Simulation free prediction intervals for a
state dependent failure process using
accellerated lifetime experiments
Christine H. Müller
Faculty of Statistics, TU Dortmund University
and
Sebastian Szugat
Faculty of Statistics, TU Dortmund University
and
Reinhard Maurer
Faculty of Architecture and Civil Engineering, TU Dortmund University
May 18, 2016
Abstract
We consider the problem of constructing prediction intervals for the time point
at which a given number of components of a system exposed to degradation fails.
The failure process with respect to the failure times of the components is modeled
by a state dependent point process which is an alternative to the nonhomogeneous
Poisson process often used in failure analysis. Several failure processes observed at
different usually higher stress conditions are incorporated by a link function. Two
new simulation-free prediction intervals are proposed. One is constructed with the δmethod and the implicit function theorem applied to the hypoexponential distribution
and does not need the construction of confidence sets for the unknown parameters.
The other is based on data depth using a recent result for constructing outlier robust
confidence sets for general regression. The two new methods are compared with
two methods based on classical confidence sets for generalized linear models. The
comparison is done by leave-one-out analysis of data coming from failure processes
observed at prestressed concrete beams exposed to different cyclic loading where the
time points of breaking tension wires were reported.
Keywords: Point process, Poisson process, birth process, hypoexponential distribution,
data depth.
technometrics tex template (do not remove)
1
1
Introduction
The lefthand side of Figure 1 shows two growth curves of the crack width of an initial
crack in two prestressed concrete beams exposed to cyclic loadings with two different stress
levels. The jumps in the growth curves are caused by the fatigue of the material when the
tension wires shown on the righthand side of Figure 1 break. Since there are 35 tension
wires in each beam, up to 35 jumps could be observed. However, usually a much smaller
1.5
1.0
0.0
0.5
Rissweite [mm]
2.0
2.5
number of breaks are observed, since then the failure of the beam happens.
4.0
4.5
5.0
5.5
6.0
6.5
Schwingspiele [log(N)]
Figure 1: Crack growth curves of two experiments (left) and broken tension wires (right)
The jumps can be treated as outliers and the remaining process can be analyzed by
models derived from the Paris-Erdogan equation, see Kustosz and Müller (2014); Kustosz
et al. (2016a,b). However, these jumps are innovation outliers and the dynamic of this
process is mainly caused by these jumps. Moreover, the time points of these jumps can
be detected quite exactly by acoustical measurements. That is why, in this paper, we will
consider only these time points of jumps.
Hence, we deal with a system of I components and the system fails if a critical number
Ic ≤ I of components fails. Such situations appear not only for the example above concerning prestressed concrete beams. Many other examples are given by Kvam and Peña (2005)
and include systems of electrical components or batteries in a parallel arrangement. Typical for these situations is that the load of the system is equally distributed over all working
components, so that the load of a working component increases if other components have
failed. This is also known as load redistribution in a load sharing system.
2
The previously described experimental setting is different to the situation of systems of
I independent components often considered in reliability analysis and usually called k-outof-n systems with k = Ic and n = I in our notation. In such situations the life time of the
components can be assumed to be independent and identically distributed. For several life
time distributions, prediction intervals for the time of the Ic ’th failure are constructed when
I0 < Ic ≤ I failures of I components were observed. This is usually done by predicting the
Ic ’th order statistic based on the I0 ’th order statistic, see e.g. Patel (1989), Hsieh (1996),
Barakat et al. (2011). Nordman and Meeker (2002) also construct a prediction interval for
the additional number of failures in the I components at time tw based on the number of
failures at time t < tw .
In the situation of load redistribution in a load sharing system, the life time distribution
changes with each new failure of a component. In particular, the larger the number of failed
components is the higher is the probability that the next component fails. An approach for
this situation is given by Cramer and Kamps (1996) based on sequential order statistics.
See also Burkschat (2009) and Beutner (2010) and the references in these papers. Another
approach is given by Kvam and Peña (2005) using a semiparametric model. In both
approaches, I or I − 1 parameters must be estimated. This is only possible if the number
I of components is not too large and several independent and identical systems under
same conditions can be observed. Moreover, no prediction intervals are provided in both
approaches.
Here, we will consider systems with more than 30 components and the systems are observed under different stress conditions. The aim is to provide prediction intervals for the
time that a critical number Ic of components has failed in a system under a specific stress
level s0 . The particular interest lies in prediction intervals for accelerated lifetime experiments where the majority of systems are observed under stress levels much higher than s0 ,
the level of the system of interest. Accelerated lifetime experiments are the only possible
approach when the experiments are expensive and last long under realistic conditions which
is the case for prestressed concrete beams, for example.
We present here a approach based on a nonlinear birth processes where the stress level
is incorporated by a link function. Birth processes are state dependent point processes and
3
are special cases of the wide class of self-exciting point processes, which include many point
processes often used in reliability analysis as the homogeneous or inhomogeneous Poisson
process or renewal processes, see e.g. Snyder and Miller (1991), Sobczyk and Spencer
(1992), and Sánchez-Silva and Klutke (2016). For example, Sobczyk and Spencer (1992)
treat linear birth processes for fatigue accumulation in Chapter 18 and use birth processes
with time-varying intensity λi = iλ(t) for crack growth in Chapter 26. Many point processes for life-cycle analysis of deterioratings system are regarded in Sánchez-Silva and
Klutke (2016) but they regard mainly Poisson processes, renewal processes and marked
point processes without statistical analysis. Although a combination of a nonhomogeneous
Poisson process with a birth process would be very adequate to model damage accumalation and load redistribution, we are not aware of a statistical analysis for such models.
There is only the paper of Lawless and Thiagarajah (1996) who consider a combination
of the (nonhomogeneous) Poisson process and the renewal processes. They derive maximum likelihood estimators and apply a Wald test for a nonhomogeneous Poisson process
of airplane air-condition failure data. Dachian and Kutoyants (2006) also developed a test
for testing a stationary Poisson process against a stationary self-exciting point processes.
Other extensions of the nonhomegeneous Poisson process and renewal processes are considered for example in Engelhardt and Bain (1978); Lee and Lee (1978); Berg and Haberman
(1994); Franz et al. (2014).
Statistical inference for the more general class of self-exciting processes is treated mainly
for other fields of application as market analysis (see e.g. Kopperschmidt and Stute, 2013),
analysis of financial data (see e.g. Chan et al., 2014; Grothe et al., 2014; Hardiman and
Bouchaud, 2014; Rambaldi et al., 2015; Wheatley et al., 2016), neuron firing data (see e.g.
Kazemipour et al., 2014), cancer data (see e.g. Bai et al., 2015) or earthquake data (see e.g.
Guo and Luk, 2014). These papers treat estimation and testing of unknown parameters of
the model. The only exception is the paper of Chan et al. (2014) which provides prediction
intervals for the terminal prices in online auctions. However, the prediction intervals are
obtained by simulations.
