Asymptotically Efficient Estimation of the Derivative of the Invariant Density

The problem of estimation of the derivative of the invariant density is considered for a one-dimensional ergodic diffusion process. The lower minimax bound on the L ²-type risk of all estimators is proposed and an asymptotically efficient (up to the constant) in the sense of this bound kernel-type estimator is constructed.     

Abstract Inverse Estimation with Application to Deconvolution on Locally Compact Abelian Groups

Recovery of the unknown parameter in an abstract inverse estimation model can be based on regularizing the inverse of the operator defining the model. Such regularized-inverse type estimators are constructed with the help of a version of the spectral theorem due to Halmos, after suitable preconditioning. A lower bound to the minimax risk is obtained exploiting the van Trees inequality. The proposed estimators are shown to be asymptotically optimal in the sense that their risk converges to zero, as the sample size tends to infinity, at the same rate as this lower bound. The general theory is applied to deconvolution on locally compact Abelian groups, including both indirect density and indirect regression function estimation.     

On Mean Change-Point Detection Based on Empirical Euclidean Likelihood

Mean chage-point detection is the groundwork in statistics. The trimmed empirical Euclidean likelihood ratio function is constructed based on the features of change-point in mean model. And the explicit expression is derived. The null limit distribution of the test statistic is investigated with extreme value distribution. And the change-point detection is made. if the change-point exists, it's location and consistency are discussed. Simulations and real analysis of Nile River data show that our proposed method is practicable and effective.     

Empirical likelihood inference for estimating equation with missing data

In this article, empirical likelihood inference for estimating equation with missing data is considered. Based on the weighted-corrected estimating function, an empirical log-likelihood ratio is proved to be a standard chi-square distribution asymptotically under some suitable conditions. This result is different from those derived before. So it is convenient to construct confidence regions for the parameters of interest. We also prove that our proposed maximum empirical likelihood estimator θ is asymptotically normal and attains the semiparametric efficiency bound of missing data. Some simulations indicate that the proposed method performs the best.     

Robust H ∞ Filtering for Uncertain Time-Delay Systems: Matrix Inequality Approach

In this paper, a new robust H _∞ filtering problem for uncertain time-delay systems is considered. Based on the Lyapunov method, a design criterion of the robust H _∞ filter, in which the filtering process remains asymptotically stable for all admissible uncertainties and the transfer function from the disturbance inputs to error state outputs satisfies the prespecified H _∞ norm upper bound constraint, is derived in terms of matrix inequalities. The inequalities can be solved easily by efficient convex optimization algorithms. A numerical example is included to illustrate the validity of the proposed design approach.     

Generalized faber polynomials and an optimal error recovery algorithm

This paper deals with the problem of approximate evaluation of a certain class of analytic functions. The choice of this class is motivated by the problem of the summation of moment sequences. By assuming that the information about the function is given by its Taylor coefficients, we are able to establish a lower bound on the error of an arbitrary algorithm. We present also an algorithm whose error is asymptotically at most twice the lower bound, thereby showing that our estimate is asymptotically sharp.     

Stationary Distribution Function Estimation for Ergodic Diffusion Process

The problem of nonparametric stationary distribution function estimation by the observations of an ergodic diffusion process is considered. The local asymptotic minimax lower bound on the risk of all the estimators is found and it is proved that the empirical distribution function is asymptotically efficient in the sense of this bound.     

Stationary distribution function estimation for ergodic diffusion process

The problem of nonparametric stationary distribution function estimation by the observation of an ergodic diffusion process is considered. The local asymptotic minimax lower bound on the risk of all the estimators is found, and it is proved that the empirical distribution function is asymptotically efficient in the sense of this bound.     

Nonparametric additive model-assisted estimation for survey data

An additive model-assisted nonparametric method is investigated to estimate the finite population totals of massive survey data with the aid of auxiliary information. A class of estimators is proposed to improve the precision of the well known Horvitz-Thompson estimators by combining the spline and local polynomial smoothing methods. These estimators are calibrated, asymptotically design-unbiased, consistent, normal and robust in the sense of asymptotically attaining the Godambe-Joshi lower bound to the anticipated variance. A consistent model selection procedure is further developed to select the significant auxiliary variables. The proposed method is sufficiently fast to analyze large survey data of high dimension within seconds. The performance of the proposed method is assessed empirically via simulation studies.     

