Estimacion de la funcion cuantil y cuantil-densidad mediante polinomios de Kantorovic

Resumen En este trabajo se propone un estimador para la función cuantil, basado en polinomios de Kantorovic, como estimador natural, y se prueba que su error absoluto medio es un infinitésimo de orden n^−1/2. Mediante simulación se pone de manifiesto que dicho estimador conduce a una reducción sustancial del error absoluto medio frente a la función cuantil muestral y, por otra parte, se compara con el estimador basado en polinomios de Bernstein. Asimismo, se propone un estimador basado en polinomios de Kantorovic para estimar la función cuantil-densidad, que es la derivada del estimador de la función cuantil basado en polinomios de Bernstein.

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In this paper, we propose an estimator to the quantile function based on Kantorovic polynomials which is the natural estimator and it is shown thatE|K _n−Q|∼O(n^−1/2). By Monte Carlo simulation it is shown that this estimator conduce to substantial reduction for mean absolute error respect to the conventional empirical quantile function and it is compared with the estimator bases on Bernstein polinomials. Also, we propose an estimator based on Kantorovic polinomial for quantile-density function which is the derivative of estimator based on Bernstein polinomials for quantile function.

     

The Variance of an Estimator in Variables Sampling

Let p be the proportion of a normal population in some tolerance region. The equivalence of two different forms of the minimum variance unbiased estimator of p given in the literature is indicated, and a third form is derived and used to obtain the variance of the estimator. For sample sizes of four and six the variance of the estimator is given explicitly and finally estimation of the variance for larger sample sizes is considered.     

Semi-parametric second-order reduced-bias high quantile estimation

In many areas of application, like, for instance, Climatology, Hydrology, Insurance, Finance, and Statistical Quality Control, a typical requirement is to estimate a high quantile of probability 1−p, a value high enough so that the chance of an exceedance of that value is equal to p, small. The semi-parametric estimation of high quantiles depends not only on the estimation of the tail index or extreme value index γ, the primary parameter of extreme events, but also on the adequate estimation of a scale first order parameter. Recently, apart from new classes of reduced-bias estimators for γ>0, new classes of the scale first order parameter have been introduced in the literature. Their use in quantile estimation enables us to introduce new classes of asymptotically unbiased high quantiles’ estimators, with the same asymptotic variance as the (biased) “classical” estimator. The asymptotic distributional properties of the proposed classes of estimators are derived and the estimators are compared with alternative ones, not only asymptotically, but also for finite samples through Monte Carlo techniques. An application to the log-exchange rates of the Euro against the Sterling Pound is also provided.     

A simple stress error estimator for Hybrid-Trefftz p-version elements

An a posteriori error estimator is presented which allows a good pointwise evaluation of the error in predicted stresses and can easily be implemented in existing FE codes. Although this estimator has especially been developed for and tested on p-version Hybrid-Trefftz (HT) elements, it is anticipated that it can also be applied to conventional conforming p-version elements. The practical efficiency of the estimator is illustrated through the solution of various plate bending problems by using the HT p-version Kirchhoff plate elements.²     

An a posteriori error estimator for the p‐ and hp‐versions of the finite element method

An a posteriori error estimator is proposed in this paper for the p‐ and hp‐versions of the finite element method in two‐dimensional linear elastostatic problems. The local error estimator consists in an enhancement of an error indicator proposed by Bertóti and Szabó (Int. J. Numer. Meth. Engng. 1998; 42 :561–587), which is based on the minimum complementary energy principle. In order to obtain the local error estimate, this error indicator is corrected by a factor which depends only on the polynomial degree of the element. The proposed error estimator shows a good effectivity index in meshes with uniform and non‐uniform polynomial distributions, especially when the global error is estimated. Furthermore, the local error estimator is reliable enough to guide p‐ and hp‐adaptive refinement strategies. Copyright © 2004 John Wiley & Sons, Ltd.     

