Stein–type shrinkage quantile estimation

The simultaneous asymptotic estimation theory of quantiles is considered for an arbitrary population. The Stein–type estimator and its positive version are considered. The relative merits of the proposed estimators are compared with those of the usual estimator using asymptotic quadratic distributional risk those of the usual estimator using asymptotic quadratic distributional risk under local alternatives. It is shown that both proposed estimators are asymptotically superior to the classical estimator. Further, it is demonstrated that the Stein-type estimator is dominated by its positive part     

Asymptotic normality of Powell’s kernel estimator

We establish asymptotic normality of Powell’s kernel estimator for the asymptotic covariance matrix of the quantile regression estimator for both i.i.d. and weakly dependent data. As an application, we derive the optimal bandwidth that minimizes the approximate mean squared error of the kernel estimator. We also derive the corresponding results to censored quantile regression.     

Bahadur–Kiefer representations for time dependent quantile processes

We define a time dependent empirical process based on n independent fractional Brownian motions and describe strong approximations to it by Gaussian processes. They lead to strong approximations and functional laws of the iterated logarithm for the quantile or inverse of this empirical process. They are obtained via time dependent Bahadur–Kiefer representations.     

Bootstrapping the Kaplan–Meier estimator on the whole line

This article is concerned with proving the consistency of Efron’s bootstrap for the Kaplan–Meier estimator on the whole support of a survival function. While previous works address the asymptotic Gaussianity of the Kaplan–Meier estimator without restricting time, we enable the construction of bootstrap-based time-simultaneous confidence bands for the whole survival function. Other practical applications include bootstrap-based confidence bands for the mean residual lifetime function or the Lorenz curve as well as confidence intervals for the Gini index. Theoretical results are complemented with a simulation study and a real data example which result in statistical recommendations.

     

Optimal linear estimator of origin-destination flows with redundant data

Suppose given a network endowed with a multiflow. We want to estimate some quantities connected with this multiflow, for instance the value of an s–t flow for one of the sources–sinks pairs s–t, but only measures on some arcs are available, at least on one s–t cocycle (set of arcs having exactly one endpoint in a subset X of vertices with s∈X and t?X). These measures, supposed to be unbiased, are random variables whose variances are known. How can we combine them optimally in order to get the best estimator of the value of the s–t flow?This question arises in practical situations when the OD matrix of a transportation network must be estimated. We will give a complete answer for the case when we deal with linear combinations, not only for the value of an s–t flow but also for any quantity depending linearly from the multiflow. Interestingly, we will see that the Laplacian matrix of the network plays a central role.     

An improved Afriat–Diewert–Parkan nonparametric production function estimator

Recent developments in the production frontier literature include nonparametric estimators with shape constraints. A few of these estimators rely on the Afriat inequalities to provide piecewise linear approximations to the production function/frontier. We show in this paper that these Afriat–Diewert–Parkan (ADP) estimators have deficiencies in the presence of moderate statistical noise including overfitting and a relatively high estimator variance. We propose new estimators with lower variance and a relatively low bias. We consider such alternative estimators based on (weighted) averages of random hinge functions with parameter restrictions. Small sample properties of the estimators are presented that show our new estimators outperform the existing ADP estimators when moderate to large amounts of noise are present.     

Ratio of Generalized Hill’s estimator and its asymptotic normality theory

We present a statistical process depending on a continuous time parameter τ whose each margin provides a Generalized Hill’s estimator. In this paper, the asymptotic normality of the finite-dimensional distributions of this family are completely characterized for τ > 1/2 when the underlying distribution function lies on the maximum domain of attraction. The ratio of two different margins of the statistical process characterizes entirely the whole domain of attraction. Its asymptotic normality is also studied. The results permit in general to build a new family of estimators for the extreme value index whose asymptotic properties can be easily derived. For example, we give a new estimate of the Weibull extreme value index and we study its consistency and its asymptotic normality.      

