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We consider the problem of estimating the conditional mean of a real gaussian variable Y=∑i=1pθiXi+ɛ where the vector of the covariates (Xi)1≤i≤p follows a joint gaussian distribution. This issue often occurs when one aims at estimating the graph or the distribution of a gaussian graphical model. We introduce a general model selection procedure which is based on the minimization of a penalized least squares type criterion. It handles a variety of problems such as ordered and complete variable selection,...
Nonsensitiveness regions for estimators of linear functions, for confidence ellipsoids, for the level of a test of a linear hypothesis on parameters and for the value of the power function are investigated in a linear model with variance components. The influence of the design of an experiment on the nonsensitiveness regions mentioned is numerically demonstrated and discussed on an example.
If a nonlinear regression model is linearized in a non-sufficient small neighbourhood of the actual parameter, then all statistical inferences may be deteriorated. Some criteria how to recognize this are already developed. The aim of the paper is to demonstrate the behaviour of the program for utilization of these criteria.
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...
In linear regression models the estimator of variance components needs a suitable choice of a starting point for an iterative procedure for a determination of the estimate. The aim of this paper is to find a criterion for a decision whether a linear regression model enables to determine the estimate reasonably and whether it is possible to do so when using the given data.
In this paper, we extend the traditional linear regression methods to the (numerical input)-(interval output) data case assuming both the observation/measurement error and the indeterminacy of the input-output relationship. We propose three different models based on three different assumptions of interval output data. In each model, the errors are defined as intervals by solving the interval equation representing the relationship among the interval output, the interval function and the interval...
Testing that some regression coefficients are equal to zero is an important problem in many applications. Homoscedasticity is not necessarily a realistic condition in this setting and, as a consequence, no frequentist test there exist. Approximate tests have been proposed. In this paper a Bayesian analysis of this problem is carried out, from a default Bayesian model choice perspective. Explicit expressions for intrinsic priors are provided, and it is shown that the corresponding Bayes factor is...
The estimation procedures in the multiepoch (and specially twoepoch) linear regression models with the nuisance parameters that were described in [2], Chapter 9, frequently need finding the inverse of a partitioned matrix. We use different kinds of such inversion in dependence on simplicity of the result, similarly as in well known Rohde formula for partitioned matrix. We will show some of these formulas, also methods how to get the other formulas, and then we applicate the formulas in estimation...
We derive the two-sample Kolmogorov-Smirnov type test when a nuisance linear regression is present. The test is based on regression rank scores and provides a natural extension of the classical Kolmogorov-Smirnov test. Its asymptotic distributions under the hypothesis and the local alternatives coincide with those of the classical test.
We study functional regression with random subgaussian design and real-valued response. The focus is on the problems in which the regression function can be well approximated by a functional linear model with the slope function being “sparse” in the sense that it can be represented as a sum of a small number of well separated “spikes”. This can be viewed as an extension of now classical sparse estimation problems to the case of infinite dictionaries. We study an estimator of the regression function...
Consistency of LSE estimator in linear models is studied assuming that the error vector has radial symmetry. Generalized polar coordinates and algebraic assumptions on the design matrix are considered in the results that are established.
A linear model in which the mean vector and covariance matrix depend on the same parameters is connected. Limit results for these models are presented. The characteristic function of the gradient of the score is obtained for normal connected models, thus, enabling the study of maximum likelihood estimators. A special case with diagonal covariance matrix is studied.
In this paper the likelihood function is considered to be the primary source of the objectivity of a Bayesian method. The necessity of using the expected behaviour of the likelihood function for the choice of the prior distribution is emphasized. Numerical examples, including seasonal adjustment of time series, are given to illustrate the practical utility of the common-sense approach to Bayesian statistics proposed in this paper.
The paper deals with the linear comparative calibration problem, i. e. the situation when both variables are subject to errors. Considered is a quite general model which allows to include possibly correlated data (measurements). From statistical point of view the model could be represented by the linear errors-in-variables (EIV) model. We suggest an iterative algorithm for estimation the parameters of the analysis function (inverse of the calibration line) and we solve the problem of deriving the...
Linear conform transformation in the case of non-negligible errors in both coordinate systems is investigated. Estimation of transformation parameters and their statistical properties are described. Confidence ellipses of transformed nonidentical points and cross covariance matrices among them and identical points are determined. Some simulation for a verification of theoretical results are presented.
In mixed linear statistical models the best linear unbiased estimators need a known covariance matrix. However, the variance components must be usually estimated. Thus a problem arises what is the covariance matrix of the plug-in estimators.
A linear model with approximate variance components is considered. Differences among approximate and actual values of variance components influence the proper position and the shape of confidence ellipsoids, the level of statistical tests and their power function. A procedure how to recognize whether these diferences can be neglected is given in the paper.
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