Michal Černý and Milan Hladík. The regression tolerance quotient in data analysis. In CD-ROM Proceedings 28-th Int. Conf. Mathematical Methods in Economics MME 2010, Part~I, České Budějovice, pp. 98–104, 2010.
Let E(y)=Xb' be the traditional linear regression model and let b be an estimate of the unknown vector of regression parameters b. The tolerance quotient d*, defined and studied by Hladik and Cerny in (Interval regression by tolerance analysis approach, preprint in KAM-DIMATIA Series 963, 2010) is the least d*>=0 such that for any i, the equation yi = Xib', where yi is the i-th observation of the dependent variable and Xi is the i-th row of X, is satisfied with some b' in [b-d*|b|,b+d*|b|]. The tolerance quotient d* measures the relative perturbation rate, i.e. how much it is necessary to perturb the estimated regression coefficients b to satisfy each of the equations yi = Xib', and hence is a measure of goodness of fit of the model. We demonstrate the usage of the quotient in analysis of both crisp and interval data and, in particular, interval data arising in econometrics and finance. We show a method to study probabilistic properties of the tolerance quotient: we derive the distribution of and, under certain assumptions, we present a method for construction of a confidence interval for d*.
@InProceedings{CerHla2010,
author = "Michal {\v{C}}ern\'{y} and Hlad{\'\i}k, Milan",
editor = "Michal Houda and Jana Friebelov\'{a}",
title = "The regression tolerance quotient in data analysis",
booktitle = "CD-ROM Proceedings 28-th Int. Conf. Mathematical Methods in Economics MME 2010, Part~I, \v{C}esk\'{e} Bud\v{e}jovice",
pages = "98-104",
year = "2010",
bib2html_dl_pdf = "https://kam.mff.cuni.cz/~hladik/doc/2010-proc-MME-RegTolQuoDatAnal.pdf",
abstract = "Let E(y)=Xb' be the traditional linear regression model and let b be an estimate of the unknown vector of regression parameters b. The tolerance quotient d*, defined and studied by Hladik and Cerny in (Interval regression by tolerance analysis approach, preprint in KAM-DIMATIA Series 963, 2010) is the least d*>=0 such that for any i, the equation yi = Xib', where yi is the i-th observation of the dependent variable and Xi is the i-th row of X, is satisfied with some b' in [b-d*|b|,b+d*|b|]. The tolerance quotient d* measures the relative perturbation rate, i.e. how much it is necessary to perturb the estimated regression coefficients b to satisfy each of the equations yi = Xib', and hence is a measure of goodness of fit of the model. We demonstrate the usage of the quotient in analysis of both crisp and interval data and, in particular, interval data arising in econometrics and finance. We show a method to study probabilistic properties of the tolerance quotient: we derive the distribution of and, under certain assumptions, we present a method for construction of a confidence interval for d*.
",
keywords = "tolerance quotient, interval regression, confidence interval",
}
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