On construction of improved estimators in multiple-design multivariate linear models under general restriction

Consider a set ofp equations Y_i = X_ii + _i,i=1,...,p, where the rows of the random error matrix (₁,..., _p):n × p are mutually independent and identically distributed with ap-variate distribution functionF(x) having null mean and finite positive definite variance-covariance matrix . We are mainly interested in an improvement upon a feasible generalized least squares estimator (FGLSE) for = ( ₁ ,..., _p ) when it is a priori suspected thatC=c_o may hold. For this problem, Saleh and Shiraishi (1992,Nonparametric Statistics and Related Topics (ed. A. K. Md. E. Saleh), 269–279, North-Holland, Amsterdam) investigated the property of estimators such as the shrinkage estimator (SE), the positive-rule shrinkage estimator (PSE) in the light of their asymptotic distributional risks associated with the Mahalanobis loss function. We consider a general form of estimators and give a sufficient condition for proposed estimators to improve on FGLSE with respect to their asymptotic distributional quadratic risks (ADQR). The relative merits of these estimators are studied in the light of the ADQR under local alternatives. It is shown that the SE, the PSE and the Kubokawa-type shrinkage estimator (KSE) outperform the FGLSE and that the PSE is the most effective among the four estimators considered underC=c_o. It is also observed that the PSE and the KSE fairly improve over the FGLSE. Lastly, the construction of estimators improved on a generalized least squares estimator is studied, assuming normality when is known.     

On nonlinear weighted total least squares parameter estimation problem for the three-parameter Weibull density

The three-parameter Weibull density function is widely employed as a model in reliability and lifetime studies. Estimation of its parameters has been approached in the literature by various techniques, because a standard maximum likelihood estimate does not exist. In this paper we consider the nonlinear weighted total least squares fitting approach. As a main result, a theorem on the existence of the total least squares estimate is obtained, as well as its generalization in the total _lq$l_q$ norm (q?1$q?1$). Some numerical simulations to support the theoretical work are given.     

New algorithm for optimal parameter estimation with linear constraints

This paper presents a new algorithm for optimal parameter estimation problems with linear constraints. The algorithm developed is based on least absolute-value approximations. The problem is solved first using a least-error-square technique, where we add to the cost function the equality constraints via Lagrange multipliers, to obtain a good estimate for the residuals of the measurements, having gained this information, we choose a number of measurements with the smallest residuals. This number equals the number of parameters to be estimated minus the number of constraints. Using these measurements together with the constraints, we obtain a number of observations equal to the number of parameters to be estimated. By using this technique, we show that there is no need to either iterate or use linear programming to obtain the estimation.This work was supported by the Natural Sciences and Engineering Research Council of Canada, Grant A4146.     

A review of existence criteria for parameter estimation of the Michaelis–Menten regression model

In this paper we consider the least squares (LS) and total least squares (TLS) problems for a Michaelis–Menten enzyme kinetic model f(x; a, b) = ax/(b + x), a, b > 0. In various applied research such as biochemistry, pharmacology, biology and medicine there are lots of different applications of this model. We will systematize some of our results pertaining to the existence of the LS and TLS estimate, which were proved in Hadeler et al. (Math Method Appl Sci 30:1231–1241, 2007) and Jukić et al. (J Comput Appl Math 201:230–246, 2007). Finally, we suggest a choice of good initial approximation and give one numerical example.      

Parameter estimation in linear static systems based on weighted least-absolute value estimation

In this paper, we present a new method for estimating a parameter vector of which measurements are carried out; however, these measurements are subjected to noise. First, we briefly consider least-square estimation of such vector to obtain some well-known results. Then, we proceed to formulate the problem in the least-absolute value (LAV) sense and show that we can obtain a set of overdetermined equations for the components of the unknown vector. These equations are solved using the least-square approach to ascertain which points give the least residuals. Having gained that information, we set to zero a number of residuals equal to the rank of the matrixH. Let this rank bek; then, the number of points which satisfy the LAV solution identically isk; this is a requirement that the LAV solution must satisfy (Refs. 1, 2). Several examples are presented in the paper.This work was supported by the Natural Science and Engineering Research Council of Canada, Grant A4146.     