Adaptive selection of polynomial degrees on a finite element mesh

The problem of finding a nearly optimal distribution of polynomial degrees on a fixed finite element mesh is discussed. An a posteriori error estimator based on the minimum complementary energy principle is proposed which utilizes the displacement vector field computed from the finite element solution. This estimator, designed for p- and hp-extensions, is conceptually different from estimators based on residuals or patch recovery which are designed for h-extension procedures. The quality of the error estimator is demonstrated by examples. The results show that the effectivity index is reasonably close to unity and the sequences of p-distributions obtained with the error indicators closely follow the optimal trajectory. © 1998 John Wiley & Sons, Ltd.     

Estimation of Cpm when Measurement Error is Present

In this paper we study the properties of the estimator of C_pm when the observations are affected by measurement errors. We compare the performances of the estimator in the error case with those of the estimator in the error‐free case. The results indicate that the presence of measurement errors in the data leads to different behavior of the estimator according to the entity of the error variability. We finally show how to use our results in practice. Copyright © 2005 John Wiley & Sons, Ltd.     

A p‐hierarchical adaptive procedure for the scaled boundary finite element method

This paper describes a p‐hierarchical adaptive procedure based on minimizing the classical energy norm for the scaled boundary finite element method. The reference solution, which is the solution of the fine mesh formed by uniformly refining the current mesh element‐wise one order higher, is used to represent the unknown exact solution. The optimum mesh is assumed to be obtained when each element contributes equally to the global error. The refinement criteria and the energy norm‐based error estimator are described and formulated for the scaled boundary finite element method. The effectivity index is derived and used to examine quality of the proposed error estimator. An algorithm for implementing the proposed p‐hierarchical adaptive procedure is developed. Numerical studies are performed on various bounded domain and unbounded domain problems. The results reflect a number of key points. Higher‐order elements are shown to be highly efficient. The effectivity index indicates that the proposed error estimator based on the classical energy norm works effectively and that the reference solution employed is a high‐quality approximation of the exact solution. The proposed p‐hierarchical adaptive strategy works efficiently. Copyright © 2007 John Wiley & Sons, Ltd.     

Moderate deviations for M-estimators

General sufficient conditions for the moderate deviations of M-estimators are presented. These results are applied to many different types of M-estimators such as thep-th quantile, the spatial median, the least absolute deviation estimator in linear regression, maximum likelihood estimators and other location estimators. Moderate deviations theorems from empirical processes are applied.     

Multiphysics model reduction of symmetric vibro-acoustic formulation with a priori error estimation criteria

A new multiphysics mode synthesis (MMS) is presented for the construction of reduced-order models of acoustic fluid-structure interaction systems. The present acoustic-structure interaction model is symmetric and consists of the fluid pressure p, the fluid displacement potential φ , and the structural displacement u. In carrying out MMS, the structure is first reduced, which is then applied to the coupling terms in the acoustic (p, φ )-equations. The fluid parts (p, φ ) are then reduced while preserving the coupling effects, resulting in improved accuracy. A combination of a priori modal contribution indicator and cumulative error estimator are derived from the moment matching approach that provides rational criteria for how many structural and fluid modes need to be retained for the construction of reduced models. An iterative MMS algorithm is then proposed by combining the modal indicators and error estimators. The performance of the proposed MMS method is illustrated via numerical examples.     

Asymptotic properties in partial linear models under dependence

Consider the regression modely _i =ζ _i ^T β+m(t _i )+ε _i fori=1,...,n. Here (ζ _i ^T ,t _i ) ^T ∈ℝ ^p ×[0,1] are design points, β is an unknownp×1 vector of parameters,m is an unknown smooth function from [0,1] to ℝ andε _i are the unobserved errors. We will assume that these errors are not independent. Under suitable assumptions, we obtain expansions for the bias and the variance of a Generalized Least Squares (GLS) type regression estimator, and for an estimator of the nonparametric functionm(·). Furthermore, we prove the asymptotic normality of the first estimator. The obtained results are a generalization of those contained in Speckman (1988), who studied a similar model with i.i.d. error variables. Research supported by the Xunta de Galicia (Spain) and the DGES (Spain) under research projects XUGA 10503A98 and PB98-0182-c02-01, respectively.     

