Maximum-likelihood estimation of the parameters of a multivariate normal distribution

This paper provides an exposition of alternative approaches for obtaining maximum- likelihood estimators (MLE) for the parameters of a multivariate normal distribution under different assumptions about the parameters. A central focus is on two general techniques, namely, matrix differentiation and matrix transformations. These are systematically applied to derive the MLE of the means under a rank constraint and of the covariances when there are missing observations. Derivations using induction and inequalities are also included to illustrate alternative methods. Other examples, such as a connection with an econometric model, are included. Although the paper is primarily expository, some of the proofs are new.     

On inf-sup conditions and projections in the theory of mixed finite-element methods

We indicate constraints on the space of finite elements providing the validity of discrete inf-sup conditions and the existence of projections specific for mixed finite-element methods. We consider both conformal and nonconformal approximations. We suggest a definition of special projections onto the vector space of finite elements which provides their existence under quite general conditions without determining the degrees of freedom of the elements.     

Nonparametric estimation of the location and scale parameters based on density estimation

Summary By representing the location and scale parameters of an absolutely continuous distribution as functionals of the usually unknown probability density function, it is possible to provide estimates of these parameters in terms of estimates of the unknown functionals. Using the properties of well-known methods of density estimates, it is shown that the proposed estimates possess nice large sample properties and it is indicated that they are also robust against dependence in the sample. The estimates perform well against other estimates of location and scale parameters.     

Efficient on-line estimation of autoregressive parameters

New procedures for estimating autoregressive parameters in AR(m) models are proposed. The proposed method allows for incorporation of auxiliary information into the estimation process and produces estimation procedures, which are consistent and asymptotically efficient under certain regularity conditions. Also, these procedures are naturally on-line and do not require storing all the data. Theoretical results are presented in the case when m = 1. Two important particular cases are considered in detail: linear procedures and likelihood procedures with the LS truncations. A specific example is also presented to briefly discuss some practical aspects of applications of the procedures of this type.     

Some aspects in practical estimation of groundwater contaminant concentrations

Spatial distribution of groundwater contaminant concentration has special characteristics such as approximate symmetric profile, for example, in the transversal direction to groundwater flow direction, a certain ratio in directional propagation distances, etc. To obtain a geophysically appropriate semivariogram which is a key factor in estimation of groundwater contaminant concentration at desired locations, these special characteristics should be considered. Specifically, the concepts of symmetry and ratio are considered in this paper. By applying these two concepts, significant improvement of semivariograms, estimation variances, and final estimation results compared with the ones by conventional approaches which usually do not account for symmetry and ratio are shown using field experimental data.     

Simultaneous estimation of parameters in exponential families

Summary The paper considers estimation of the natural parameter vector or the mean vector from independent distributions each belonging to the one-parameter discrete or absolutely continuous exponential family. The usual estimators (maximum likelihood, minimum variance unbiased or best invariant) are improved simultaneously under various weighted squared error losses. Research supported by the NSF Grant Number MCS-8202116.     

Bayes minimax estimation of multiple Poisson parameters

For the p-variate Poisson mean, under the sum of weighted squared error losses, weights being reciprocals of variances, a class of proper Bayes minimax estimates dominating the usual estimate, namely the sample mean is produced. An example is given to illustrate this. The interrelation of our results with those of Clevenson and Zidek is pointed out.     

Simultaneous estimation of negative binomial dispersion parameters

Simultaneous estimators of the dispersion parameters in m negative binomial distributions with known success probabilities are shown to dominate the usual estimator under squared error loss by shrinking it toward either a fixed or a data-driven point.     

