Estimating linear functionals in nonlinear regression with responses missing at random

We consider regression models with parametric (linear or nonlinear) regression function and allow responses to be ``missing at random.'' We assume that the errors have mean zero and are independent of the covariates. In order to estimate expectations of functions of covariate and response we use a fully imputed estimator, namely an empirical estimator based on estimators of conditional expectations given the covariate. We exploit the independence of covariates and errors by writing the conditional expectations as unconditional expectations, which can now be estimated by empirical plug-in estimators. The mean zero constraint on the error distribution is exploited by adding suitable residual-based weights. We prove that the estimator is efficient (in the sense of H\'{a}jek and Le Cam) if an efficient estimator of the parameter is used. Our results give rise to new efficient estimators of smooth transformations of expectations. Estimation of the mean response is discussed as a special (degenerate) case.Comment: Published in at http://dx.doi.org/10.1214/08-AOS642 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Inference about the slope in linear regression: an empirical likelihood approach

We present a new, efficient maximum empirical likelihood estimator for the slope in linear regression with independent errors and covariates. The estimator does not require estimation of the influence function, in contrast to other approaches, and is easy to obtain numerically. Our approach can also be used in the model with responses missing at random, for which we recommend a complete case analysis. This suffices thanks to results by Müller and Schick (Bernoulli 23:2693–2719, 2017), which demonstrate that efficiency is preserved. We provide confidence intervals and tests for the slope, based on the limiting Chi-square distribution of the empirical likelihood, and a uniform expansion for the empirical likelihood ratio. The article concludes with a small simulation study

Efficient prediction for linear and nonlinear autoregressive models

Conditional expectations given past observations in stationary time series are usually estimated directly by kernel estimators, or by plugging in kernel estimators for transition densities. We show that, for linear and nonlinear autoregressive models driven by independent innovations, appropriate smoothed and weighted von Mises statistics of residuals estimate conditional expectations at better parametric rates and are asymptotically efficient. The proof is based on a uniform stochastic expansion for smoothed and weighted von Mises processes of residuals. We consider, in particular, estimation of conditional distribution functions and of conditional quantile functions.Comment: Published at http://dx.doi.org/10.1214/009053606000000812 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Optimality of estimators for misspecified semi-Markov models

Suppose we observe a geometrically ergodic semi-Markov process and have a parametric model for the transition distribution of the embedded Markov chain, for the conditional distribution of the inter-arrival times, or for both. The first two models for the process are semiparametric, and the parameters can be estimated by conditional maximum likelihood estimators. The third model for the process is parametric, and the parameter can be estimated by an unconditional maximum likelihood estimator. We determine heuristically the asymptotic distributions of these estimators and show that they are asymptotically efficient. If the parametric models are not correct, the (conditional) maximum likelihood estimators estimate the parameter that maximizes the Kullback--Leibler information. We show that they remain asymptotically efficient in a nonparametric sense.Comment: To appear in a Special Volume of Stochastics: An International Journal of Probability and Stochastic Processes (http://www.informaworld.com/openurl?genre=journal%26issn=1744-2508) edited by N.H. Bingham and I.V. Evstigneev which will be reprinted as Volume 57 of the IMS Lecture Notes Monograph Series (http://imstat.org/publications/lecnotes.htm

On shape variation of confined triatomics of XY2-type

Harmonic confinement of initially isolated symmetric triatomic molecules can induce a transition from a bent, directed bond-type structure to helium-like angular correlation of the two equal particles. In an exactly solvable modification of the Hooke-Calogero model it is demonstrated that there is a well-defined transition between the two cases if the confinement strength is increased. Furthermore confinement is shown to reduce the system's effective diameter, which at the transition point has shrunk by 26% in comparison to the isolated syste

On the transition between directed bonding and helium-like angular correlation in a modified Hooke-Calogero model

Angular correlation in three-body systems varies between the limiting cases of slightly perturbed equi-distribution, as in the electronic ground state of helium and directed bond-type bent structure, as in the isolated water molecule. In an exactly solvable modification of the Hooke-Calogero model, it is shown that there is a sharp transition between the two cases if the particles' masses are suitably varied. In the Hooke-Calogero model attraction between different particles is harmonic and the repulsion between equal particles is given by a 1/r 2 potential. The bent structure appears in the angular distribution function if the masses of the two equal particles are below a critical value, which depends on the mass of the third particle. Above the critical value, the angular correlation is of helium type and exhibits a minimum at 0° corresponding to the Coulomb hole and a maximum at 180°. The model thus demonstrates the modulating role of mass in the transition between semi-rigid structure and more diffuse nuclear state

Variable Metric Random Pursuit

We consider unconstrained randomized optimization of smooth convex objective functions in the gradient-free setting. We analyze Random Pursuit (RP) algorithms with fixed (F-RP) and variable metric (V-RP). The algorithms only use zeroth-order information about the objective function and compute an approximate solution by repeated optimization over randomly chosen one-dimensional subspaces. The distribution of search directions is dictated by the chosen metric. Variable Metric RP uses novel variants of a randomized zeroth-order Hessian approximation scheme recently introduced by Leventhal and Lewis (D. Leventhal and A. S. Lewis., Optimization 60(3), 329--245, 2011). We here present (i) a refined analysis of the expected single step progress of RP algorithms and their global convergence on (strictly) convex functions and (ii) novel convergence bounds for V-RP on strongly convex functions. We also quantify how well the employed metric needs to match the local geometry of the function in order for the RP algorithms to converge with the best possible rate. Our theoretical results are accompanied by numerical experiments, comparing V-RP with the derivative-free schemes CMA-ES, Implicit Filtering, Nelder-Mead, NEWUOA, Pattern-Search and Nesterov's gradient-free algorithms.Comment: 42 pages, 6 figures, 15 tables, submitted to journal, Version 3: majorly revised second part, i.e. Section 5 and Appendi

Fast high--voltage amplifiers for driving electro-optic modulators

We describe five high-voltage (60 to 550V peak to peak), high-speed (1-300ns rise time; 1.3-300MHz bandwidth) linear amplifiers for driving capacitive or resistive loads such as electro-optic modulators. The amplifiers use bipolar transistors in various topologies. Two use electron tubes to overcome the speed limitations of high-voltage semiconductors. All amplifiers have been built. Measured performance data is given for each.Comment: 9pages, 6figures, 6tables, to appear in Review of Scientific Instrument