A pollution controlled hybrid interior error estimator for linear elastostatics

A hybrid error estimator using a priori interior region estimates in an a posteriori framework is presented for linear elastostatics problems in FEA. It is shown that local rates of convergence are augmented by this technique and global rates are not adversely affected. The effects of pollution for this estimator are explained and a pollution error estimator is derived using the concept of error loads. It is shown that pollution error estimation can improve the performance of both the conventional a posteriori and the hybrid techniques. A series of numerical results are presented which demonstrate the superior performance of the proposed method over previously published interior error estimation techniques. © 1998 John Wiley & Sons, Ltd.     

A MODEL STUDY OF THE QUALITY OF A POSTERIORI ERROR ESTIMATORS FOR FINITE ELEMENT SOLUTIONS OF LINEAR ELLIPTIC PROBLEMS,WITH PARTICULAR REFERENCE TO THE BEHAVIOR NEAR THE BOUNDARY

In References 1–3 we presented a computer-based theory for analysing the asymptotic accuracy (quality of robustness) of error estimators for mesh-patches in the interior of the domain. In this paper we review the approach employed in References 1–3 and extend it to analyse the asymptotic quality of error estimators for mesh-patches at or near a domain boundary. We analyse two error estimators which were found in References 1–3 to be robust in the interior of the mesh (the element residual with p-order equilibrated fluxes and (p+1)) degree bubble solution or (p+1) degree polynomial solution (ERpB or ERpPp+1; see References 1–3) and the Zienkiewicz–Zhu Superconvergent Patch Recovery (ZZ-SPR; see References 4–7) and we show that the robustness of these estimators for elements adjacent to the boundary can be significantly inferior to their robustness for interior elements. This deterioration is due to the difference in the definition of the estimators for the elements in the interior of the mesh and the elements adjacent to the boundary. In order to demonstrate how our approach can be employed to determine the most robust version of an estimator we analysed the versions of the ZZ estimator proposed in References 9–12. We found that the original ZZ-SPR proposed in References 4–7 is the most robust one, among the various versions tested, and some of the proposed ‘enhancements’ can lead to a significant deterioration of the asymptotic robustness of the estimator. From the analyses given in References 1–3 and in this paper, we found that the original ZZ estimator (given in References 4–7) is the most robust among all estimators analysed in References 1–3 and in this study. © 1997 John Wiley & Sons, Ltd.     

Generalized finite element method in structural nonlinear analysis – a <Emphasis Type="Italic">p</Emphasis>-adaptive strategy

This paper is concerned with an extension of the generalized finite element method, GFEM, to nonlinear analysis and to the proposition of a p-adaptive strategy. The p-adaptivity is considered due to the nodal enrichment scheme of the method. Here, such scheme consists of multiplying the partition of unity functions by a set of polynomials. In a first part, the performance of the method in nonlinear analysis of a reinforced concrete beam with progressive damage is presented. The adaptive strategy is then proposed on basis of a control over the approximation error. Aiming to estimate the approximation error, the equilibrated element residual method is adapted to the GFEM and to the nonlinear approach. Then, global and local error measures are defined. A numerical example is presented outlining the effectivity index of the error estimator proposed. Finally, a p-adaptive procedure is described and its good performance is illustrated by a numerical example.The authors gratefully acknowledge the Conselho Nacional de Desenvolvimento Cientìifico e Tecnológico (CNPq) at Brazil.     

Some Tests for Variance Components Using Generalized p Values

The concept of a generalized p value, introduced by Tsui and Weerahandi, is applied for testing hypotheses in two situations, testing the significance of a variance component in a general balanced mixed model when an exact F test does not exist and comparing randomeffects variance components in two independent balanced mixed models. Extensions to the unbalanced cases are also indicated. The proposed test is compared with the test based on the Satterthwaite approximation through their simulated Type I error probabilities. The simulations indicate that the test based on the generalized p value hasType I error probabilities less than the chosen significance level most of the time, whereas the Type I error probabilities of the Satterthwaite approximate test can be much larger than the significance level. The results are illustrated using two examples.     

