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Statistical Methods for Fuzzy Data by Reinhard Viertl

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19.2 Linear regression models with Gaussian dependent variables

Assuming Gaussian distributions for the dependent variables Yi = Yxi the estimators from the Gauss–Markov theorem have additional properties. These can be used to construct confidence intervals for the parameters. In order to obtain the confidence intervals some results are necessary. These are given in the following theorems.

Theorem 19.2

Assuming a linear regression model (19.4) and all conditions from Theorem 19.1 and additionally the stochastic quantities Y1,…,Yn to have Gaussian distributions, i.e. Unnumbered Display Equation has multivariate Gaussian distribution where In is the n × n identity matrix

Unnumbered Display Equation

then the following holds:

1. The estimator is also the maximum likelihood estimator for θ.

2. The distribution of the estimator is a multivariate normal distribution

Unnumbered Display Equation

3. The pivotal quantity has chi-square distribution with nk − 1 degrees of ...

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