An Improved Estimation of Parameter of Morgenstern-Type Bivariate Exponential Distribution Using Ranked Set Sampling

In this chapter, the authors suggest some improved versions of estimators of Morgenstern type bivariate exponential distribution (MTBED) based on the observations made on the units of ranked set sampling (RSS) regarding the study variable Y, which is correlated with the auxiliary variable X, where (X,Y) follows a MTBED. In this chapter, they firstly suggested minimum mean squared error estimator for estimation of 𝜃2 based on censored ranked set sample and their special case; further, they have suggested minimum mean squared error estimator for best linear unbiased estimator of 𝜃2 based on censored ranked set sample and their special cases; they also suggested minimum mean squared error estimator for estimation of 𝜃2 based on unbalanced multistage ranked set sampling and their special cases. Efficiency comparisons are also made in this work.

FAST ARRAY ALGORITHMS FOR FILTERING OF MARKOVIAN JUMP LINEAR SYSTEMS WITH STRUCTURED TIME-VARIANT PARAMETERS

In this paper were developed fast array algorithms for the linear minimum mean square error estimator for a class of Markovian jump linear systems with structured time-variant parameters. The fast array algorithms for systems with structured time-variant parameters arises as an alternative to calculate this type algorithm for some variation in the time of the parameters. Numerical example to show the advantage of using fast array algorithm to filter this class of systems are provided.

An equilibrated a posteriori error estimator for an Interior Penalty Discontinuous Galerkin approximation of the p-Laplace problem

We are concerned with an Interior Penalty Discontinuous Galerkin (IPDG) approximation of the p-Laplace equation and an equilibrated a posteriori error estimator. The IPDG method can be derived from a discretization of the associated minimization problem involving appropriately defined reconstruction operators. The equilibrated a posteriori error estimator provides an upper bound for the discretization error in the broken W 1,p norm and relies on the construction of an equilibrated flux in terms of a numerical flux function associated with the mixed formulation of the IPDG approximation. The relationship with a residual-type a posteriori error estimator is established as well. Numerical results illustrate the performance of both estimators.

Robust Estimation of the Theil Index and the Gini Coeffient for Small Areas

Small area estimation is receiving considerable attention due to the high demand for small area statistics. Small area estimators of means and totals have been widely studied in the literature. Moreover, in the last years also small area estimators of quantiles and poverty indicators have been studied. In contrast, small area estimators of inequality indicators, which are often used in socio-economic studies, have received less attention. In this article, we propose a robust method based on the M-quantile regression model for small area estimation of the Theil index and the Gini coefficient, two popular inequality measures. To estimate the mean squared error a non-parametric bootstrap is adopted. A robust approach is used because often inequality is measured using income or consumption data, which are often non-normal and affected by outliers. The proposed methodology is applied to income data to estimate the Theil index and the Gini coefficient for small domains in Tuscany (provinces by age groups), using survey and Census micro-data as auxiliary variables. In addition, a design-based simulation is carried out to study the behaviour of the proposed robust estimators. The performance of the bootstrap mean squared error estimator is also investigated in the simulation study.

Combining spectral and POD modes to improve error estimation of numerical model reduction for porous media

Numerical model reduction (NMR) is used to solve the microscale problem that arises from computational homogenization of a model problem of porous media with displacement and pressure as unknown fields. The reduction technique and an associated error estimator for the NMR error have been presented in prior work, where both spectral decomposition (SD) and proper orthogonal decomposition (POD) were used to construct the reduced basis. It was shown that the POD basis performs better w.r.t. minimizing the residual, but the SD basis has some advantageous properties for the estimator. Since it is the estimated error that will govern the error control, the most efficient procedure is the one that results in the lowest error bound. The main contribution of this paper is further development of the previous work with a proposed combined basis constructed using both SD and POD modes together with an adaptive mode selection strategy. The performance of the combined basis is compared to (i) the pure SD basis and (ii) the pure POD basis via numerical examples. The examples show that it is possible to find a combination of SD/POD modes which is improved, i.e. it yields a smaller estimate, compared to the cases of pure SD or pure POD.