Here, we present different methods for simulation free prediction intervals for the time
until the Ic ’th failure (event) in a new system under stress s0 when other systems exposed
4
to different stress levels have been observed before. We include also the possibility that
some failures of the new system have already been detected. As birth processes we use
point processes where the interarrival times are independent with exponential distribution
depending on the number of past failures (events) and the stress levels.
Although there are many methods for getting prediction intervals for exponential and
other univariate life time distributions (see e.g. Fertig et al., 1980; Engelhardt and Bain,
1982; Patel, 1989; Lawless and Fredette, 2005; Krishnamoorthy et al., 2009; Frey, 2012, and
the references in these papers), not much is published for accelerated lifetime experiments.
Patel (1989) mentions only a prediction interval for acceleated lifetime experiments with
normal distribution while Xiong and Milliken (2002) provide simulation based prediction
intervals for exponential lifetimes for accelerated life testing. Hong and Meeker (2013)
provide a prediction method based on dynamic covariate information, but no prediction
intervals are given. There are also some results for predicting degradation as crack growth
based on mixed models (see e.g. Meeker et al., 1998; De Oliveira and Colosimo, 2004; Wang
and Xu, 2010).
The main key of our approach for birth processes with exponential distributed interarrival times is that the distribution of a sum of random variables with different exponential
distributions has a hypoexponential distribution and that recently Gertsbakh et al. (2015)
provided an efficient and more stable algorithm for calculating the hypoexponential distribution. This leads at once to more stable quantiles of the future distribution and only
the uncertainty of the unknown parameters of the model must be incorporated. A simple
possibility is to use classical confidence sets for generalized linear models based for example
on the Wald or the likelihood ratio test, see e.g. Mood et al. (1974), pp. 409. Here, we propose two alternatives. One alternative is based on the approach of Kustosz et al. (2016a,b)
using simplicial depth for getting outlier robust confidence sets. Since all methods based
on confidence sets have the shortcoming that they require a grid search or another form of
candidate generation to evaluate the respective test, we also use the δ-method to construct
confidence intervals directly as a second alternative.
Section 2 introduces the model based on birth processes. After providing the classical
confidence sets for this model in Section 3, we derive the confidence sets based on simplicial
5
depth in Section 4. In Section 5, the δ-method is already used to obtain asymptotic
confidence intervals for the expected time until the failure of the Ic ’th component. Using
the former results, we provide the prediction intervals for the time until the failure of the
Ic ’th component in Section 6. In Section 7, we apply the new prediction methods to data of
experiments with prestressed concrete beams and compare the performance of the different
prediction intervals by cross-validation. We summarize and discuss our results in Section
8.
2
The model
We assume that the failure times of the components of a system are given by a point
process. For that, let 0 = T0 < T1 < T2 < . . . < TIc < . . . < TI be the time points of
failing components. It can be assumed that all these time points are different although in
reality several failures may be observed at one time since the time is measured at discrete
scale. Although the number of possible time points are finite, we can model 0 = T0 < T1 <
T2 < . . . by a simple point process with explosion (see e.g. Jacobsen, 2006, p. 10). Set
Wi = Ti − Ti−1 , i ∈ N, for the interarrival or waiting times between the failures (events)
and let be
Nt =
∞
X
1[0,t] (Ti )
i=1
the corresponding counting process and N (s, t) = Nt − Ns . The general self-exciting point
process is given by the intensity process (see e.g. Snyder and Miller, 1991, pp. 288)
1
P (N (t, t + ∆t) = 1|Nt ), for Nt = 0,
∆t↓0 ∆t
1
µ(t; Nt , W1 , . . . , WNt 1 ) = lim
P (N (t, t + ∆t) = 1|Nt , W1 , . . . , WNt ), for Nt ≥ 1.
∆t↓0 ∆t
µ(t; 0) = lim
Special cases are the homogeneous and the inhomogeneous Poisson process given by
µ(t; Nt , W1 , . . . , WNt ) = λ and µ(t; Nt , W1 , . . . , WNt ) = λ(t), respectively, where no selfexciting appears. The inhomogeneous Poisson process is often used to model aging and
damage accumulation, see e.g. Sobczyk and Spencer (1992) and Sánchez-Silva and Klutke
(2016). Other special cases are the birth processes given by µ(t; Nt , W1 , . . . , WNt ) = λ(Nt ),
6
P t
the renewal processes given by µ(t; Nt , W1 , . . . , WNt ) = λ(t − N
i=1 Wi ) and the Hawkes
P
processes given by µ(t; Nt , W1 , . . . , WNt ) = µ + η i:Ti ≤t h(t − Ti ) (Hawkes, 1971a,b).
Here we assume that only the number of past failures (events) Nt influences the intensity
so that we have a state dependent point process, i.e. a birth process. In particular, we
regard nonlinear birth processes given by
µ(t; Nt , W1 , . . . , WNt )) = λ(Nt ) = h
I
I − Nt
.
This means that the interarrival or waiting times Wi+1 have an exponential distribution
with parameter λi = h (I/(I − i)). This is different to the linear birth processes given by
λi = iλ for fatigue accumulation mentioned in Sobczyk and Spencer (1992). However, for
load redistribution, this is very reasonable since the load for the remaining I/2 components
is doubled if half of the components, i.e. i = I/2, has failed. If, for example, i = 3/4 · I
components has failed, then the load for the remaining I/4 components is four times higher
than at the beginning. Hence, the load increases nonlinearly. Note that this load sharing
model has previously been used by Kim and Kvam (2004) but without the function h.
We are aware that this model does not include damage accumulation and aging. This is
only possible by the more general self-exciting point processes. However, the derivation
of simulation free prediction intervals will be difficult then. Hence, we keep the model as
simple as possible. The results for the real data set in Section 7 indicate that this model
is realistic enough to provide good prediction intervals.
The system fails if a critical number Ic of the I components has failed. Thereby Ic could
be I but a smaller value is possible as well.
Since we have several independent systems under different stress levels, we incorporate
the stress by
λθ (i, s) := hθ
I
s·
I −i
where hθ is a given link function between stress and lifetime and θ ∈ Θ ⊂ Rp is an
unknown parameter. This means that the waiting times are independent and exponentially
distributed, i.e.
Wi ∼ Exp(λθ (i − 1, s)).
7
An example for the function hθ is
hθ (x) := exp (−θ1 + θ2 ln(x))
with θ = (θ1 , θ2 )> ∈ Θ = [0, ∞)2 so that
1
I
log(E(Wi )) = log
= θ1 − θ2 ln s ·
λθ (i − 1, s)
I − (i − 1)
is strictly decreasing in s and i. It is the often used linear link function proposed by Basquin
(1910). However, other linear and nonlinear link functions as
hθ (x) := exp −θ1 + θ2 · x − θ3 · x−θ4
with θ = (θ1 , θ2 , θ3 , θ4 )> ∈ Θ = [0, ∞)4
are possible.
