AVERAGE DERIVATIVE ESTIMATION UNDER MEASUREMENT ERROR

In this paper, we derive the asymptotic properties of the density-weighted average derivative estimator when a regressor is contaminated with classical measurement error and the density of this error must be estimated. Average derivatives of conditional mean functions are used extensively in economics and statistics, most notably in semiparametric index models. As well as ordinary smooth measurement error, we provide results for supersmooth error distributions. This is a particularly important class of error distribution as it includes the Gaussian density. We show that under either type of measurement error, despite using nonparametric deconvolution techniques and an estimated error characteristic function, we are able to achieve a $\sqrt {n}$ -rate of convergence for the average derivative estimator. Interestingly, if the measurement error density is symmetric, the asymptotic variance of the average derivative estimator is the same irrespective of whether the error density is estimated or not. The promising finite sample performance of the estimator is shown through a Monte Carlo simulation.

Fractional-order Kalman filters for continuous-time fractional-order systems involving correlated and uncorrelated process and measurement noises

This paper presents fractional-order Kalman filters using the fractional-order average derivative method for linear fractional-order systems involving process and measurement noises. By using the fractional-order average derivative method, a difference equation model is obtained by discretizing the investigated continuous-time fractional-order system, and the accuracy of state estimation is improved. Meanwhile, compared with the Tustin generating function, the fractional-order average derivative method proposed in this paper can reduce computation load and save calculation time. Two kinds of fractional-order Kalman filters are given, for the correlated and uncorrelated cases, in terms of the process and measurement noises for linear fractional-order systems, respectively. Finally, simulation results are illustrated to verify the effectiveness of the proposed Kalman filters using the fractional-order average derivative method.

A generalized average-derivative optimal finite-difference scheme for 2D frequency-domain acoustic-wave modeling on continuous nonuniform grids

The frequency-domain finite-difference (FDFD) method is an effective tool for implementing frequency-domain seismic modeling, inversion, and migration. However, the computational cost for the FDFD method dealing with large models is prohibitive, limiting its application. As a common strategy to improve the computational efficiency, a nonuniform grid is usually adopted in the time-domain finite-difference method instead of the FDFD method. We have developed a generalized average-derivative optimal scheme (GADOS) that can perform frequency-domain acoustic-wave modeling on continuous nonuniform grids in the vertical and horizontal directions. Before we begin the calculations, we optimize numerous stencils in which the grid spacing ratios are different to obtain a large dictionary composed of many groups of optimal coefficients. We consider the continuous nonuniform grids as a gathering of nonuniform nine-point stencils (i.e., the stencil of the GADOS) and select the proper weighted coefficients for every stencil to ensure that the numerical dispersion is minimal in the global area. All the phase-velocity errors of the GADOS for different grid spacing ratios are less than [Formula: see text] even if the number of grid points per wavelength is as small as four after the weighted coefficients are optimized by minimizing the numerical dispersion. Compared with the average-derivative optimal scheme (ADOS), simulating seismic waves with the GADOS on nonuniform grids reduce the computational cost with the premise of ensuring sufficient accuracy. Several numerical examples are presented to illustrate the feasibility and efficiency of the GADOS on continuous nonuniform grids.