The regression tolerance quotient in data analysis

Let E(y) = Xb be the traditional linear regression model and let b̂ be an estimate of the unknown vector of regression parameters b. The tolerance quotient δ∗, defined and studied by Hlad́ık and Černý in (Interval regression by tolerance analysis approach, preprint in KAM-DIMATIA Series 963, 2010) is the least δ ≥ 0 such that for any i, the equation yi = Xiβ, where yi is the i-th observation of the dependent variable and Xi is the i-th row of X, is satisfied with some β ∈ [̂b−δ · |̂b|, b̂+δ · |̂b|]. The tolerance quotient δ∗ measures the relative perturbation rate, i.e. how much it is necessary to perturb the estimated regression coefficients b̂ to satisfy each of the equations yi = Xiβ, 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 δ∗ and, under certain assumptions, we present a method for construction of a confidence interval for δ∗.