We assume that we have processes coming from different systems j = 1, . . . , J which
were run under different stress levels s1 , . . . , sJ . Each system j is observed up to the Ij ’th
failure which could be the critical number Ic of failures but other values are possible as
well. A smaller value Ij < Ic usually appears if the experiment with the system j lasts too
long. Larger values Ij > Ic are possible if the system is still working with more than Ic
failures and Ic is only the minimum number of failures from which some safety condition
of the system is not ensured. I.e. we observe realizations wi,j , i = 1, . . . , Ij , j = 1, . . . , J, of
Wi,j ∼ Exp(λθ (i − 1, sj )), i = 1, . . . , Ij ≤ I, j = 1, . . . , J.
Additionally, we have realisations wi,0 , i = 1, . . . , I0 , of
Wi,0 ∼ Exp(λθ (i − 1, s0 )), i = 1, . . . , I0 ,
from a new experiment, where I0 is much smaller than I and is set equal to 0 if no failure
of the new process was observed. The aim is now to predict
w1,0 + . . . + wI0 ,0 + WI0 +1,0 + . . . + WIc ,0 , Ic > I0 ,
(1)
i.e. the time up to the failure of the Ic ’th component of the new system. If we can predict
the time W := WI0 +1,0 + . . . WIc ,0 , then of course we can also predict (1).
Remark on the related counting process (Nt )t≥0 . Being a pure jump process with
exponential waiting times, the process (Nt )t≥0 is Markovian. Furthermore, it is a continuous
8
time random-walk (CTRW, cf. Meerschaert et al., 2009) and it is the canonical counting
process corresponding to a simple point process with explosion (cf. Jacobsen, 2006, Example
3.1.4). Finally the process is a semimartingale with killing in the sense of Schnurr (2012).
Let us mention that the construction above works perfectly analog for other waiting
time distributions, that is, parametric families of probability distributions with support in
[0, ∞). However, only the exponential waiting times yield a Markov process (cf. Cinlar,
1975, Theorem 8.2.9).
3
Classical confidence sets for the parameter
Since W1,0 , . . . , WI0 ,0 , . . . , W1,J , . . . , WIJ ,J are independent, we can easily estimate θ by the
maximum likelihood principle, i.e.
b ∈ arg max
θ
θ∈Θ
Ij
J Y
Y
fλθ (i−1,sj ) (wi,j )
j=0 i=1
where fλ (w) = λe−λw is the density of the exponential distribution.
Confidence sets for the parameter θ can only be obtained approximately via an asymptotic law. Therefore we need additional assumptions for the design of the stress levels.
P
Let dJ := (s0 , s1 , . . . , sJ ) ∈ [0, smax ]J+1 be the concrete design and δJ := Jj=0 esj the
corresponding design measure on [0, smax ], where es denotes the Dirac measure on s. Then
we assume
δJ −→ δ weakly for J −→ ∞.
(2)
Additionally, set d̃J := ((1, s0 ), . . . , (I0 , s0 ), . . . , (1, sJ ), . . . , (IJ , sJ )) with corresponding design measure δ̃J on {1, . . . , I} × [0, smax ]. Since {1, . . . , I} is finite, the assumption (2)
immediately implies
δ̃J −→ δ̃ weakly for J −→ ∞,
(3)
where δ̃ is a design measure on {1, . . . , I} × [0, smax ].
Property (3) implies the following central limit theorem for the maximum likelihood
estimator (see e.g. Schervish, 1995, p. 421-428)
p
b − θ) −→ N 0, I θ (δ̃)−1 ,
N (J) (θ
9
(4)
where N (J) = I0 + . . . + IJ are the observed events or failures in the J + 1 systems and
> !
Z
∂
∂
ln fλθ (i−i,s) (W̃i,s )
ln fλθ (i−i,s) (W̃i,s )
δ̃(d(i, s))
Iθ (δ̃) := Eθ
∂θ
∂θ
with W̃i,s ∼ Exp(λθ (i − 1, s)) being the information matrix. Since we have exponential
distribution, the information matrix equals (see. e.g. Müller, 2013)
Z
1
I θ (δ̃) =
λ̇θ (i − 1, s) λ̇θ (i − 1, s)> δ̃(d(i, s))
2
λθ (i − 1, s)
with λ̇θ (i, s) :=
∂
λ (i, s).
∂θ θ
p
The convergence in (4) is equivalent to
b − θ) −→ N (0, I p ) ,
N (J) I θ (δ̃)1/2 (θ
where A1/2 ∈ Rp×p denotes the matrix with A = (A1/2 )> A1/2 , I p is the p-dimensional
identity matrix, and p is the dimension of the parameter vector θ. I θ (δ̃) is estimated by
b where
1/N (J) · I(θ),
I(θ) :=
Ij
J X
X
j=0 i=1
1
λ̇θ (i − 1, sj ) λ̇θ (i − 1, sj )> .
2
λθ (i − 1, sj )
(5)
Then we have
1
b −→ I θ (δ̃) in probability
I(θ)
N (J)
and the Lemma of Slutzky provides
b − θ)>
N (J) (θ
1
b (θ
b − θ) = (θ
b − θ)> I(θ)
b (θ
b − θ) −→ χ2 ,
I(θ)
p
N (J)
where χ2p is the χ2 -distribution with p degrees of freedom.
This yields the following asymptotic confidence set for θ given by
b − θ)> I(θ)
b (θ
b − θ) ≤ χ2
{θ; (θ
p,1−α }.
(6)
Thereby χ2p,1−α denotes the (1−α)-quantile of the χ2 -distribution with p degrees of freedom.
The related test is widely known as the Wald test (see e.g. Rao, 1965, p. 349). Using an
additional approximation with the Taylor expansion as in Schervish (1995), p. 459-461, for
the i.i.d. case, leads to the likelihood-ratio test based confidence set given by
(
!
)
QJ QIj
f
(w
)
i,j
λ
(i−1,s
)
i=1
j
θ
j=0
θ; −2 ln QJ QIj
≤ χ2p,1−α .
i=1 fλθb (i−1,sj ) (wi,j )
j=0
10
(7)
This is much easier to calculate than the information based confidence set given by (6)
since no information matrix must be calculated. It includes another approximation error
though.
4
Confidence sets based on simplicial depth
Other confidence sets can be derived by the depth tests proposed in Kustosz et al. (2016a,b).
