Cattaneo, Marco E. G. V. and Wiencierz, Andrea
(30. June 2012):
On the implementation of LIR: the case of simple linear regression with interval data.
Department of Statistics: Technical Reports, No.127

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Abstract
This paper considers the problem of simple linear regression with intervalcensored data. That is, n pairs of intervals are observed instead of the n pairs of precise values for the two variables (dependent and independent). Each of these intervals is closed but possibly unbounded, and contains the corresponding (unobserved) value of the dependent or independent variable. The goal of the regression is to describe the relationship between (the precise values of) these two variables by means of a linear function.
Likelihoodbased Imprecise Regression (LIR) is a recently introduced, very general approach to regression for imprecisely observed quantities. The result of a LIR analysis is in general setvalued: it consists of all regression functions that cannot be excluded on the basis of likelihood inference. These regression functions are said to be undominated.
Since the interval data can be unbounded, a robust regression method is necessary. Hence, we consider the robust LIR method based on the minimization of the residuals' quantiles. For this method, we prove that the set of all the interceptslope pairs corresponding to the undominated regression functions is the union of finitely many polygons. We give an exact algorithm for determining this set (i.e., for determining the setvalued result of the robust LIR analysis), and show that it has worstcase time complexity O(n^3 log n). We have implemented this exact algorithm as part of the R package linLIR.