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Statistical Methods for Fuzzy Data by Reinhard Viertl

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19.1 Regression models

Standard regression models consider k independent deterministic variables x1,…,xk and a dependent variable y which is modeled by a stochastic quantity Y whose distribution depends on the values x1,…,xk of the independent variables.

Using the notation x = (x1,…,xk) the independent variables of the corresponding stochastic quantity describing the dependent variable is denoted by Yx.

A so-called parametric regression model models the expectation IEYx of Yx as a function of the independent variables and the parameters θ0, θ1,…, θk in the following way:

(19.1) Numbered Display Equation

where ψ(·) is a known function, and θ0, θ1,…, θk are unknown constants called regression parameters.

Usually the variance of Yx are assumed to be the same for all xk:

(19.2) Numbered Display Equation

Condition (19.2) is called variance homogeneity or homoscedasticity.

The values of the dependent variable y are considered to be the sum of a deterministic function and a stochastic quantity U with IEU = 0 and Var U = σ2:

(19.3) Numbered Display Equation

where θ = (θ0, θ1,…, θk) denotes the vector of unknown parameters.

Based on observed data (xi1,…,xik; yi) for ...

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