Since Wi,j (j = 0, . . . , J, i = 1, . . . , Ij ) are exponentially distributed with parameter λθ (i −
1
. Set N := N (J) and
1, sj ) the median is ln(2) λθ (i−1,s
j)
w∗ := (w1,0 , . . . , wI0 ,0 , . . . , w1,J , . . . , wIJ ,J )> ∈ RN ,
>
I
I
I
I
, . . . , s0
, . . . , sJ , sJ
, . . . , sJ
∈ RN .
s∗ :=
s0 , s0
I −1
I − (I0 − 1)
I −1
I − (IJ − 1)
If s̃ := (s̃1 , . . . , s̃N )> is the ordered vector of s∗ , then let be w̃ := (w̃1 , . . . , w̃N )> the
corresponding vector obtained from w∗ . Then the residuals
rn (θ) := w̃n − ln(2)
1
, n = 1, . . . , N,
hθ (s̃n )
are realizations of independent random variables Rn (θ) which have median zero if θ is the
true parameter. Setting z ∗ = (z 1 , . . . , z N )> with z n = (s̃n , w̃n )> , the tangential depth of
θ in the sample z ∗ is
1
> ∂
2
dT (θ, z ∗ ) =
min # n ∈ {1, . . . , N } : u
rn (θ) ≤ 0 .
N |u|=1
∂θ
according to Mizera (2002). Simplicial depth of a p-dimensional parameter θ in the sample
z ∗ based on tangential depth can be defined as
dS (θ, z ∗ ) =
1
X
N
p+1
1 dT (θ, (z n1 , . . . , z np+1 ) > 0 ,
1≤n1 <n2 <...<np+1 ≤N
where 1 dT (θ, (z n1 , . . . , z np+1 ) > 0 denotes the indicator function 1A (z 1 , . . . , z p+1 ) with
A = (z̄ 1 , . . . , z̄ p+1 )> ∈ Rp+1 : dT (θ, (z̄ 1 , . . . , z̄ p+1 )) > 0 . In Kustosz et al. (2016b) it is
shown for strictly increasing regressors s̃1 < s̃2 < . . . < s̃N that the tangential depth
dT (θ, (z n1 , . . . , z np+1 ) is greater than 0 if and only if either the residuals rn1 (θ), . . . , rnp+1 (θ)
have alternating signs or at least rni (θ) = 0 for one i ∈ {1, . . . , p + 1} which can be used
11
to simplify the computation of 1 dT (θ, (z n1 , . . . , z np+1 ) > 0 . If there are systems k and l
with the same initial stress values sk and sl , a small noise can be added to ensure the strict
monotonicity of the regressors. Furthermore, Kustosz et al. (2016a) show that for p = 2
N
1
dS (θ, Z ∗ ) −
4
3
3 3
−→ + X22 (0) −
4 4
2
d
Z2
X12 (t)dt,
−2
where X = (X1 , X2 )> is a centered Gaussian process on [−2, 2] with continuous paths and
the covariance structure
R
Cov(X(s), X(t)) = 
1
0
1(−0.5,0.5] (x − s)1(−0.5,0.5] (x − t)dx
R1
1(−0.5,0.5] (x − t)dx
0
R1
0
1(−0.5,0.5] (x − s)dx
1

.
The hypothesis H0 : θ ∈ Θ0 can be rejected, if (see Müller, 2005)
1
sup N dS (θ, z ∗ ) −
< qαG ,
4
θ∈Θ0
where qαG is the α-quantile from the asymptotic distribution which can be simulated and
is given for several values of α in Kustosz et al. (2016a). Hence, the confidence set for θ
based on simplicial depth is given by
1
G
θ; N dS (θ, z ∗ ) −
≥ qα .
4
(8)
For the case p > 2, the asymptotic distribution of dS (θ, Z ∗ ) is not known yet. That is why
in this case we consider the simplified simplicial depth
dsS (θ, z ∗ )
!
p+1
N −p p+1
Y
1 X Y k
k+1
1 rn−1+k (θ)(−1) > 0 +
1 rn−1+k (θ)(−1)
>0
.
=
N − p n=1 k=1
k=1
Kustosz et al. (2016b) show that
p
TN (θ) := N − p r
p
dsS (θ, Z ∗ ) − 12
1 p
1 p−1
·
3
−
·p−3·
2
2
1 p
2
−→ N (0, 1).
Hence, a confidence set for θ based on simplified simplicial depth is given by
θ; TN (θ) ≥ qαN ,
12
(9)
where qαN is the α-quantile of the standard normal distribution.
Note that both, the original simplicial depth and the simplified simplicial depth, are
basing only on the signs of the residuals. Hence, tests based on them can be considered as
a generalization of the sign test so that they are outlier robust. However, Kustosz et al.
(2016a,b) have shown that these tests are much more efficient than the sign test. For this
reason, the proposed confidence sets are outlier robust and more efficient than confidence
sets based on the sign test.
5
Confidence intervals for the expected time of the
failure of the Ic’th component
b for θ, then the estimated expected additional time Eθ (W ) until the
Having an estimate θ
Ic ’th failure is
Eθb (W ) = Eθb (WI0 +1,0 + . . . + WIc ,0 ) =
1
1
b
+ ... +
= g(θ)
λθb (I0 , s0 )
λθb (Ic − 1, s0 )
with
g(θ) := Eθ (W ) =
1
1
+ ... +
.
λθ (I0 , s0 )
λθ (Ic − 1, s0 )
(10)
Since w1,0 , . . . , wI0 ,0 are already observed, we get at once that
w1,0 + . . . + wI0 ,0 +
1
1
+ ... +
λθb (I0 , s0 )
λθb (Ic − 1, s0 )
is the estimated expected time up to failure of the Ic ’th component.
b for θ like those given by (6), (7), (8)
Having an asymptotic (1 − α)-confidence set C
and (9), then an asymptotic (1 − α)-confidence interval for g(θ) is at once given by
n
o
b .
g(θ); θ ∈ C
b
This simple confidence interval has the disadvantage that the parameter vectors θ ∈ C
must be obtained by grid search or a similar method. In particular this is a problem when
b can be avoided
the dimension of the parameter space is high. The calculation of all θ ∈ C
using the δ-method. This leads to another type of confidence interval which is presented
in the following.
13
Although special interest is here the confidence interval for the expected time g(θ) defined by (10), in Section 6 confidence intervals of other aspects of θ are necessary. Therefore
let be g : Θ → R any function of θ so that ġ(θ) :=
∂
g(θ)
∂θ
exists.
The δ-method applied to (4) provides then
p
N (J)
b − g(θ) −→ N 0, ġ(θ)> I θ (δ̃)−1 ġ(θ)
g(θ)
or, respectively,
p
b − g(θ))
N (J) (g(θ)
q
−→ N (0, 1).
ġ(θ)> I θ (δ̃)−1 ġ(θ)
Using again the estimate
>
b
ġ(θ)
1
b
I(θ)
N (J)
with I(θ) given by (5), we have
−1
1
b
b −→ ġ(θ)> I θ (δ̃)−1 ġ(θ) in probability
I(θ)
ġ(θ)
N (J)
for J → ∞ and Lemma of Slutzky provides
p
b − g(θ))
b − g(θ)
N (J) (g(θ)
g(θ)
q
=r
−→ N (0, 1).
