Prediction of record values by using quantile regression curves and distortion functions

Abstract

The purpose of the paper is to provide a general method based on conditional quantile curves to predict record values from preceding records. The predictions are based on conditional median (or median regression) curves. Moreover, conditional quantiles curves are used to provide confidence bands for these predictions. The method is based on the recently introduced concept of multivariate distorted distributions that are used instead of copulas to represent the dependence structure. This concept allows us to compute the conditional quantile curves in a simple way. The theoretical findings are illustrated with a non-parametric model (standard uniform), two parametric models (exponential and Pareto), and a non-parametric procedure for the general case. A real data set and a simulated case study in reliability are analysed.

Predicting future failure times by using quantile regression

“…Actually, the results for the case n = 2, r = 1 and s = 2 (paired ordered data) were already obtained there. Similar results were obtained in Navarro (2022b) for record values. The procedure is similar but, the expressions obtained here for n = 4 will be more complicated.…”

On sums of dependent random lifetimes under the time-transformed exponential model

“…Actually, in some situations, we can choose a common univariate distribution H = H 1 = H 2 . Some examples will be provided later (see also Navarro (2021); Navarro et al (2022)). Moreover, if D has uniform univariate marginal distributions over the interval (0, 1), then D is a copula, H 1 and H 2 are the marginal distributions and (2.4) is the same as the copula representation (2.1) (but only in this case).…”

“…This model is an extension of the well-known Schur-constant model which is obtained when α = β (see Caramellino and Spizzichino (1994), and references therein). Properties of the Schur-constant model and of the corresponding sum X 1 + X 2 are studied also in Pellerey and Navarro (2021). The GK-model represents distributions satisfying the so-called law of uniform seniority of dependent lives (see Genest and Kolev 2021).…”

Weighted Mean Inactivity Time Function with Applications

“…Yaari [18]). As an example, for some recent applications see Hu et al [19] (in risk theory), and Navarro [20] and Navarro et al [21] (in statistics). (i) If the cumulative weight function is given by ψ(x) = F 0 (x), then due to (8) the WMIT function of X can be expressed as…”

“…Hence, we have V(X) = Var(ψ(X)) = Var(− log f (X)), so that, recalling Example 4, due to Equations ( 19) and ( 25), we obtain another representation of the varentropy in terms of the past entropy (20) and the reversed hazard rate (3) of X as follows:…”