Adaptive discrete thin plate spline smoother

The discrete thin plate spline smoother fits smooth surfaces to large data sets efficiently. It combines the favourable properties of the finite element surface fitting and thin plate splines. The efficiency of its finite element grid is improved by adaptive refinement, which adapts the precision of the solution. It reduces computational costs by refining only in sensitive regions, which are identified using error indicators. While many error indicators have been developed for the finite element method, they may not work for the discrete smoother. In this article we show three error indicators adapted from the finite element method for the discrete smoother. A numerical experiment is provided to evaluate their performance in producing efficient finite element grids. References F. L. Bookstein. Principal warps: Thin-plate splines and the decomposition of deformations. IEEE Trans. Pat. Anal. Mach. Int. 11.6 (1989), pp. 567–585. doi: 10.1109/34.24792. C. Chen and Y. Li. A robust method of thin plate spline and its application to DEM construction. Comput. Geosci. 48 (2012), pp. 9–16. doi: 10.1016/j.cageo.2012.05.018. L. Fang. Error estimation and adaptive refinement of finite element thin plate spline. PhD thesis. The Australian National University. http://hdl.handle.net/1885/237742. L. Fang. Error indicators and adaptive refinement of the discrete thin plate spline smoother. ANZIAM J. 60 (2018), pp. 33–51. doi: 10.21914/anziamj.v60i0.14061. M. F. Hutchinson. A stochastic estimator of the trace of the influence matrix for laplacian smoothing splines. Commun. Stat. Simul. Comput. 19.2 (1990), pp. 433–450. doi: 10.1080/0361091900881286. W. F. Mitchell. A comparison of adaptive refinement techniques for elliptic problems. ACM Trans. Math. Soft. 15.4 (1989), pp. 326–347. doi: 10.1145/76909.76912. R. F. Reiniger and C. K. Ross. A method of interpolation with application to oceanographic data. Deep Sea Res. Oceanographic Abs. 15.2 (1968), pp. 185–193. doi: 10.1016/0011-7471(68)90040-5. S. Roberts, M. Hegland, and I. Altas. Approximation of a thin plate spline smoother using continuous piecewise polynomial functions. SIAM J. Numer. Anal. 41.1 (2003), pp. 208–234. doi: 10.1137/S0036142901383296. D. Ruprecht and H. Muller. Image warping with scattered data interpolation. IEEE Comput. Graphics Appl. 15.2 (1995), pp. 37–43. doi: 10.1109/38.365004. E. G. Sewell. Analysis of a finite element method. Springer, 2012. doi: 10.1007/978-1-4684-6331-6. L. Stals. Efficient solution techniques for a finite element thin plate spline formulation. J. Sci. Comput. 63.2 (2015), pp. 374–409. doi: 10.1007/s10915-014-9898-x. O. C. Zienkiewicz and J. Z. Zhu. A simple error estimator and adaptive procedure for practical engineerng analysis. Int. J. Numer. Meth. Eng. 24.2 (1987), pp. 337–357. doi: 10.1002/nme.1620240206.

Adaptive space-time finite element methods for parabolic optimal control problems

We present, analyze, and test locally stabilized space-time finite element methods on fully unstructured simplicial space-time meshes for the numerical solution of space-time tracking parabolic optimal control problems with the standard L 2-regularization. We derive a priori discretization error estimates in terms of the local mesh-sizes for shape-regular meshes. The adaptive version is driven by local residual error indicators, or, alternatively, by local error indicators derived from a new functional a posteriori error estimator. The latter provides a guaranteed upper bound of the error, but is more costly than the residual error indicators. We perform numerical tests for benchmark examples having different features. In particular, we consider a discontinuous target in form of a first expanding and then contracting ball in 3d that is fixed in the 4d space-time cylinder.