On guaranteed parameter estimation of a multiparameter linear regression process

Stochastic evolution equations with general white noise disturbance

This paper develops a Rao-Blackwellised particle filtering algorithm for blind system identification. The state space model under consideration uses a time-varying autoregressive (AR) model for the sources, and a time-varying finite impulse response (FIR) model for the channel. The multi-sensor measurements result from the convolution of the sources with the channels in the presence of additive noise. A numerical approximation to the optimal Bayesian solution for the nonlinear sequential state estimation problem is implemented using sequential Monte Carlo (SMC) methods. The Rao-Blackwellisation technique is applied to improve the efficiency of the particle filter by marginalizing out the AR and FIR coefficients from the joint posterior distribution. Simulation results are given to verify the performance of the proposed method.

Sequential estimation problems for. the exponential class of processes with independent increments have been dealt with in [3], [4] and, particularly, the concept of efficiency has been discussed in the sense of CRAMER-RAO inequalities. The aim of this note is to show that efficient sequential estimators turn out maximum likelihood estimators too. Using the methods suggested in [1] these estimators can be shown to be asymptotically normal!v distributed

Modelling and simulation: An overview

This paper considers the problem of sequential point estimation and fixed accuracy confidence set procedures of autoregressive parameters in a ρ-th order stationary autoregressive model. The sequential estimator proposed here is based on the least squares estimator and is shown to be risk efficient as the cost of estimation error tends to infinity. Furthermore, the proposed procedure for fixed-width confidence set is shown to be both asymptotically consistent and asymptotically efficient as the width approaches zero.

The problem of sequential estimation of the mean with quadratic loss and fixed cost per observation is considered within the Bayesian framework. A robust sequential procedure, not depending on the distributions of outcome variables and the prior, in the Bayes sequential estimation is investigated. In the present article, the second-order approximations to the expected sample size and the Bayes risk of the robust sequential procedure are obtained for the arbitrary distributions of outcome variables and the prior. The second-order efficiency is further discussed in an example.

We consider the sequential point estimation problem of the mean of a normal distribution N(μ, σ2) when the loss function is squared error plus linear cost. It is shown that a two-stage procedure has the asymptotic efficiency of which the order is higher than second order, provided the standard deviation has a known lower bound. We also give a higher than second-order approximation to the risk.

This paper presents a method for recursively estimating the static parameters of linear or nonlinear stochastic dynamic systems given the systems' inputs and outputs. The paper accomplishes this objective by combining polynomial chaos theory with maximum likelihood estimation. The parameter estimates are calculated in a recursive or iterative manner. To the best of the author's knowledge, this is the first paper to address recursive maximum likelihood parameter estimation using polynomial chaos theory. The proposed approach is demonstrated on two systems: a linear 2nd order system with unknown damping and natural frequency, and a nonlinear Van der Pol oscillator with an unknown nonlinear damping coefficient. Because this recursive estimator is applicable to nonlinear systems, the authors portend that this novel formulation will be useful for a broad range of estimation problems.

Data-driven control strategies for dynamical systems with unknown parameters are popular in theory and applications. An essential problem is to prevent stochastic linear systems becoming destabilized, due to the uncertainty of the decision-maker about the dynamical parameter. Two randomized algorithms are proposed for this problem, but the performance is not sufficiently investigated. Further, the effect of key parameters of the algorithms such as the magnitude and the frequency of applying the randomizations is not currently available. This work studies the stabilization speed and the failure probability of data-driven procedures. We provide numerical analyses for the performance of two methods: stochastic feedback, and stochastic parameter. The presented results imply that as long as the number of statistically independent randomizations is not too small, fast stabilization is guaranteed.

Abtsrcat The purely sequential minimum risk point estimation procedure for the unknown parameter θ (> 0) in a Uniform (0; θ) population has been discussed in this article. This is developed under a squared error loss plus a linear cost function of sampling. The unknown parameter θ is estimated by means of four different estimators in the stopping rule, where as in the loss function two different unbiased estimators of θ are proposed. The unbiased estimators are randomly stopped versions of Χ n: n and X¯ n in either loss function. Performances of such estimators are compared. Clearly, using a randomly stopped version of X¯ n would amount to some loss of in- formation when compared with a corresponding randomly stopped largest sample order statistic under both loss functions and the stopping rules. In this paper, we explore a novel approach to recover any loss of information by fine-tuning the loss function and then properly tailoring the associated sequential methodologies. We examine how the sequential risks of our newly proposed methodologies would compare with those associated with the ex- isting sequential estimators and illustrate our methodologies with the help of a real dataset.

Estimating “tree” logit models

The authors introduce a new estimation procedure, Augmented Kalman Filter with Continuous State and Discrete Observations (AKF(C-D)), for estimating diffusion models. This method is directly applicable to differential diffusion models without imposing constraints on the model structure or the nature of the unknown parameters. It provides a systematic way to incorporate prior knowledge about the likely values of unknown parameters and updates the estimates when new data become available. The authors compare AKF(C-D) empirically with five other estimation procedures, demonstrating AKF(C-D)'s superior prediction performance. As an extension to the basic AKF(C-D) approach, they also develop a parallel-filters procedure for estimating diffusion models when there is uncertainty about diffusion model structure or prior distributions of the unknown parameters.

An adaptive approach is presented for optimal estimation of a sampled stochastic process with finite-state unknown parameters. It is shown that for processes with an implicit generalized Markov property that the optimal (conditional mean) state estimates can be formed from (i) a set of optimal estimates based on known parameters, and (ii) a set of "learning" statistics which are recursively updated. The formulation thus provides a separation technique which simplifies the optimal solution of this class of nonlinear estimation problems. Examples of the separation technique are given for prediction of a non-Gaussian Markov process with unknown parameters and for filtering the state of a Gauss-Markov process with unknown parameters. General results are given on the convergence of optimal estimation systems operating in the presence of unknown parameters. Conditions are given under which a Bayes optimal (conditional mean) adaptive estimation system will converge in performance to an optimal system which is "told" the value of unknown parameters.

Sequential estimation for dependent observations with an application to non-standard autoregressive processes

We propose an algorithm for recursive estimation of structure and motion in rigid body perspective dynamic systems, based on the novel concept of continuous-differential matching constraints for the estimation of the velocity parameters. The parameter estimation procedure is fused with a continuous-discrete extended Kalman filter for the state estimation. Also, the structure and motion estimation processes are connected by a reprojection error constraint, where feedback of the structure estimates is used to recursively obtain corrections to the motion parameters, leading to more accurate estimates and a more robust performance of the method. The main advantages of the presented algorithm are that after initialization, only three observed object point correspondences between consecutive pairs of views are required for the sequential motion estimation, and that both the parameter update and the correction step are performed using linear constraints only. Simulated experiments are provided to demonstrate the performance of the method.