The set of stable switching sequences for discrete-time linear switched systems

In this paper we study the characterization of the asymptotical stability for discrete-time switched linear systems. We first translate the system dynamics into a symbolic setting under the framework of symbolic topology. Then by using the ergodic measure theory, a lower bound estimate of Hausdorff dimension of the set of asymptotically stable sequences is obtained. We show that the Hausdorff dimension of the set of asymptotically stable switching sequences is positive if and only if the corresponding switched linear system has at least one asymptotically stable switching sequence. The obtained result reveals an underlying fundamental principle: a switched linear system either possesses uncountable numbers of asymptotically stable switching sequences or has none of them, provided that the switching is arbitrary. We also develop frequency and density indexes to identify those asymptotically stable switching sequences of the system.     

Robust Stability Criteria for Uncertain Neutral Systems with Interval Time-Varying Delay

In this paper, we consider the problem of robust stability of a class of linear uncertain neutral systems with interval time-varying delay under (i) nonlinear perturbations in state, and (ii) time-varying parametric uncertainties using Lyapunov-Krasovskii approach. By constructing a candidate Lyapunov-Krasovskii (LK) functional, that takes into account the delay-range information appropriately, less conservative robust stability criteria are proposed in terms of linear matrix inequalities (LMI) to compute the maximum allowable bound for the delay-range within which the uncertain neutral system under consideration remains asymptotically stable. The reduction in conservatism of the proposed stability criterion over recently reported results is attributed to the fact that time-derivative of the LK functional is bounded tightly without neglecting any useful terms using a minimal number of slack matrix variables. The analysis, subsequently, yields a stability condition in convex LMI framework, that can be solved non-conservatively at boundary conditions using standard LMI solvers. The effectiveness of the proposed stability criterion is demonstrated through standard numerical examples.     

An Analytic Expression for the Distribution of the Generalized Shiryaev–Roberts Diffusion

We consider the quickest change-point detection problem where the aim is to detect the onset of a pre-specified drift in “live”-monitored standard Brownian motion; the change-point is assumed unknown (nonrandom). The topic of interest is the distribution of the Generalized Shryaev–Roberts (GSR) detection statistic set up to “sense” the presence of the drift. Specifically, we derive a closed-form formula for the transition probability density function (pdf) of the time-homogeneous Markov diffusion process generated by the GSR statistic when the Brownian motion under surveillance is “drift-free”, i.e., in the pre-change regime; the GSR statistic’s (deterministic) nonnegative headstart is assumed arbitrarily given. The transition pdf formula is found analytically, through direct solution of the respective Kolmogorov forward equation via the Fourier spectral method to achieve separation of the spacial and temporal variables. The obtained result generalizes the well-known formula for the (pre-change) stationary distribution of the GSR statistic: the latter’s stationary distribution is the temporal limit of the distribution sought in this work. To conclude, we exploit the obtained formula numerically and briefly study the pre-change behavior of the GSR statistic versus three factors: (a) drift-shift magnitude, (b) time, and (c) the GSR statistic’s headstart.     

Based on interval type-2 fuzzy-neural network direct adaptive sliding mode control for SISO nonlinear systems

In this paper, a novel direct adaptive interval type-2 fuzzy-neural tracking control equipped with sliding mode and Lyapunov synthesis approach is proposed to handle the training data corrupted by noise or rule uncertainties for nonlinear SISO nonlinear systems involving external disturbances. By employing adaptive fuzzy-neural control theory, the update laws will be derived for approximating the uncertain nonlinear dynamical system. In the meantime, the sliding mode control method and the Lyapunov stability criterion are incorporated into the adaptive fuzzy-neural control scheme such that the derived controller is robust with respect to unmodeled dynamics, external disturbance and approximation errors. In comparison with conventional methods, the advocated approach not only guarantees closed-loop stability but also the output tracking error of the overall system will converge to zero asymptotically without prior knowledge on the upper bound of the lumped uncertainty. Furthermore, chattering effect of the control input will be substantially reduced by the proposed technique. To illustrate the performance of the proposed method, finally simulation example will be given.     

Multi-stage portfolio selection problem with dynamic stochastic dominance constraints

We study the multi-stage portfolio selection problem where the utility function of an investor is ambiguous. The ambiguity is characterized by dynamic stochastic dominance constraints, which are able to capture the dynamics of the random return sequence during the investment process. We propose a multi-stage dynamic stochastic dominance constrained portfolio selection model, and use a mixed normal distribution with time-varying weights and the K-means clustering technique to generate a scenario tree for the transformation of the proposed model. Based on the scenario tree representation, we derive two linear programming approximation problems, using the sampling approach or the duality theory, which provide an upper bound approximation and a lower bound approximation for the original nonconvex problem. The upper bound is asymptotically tight with infinitely many samples. Numerical results illustrate the practicality and efficiency of the proposed new model and solution techniques.

