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Abstract

The paper deals with the comparative calibration model, i.e. with a situation when both variables are subject to errors. The calibration function is supposed to be a polynomial. From the statistical point of view, the model after linearization could be represented by the linear errors-in-variables (EIV) model. There are two different ways of using the Kenward and Roger’s type approximation to obtain the confidence region for calibration function coefficients. These two confidence regions are compared on a small simulation study. Calibration process and process of measuring with calibrated device are described under the assumption that the measuring errors are normally distributed.

1. Introduction Not all linear functions of parameters are unbiasedly estimable in singular linear models. However if the observation vector of the model is normally distributed, then there is no essential problem to construct a confidence region for a group of linear unbiasedly estimable linear functions. In the case of nonlinearity of the model a construction of such region is more complicated. One way how to proceed in a solution of the problem in weakly nonlinear models is described in the paper. The procedure is based on measures of nonlinearity which enable

are applied to generate random variates for the measurements. By assuming the normal distribution, the densities of the measurements are determined and lead to confidence ellipsoids. The uniform multivariate distribution is adopted for the systematic effects. The Monte Carlo estimate then gives the expectations and the covariance matrix of the measurements distorted by these effects. This leads to the confidence region for the measurements with systematic effects. The densities for the measurements plus systematic effects may be determined by the kernel density

-coefficient model, single-index model, partial linear single-index model, single-index varying-coefficient model, etc. The linear component θ 0 τ Z $\begin{array}{} \displaystyle \theta_0^{\tau}Z \end{array}$ provides a simple summary of covariates effects which are of the main scientific interest. The index β 0 τ U $\begin{array}{} \displaystyle \beta_0^{\tau}U \end{array}$ enables us to simplify the treatment of the multiple auxiliary variables, and the functions g 0 (⋅) s enrich model flexibility. It is well known that in order to construct the confidence region

Abstract

The Bayesian approach allows an intuitive way to derive the methods of statistics. Probability is defined as a measure of the plausibility of statements or propositions. Three rules are sufficient to obtain the laws of probability. If the statements refer to the numerical values of variables, the so-called random variables, univariate and multivariate distributions follow. They lead to the point estimation by which unknown quantities, i.e. unknown parameters, are computed from measurements. The unknown parameters are random variables, they are fixed quantities in traditional statistics which is not founded on Bayes’ theorem. Bayesian statistics therefore recommends itself for Monte Carlo methods, which generate random variates from given distributions. Monte Carlo methods, of course, can also be applied in traditional statistics. The unknown parameters, are introduced as functions of the measurements, and the Monte Carlo methods give the covariance matrix and the expectation of these functions. A confidence region is derived where the unknown parameters are situated with a given probability. Following a method of traditional statistics, hypotheses are tested by determining whether a value for an unknown parameter lies inside or outside the confidence region. The error propagation of a random vector by the Monte Carlo methods is presented as an application. If the random vector results from a nonlinearly transformed vector, its covariance matrix and its expectation follow from the Monte Carlo estimate. This saves a considerable amount of derivatives to be computed, and errors of the linearization are avoided. The Monte Carlo method is therefore efficient. If the functions of the measurements are given by a sum of two or more random vectors with different multivariate distributions, the resulting distribution is generally not known. TheMonte Carlo methods are then needed to obtain the covariance matrix and the expectation of the sum.

concepts. Key Words: Asymptotic distribution, consistent estimator, confidence region, failure rate, score test, TTT-transform. 1 Introduction The literature about change point problems is quite rich. Historically, change point problems originated in the context of quality control, where one typically observes the output of a produc- tion process and would wish to signal deviation from an acceptable average output level while observing the data. However, in this review paper we are concerned with inferential problems involving change points when there is a sudden change

is frequently used instead of the standard uncer- tainty. The length of the interval defined in that way is re- garded as a measure of the measurement precision. This interpretation, however, is notdirectly transferable tomul- tivariate distributions and is even for univariate cases not always reasonable. Therefore, a so-called credible region is introduced here, which can serve as a measure of the measurement precision. Keywords: Credible region, confidence region, cover- age region, multivariate, multimodal, probability density function. DOI 10.1515/teme-2014

Treatment of Experimental Data with Discordant Observations: Issues in Empirical Identification of Distribution

Performances of several methods currently used for detection of discordant observations are reviewed, considering a set of absolute measurements of gravity acceleration exhibiting some peculiar features. Along with currently used methods, a criterion based upon distribution of extremes is also relied upon to provide references; a modification of a simple, broadly used method is mentioned, improving performances while retaining inherent ease of use. Identification of distributions underlying experimental data may entail a substantial uncertainty component, particularly when sample size is small, and no mechanistic models are available. A pragmatic approach is described, providing estimation to a first approximation of overall uncertainty, covering both estimation of parameters, and identification of distribution shape.

Abstract

The multivariate model, where not only parameters of the mean value of the observation matrix, but also some other parameters occur in constraints, is considered in the paper. Some basic inference is presented under the condition that the covariance matrix is either unknown, or partially unknown, or known.

Abstract

The aim of the paper is to present explicit formulae for parameter estimators and confidence regions in multivariate regression model with different kind of constraints and to give some comments to it. The covariance matrix of observation is either totally known, or some unknown parameters of it must be estimated, or the covariance matrix is totally unknown.