE is compared with other estimators in a small simulation
study.Comment: 24 pages, 6 figure
Estimating a concave distribution function from data corrupted with additive noise
We consider two nonparametric procedures for estimating a concave
distribution function based on data corrupted with additive noise generated by
a bounded decreasing density on . For the maximum likelihood (ML)
estimator and least squares (LS) estimator, we state qualitative properties,
prove consistency and propose a computational algorithm. For the LS estimator
and its derivative, we also derive the pointwise asymptotic distribution.
Moreover, the rate achieved by the LS estimator is shown to be
minimax for estimating the distribution function at a fixed point.Comment: Published in at http://dx.doi.org/10.1214/07-AOS579 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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