### ABSTRACT - Models with a Low Nonlinearity.

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In this paper we consider an exploratory canonical analysis approach for multinomial population based on the $\phi $-divergence measure. We define the restricted minimum $\phi $-divergence estimator, which is seen to be a generalization of the restricted maximum likelihood estimator. This estimator is then used in $\phi $-divergence goodness-of-fit statistics which is the basis of two new families of statistics for solving the problem of selecting the number of significant correlations as well as the appropriateness...

We derive expressions for the asymptotic approximation of the bias of the least squares estimators in nonlinear regression models with parameters which are subject to nonlinear equality constraints. The approach suggested modifies the normal equations of the estimator, and approximates them up to ${o}_{p}\left({N}^{-1}\right)$, where $N$ is the number of observations. The “bias equations” so obtained are solved under different assumptions on constraints and on the model. For functions of the parameters the invariance of the approximate...

General results giving approximate bias for nonlinear models with constrained parameters are applied to bilinear models in anova framework, called biadditive models. Known results on the information matrix and the asymptotic variance matrix of the parameters are summarized, and the Jacobians and Hessians of the response and of the constraints are derived. These intermediate results are the basis for any subsequent second order study of the model. Despite the large number of parameters involved,...

In this paper we present a simulation study to analyze the behavior of the $\phi $-divergence test statistics in the problem of goodness-of-fit for loglinear models with linear constraints and multinomial sampling. We pay special attention to the Rényi’s and ${I}_{r}$-divergence measures.

Employing recently derived asymptotic representation of the least trimmed squares estimator, the combinations of the forecasts with constraints are studied. Under assumption of unbiasedness of individual forecasts it is shown that the combination without intercept and with constraint imposed on the estimate of regression coefficients that they sum to one, is better than others. A numerical example is included to support theoretical conclusions.

En este trabajo se estudian problemas de test de hipótesis para el vector de las medidas de poblaciones normales independientes con varianzas conocidas, cuando ambas, la hipótesis nula y la alternativa, imponen restricciones de orden sobre los parámetros.Se demuestra que para las hipótesis planteadas el test de razón de verosimilitud (TRV) está dominado por otro TRV para hipótesis que desprecian parte de la información de que se dispone.

The linear regression model in which the vector of the first order parameter is divided into two parts: to the vector of the useful parameters and to the vector of the nuisance parameters is considered. The type I constraints are given on the useful parameters. We examine eliminating transformations which eliminate the nuisance parameters without loss of information on the useful parameters.

In modelling a measurement experiment some singularities can occur even if the experiment is quite standard and simple. Such an experiment is described in the paper as a motivation example. It is presented in the papar how to solve these situations under special restrictions on model parameters. The estimability of model parameters is studied and unbiased estimators are given in explicit forms.

In many econometric applications there is prior information available for some or all parameters of the underlying model which can be formulated in form of inequality constraints. Procedures which incorporate this prior information promise to lead to improved inference. However careful application seems to be necessary. In this paper we will review some methods proposed in the literature. Among these there are inequality constrained least squares (ICLS), constrained maximum likelihood (CML) and...

For inferences from random-effect models Lee and Nelder (1996) proposed to use hierarchical likelihood (h-likelihood). It allows influence from models that may include both fixed and random parameters. Because of the presence of unobserved random variables h-likelihood is not a likelihood in the Fisherian sense. The Fisher likelihood framework has advantages such as generality of application, statistical and computational efficiency. We introduce an extended likelihood framework and discuss why...

The properties of the regular linear model are well known (see [1], Chapter 1). In this paper the situation where the vector of the first order parameters is divided into two parts (to the vector of the useful parameters and to the vector of the nuisance parameters) is considered. It will be shown how the BLUEs of these parameters will be changed by constraints given on them. The theory will be illustrated by an example from the practice.

Least-Squares Solution (LSS) of a linear matrix equation and Ordinary Least-Squares Estimator (OLSE) of unknown parameters in a general linear model are two standard algebraical methods in computational mathematics and regression analysis. Assume that a symmetric quadratic matrix-valued function Φ(Z) = Q − ZPZ0 is given, where Z is taken as the LSS of the linear matrix equation AZ = B. In this paper, we establish a group of formulas for calculating maximum and minimum ranks and inertias of Φ(Z)...

The paper deals with modified minimax quadratic estimation of variance and covariance components under full ellipsoidal restrictions. Based on the, so called, linear approach to estimation variance components, i. e. considering useful local transformation of the original model, we can directly adopt the results from the linear theory. Under normality assumption we can can derive the explicit form of the estimator which is formally find to be the Kuks–Olman type estimator.