−1
b > I(θ)
b −1 ġ(θ)
b
1
>
ġ(θ)
b
b
b
ġ(θ) N (J) I(θ)
ġ(θ)
Hence, an asymptotic (1 − α)-confidence interval is
q
q
N
N
>
−1
>
−1
b
b
b
b
b
b
b
b
g(θ) − q1−α/2 ġ(θ) I(θ) ġ(θ), g(θ) + q1−α/2 ġ(θ) I(θ) ġ(θ) ,
where qαN is the α-quantile of the standard normal distribution.
6
Prediction intervals for the time of the failure of the
Ic’th component
Here we are going to construct asymptotic (1 − α) prediction intervals Ibpred for the time
WI0 +1,0 + . . . WIc ,0 based on W1,0 , . . . , WI0 ,0 , . . . , W1,J , . . . , WI ,J , i.e. an interval Ibpred so
J
that
lim Pθ (WI0 +1,0 + . . . WIc ,0 ∈ Ibpred ) ≥ 1 − α
J→∞
14
for all θ ∈ Θ. For that, we need the distribution of W := WI0 +1,0 + . . . WIc ,0 .
Since λθ (I0 , s0 ), . . . , λθ (Ic − 1, s0 ) are pairwise different, the density of W is given by
(see e.g. Ross, 2014)
fW,θ (w) :=
IX
c −1
λθ (i, s0 ) e−w λθ (i,s0 )
i=I0
=
IX
c −1
IY
c −1
k=I0
λθ (k, s0 )
λθ (k, s0 ) − λθ (i, s0 )
,k6=i
λθ (i, s0 ) e−w λθ (i,s0 ) ai (θ)
i=I0
with ai (θ) :=
QIc −1
λθ (k,s0 )
k=I0 ,k6=i λθ (k,s0 )−λθ (i,s0 ) .
FW,θ (w) :=
Hence, the cumulative distribution function is
IX
c −1
ai (θ) 1 − e−w λθ (i,s0 ) .
(11)
i=I0
Expression (11) can be computed efficiently (see R package sdprisk from Baumgartner
and Gatto, 2014), but Botev et al. (2013) show that it suffers from numerical instability
when λθ (I0 , s0 ), . . . , λθ (Ic − 1, s0 ) do not differ much. Since in our data, this case can
appear, we need a more stable algorithm. To achieve this, the cumulative distribution
function can be written using a matrix exponential as
FW,θ (w) = 1 −
e>
1
exp(Dw)1Ic −I0 = 1 −
e>
1
∞
X
Dk wk
k=0
k!
1Ic −I0 ,
(12)
where e1 = (1, 0, . . . , 0)> is the first (Ic − I0 ) × 1 unit vector, 1Ic −I0 = (1, 1, . . . , 1)> is an
(Ic − I0 ) × 1 vector of ones and





D=




−λθ (I0 , s0 )
0
..
.
λθ (I0 , s0 )
0
−λθ (I0 + 1, s0 ) λθ (I0 + 1, s0 )
..
..
.
.
0
···
0
0
···
0
···
0








−λθ (Ic − 2, s0 ) λθ (Ic − 2, s0 ) 

0
−λθ (Ic − 1, s0 )
···
..
.
0
..
.
is a (Ic − I0 ) × (Ic − I0 ) matrix. Gertsbakh et al. (2015) provide an implementation to
compute expression (12) which is considerably more stable than the direct computation of
P
Dk wk
1Ic −I0 is computed via the Matrix R package
expression (11). The infinite series ∞
k=0
k!
from Bates and Maechler (2016) using Ward’s diagonal Padé approximation. Since this
15
implementation takes more computation time it is advisable to only use expression (12)
when the problems turns out to be affected by the mentioned instability. This can be
checked by computing
c=
IX
c −1
ai (θ).
i=I0
If c = 1, expression (11) is a valid distribution function. Otherwise the problem can be
considered as numerically instable and expression (12) should be used.
An α-quantile bα (θ) of this distribution can be given only implicitly, namely as
Hα (θ, bα (θ)) = 0
where
Hα (θ, b) :=
IX
c −1
ai (θ) 1 − e−b λθ (i,s0 ) − α
i=I0
or
Hα (θ, b) = 1 −
e>
1
∞
X
D k bk
k=0
k!
1Ic −I0 − α,
respectively.
b for θ like those given by (6), (7), (8)
Having an asymptotic (1 − α)-confidence set C
and (9), then an asymptotic (1 − 2α)-prediction interval is at once given by
i
[
h
b max{b1−α/2 (θ); θ ∈ C}
b .
bα/2 (θ), b1−α/2 (θ) ⊂ min{bα/2 (θ); θ ∈ C},
(13)
b
θ∈C
Again, with the δ-method another type of prediction interval can be constructed. The
∂
implicit function theorem provides for ḃα (θ) := ∂θ
bα (θ)
!
−1
∂
∂
ḃα (θ) = −
Hα (θ̃, b̃)
Hα (θ̃, b̃)
,
∂ b̃
∂ θ̃
(θ̃,b̃)=(θ,bα (θ))
(θ̃,b̃)=(θ,bα (θ))
which can be calculated explicitly. Setting g(θ) = bα/2 (θ) and g(θ) = b1−α/2 (θ), respectively, in Section 5 provide that
q
N
>
−1
b
b
b
b
bα/2 (θ) − q1−α/2 ḃα/2 (θ) I(θ) ḃα/2 (θ), ∞
(14)
and
q
N
>
−1
b
b
b
b
−∞, b1−α/2 (θ) + q1−α/2 ḃ1−α/2 (θ) I(θ) ḃ1−α/2 (θ)
are one-sided 1 − α2 -confidence intervals for bα/2 (θ) and b1−α/2 (θ), respectively.
16
(15)
Theorem 6.1. The interval Ibpred given by
h
i
b − vb1 , b1−α/2 (θ)
b + vb2 ,
Ibpred := bα/2 (θ)
(16)
where
q
b > I(θ)
b −1 ḃα/2 (θ)
b
ḃα/2 (θ)
q
N
b > I(θ)
b −1 ḃ1−α/2 (θ),
b
vb2 := q1−α/2
ḃ1−α/2 (θ)
N
vb1 := q1−α/2
is an asymptotic (1 − α)2 -prediction interval for W = WI0 +1,0 + . . . WIc ,0 .
Proof. The one-sided confidence intervals given by (14) and (15) provide
b − vb1 ≤ α
Pθ bα/2 (θ) < bα/2 (θ)
2
and
Pθ
α
b
b1−α/2 (θ) > b1−α/2 (θ) + vb2 ≤ .
