Efficient Estimation for the Derivative of Nonparametric Function by Optimally Combining Quantile Information

In this article, we focus on the efficient estimators of the derivative of the nonparametric function in the nonparametric quantile regression model. We develop two ways of combining quantile regression information to derive the estimators. One is the weighted composite quantile regression estimator based on the quantile weighted loss function; the other is the weighted quantile average estimator based on the weighted average of quantile regression estimators at a single quantile. Furthermore, by minimizing the asymptotic variance, the optimal weight vector is computed, and consequently, the optimal estimator is obtained. Furthermore, we conduct some simulations to evaluate the performance of our proposed estimators under different symmetric error distributions. Simulation studies further illustrate that both estimators work better than the local linear least square estimator for all the symmetric errors considered except the normal error, and the weighted quantile average estimator performs better than the weighted composite quantile regression estimator in most situations.

Averaging Indoor Localization System

This chapter aims at improving the accuracy of estimation the localization by using the RSS method to estimate the positions and take into account the effects of both LOS propagation. The proposed system is depending on developing a mathematical model for the noisy VLC positioning system. For improving the results, adopting the KF is combined with the proposed system, which is considered an optimal estimator. The performance of the proposed technique is determined by evaluating the positioning errors in a typical room. Also this chapter develops the accuracy of the positioning system by using different ideas with average techniques. The discussion of the results for averaging technique is displayed.

Minimax Estimation in Regression under Sample Conformity Constraints

The paper is devoted to the guaranteeing estimation of parameters in the uncertain stochastic nonlinear regression. The loss function is the conditional mean square of the estimation error given the available observations. The distribution of regression parameters is partially unknown, and the uncertainty is described by a subset of probability distributions with a known compact domain. The essential feature is the usage of some additional constraints describing the conformity of the uncertain distribution to the realized observation sample. The paper contains various examples of the conformity indices. The estimation task is formulated as the minimax optimization problem, which, in turn, is solved in terms of saddle points. The paper presents the characterization of both the optimal estimator and the set of least favorable distributions. The saddle points are found via the solution to a dual finite-dimensional optimization problem, which is simpler than the initial minimax problem. The paper proposes a numerical mesh procedure of the solution to the dual optimization problem. The interconnection between the least favorable distributions under the conformity constraint, and their Pareto efficiency in the sense of a vector criterion is also indicated. The influence of various conformity constraints on the estimation performance is demonstrated by the illustrative numerical examples.

Saturn´s Stratospheric Hazes From HST Ultraviolet Imaging

<p>We present a study on Saturn's stratospheric hazes using archived images from the Hubble Space Telescope Advanced Camera for Surveys. These observations were taken from 2005 to 2014, including the Great Storm during the years 2010 and 2011. For our research we used ultraviolet images from the Solar Blind Channel camera equipped with the F115LP and F125LP filters. At these wavelengths, the reflected spectrum is fundamentally Rayleigh-scattered, with substantial contributions from hydrocarbon absorptions and additional scattering by the aerosols in the hazes above the tropopause. The goal of this work is to analyze temporal and latitudinal changes in the characteristics of the stratospheric haze, gases and particles, analyzing the absolute reflectivity and its limb darkening. Such behavior can be reproduced using the empirical Minnaert's law. This provides nadir-viewing reflectivity and limb darkening coefficient as a function of latitude and time. This is a first approach that helps to qualitatively identify the changes occurring in the aerosol layer during this period of time, which include the massive Great White Spot of 2010. In order to quantify such aerosol changes, we use the radiative transfer code and retrieval suite NEMESIS (Non-Linear Optimal Estimator for Multivariat Spectral AnalySIS) to reproduce the observed reflectivity.  Here we will focus on the detected variations of the vertical distribution of the stratospheric particles, their integrated optical thickness and size distribution and will correlate them with the seasonal changes taken place in the atmosphere of the planet.</p>

The hybrid Cramér-Rao lower bound for simultaneous self-localization and room geometry estimation

This paper addresses the problem of tracking a moving source, e.g., a robot, equipped with both receivers and a source, that is tracking its own location and simultaneously estimating the locations of multiple plane reflectors. We assume a noisy knowledge of the robot’s movement. We formulate this problem, which is also known as simultaneous localization and mapping (SLAM), as a hybrid estimation problem. We derive the extended Kalman filter (EKF) for both tracking the robot’s own location and estimating the room geometry. Since the EKF employs linearization at every step, we incorporate a regulated kinematic model, which facilitates a successful tracking. In addition, we consider the echo-labeling problem as solved and beyond the scope of this paper. We then develop the hybrid Cramér-Rao lower bound on the estimation accuracy of both the localization and mapping parameters. The algorithm is evaluated with respect to the bound via simulations, which shows that the EKF approaches the hybrid Cramér-Rao bound (CRB) (HCRB) as the number of observation increases. This result implies that for the examples tested in simulation, the HCRB is an asymptotically tight bound and that the EKF is an optimal estimator. Whether this property is true in general remains an open question.

Finite‐Sample Optimal Estimation and Inference on Average Treatment Effects Under Unconfoundedness

We consider estimation and inference on average treatment effects under unconfoundedness conditional on the realizations of the treatment variable and covariates. Given nonparametric smoothness and/or shape restrictions on the conditional mean of the outcome variable, we derive estimators and confidence intervals (CIs) that are optimal in finite samples when the regression errors are normal with known variance. In contrast to conventional CIs, our CIs use a larger critical value that explicitly takes into account the potential bias of the estimator. When the error distribution is unknown, feasible versions of our CIs are valid asymptotically, even when n ‐inference is not possible due to lack of overlap, or low smoothness of the conditional mean. We also derive the minimum smoothness conditions on the conditional mean that are necessary for n ‐inference. When the conditional mean is restricted to be Lipschitz with a large enough bound on the Lipschitz constant, the optimal estimator reduces to a matching estimator with the number of matches set to one. We illustrate our methods in an application to the National Supported Work Demonstration.