Sequential estimation of the parameters of diffusion processes

For the parameter of a diffusion process(t), satisfying the stochastic differential equation d(t)=f (t,)dt+dw(l), we propose an effective sequential estimation plan with an unbiased and normally distributed estimate. The proposed sequential plan is discussed in detail for the example of a process (t) having a linear stochastic differential.Translated from Matematicheskie Zametki, Vol. 12, No. 5, pp. 627–638, November, 1972.In conclusion the author wishes to express his deep gratitude to A. N. Shiryaev for formulating the problem and for useful observations     

Double shrinkage estimation of ratio of scale parameters

The problems of estimating ratio of scale parameters of two distributions with unknown location parameters are treated from a decision-theoretic point of view. The paper provides the procedures improving on the usual ratio estimator under strictly convex loss functions and the general distributions having monotone likelihood ratio properties. In particular,double shrinkage improved estimators which utilize both of estimators of two location parameters are presented. Under order restrictions on the scale parameters, various improvements for estimation of the ratio and the scale parameters are also considered. These results are applied to normal, lognormal, exponential and pareto distributions. Finally, a multivariate extension is given for ratio of covariance matrices.     

Identification of plastic material parameters with error estimation

In recent years, inverse analysis has become a common approach to typical engineering problems such as model identification. In this contribution, the inverse problem is discussed in light of taking experimental uncertainties into account. This involves in particular the propagation of experimental errors and the analysis of the sensitivity of the model response to variations in the model parameters to be determined. The method is applied to an elasto-viscoplastic material model which is used in the context of electromagnetic high-speed forming. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On estimation of parameters for spatial autoregressive model

In the paper asymptotic properties of the estimated coefficients of multi-indexed autoregressive model are investigated. Considering Least Squares estimates we use some kind of self-normalization and obtain limit normal law independent of unknown parameters. In the proof of this result we use recent result on the Central Limit Theorem for dependent summands.      

Circuit realization,chaos synchronization and estimation of parameters of a hyperchaotic system with unknown parameters

In this article, the adaptive chaos synchronization technique is implemented by an electronic circuit and applied to the hyperchaotic system proposed by Chen et al. We consider the more realistic and practical case where all the parameters of the master system are unknowns. We propose and implement an electronic circuit that performs the estimation of the unknown parameters and the updating of the parameters of the slave system automatically, and hence it achieves the synchronization. To the best of our knowledge, this is the first attempt to implement a circuit that estimates the values of the unknown parameters of chaotic system and achieves synchronization. The proposed circuit has a variety of suitable real applications related to chaos encryption and cryptography. The outputs of the implemented circuits and numerical simulation results are shown to view the performance of the synchronized system and the proposed circuit.     

The nonsteady-state planar oscillations of a plate and oscillator on a two-layer base

We consider a plane dynamic contact problem for an inhomogeneous base of the following form: a soft layer on a rigid layer of an elastic half-plane. The layer is represented by a Winkler model, corresponding to the long-wave asymptotics of the equations of elasticity theory. The problem is reduced to a system of integro-differential equations that is solved numerically. We present the results of the computations of dynamic characteristics describing the oscillations of a rigid body and an oscillator on this base.Translated fromDinamicheskie Sistemy, No. 6, 1987, pp. 64–68.     

Identification and estimation of parameters defining a class of hybrid systems

In this paper, we consider the problem of parameter estimation in an air brake system. In an air brake system, the pressure of air in the brake chamber and the displacement of the pushrod and their derivatives form a set of states that characterize the system. The position of a valve or mass flow rate of air is an input and the pressure is the measured variable or the output. The pressure acting on the pushrod of the brake chamber causes motion, and the mode in which the system operates depends on the displacement of the pushrod. The mode-dependent nature of the system is a result of different sets of spring compliances associated with the piston in different ranges of its displacement. The mode to mode transition in the air brake system is governed by a parameter which is the clearance between the brake pads and the drum. The clearance between the brake pads and the drum can vary due to a variety of factors — for example, brake pad wear or brake fade. In these applications, characterizing the transition from one mode to another requires a lot of constitutive assumptions, and it can be difficult to calibrate the parameters associated with the constitutive assumptions. We therefore treat the air brake system as a system in which the parameter governing the transition from one mode to another (clearance between the brake pads and the drum) is not known exactly. Clearly, this parameter dictates the time delay and lag between the command and delivery of the brake torque at the wheels and affects the stopping distance of the vehicles considerably. The problem of identification considered in this paper is as follows. Suppose that the pressure of the fluid were to be measured and that the motion of the piston is not measured. Is it possible to estimate the final displacement of the piston without knowing the parameters that govern the system to transition from one mode to another?