A Kiefer-Wolfowitz type of result in a general setting, with an application to smooth monotone estimation

We consider Grenander type estimators for monotone functions $f$ in a very general setting, which includes estimation of monotone regression curves, monotone densities, and monotone failure rates. These estimators are defined as the left-hand slope of the least concave majorant $\hat{F}_n$ of a naive estimator $F_n$ of the integrated curve $F$ corresponding to $f$. We prove that the supremum distance between $\hat{F}_n$ and $F_n$ is of the order $O_p(n^{-1}\log n)^{2/(4-\tau)}$, for some $\tau\in[0,4)$ that characterizes the tail probabilities of an approximating process for $F_n$. In typical examples, the approximating process is Gaussian and $\tau=1$, in which case the convergence rate is $n^{-2/3}(\log n)^{2/3}$ is in the same spirit as the one obtained by Kiefer and Wolfowitz (1976) for the special case of estimating a decreasing density. We also obtain a similar result for the primitive of $F_n$, in which case $\tau=2$, leading to a faster rate $n^{-1}\log n$, also found by Wang and Woodfroofe (2007). As an application in our general setup, we show that a smoothed Grenander type estimator and its derivative are asymptotically equivalent to the ordinary kernel estimator and its derivative in first order

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

Estimating the joint distribution of independent categorical variables via model selection

Assume one observes independent categorical variables or, equivalently, one observes the corresponding multinomial variables. Estimating the distribution of the observed sequence amounts to estimating the expectation of the multinomial sequence. A new estimator for this mean is proposed that is nonparametric, non-asymptotic and implementable even for large sequences. It is a penalized least-squares estimator based on wavelets, with a penalization term inspired by papers of Birg\'{e} and Massart. The estimator is proved to satisfy an oracle inequality and to be adaptive in the minimax sense over a class of Besov bodies. The method is embedded in a general framework which allows us to recover also an existing method for segmentation. Beyond theoretical results, a simulation study is reported and an application on real data is provided.Comment: Published in at http://dx.doi.org/10.3150/08-BEJ155 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

On the Lp\mathbb{L}_p-error of monotonicity constrained estimators

We aim at estimating a function $\lambda:[0,1]\to \mathbb {R}$, subject to the constraint that it is decreasing (or increasing). We provide a unified approach for studying the $\mathbb {L}_p$-loss of an estimator defined as the slope of a concave (or convex) approximation of an estimator of a primitive of $\lambda$, based on $n$ observations. Our main task is to prove that the $\mathbb {L}_p$-loss is asymptotically Gaussian with explicit (though unknown) asymptotic mean and variance. We also prove that the local $\mathbb {L}_p$-risk at a fixed point and the global $\mathbb {L}_p$-risk are of order $n^{-p/3}$. Applying the results to the density and regression models, we recover and generalize known results about Grenander and Brunk estimators. Also, we obtain new results for the Huang--Wellner estimator of a monotone failure rate in the random censorship model, and for an estimator of the monotone intensity function of an inhomogeneous Poisson process.Comment: Published at http://dx.doi.org/10.1214/009053606000001497 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Development of a Time-Resolved Laser-Induced Fluorescence Technique for Nonperiodic Oscillations

Time-resolved measurements of ion dynamics could be key to understanding the physics of instabilities, electron transport, and erosion in Hall thrusters. Traditional measurements of the ion velocity distribution in Hall thrusters using laser-induced fluorescence (LIF) are time-averaged since lock-in amplifiers must average over a long time constant for a reasonable signal-to-noise ratio. Over about the past decade, at least four other time-resolved LIF techniques have been developed and applied to Hall thrusters or similar plasma devices. One limitation of these techniques is the implicit assumption of periodic oscillations in the averaging scheme. There is a need for a more general technique since Hall thrusters can operate with nonperiodic oscillations that vary unpredictably. This dissertation presents the development of a time-resolved LIF (TRLIF) technique that addresses this need. This system averages the signal using a combination of electronic filtering, phase-sensitive detection, and Fourier analysis. A transfer function is measured to map an input signal (such as discharge current) to an output signal (TRLIF signal). The implicit assumption of this technique is that the input is related to the output by a time-invariant linear system, a more general assumption than periodicity. The system was validated using a hollow cathode with both periodic and random discharge current oscillations. A series of benchmark tests was developed to validate the signal by verifying that it satisfies theoretical expectations. The first campaign with the H6 Hall thruster demonstrated signal recovery in both periodic and nonperiodic modes. Measurements of the evolution of the ion flow downstream show that kinematic compression explains the width of the ion velocity distribution only at certain phases of the oscillation. A distinct change in ion dynamics was detected as the magnetic field magnitude increased: a high-amplitude, relatively periodic oscillation in the ion velocity distribution gave way to a low-amplitude, chaotic oscillation. High amplitude oscillations of the mean ion velocity suggest that the bimodal distributions detected at many operating conditions (with time-averaged measurements) are the result of oscillations.PhDApplied PhysicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/133454/1/durot_1.pd