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Least squares estimation of linear regression models for convex compact random sets. (English) Zbl 1131.62058

Summary: Simple and multiple linear regression models are considered between variables whose “values” are convex compact random sets in \({\mathbb{R}^p}\) (that is, hypercubes, spheres, and so on). We analyze such models within a set-arithmetic approach. Contrary to what happens for random variables, the least squares optimal solutions for the basic affine transformation model do not produce suitable estimates for the linear regression model. First, we derive least squares estimators for the simple linear regression model and examine them from a theoretical perspective. Moreover, the multiple linear regression model is dealt with and a stepwise algorithm is developed in order to find the estimates in this case. The particular problem of the linear regression with interval-valued data is also considered and illustrated by means of a real-life example.

MSC:

62J05 Linear regression; mixed models
60D05 Geometric probability and stochastic geometry
65C60 Computational problems in statistics (MSC2010)
65K05 Numerical mathematical programming methods
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