Empirical Likelihood Confidence Bands in Density Estimation

Abstract

Empirical likelihood methods are developed for constructing confidence bands in problems of nonparametric density estimation. These techniques have an advantage over more conventional methods in that the shape of the bands is determined solely by the data. We show how to construct an empirical likelihood functional, rather than a function, and contour it to produce the confidence bands. Analogs of Wilks's theorem are established in this infinite-parameter setting and may be used to select the appropriate contour. An alternative calibration, based on the bootstrap, is also suggested. Large-sample theory is developed to show that the bands have asymptotically correct coverage, and a numerical example is presented to demonstrate the technique. Comparisons are made with the use of bootstrap replications to choose both the shape and size of the bands.     

Direct modulation and optical confinement factor modulation of semiconductor lasers

A method for modulation of semiconductor lasers is demonstrated. The method is based on the modulation of the optical confinement factor. Using this method an enhanced—3-dB bandwidth is observed in agreement with the small signal rate equation analysis. A modulation response that drops at high frequencies slower than the conventional direct current modulation response is achieved. This supports the theoretical predictions, showing an intrinsic 1/f decay rate for the optical confinement factor modulation scheme versus an intrinsic 1/f² decay for direct modulation.     

Performance of a combined magnetron pump-gauge at ultrahigh vacuum

This paper reports experimental results on 3 magnetron pump-gauges designed to combine the properties of UHV magnetron pumps and magnetron gauges into a single device with a portable (battery operated) power supply. The pump-gauges had a renewable cathode of Ti and were useful in the pressure range 7 × 10^?9 to 1.5 × 10^?4 Pa. As gauges, the sensitivities were about 1.5 × 10^?2 A Pa^?1. As pumps, the speeds were about 0.15 litre s^?1 for Kr.     

Hindered rotation about the amido N?C bond and the tautomeric equilibrium in N,N-dimethylacetoacetamide

The molecule N,N-dimethylacetoacetamide, which is subject to a keto–enol equilibrium process in solution, also exhibits hindered rotation about the amido N? C bond. The hindered rotation rates have been studied by lineshape fit methods of the nuclear magnetic resonance spectra. In spite of some overlap of the keto and enol N-methyl proton signals, the simultaneous measurement of the two distinct energy barriers in the two forms is possible as well as a determination of the keto–enol equilibrium. The differences in free energy of activation between keto and enol forms for the rotation barrier can be related to the conjugation energy of the N? C π system with the enolic hydrogen bonded ring. Appeal to the model compound acetylacetone reveals a consistent set of energies for the keto and enol forms in the ground and transition states for internal rotation. The opportunity has been taken to reexamine and compare the keto–enol system ethylacetoacetate. Long range, solvent, concentration and temperature sensitive scalar couplings ⁴J(HH) between the enolic –OH and the adjacent methyl group in acetoacetic ester have not hitherto been reported.     

Untangling the excited states of DR1 in solution: an experimental and theoretical study

We report the experimental observations and the theoretical predictions of the fully separated n-pi* and pi-pi* bands of Disperse Red One in acidified methanol solution. The analysis of the linear and two-photon absorption spectra is presented in four specific solvents. We demonstrate the possibility to establish the position of the first two excited states combining linear and nonlinear spectroscopy. Calculations using density functional theory at TD-B3LYP/6-31G*//HF/6-31G* level accurately predict the spectral region of the n-pi* and pi-pi* transitions of DR1 in all solvents.     

Applicability of hybrid density functional theory methods to calculation of molecular hyperpolarizability

The donor/acceptor (D/A) substituted pi-conjugated organic molecules possess extremely fast nonlinear optical (NLO) response time that is purely electronic in origin. This makes them promising candidates for optoelectronic applications. In the present study, we utilized four hybrid density functionals (B3LYP, B97-2, PBE0, BMK), Hartree-Fock, and second order Moller-Plesset correlation energy correction, truncated at second-order (MP2) methods with different basis sets to estimate molecular first hyperpolarizability (beta) of D/A-substituted benzenes and stilbenes (D=OMe, OH, NMe(2), NH(2); A=NO(2), CN). The results of density functional theory (DFT) calculations are compared to those of MP2 method and to the experimental data. We addressed the following questions: (1) the accurate techniques to compare calculated results to each other and to experiment, (2) the choice of the basis set, (3) the effect of molecular planarity, and (4) the choice of the method. Comparison of the absolute values of hyperpolarizabilities obtained computationally and experimentally is complicated by the ambiguities in conventions and reference values used by different experimental groups. A much more tangible way is to compare the ratios of beta's for two (or more) given molecules of interest that were calculated at the same level of theory and measured at the same laboratory using the same conventions and reference values. Coincidentally, it is the relative hyperpolarizabilities rather than absolute ones that are of importance in the rational molecular design of effective NLO materials. This design includes prediction of the most promising candidates from particular homologous series, which are to be synthesized and used for further investigation. In order to accomplish this goal, semiquantitative level of accuracy is usually sufficient. Augmentation of the basis set with polarization and diffuse functions changes beta by 20%; however, further extension of the basis set does not have significant effect. Thus, we recommend 6-31+G(*) basis set. We also show that the use of planar geometry constraints for the molecules, which can somewhat deviate from planarity in the gas phase, leads to sufficient accuracy (with an error less than 10%) of predicted values. For all the molecules studied, MP2 values are in better agreement with experiment, while DFT hybrid methods overestimate beta values. BMK functional gives the best agreement with experiment, with systematic overestimation close to the factor of 1.4. We propose to use the scaled BMK results for prediction of molecular hyperpolarizability at semiquantitative level of accuracy.     

Outlets of 2D invasion percolation and multiple-armed incipient infinite clusters

We study invasion percolation in two dimensions, focusing on properties of the outlets of the invasion and their relation to critical percolation and to incipient infinite clusters (IICs). First we compute the exact decay rate of the distribution of both the weight of the kth outlet and the volume of the kth pond. Next we prove bounds for all moments of the distribution of the number of outlets in an annulus. This result leads to almost sure bounds for the number of outlets in a box B(2 ⁿ ) and for the decay rate of the weight of the kth outlet to p _c . We then prove existence of multiple-armed IIC measures for any number of arms and for any color sequence which is alternating or monochromatic. We use these measures to study the invaded region near outlets and near edges in the invasion backbone far from the origin.