2
b are independent, we obtain
Since W and θ
lim Pθ (W ∈ Ibpred )
b
b
= lim Pθ W ≥ bα/2 (θ) − vb1 and W ≤ b1−α/2 (θ) + vb2
J→∞
b − vb1 and
≥ lim Pθ W ≥ bα/2 (θ) and bα/2 (θ) ≥ bα/2 (θ)
J→∞
b
W ≤ b1−α/2 (θ) and b1−α/2 (θ) ≤ b1−α/2 (θ) + vb2
= lim Pθ bα/2 (θ) ≤ W ≤ b1−α/2 (θ)
J→∞
b − vb1 or b1−α/2 (θ) > b1−α/2 (θ)
b + vb2
· 1 − Pθ bα/2 (θ) < bα/2 (θ)
α α
= (1 − α)2 .
≥ (1 − α) · 1 − −
2
2
J→∞
Note that bα (θ) can be explicitly given if Ic = I0 + 1, hence we want to predict only the
next failure. Then we have
1
ln 1−α
ln(1 − α)
1
bα (θ) = −
=
= ln
g(θ),
λθ (I0 , s0 )
λθ (I0 , s0 )
1−α
where g(θ) is the same as in (10) for the case Ic = I0 + 1. Setting
q
N
b > I(θ)
b −1 ġ(θ),
b
vb := q1−α/2 ġ(θ)
17
the (1 − α)2 -prediction interval is then
2
2
b
b
b
g(θ) − vb , ln
g(θ) + vb .
Ipred = ln
2−α
α
Table 1: The experiments with the stress levels and the observed number of breaks and
load cycles until failure in millions
stress range s
number of
number of load cycles until failure
in MPa
observed breaks
in millions
TR01
200
15
3.39
TR02
455
9
0.21
TR03
200
12
3.47
TR04
150
7
5.45
TR05
98
4
15.07
SB01
200
17
5.66
SB02
100
18
16.19
SB03
60
18
85.16
SB04
80
20
21.63
SB05
80
19
66.47
beam
7
Performance of the different prediction intervals in
the application
J = 10 experiments are used to study the performance of the different prediction intervals
for finite sample size. Table 1 shows the different stress levels, the number of observed
breaks and the number of load cycles until failure. Experiment TR05 lasted so long that
after 13 weeks the experiment was stopped without observing the failure. In a second set
of experiments denoted by SB, the frequency of load cycles was increased so that a failure
of the beams could be observed for low stress levels as well.
Figure 2 shows the logarithmic waiting times between the breaks of all experiments together with the estimated function log10 (hθ (·)−1 ) with hθ (x) = exp (−θ1 + θ2 · x − θ3 · ln(x))
18
10
Fitted expectation of the exponential distribution
Pointwise 90% prediction interval using δ−method
Pointwise 90% prediction interval using data depth
4
0
2
log10(wij)
6
8
200 MPa
455 MPa
200 MPa
150 MPa
98 MPa
200 MPa
100 MPa
60 MPa
80 MPa
80 MPa
0
100
200
300
400
500
600
700
s ⋅ Imax (Imax − i)
Figure 2: Logarithmic waiting times between the breaks of all experiments with pointwise
90% prediction interval for the next wire break using data depth and the δ-method with
link function hθ (x) = exp (−θ1 + θ2 · x + θ3 · ln(x))
and the pointwise prediction intervals for the logarithmic waiting time of the next break
based on the two new prediction methods. For all prediction intervals a level of 90% was
chosen, which is a common choice for applications in construction engineering. That means
α = 0.05 for the confidence set based methods and α ≈ 0.0513 for the δ-method to ensure
the levels (1 − 2α) = 90% and (1 − α)2 = 90%, respectively. The majority of the data
follows quite good the estimated function. The δ-method provides smaller prediction intervals than the less efficient method based on simplified simplicial depth. However, the
intervals are very similar and in particular not much influenced by the outliers with very
small waiting times.
Figure 3 shows the logarithmized prediction intervals for the time of the 20th break in
Experiment SB04 using no break, the first, the first two, . . . , the first 19 breaks of this
19
experiment and all observed breaks of the other experiments. All prediction intervals using
the δ-method contain the true time of the 20th break of this experiment when hθ (x) =
exp (−θ1 + θ2 ln(x)) is used. Here the more efficient full simplicial depth can be used so that
the prediction intervals based on depth are now smaller than those based on the δ-method.
Hence, the true time of the 20th break is only contained in prediction intervals based on
full depth when at least the first two jumps of the experiment have been observed. The
predictions intervals in general become smaller the more breaks are observed. There is a
noticeable change in the prediction intervals from 14 to 15 observed jumps. This change
is due to a long waiting time between the 14th and the 15th break in this experiment
8.0
compared to the other waiting times.
7.8
7.6
7.4
7.2
Log. prediction interval for event 20
observed time of event 20
δ−method
depth method
0
5
10
15
20
Number of observed events
Figure 3: Prediction intervals using for the time of the 20th break in Experiment SB04
using the first 0, 1, . . . , 19 breaks of this experiment and all observed breaks of the other
experiments and using hθ (x) = exp (−θ1 + θ2 ln(x))
A throughout comparison of all prediction methods can be performed with the leave20
one-out cross-validation. The following scenarios for leave-one-out cross-validation are considered:
For all j = 1, . . . , J and K = 0, . . . , Ij − 1,
Next :
the waiting time WK+1,j for the next event is predicted using
w1,j , . . . , wK,j and wi,l for i = 1, . . . , Il , l 6= j,
For all j = 1, . . . , J and K = 0, . . . , Ij − 5,
Next 5 :
the waiting time w1,j + . . . + wK,j + WK+1,j + . . . + WK+5,j
until the future five events is predicted using
w1,j + . . . + wK,j and wi,l for i = 1, . . . , Il , l 6= j.
The chosen j = 1, . . . , J takes the role of the new series so that K = I0 . Again K = 0 means
that no observations of the new series are used. The leave-one-out analysis is implemented
using the R package BatchExperiments from Bischl et al. (2015) for parallel computation.
To measure the performance of the prediction intervals, the interval score of Gneiting and
Raftery (2007) is used. It is given by
S([l, u], w0 ) := (u − l) +
2
2
(l − w0 )1{w0 < l} + (w0 − u)1{w0 > u},
α
α
if [l, u] is the prediction interval, w0 the realized future observation, and 1{ } denotes the
indicator function. It combines the coverage and length of the prediction interval.
Table 2 shows the median of this score in millions in the scenarios for the following
predictions intervals
Wald: information based prediction interval given by (13) based on (6),
LR: likelihood ratio based prediction interval given by (13) based on (7),
FD: full depth based prediction interval given by (13) based on (8),
SD: simplified depth based prediction interval given by (13) based on (9),
δ: δ-method based prediction interval given by (16),
21
in the following models
A:
hθ (x) := exp (−θ1 + θ2 · x) ,
B:
hθ (x) := exp (−θ1 + θ2 ln(x)) ,
C:
hθ (x) := exp (−θ1 + θ2 · x + θ3 · ln(x)) .