A residual–based error estimator for BEM–discretizations of contact problems

Summary. We develop an a posteriori error estimate for boundary element solutions of static contact problems without friction. The presented result is based on an error estimate for linear pseudodifferential equations and on a certain commutator property for pseudodifferential operators. A heuristic extension of the obtained error estimate to frictional contact problems is presented, too. Numerical examples indicate a good performance of the error estimator for both the frictionless and the frictional problem. Mathematics Subject Classification (1991):35J85, 65N38, 73T05Dedicated to Hans Grabmüller on the occasion of his 60th birthday     

The role of the isotonizing algorithm in Stein’s covariance matrix estimator

Covariance matrix estimation is central to many applications in statistics and allied fields. A useful estimator in this context was proposed by Stein which regularizes the sample covariance matrix by shrinking its eigenvalues together. This estimator can sometimes yield estimates of the eigenvalues that are negative or differ in order from the observed eigenvalues. In order to rectify this problem, Stein also proposed an ad hoc “isotonizing” procedure which pools together eigenvalue estimates in such a way that the original ordering and positivity of the estimates are enforced. From numerical studies, Stein’s “isotonized” estimator is known to have good risk properties in comparison with the maximum likelihood estimator. However, it remains unclear what role is played by the isotonizing procedure in the remarkable risk reductions achieved by Stein’s estimator. Through two distinct lines of investigations, it is established that Stein’s estimator without the isotonizing algorithm gives only modest risk reductions. In cases where the isotonizing algorithm is frequently used, however, Stein’s estimator can lead to significant risk reductions for certain domains of the parameter. In other cases, Stein’s estimator can even yield risk increases, such as when (1) the theoretical eigenvalues are well separated, and/or (2) when the sample size is moderate to large, leading to over-shrinkage.     

On designing H∞ fault estimator for switched nonlinear systems of neutral type

This paper deals with the problem of fault estimation for a class of switched nonlinear systems of neutral type. The nonlinearities are assumed to satisfy global Lipschitz conditions and appear in both the state and measured output equations. By employing a switched observer-based fault estimator, the problem is formulated as an H_∞ filtering problem. Sufficient delay-dependent existence conditions of the H_∞ fault estimator (H_∞-FE) are given in terms of certain matrix inequalities based on the average dwell time approach. In addition, by using cone complementarity algorithm, the solutions to the observer gain matrices are obtained by solving a set of linear matrix inequalities (LMIs). A numerical example is provided to demonstrate the effectiveness of the proposed approach.     

Rate of strong consistency of the maximum quasi-likelihood estimator in quasi-likelihood nonlinear models

Quasi-likelihood nonlinear models （QLNM） include generalized linear models as a special case. Under some regularity conditions, the rate of the strong consistency of the maximum quasi-likelihood estimation （MQLE） is obtained in QLNM. In an important case, this rate is O（n-^1/2（loglogn）^1/2）, which is just the rate of LIL of partial sums for i.i.d variables, and thus cannot be improved anymore.     

Asymptotic ɛ -admissibility of the sample variance as an estimator of the population variance

A quantitative estimate is given of the robustness of the characterization of the distribution with a density by the property of asymptotic -admissibility of the sample variance as an estimator of the population variance with a quadratic loss function.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 61, pp. 75–83, 1976.     

A residual-based a posteriori error estimator for a two-dimensional fluid–solid interaction problem

In this paper, we develop an a posteriori error analysis of a mixed finite element method for a fluid–solid interaction problem posed in the plane. The media are governed by the acoustic and elastodynamic equations in time-harmonic regime, respectively, and the transmission conditions are given by the equilibrium of forces and the equality of the normal displacements of the solid and the fluid. The coupling of primal and dual-mixed finite element methods is applied to compute both the pressure of the scattered wave in the linearized fluid and the elastic vibrations that take place in the elastic body. The finite element subspaces consider continuous piecewise linear elements for the pressure and a Lagrange multiplier defined on the interface, and PEERS for the stress and rotation in the solid domain. We derive a reliable and efficient residual-based a posteriori error estimator for this coupled problem. Suitable auxiliary problems, the continuous inf-sup conditions satisfied by the bilinear forms involved, a discrete Helmholtz decomposition, and the local approximation properties of the Clément interpolant and Raviart–Thomas operator are the main tools for proving the reliability of the estimator. Then, Helmholtz decomposition, inverse inequalities, and the localization technique based on triangle-bubble and edge-bubble functions are employed to show the efficiency. Finally, some numerical results confirming the reliability and efficiency of the estimator are reported.     