Least squares and Chebyshev fitting for parameter estimation in ODEs

We discuss the problem of determining parameters in mathematical models described by ordinary differential equations. This problem is normally treated by least squares fitting. Here some results from nonlinear mean square approximation theory are outlined which highlight the problems associated with nonuniqueness of global and local minima in this fitting procedure. Alternatively, for Chebyshev fitting and for the case of a single differential equation, we extend and apply the theory of [17, 18] which ensures a unique global best approximation. The theory is applied to two numerical examples which show how typical difficulties associated with mean square fitting can be avoided in Chebyshev fitting.This paper is presented as an outcome of the LMS Durham Symposium convened by Professor C.T.H. Baker on 4th-14th July 1992 with support from the SERC under Grant reference number GR/H03964.     

Parameter estimation in two-type continuous-state branching processes with immigration

We study the parameter estimation of two-type continuous-state branching processes with immigration based on low frequency observations at equidistant time points. The ergodicity of the processes is proved. The estimators are based on the minimization of a sum of squared deviation about conditional expectations. We also establish the strong consistency and central limit theorems of the conditional least squares estimators and the weighted conditional least squares estimators of the drift and diffusion coefficients based on low frequency observations.     

On the credibility of insurance claim frequency: Generalized count models and parametric estimators

We analyze the concept of credibility in claim frequency in two generalized count models–Mittag-Leffler and Weibull count models–which can handle both underdispersion and overdispersion in count data and nest the commonly used Poisson model as a special case. We find evidence, using data from a Danish insurance company, that the simple Poisson model can set the credibility weight to one even when there are only three years of individual experience data resulting from large heterogeneity among policyholders, and in doing so, it can thus break down the credibility model. The generalized count models, on the other hand, allow the weight to adjust according to the number of years of experience available. We propose parametric estimators for the structural parameters in the credibility formula using the mean and variance of the assumed distributions and a maximum likelihood estimation over a collective data. As an example, we show that the proposed parameters from Mittag-Leffler provide weights that are consistent with the idea of credibility. A simulation study is carried out investigating the stability of the maximum likelihood estimates from the Weibull count model. Finally, we extend the analyses to multidimensional lines and explain how our approach can be used in selecting profitable customers in cross-selling; customers can now be selected by estimating a function of their unknown risk profiles, which is the mean of the assumed distribution on their number of claims.     

On the Convergence of the Generalized Linear Least Squares Algorithm

This paper considers the issue of parameter estimation for biomedical applications using nonuniformly sampled data. The generalized linear least squares (GLLS) algorithm, first introduced by Feng and Ho (1993), is used in the medical imaging community for removal of bias when the data defining the model are correlated. GLLS provides an efficient iterative linear algorithm for the solution of the non linear parameter estimation problem. This paper presents a theoretical discussion of GLLS and introduces use of both Gauss Newton and an alternating Gauss Newton for solution of the parameter estimation problem in nonlinear form. Numerical examples are presented to contrast the algorithms and emphasize aspects of the theoretical discussion. AMS subject classification (2000) 65F10.R. A. Renaut: This work was partially supported by the Arizona Center for Alzheimer’s Disease Research, by NIH grant EB 2553301 and for the second author by NSF CMG-02223.Received December 2003. Revised November 2004. Communicated by Lars Eldén.     

Minimum information estimation of allocation models of different sizes

For allocation models consisting of n = 10, 6 and 4 equations, constrained generalized least squares coefficient estimates are compared with those obtained from the minimum information (MI) criterion in the sense of statistical information theory. The MI estimates are more efficient for n = 10. The bootstrap procedure is illustrated for the assessment of the variability of these estimates.     