The Scenarios Next and Next 5 show that Model B is the best model in our data example
as the median interval score is the lowest for each method. The second best model is Model
C for all methods except for SD. Due to the use of the simplified simplicial depth for the
model with three parameters we lose power compared to the Models A and B with two
parameters each. Although Table 2 shows that the prediction intervals using the simplified
depth have good coverage rates, the scores are much higher because of the very large
intervals. The other methods have good coverage rates around 90% which is good because
90% is the chosen level of the intervals. For Model A, the results are similar for the classical
methods while the full depth prediction intervals have the best scores. In Model A, the
δ-method always leads to the highest score. For Model B, the results are similar except
that the method based on depth is not anymore superior. Only for Model C, the δ-method
performs best among the considered methods. The superiority of the outlier robust method
based on depth in Model A may be due to the fact that more outliers are available in a
misspecified model. This advantage reduces for the more realistic Model B and disappears
completely for Model C since here the less efficient method based on simplified simplicial
depth must be used.
The proposed prediction methods can also be used to predict a new experiment with a
very low stress range, where no observations are available yet. Figure 4a shows the 90%
prediction intervals for such a new experiment SB06 with a stress range s0 = 50 MPa
using the depth method and the δ-method where model B was used as the link function.
The intervals for the δ-method are a little larger than for the depth method. When the
experiment was conducted at the SFB 823 in TU Dortmund, one wire breakage could be
observed after about 28 million load cycles. The experiment was stopped after about 108
million load cycles without observing a second wire failure. This results in a censored
observation indicated by the arrow in Figure 4a. Both methods imply that the experiment
22
Table 2: Median interval score in millions and coverage rates for the 90% prediction intervals
in the scenarios Next and Next 5 for different models and prediction methods
Method
Interval score
Coverage rate
& model
Next Next 5
Next Next 5
Wald & A
4.56
14.16
0.86
0.87
LR & A
4.56
14.16
0.86
0.87
δ&A
4.82
15.72
0.87
0.88
FD & A
4.29
12.59
0.84
0.94
Wald & B
3.02
9.84
0.87
0.85
LR & B
3.02
9.84
0.87
0.85
δ&B
3.35
11.26
0.88
0.91
FD & B
3.02
10.02
0.87
0.88
Wald & C
3.95
13.46
0.88
0.92
LR & C
3.95
13.80
0.88
0.92
δ&C
3.71
12.59
0.88
0.91
SD & C
5.20
22.65
0.90
0.99
Sample size
139
99
139
99
was stopped too early because the upper bound of the prediction intervals for the second
wire breakage is at about 125 million for the depth method and at about 133 million for
the delta method. That means a second wire failure could be expected within the next 25
million load cycles.
In order to check for pre-existing defects in the tension wires caused by the 108 million
load cycles, the experiment was restarted at a higher stress range of 120 MPa as SB06a. The
course of this experiment is depicted in Figure 4b along with the 90% prediction intervals
given by the two methods where no pre-existing damage is assumed. The intervals for the δmethod are again larger. None of the true failure times of the experiment SB06a are covered
by either of the methods. The wires broke earlier than our methods proposed. This makes
pre-existing defects in the wires plausible and excludes the possibility that a fatigue limit is
reached with 50 MPa. It can be assumed that further failures would have happened at the
23
SB06a with 120 MPa
5
10
15
20
depth method
δ−method
true experiment
0
1
2
3
Number of broken tension wires
depth method
δ−method
true experiment
0
Number of broken tension wires
4
SB06 with 50 MPa
0
50
100
150
0
Load cycles in millions
5
10
15
20
25
30
Load cycles in millions
(a)
(b)
Figure 4: 90% prediction intervals for a new experiment SB06 with a very low stress range
of 50 MPa in (a). The arrow indicates that the experiment was stopped after 108 million
load cycles with only one wire breakage. The restarted experiment SB06a with a higher
stress range of 120 MPa is shown in (b).
lower stress range as well when the experiment would not have been stopped prematurely.
Comparing the two methods with each other, they perform very similar. Only the depthmethod yields smaller prediction intervals than the δ-method which is appropriate for the
first event of SB06. This corresponds with the results of the leave-one-out-analysis in Table
2.
8
Conclusion
We have proposed two new prediction intervals for the time of the failure of the Ic ’th
component of a system with I components when the failure process of other systems with
I components under different stress levels has been observed before. One new prediction
interval is outlier robust because it is only based on the signs of residuals of the observed
failures. Outlier robustness is important since sometimes the interarrival times between
the failure of components are very short. Although outlier robust methods are often less
efficient than classical methods, the application to a real data set coming from experiments
24
with prestressed concrete beams showed that they behave similar or even better than the
classical methods based on the Wald test and the likelihood ratio test, at least if the
unknown parameter is two-dimensional. The problem for parameters of dimension higher
than two is that the asymptotic distribution of the underlying test statistic is not known
up to now. We used a simplified version for this test statistic which is less efficient and
leads to larger prediction intervals. We expect better results as soon as the asymptotic
distribution of the original outlier robust test statistic for dimensions higher than two is
found. An alternative could be to use the depth for the two-parameter case for higher
dimensions as well. Then, the validity and the power of the method has to be checked via
simulation studies though.
The outlier robust methods and the classical methods have the disadvantage that confidence sets for the unknown parameter must be calculated. In particular this can be very
time consuming for higher dimensions of the parameter and has always the problem of accuracy. Therefore we proposed a more accurate method via the δ-method and the implicit
function theorem where no confidence sets has to be calculated. The application to the
real data set shows also a very good behavior of this method although the used dimension
of the parameter was not high. It can be assumed that the superiority of this method
becomes higher if the parameter has a higher dimension. However, the used models with
two and three dimensional parameters provided already a good fit so that there was no
need to consider higher dimensions of parameters for our real data set.
We assumed that the number I of components is the same for all system since this was
the case for our application with experiments with prestressed concrete beams. However,
an extension to systems with different numbers of components can be done easily. Much
more complicated will be the extension of our model based on nonlinear birth processes to
a model incorporating damage accumulation and aging of the components. Then simple
state dependent processes are not anymore adequate and processes from the more general
class of self-exciting processes must be used. In such models, simulation free predictions
intervals will be difficult to derive since our simulation free prediction intervals depend
heavily on the hypoexponential distribution which is the distribution of the sum of the
interarrival times which have different exponential distributions.
25
Acknowledgments
We would like to thank Alexander Schnurr for the valuable discussions about point processes. The research was supported by the Collaborative Research Center SFB 823 Statistical modeling of nonlinear dynamic processes.
SUPPLEMENTARY MATERIAL
R-package rexpar: R-package rexpar from Kustosz and Szugat (2016) containing code
needed for data depth (.zip file)
BatchExperiment registry: BatchExperiment registry for the leave-one-out validation
named crossvali (.zip file)
R source: R scripts to reproduce all figures and tables in the article. The considered data
sets and result files for long-running code are included in .RData files (.zip file)
References
Bai, F., Chen, F., and Chen, K. (2015). Semiparametric estimation of a self-exciting regression model with an application in recurrent event data analysis. Stat. Sin., 25(4):1503–
1526.
