Trapezoidal rule and sampling designs for the nonparametric estimation of the regression function in models with correlated errors

The problem of estimating the regression function in a fixed design models with correlated observations is considered. Such observations are obtained from several experimental units, each of them forms a time series. Based on the trapezoidal rule, we propose a simple kernel estimator and we derive the asymptotic expression of its integrated mean squared error IMSE and its asymptotic normality. The problems of the optimal bandwidth and the optimal design with respect to the asymptotic IMSE are also investigated. Finally, a simulation study is conducted to study the performance of the new estimator and to compare it with the classical estimator of Gasser and M\"uller in a finite sample set. In addition, we study the robustness of the optimal design with respect to the misspecification of the autocovariance function.Comment: 36 pages, 3 figure

Angular Distribution of Auger Electrons Emitted through the Resonant Transfer and Excitation Process Following O⁵⁺+He Collisions

This Letter reports the first measurements of the angular distribution of Auger electrons emitted from the decay of the (1s2s2p2)3D O4+** doubly excited state formed predominantly through resonant transfer and excitation (RTE) in collisions of 13-MeV O5+ projectiles with He. The (1s2s2p2)3D angular distribution is strongly peaked along the beam direction, in agreement with recent calculations of the RTE angle-dependent impulse approximation. Furthermore, interference effects between the RTE and the elastic target direct-ionization channels are observed

Double Excitation of He by Fast Ions

Autoionization of He atoms following double excitation by electrons, protons, CQ+ (Q=4-6), and FQ+ (Q=7-9) ions has been studied. The electron-emission yields from the doubly excited 2s2(1S), 2s2p(1P), and 2p2(1P) states were measured at the reduced projectile energy of 1.5 MeV/nucleon for observation angles between 10°and 60°. The results indicate excitation to the 2s2(1S) and 2p2(1D) states increases as approximately Q3, while excitation to the 2s2p(1P) state varies as approximately Q2, where Q is the charge of the projectile. These charge dependences are significantly less than the Q4 dependence expected in the independent-electron model, suggesting the interaction between the two target electrons is important in creating the doubly excited states

Electron-Electron Interactions in Transfer and Excitation in F⁸⁺ →₂ Collisions

We have measured projectile Auger electrons emitted after collisions of H-like F with H2. The cross sections for emission of KLL, KLM, KLN, and KLO Auger electrons show maxima as a function of the projectile energy. One maximum in the KLL emission cross section is due to resonant transfer and excitation. A second maximum in the cross section for KLL emission as well as the maxima in the emission cross section for the higher-n Auger electrons are attributed to a new transfer and excitation process. This involves excitation of a projectile electron by one target electron accompanied by the capture of a second target electron

A Monte Carlo simulation of ion transport at finite temperatures

We have developed a Monte Carlo simulation for ion transport in hot background gases, which is an alternative way of solving the corresponding Boltzmann equation that determines the distribution function of ions. We consider the limit of low ion densities when the distribution function of the background gas remains unchanged due to collision with ions. A special attention has been paid to properly treat the thermal motion of the host gas particles and their influence on ions, which is very important at low electric fields, when the mean ion energy is comparable to the thermal energy of the host gas. We found the conditional probability distribution of gas velocities that correspond to an ion of specific velocity which collides with a gas particle. Also, we have derived exact analytical formulas for piecewise calculation of the collision frequency integrals. We address the cases when the background gas is monocomponent and when it is a mixture of different gases. The developed techniques described here are required for Monte Carlo simulations of ion transport and for hybrid models of non-equilibrium plasmas. The range of energies where it is necessary to apply the technique has been defined. The results we obtained are in excellent agreement with the existing ones obtained by complementary methods. Having verified our algorithm, we were able to produce calculations for Ar$^+$ ions in Ar and propose them as a new benchmark for thermal effects. The developed method is widely applicable for solving the Boltzmann equation that appears in many different contexts in physics.Comment: 14 page

The effect of the regularity of the error process on the performance of kernel regression estimators

This article considers estimation of regression function ff in the fixed design model Y(xi)=f(xi)+ϵ(xi),i=1, ¦,nY(xi)=f(xi)+ϵ(xi),i=1, ¦,n , by use of the Gasser and Müller kernel estimator. The point set {xi}ni=1⊂[0,1]{xi}i=1n⊂[0,1] constitutes the sampling design points, and ϵ(xi)ϵ(xi) are correlated errors. The error process ϵϵ is assumed to satisfy certain regularity conditions, namely, it has exactly kk ( =0,1,2, ¦=0,1,2, ¦ ) quadratic mean derivatives (q.m.d.). The quality of the estimation is measured by the mean squared error (MSE). Here the asymptotic results of the mean squared error are established. We found that the optimal bandwidth depends on the (2k+1)(2k+1) th mixed partial derivatives of the autocovariance function along the diagonal of the unit square. Simulation results for the model of kk th order integrated Brownian motion error are given in order to assess the effect of the regularity of this error process on the performance of the kernel estimator

Estimation of the regression operator from functional fixed-design with correlated errors

We consider the estimation of the regression operator r in the functional model: Y=r(x)+[epsilon], where the explanatory variable x is of functional fixed-design type, the response Y is a real random variable and the error process [epsilon] is a second order stationary process. We construct the kernel type estimate of r from functional data curves and correlated errors. Then we study their performances in terms of the mean square convergence and the convergence in probability. In particular, we consider the cases of short and long range error processes. When the errors are negatively correlated or come from a short memory process, the asymptotic normality of this estimate is derived. Finally, some simulation studies are conducted for a fractional autoregressive integrated moving average and for an Ornstein-Uhlenbeck error processes.Nonparametric regression operator Functional fixed-design Short memory process Long memory process Fractional process Ornstein-Uhlenbeck process Negatively associated process