Se entiende por líneas, fuera de servicios aquellas líneas que a causa del daño reportado están inactivas, tienen interrumpida la continuidad del servicio, por ende no están generando ingresos

Henceforth, for any cdf F we write F−1 _{to denote its generalized inverse given}

by F−1(u) = inf_{τ ∈ R : F (τ) ≥ u . A confidence interval for ψ}n+1 based on

quantiles of (2.24) or (2.25) is typically infeasible as these cumulative distribution functions are unknown for finite n. Here, they are infeasible because roughly they are the distribution functions of some weights which induce merging multiplied by mn θ(Xˆ 1:n) − θ0

and mnE θ(Xˆ 1:nE) − θ0
, respectively, where, in general,

the distributions of mn θ(Xˆ 1:n) − θ0
and mnE θ(Xˆ 1:nE) − θ0
are unknown in

approximation can be based on G∞with merging induced by the non-convergent

weights. In general, we also need to estimate G∞; see Examples 2.4 and 2.5 below

for common approaches. We denote estimators of (2.24) and (2.25) resulting from this approximation by F2IP

n

V

(·) and FSP L n

V

(·), respectively. In the next subsection we provide explicit expressions when G∞ is multivariate normal. For the general

construction, we refer to relations (2.42) and (2.43) in Appendix 2.A and the explanations preceding these relations. Based on the 2IP approach, we consider an interval of the form

I_γ2IP(x1:n, Y1:n) =  ψ V2IP n+1|1:n− F2IP n V−1 (1 − γ2) mn , ψ V2IP n+1|1:n− F2IP n V−1 (γ1) mn  , (2.26) where γ1, γ2 ∈ [0, 1) satisfy γ = γ1+ γ2. We typically take γ1 = γ2 = γ/2 such

that the interval is equal-tailed. Similarly, we construct the following sample-split interval: I_γSP L(xc_nP:n, X1:nE) =  ψ VSP L n+1|nP:n− FSP L n V−1 (1 − γ2) mn , ψ VSP L n+1|nP:n− FSP L n V−1 (γ1) mn  . (2.27) To achieve correct coverage, we need that Fn2IP

V

(·) and Fn2IP(·|I1:n) merge in prob-

ability and likewise for SPL. A sufficient condition for this in our setting is that we can consistently estimate the asymptotic distribution of the parameter estimator, G∞, by an appropriate estimator. This is formulated in Assumption 2.4 below.

Assumption 2.4. (CDF estimator ) Let G

V

n(·) denote a random (r-dimensional)

cdf as a function of X1:n, used to estimate G∞: i.e. R h dG

V

n(·) p

→ R h dG∞ as

n → ∞ for all h ∈ BL.

Although we did not explicitly specify in Assumption 2.4 the dependence of G

V

n

on X1:n, it should be understood to hold for any subsample of X1:n whose size

goes to infinity. The verification of Assumption 2.4 is a standard step in asymp- totic analysis. The two examples below provide common methods for verifying Assumption 2.4.

Example 2.4. Suppose that G∞ belongs to some parametric family {Gθ,ξ|θ ∈

2.3 Asymptotic Justification some consistent estimators ˆθ(X1:n) and ˆξ(X1:n) for θ0 and ξ0 respectively, it fol-

lows from the continuous mapping theorem that G

V

n = G_{θ(X1:n), ˆ}ˆ _ξ(X1:n) satisfies

Assumption 2.4 if Gθ,ξ is continuous in θ and ξ.

Example 2.5. If ˆGn is based on a consistent bootstrap procedure for G∞ then

Assumption 2.4 clearly holds.

The following theorem states the intervals’ asymptotic validity. Theorem 2.2. (Asymptotic coverage)

1. (i) Under Assumption 2.1, 2.2 and 2.4, F_n2IP(·|I1:n) and Fn2IP

V

(·) merge in probability.

(ii) If in addition Fn2IP

V

(·) is stochastically uniformly equicontinuous, then P h Iγ2IP(x1:n, Y1:n) 3 ψn+1   I1:n i _p → 1 − γ. (2.28) 2. (i) Under Assumption 2.1, 2.3 and 2.4, FSP L

n (·|InP:n) and F SP L n V (·) merge in probability. (ii) If in addition FnSP L V

(·) is stochastically uniformly equicontinuous, then P h I_γSP L(xc_nP:n, X1:nE) 3 ψn+1   InP:n i _p → 1 − γ. (2.29) However, the standard approach, motivated by I_γ2IP as in (2.28), computes an interval of the form IST A

γ (x1:n, x1:n) = Iγ2IP(x1:n, x1:n) as only one sample realiza-

tion is available. This, of course, strongly violates the independence assumption of {Xt} and {Yt}. Specifically, replacing Y1:nby X1:nin equation (2.26), leads to

I_γST A∗(x1:n, X1:n) =  ψ VST A∗ n+1|1:n− FST A n V−1 (1 − γ2) mn , ψ VST A∗ n+1|1:n− FST A n V−1 (γ1) mn  , (2.30) where FST A n V

(·) is defined in relation (2.44) and the text preceding it. Whereas it is difficult to justify a conditional confidence interval like IST A

γ (x1:n, X1:n) directly

due to the lack of randomness, we can provide a justification by characterizing how closely the interval resembles the SPL interval. We establish the asymptotic equivalence, defined in terms of location and (scaled) length, between the two

intervals in the following theorem. Note that, as our characterization of equivalence is probabilistic, we need to introduce the “doubly random” versions of the STA and SPL estimators, where the sample we condition on is considered random. These estimators are denoted as ˆψn+1(X1:n, X1:n) and ˆψn+1(XcnP:n, X1:nE) respectively.

Theorem 2.3. (Asymptotic equivalence of confidence intervals) 1. (Location) If Assumptions 2.1 to 2.3 hold, then

ˆ

ψn+1(X1:n, X1:n) − ˆψn+1(XcnP:n, X1:nE) p

→ 0.

2. (Length) Under the assumptions of Theorem 2.1 and 2.2 and FSP L n

V−1

(·) being stochastically pointwise continuous at u = γ1, 1 − γ2, we have

F_nST A

V−1

(u) − F_nSP L

V−1

(u)→ 0.p (2.31) The first implication states that the locations of the two intervals coincide asymptotically. The second statement establishes asymptotic closeness of the se- lected quantiles such that the scaled lengths of the intervals in (2.27) and (2.30) coincide asymptotically. As such, our sample-split interval coincides asymptoti- cally with the standard interval, meaning that the standard interval can be substi- tuted for an (asymptotically) equivalent interval which has a formal justification in terms of conditional coverage. As such, this provides a justification for the intervals commonly constructed in practice without having to rely on the two- independent-processes assumption.