     

Change-point detection of failure mechanism for electronic devices based on Arrhenius model

In this paper, we consider the abrupt change-point and continuous change-point failure mechanisms in the accelerated life tests of electronic devices and propose corresponding detection methods based on the Weibull distribution and the Arrhenius model. The new methods can handle the detection of failure mechanism change-point with censored data and small samples. Comprehensive simulation studies are conducted to test the performance of the proposed methods and investigate the influences of the change location, the parameters and the censored ratios. Results of the simulation studies show that the proposed methods perform well in terms of type I errors and powers, and the change of parameters has a stronger impact on the performance of the new methods compared to the change location and censored ratios. In the end, we provide two real-world examples. The new class H insulation example shows that the proposed methods can predict the reliability reasonably well by detecting the change point, and the metal oxide semiconductor transistor example shows that the new methods can greatly reduce the experimental time and provide accurate reliability predictions.     

On the lower bound in second order estimation for Poisson processes: Asymptotic efficiency

In the estimation problem of the mean function of an inhomogeneous Poisson process there is a class of kernel type estimators that are asymptotically efficient alongside with the empirical mean function. We start by describing such a class of estimators which we call first order efficient estimators. To choose the best one among them we prove a lower bound that compares the second order term of the mean integrated square error of all estimators. The proof is carried out under the assumption on the mean function Λ(·) that Λ(τ) = S, where S is a known positive number. In the end, we discuss the possibility of the construction of an estimator which attains this lower bound, thus, is asymptotically second order efficient.     

Efficient Density Estimation for Ergodic Diffusion Processes

The problem of asymptotically efficient estimation of the density of invariant measure of a diffusion process is considered. The efficient estimator is defined with the help of the minimax lower bound on the risk of all estimators. We show that the local–time and kernel–type estimators are asymptotically efficient for the loss functions with polynomial majorants. The asymptotic behavior of a wide class of unbiased estimators with the same limit variances is also discussed. This revised version was published online in June 2006 with corrections to the Cover Date.     

Exploring the latent segmentation space for the assessment of multiple change-point models

This paper addresses the retrospective or off-line multiple change-point detection problem. Multiple change-point models are here viewed as latent structure models and the focus is on inference concerning the latent segmentation space. Methods for exploring the space of possible segmentations of a sequence for a fixed number of change points may be divided into two categories: (i) enumeration of segmentations, (ii) summary of the possible segmentations in change-point or segment profiles. Concerning the first category, a dynamic programming algorithm for computing the top $N$ most probable segmentations is derived. Concerning the second category, a forward-backward dynamic programming algorithm and a smoothing-type forward-backward algorithm for computing two types of change-point and segment profiles are derived. The proposed methods are mainly useful for exploring the segmentation space for successive numbers of change points and provide a set of assessment tools for multiple change-point models that can be applied both in a non-Bayesian and a Bayesian framework. We show using examples that the proposed methods may help to compare alternative multiple change-point models (e.g. Gaussian model with piecewise constant variances or global variance), predict supplementary change points, highlight overestimation of the number of change points and summarize the uncertainty concerning the position of change points.     

A generalization of the growth curve model which allows missing data

This study presents methods for estimating and testing hypotheses about linear functions of the unknown parameters in a generalization of the growth curve model which allows missing data. The estimators proposed are best asymptotically normal (BAN). A testing method for large samples is described which uses a test criterion given in general form by Wald. The asymptotic null distribution of the test statistic is a central chi-square variable. A BAN estimator of a linear vector function of the unknown parameters of the expectation model and consistent estimators of the variance-covariance parameters are required for computation.     

On nonlinear TAR processes and threshold estimation

We consider the problem of threshold estimation for autoregressive time series with a ??space switching?? in the situation when the regression is nonlinear and the innovations have a smooth, possibly non-Gaussian, probability density. Assuming that the unknown threshold parameter is sampled from a continuous positive prior density, we find the asymptotic distribution of the Bayes estimator. As is usual in the singular estimation problems, the sequence of Bayes estimators is asymptotically efficient, attaining the minimax risk lower bound.