广义线性模型中一类推广的拟似然估计与异方差性诊断检验

本文考虑了随机设计情形下一类普通的异方差回归模型，在这个模型中，假定回归函数与方差函数之间的关系服从推广的广义非线性模型，该模型在实际中很常见，广义线性模型便是其特例，首先，我们导出了均值函数的局部加权拟似然估计，然后，用它来得到方差函数的估计，并且证明了这些估计有较好的性质，最后，建立了异方差检验统计量，文中的方法很吸引人。     

Parameter estimation of varying coefficients structural EV model with time series

In this paper, the parameters of a p-dimensional linear structural EV (error-in-variable) model are estimated when the coefficients vary with a real variable and the model error is time series. The adjust weighted least squares (AWLS) method is used to estimate the parameters. It is shown that the estimators are weakly consistent and asymptotically normal, and the optimal convergence rate is also obtained. Simulations study are undertaken to illustrate our AWLSEs have good performance.     

Efficient robust estimation of parameter in the random censorship model

For an open set Θ of ^k, let \s{P_θ: θ Θ\s} be a parametric family of probabilities modeling the distribution of i.i.d. random variables X₁,…, X_n. Suppose X_i's are subject to right censoring and one is only able to observe the pairs (min(X_i, Y_i), [X_i Y_i]), i = 1,…, n, where [A] denotes the indicator function of the event A, Y₁,…, Y_n are independent of X₁,…, X_n and i.i.d. with unknown distribution Q₀. This paper investigates estimation of the value θ that gives a fitted member of the parametric family when the distributions of X₁ and Y₁ are subject to contamination. The constructed estimators are adaptive under the semi-parametric model and robust against small contaminations: they achieve a lower bound for the local asymptotic minimax risk over Hellinger neighborhoods, in the Hájel—Le Cam sense. The work relies on Beran (1981). The construction employs some results on product-limit estimators.     

Double Shrinkage Estimation of Common Coefficients in Two Regression Equations with Heteroscedasticity

The problem of estimating the common regression coefficients is addressed in this paper for two regression equations with possibly different error variances. The feasible generalized least squares (FGLS) estimators have been believed to be admissible within the class of unbiased estimators. It is, nevertheless, established that the FGLS estimators are inadmissible in light of minimizing the covariance matrices if the dimension of the common regression coefficients is greater than or equal to three. Double shrinkage unbiased estimators are proposed as possible candidates of improved procedures.     

Asymptotically optimal sequential estimation of regular functionals of several distributions based on generalized U-statistics

For the several sample problem, a vector of estimable parameters is considered. For a fixed total sample size, a multistage (sequential) procedure based on generalized U-statistics is developed for choosing a partition of this sample size into individual sample size for which the generalized variance of the estimator of the parameter vector is asymptotically minimized.     

Weighted estimation of single index models with right censored responses

In this paper, the unknown link function, the direction parameter, and the heteroscedastic variance in single index models are estimated by the random weight method under the random censorship, respectively. The central limit theory and the convergence rate of the law of the iterated logarithm for the estimator of the direction parameter are derived, respectively. The optimal convergence rates for the estimators of the link function and the heteroscedastic variance are obtained. Simulation results support t...     

聚集数据线性模型广义聚集双参数估计的相对效率

对于聚集数据的线性模型,提出了广义聚集双参数估计的概念,给出了广义聚集双参数估计的两种相对效率,并得到了这两种相对效率的上界.     

A new method for parameter sensitivity estimation in structural reliability analysis

For the parameter sensitivity estimation with implicit limit state functions in the time-invariant reliability analysis, the common Monte Carlo simulation based approach involves multiple trials for each parameter being varied, which will increase associated computational cost and the cost may become inevitably high especially when many random variables are involved. Another effective approach for this problem is featured as constructing the equivalent limit state function (usually called response surface) and performing the estimation in FORM/SORM. However, as the equivalent limit state function is polynomial in the traditional response surface method, it is not a good approximation especially for some highly non-linear limit state functions. To solve the above two problems, a new method, support vector regression based response surface method, is therefore presented in this paper. The support vector regression algorithm is employed to construct the equivalent limit state function and FORM/SORM is used in the parameter sensitivity estimation, and then two illustrative examples are given. It is shown that the computational cost of the sensitivity estimation can be greatly reduced and the accuracy can be retained, and results of the sensitivity estimation obtained by the proposed method are in satisfactory agreement with those computed by the conventional Monte Carlo methods.