Element‐wise a posteriori estimates based on hierarchical bases for non‐linear parabolic problems

We show that the issue of a posteriori estimate the errors in the numerical simulation of non‐linear parabolic equations can be reduced to a posteriori estimate the errors in the approximation of an elliptic problem with the right‐hand side depending on known data of the problem and the computed numerical solution. A procedure to obtain local error estimates for the p version of the finite element method by solving small discrete elliptic problems with right‐hand side the residual of the p‐FEM solution is introduced. The boundary conditions are inherited by those of the space of hierarchical bases to which the error estimator belongs. We prove that the error in the numerical solution can be reduced by adding the estimators that behave as a locally defined correction to the computed approximation. When the error being estimated is that of a elliptic problem constant free local lower bounds are obtained. The local error estimation procedure is applied to non‐linear parabolic differential equations in several space dimensions. Some numerical experiments for both the elliptic and the non‐linear parabolic cases are provided. Copyright © 2005 John Wiley & Sons, Ltd.     

Performance analysis of the maximum a posteriori probability direction finding

Abstract

In this paper, the mean square error performance of the maximum a posteriori (MAP) probability direction finding by sensor array in terms of its Cramer‐Rao lower bound (CRLB) is analyzed. Based on the principle of Bayesian estimator, a log posteriori probability function is formed when the a priori knowledge of location is given. The Fisher information matrix (FIM) is found accordingly. It shows that the CRLB of the MAP estimator is much lower than that of maximum likelihood technique, especially when the element SNR is low and/or the number of snapshots is small. In addition, the CRLB remains at a relatively low level in terms of the variance of DOA of sources. It also shows that the location variance dominates the behavior of the MAP direction finder when locations of sources are Gaussian distributed.     

A priori error estimator of the generalized‐α method for structural dynamics

An a priori error estimator for the generalized‐α time‐integration method is developed to solve structural dynamic problems efficiently. Since the proposed error estimator is computed with only information in the previous and current time‐steps, the time‐step size can be adaptively selected without a feedback process, which is required in most conventional a posteriori error estimators. This paper shows that the automatic time‐stepping algorithm using the a priori estimator performs more efficient time integration, when compared to algorithms using an a posteriori estimator. In particular, the proposed error estimator can be usefully applied to large‐scale structural dynamic problems, because it is helpful to save computation time. To verify efficiency of the algorithm, several examples are numerically investigated. Copyright © 2003 John Wiley & Sons, Ltd.     

An adaptive finite element approach for frictionless contact problems

The derivation of an a posteriori error estimator for frictionless contact problems under the hypotheses of linear elastic behaviour and infinitesimal deformation is presented. The approximated solution of this problem is obtained by using the finite element method. A penalization or augmented‐Lagrangian technique is used to deal with the unilateral boundary condition over the contact boundary. An a posteriori error estimator suitable for adaptive mesh refinement in this problem is proposed, together with its mathematical justification. Up to the present time, this mathematical proof is restricted to the penalization approach. Several numerical results are reported in order to corroborate the applicability of this estimator and to compare it with other a posteriori error estimators. Copyright © 2001 John Wiley & Sons, Ltd.     

Joint Monitoring of Process Location and Dispersion Based on CUSUM Procedure and Generalized Likelihood Ratio in the Presence of Measurement Errors

In this paper, we propose a new methodology based on the combination of cumulative sum procedure and generalized likelihood ratio statistic for joint monitoring of the process location and dispersion. Then, we explore the effect of measurement errors on detecting ability of the proposed control chart when (i) the variance of measurement error is constant (ii) the variance of measurement error increases linearly as the level of the process mean increases. We also utilize multiple measurements on each sample point in order to decrease the adverse effects of measurement errors on the performance of the proposed control charts. Two numerical examples based on simulation studies are given to evaluate the ability of the proposed methods in terms of average run length, median run length, standard deviation of run length, and the first and third quantile points of the run length distribution (Q₁ and Q₃). Finally, a real life example is given to illustrate the application of the proposed method. Copyright © 2017 John Wiley & Sons, Ltd.