An efficient asymptotically correct error estimator for collocation solutions to singular index-1 DAEs

A computationally efficient a posteriori error estimator is introduced and analyzed for collocation solutions to linear index-1 DAEs (differential-algebraic equations) with properly stated leading term exhibiting a singularity of the first kind. The procedure is based on a modified defect correction principle, extending an established technique from the context of ordinary differential equations to the differential-algebraic case. Using recent convergence results for stiffly accurate collocation methods, we prove that the resulting error estimate is asymptotically correct. Numerical examples demonstrate the performance of this approach. To keep the presentation reasonably self-contained, some arguments from the literature on DAEs concerning the decoupling of the problem and its discretization, which is essential for our analysis, are also briefly reviewed. The appendix contains a remark about the interrelation between collocation and implicit Runge-Kutta methods for the DAE case.     

Local linear estimator for stochastic differential equations driven by α-stable Lévy motions

We study the local linear estimator for the drift coefficient of stochastic differential equations driven by α-stable Lévy motions observed at discrete instants. Under regular conditions, we derive the weak consistency and central limit theorem of the estimator. Compared with Nadaraya-Watson estimator, the local linear estimator has a bias reduction whether the kernel function is symmetric or not under different schemes. A simulation study demonstrates that the local linear estimator performs better than Nadaraya-Watson estimator, especially on the boundary.     

Local linear estimator for stochastic diferential equations driven by α-stable Lvy motions

We study the local linear estimator for the drift coefcient of stochastic diferential equations driven byα-stable L′evy motions observed at discrete instants.Under regular conditions,we derive the weak consistency and central limit theorem of the estimator.Compared with Nadaraya-Watson estimator,the local linear estimator has a bias reduction whether the kernel function is symmetric or not under diferent schemes.A simulation study demonstrates that the local linear estimator performs better than Nadaraya-Watson estimator,especially on the boundary.     

Event-triggered non-fragile state estimator design for interval type-2 Takagi–Sugeno fuzzy systems with bounded disturbances

In this paper, the state estimator design problem of interval type-2 Takagi–Sugeno fuzzy systems suffering from bounded disturbances is studied. To enhance the resilience of the estimator, a non-fragile design scheme is proposed to tackle the estimator gain variations. Meanwhile, an event-triggered communication mechanism is introduced for relieving the transmission burden over networks. To settle down the non-fragile estimator design issue subject to bounded disturbances and event-induced error, we propose a new definition of quadratic boundedness via the multiple Lyapunov functions. Based on this definition, a novel co-design method of estimator and event generator for fuzzy system models in the presence of both measurable and immeasurable premise variables is presented. In virtue of quadratic boundedness framework, less conservative conditions of the existence and quadratic stability of the fuzzy estimators are obtained, and the upper bound of estimation error is given explicitly. The desired estimator gains are determined by convex optimization technique using slack matrices. Two illustrative examples are exploited to validate the availability and superiority of the addressed design approach.     

A general estimator for the right endpoint with an application to supercentenarian women’s records

We extend the setting of the right endpoint estimator introduced in Fraga Alves and Neves (Statist. Sinica 24, 1811–1835, 2014) to the broader class of light-tailed distributions with finite endpoint, belonging to some domain of attraction induced by the extreme value theorem. This stretch enables a general estimator for the finite endpoint, which does not require estimation of the (supposedly non-positive) extreme value index. A new testing procedure for selecting max-domains of attraction also arises in connection with the asymptotic properties of the general endpoint estimator. The simulation study conveys that the general endpoint estimator is a valuable complement to the most usual endpoint estimators, particularly when the true extreme value index stays above ?1/2, embracing the most common cases in practical applications. An illustration is provided via an extreme value analysis of supercentenarian women data.