Barakat, H., El-Adll, M. E., and Aly, A. E. (2011). Exact prediction intervals for future
exponential lifetime based on random generalized order statistics. Comput. Math. Appl.,
61(5):1366–1378.
Basquin, O. (1910). The exponential law of endurance tests. Technical report, ASTM,
Philadelphia, PA.
Bates, D. and Maechler, M. (2016). Matrix: Sparse and Dense Matrix Classes and Methods.
R package version 1.2-4.
Baumgartner, B. and Gatto, R. (2014). Value at ruin and tail value at ruin of the compound
poisson process with diffusion and efficient computational methods. Methodology and
Computing in Applied Probability, 16(3):561–582.
26
Berg, M. P. and Haberman, S. (1994). Trend analysis and prediction procedures for time
nonhomogeneous claim processes. Insur. Math. Econ., 14(1):19–32.
Beutner, E. (2010). Nonparametric model checking for k-out-of-n systems. J. Stat. Plann.
Inference, 140(3):626–639.
Bischl, B., Lang, M., Mersmann, O., Rahnenführer, J., and Weihs, C. (2015). BatchJobs
and BatchExperiments: Abstraction mechanisms for using R in batch environments.
Journal of Statistical Software, 64(11):1–25.
Botev, Z. I., L’Ecuyer, P., Rubino, G., Simard, R., and Tuffin, B. (2013). Static network
reliability estimation via generalized splitting. INFORMS J. on Computing, 25(1):56–71.
Burkschat, M. (2009). Systems with failure-dependent lifetimes of components. J. Appl.
Probab., 46(4):1052–1072.
Chan, N. H., Li, Z. R., and Yau, C. Y. (2014). Forecasting online auctions via self-exciting
point processes. Journal of Forecasting, 33(7):501–514.
Cinlar, E. (1975). Introduction to stochastic processes. Englewood Cliffs, N. J.: PrenticeHall.
Cramer, E. and Kamps, U. (1996). Sequential order statistics and k-out-of-n systems with
sequentially adjusted failure rates. Ann. Inst. Stat. Math., 48(3):535–549.
Dachian, S. and Kutoyants, Y. A. (2006). Hypotheses testing: Poisson versus self-exciting.
Scandinavian Journal of Statistics, 33(2):391–408.
De Oliveira, V. R. B. and Colosimo, E. A. (2004). Comparison of methods to estimate
the time-to-failure distribution in degradation tests. Quality & Reliability Engineering
International, 20(4):363 – 373.
Engelhardt, M. and Bain, L. J. (1978). Prediction intervals for the Weibull process. Technometrics, 20:167–169.
Engelhardt, M. and Bain, L. J. (1982). On prediction limits for samples from a weibull or
extreme-value distribution. Technometrics, 24(2):147–150.
27
Fertig, K. W., Meyer, M. E., and Mann, N. R. (1980). On constructing prediction intervals
for samples from a weibull or extreme value distribution. Technometrics, 22(4):567–573.
Franz, J., Jokiel-Rokita, A., and Magiera, R. (2014). Prediction in trend-renewal processes
for repairable systems. Stat. Comput., 24(4):633–649.
Frey, J. (2012). Prediction bands for the EDF of a future sample. J. Stat. Plann. Inference,
142(2):506–515.
Gertsbakh, I., Neuman, E., and Vaisman, R. (2015). Monte carlo for estimating exponential
convolution. Communications in Statistics - Simulation and Computation, 44(10):2696–
2704.
Gneiting, T. and Raftery, A. E. (2007). Strictly proper scoring rules, prediction, and
estimation. J. Am. Stat. Assoc., 102(477):359–378.
Grothe, O., Korniichuk, V., and Manner, H. (2014). Modeling multivariate extreme events
using self-exciting point processes. Journal of Econometrics, 182(2):269 – 289.
Guo, C. and Luk, W. (2014). Accelerating parameter estimation for multivariate selfexciting point processes. In Proceedings of the 2014 ACM/SIGDA International Symposium on Field-programmable Gate Arrays, FPGA ’14, pages 181–184, New York, NY,
USA. ACM.
Hardiman, S. J. and Bouchaud, J.-P. (2014). Branching-ratio approximation for the selfexciting hawkes process. Phys. Rev. E, 90:062807.
Hawkes, A. G. (1971a). Point spectra of some mutually exciting point processes. Journal
of the Royal Statistical Society. Series B (Methodological), 33(3):438–443.
Hawkes, A. G. (1971b). Spectra of some self-exciting and mutually exciting point processes.
Biometrika, 58(1):83–90.
Hong, Y. and Meeker, W. Q. (2013). Field-failure predictions based on failure-time data
with dynamic covariate information. Technometrics, 55(2):135–149.
28
Hsieh, H. (1996). Prediction intervals for weibull observations, based on early-failure data.
Reliability, IEEE Transactions on, 45(4):666–670.
Jacobsen, M. (2006). Point process theory and applications. Marked point and picewise
deterministic processes. Boston: Birkhäuser.
Kazemipour, A., Babadi, B., and Wu, M. (2014). Sparse estimation of self-exciting point
processes with application to lgn neural modeling. In Signal and Information Processing
(GlobalSIP), 2014 IEEE Global Conference on, pages 478–482.
Kim, H. and Kvam, P. H. (2004). Reliability estimation based on system data with an
unknown load share rule. Lifetime Data Analysis, 10(1):83–94.
Kopperschmidt, K. and Stute, W. (2013). The statistical analysis of self exciting point
processes. Stat. Sin., 23(3):1273–1298.
Krishnamoorthy, K., Lin, Y., and Xia, Y. (2009). Confidence limits and prediction limits
for a Weibull distribution based on the generalized variable approach. J. Stat. Plann.
Inference, 139(8):2675–2684.
Kustosz, C. and Szugat, S. (2016). rexpar: Simplicial Depth for Explosive Autoregressive
Processes. R package version 1.1.
Kustosz, C. P., Leucht, A., and Müller, C. H. (2016a). Tests based on simplicial depth for
AR(1) models with explosion. Journal of Time Series Analysis.
Kustosz, C. P. and Müller, C. H. (2014). Analysis of crack growth with robust, distributionfree estimators and tests for non-stationary autoregressive processes. Statistical Papers,
55(1):125–140.
Kustosz, C. P., Müller, C. H., and Wendler, M. (2016b). Simplified simplicial depth for
regression and autoregressive growth processes. Journal of Statistical Planning and Inference, 173:125–146.
Kvam, P. H. and Peña, E. A. (2005). Estimating load-sharing properties in a dynamic
reliability system. J. Am. Stat. Assoc., 100(469):262–272.
29
Lawless, J. and Fredette, M. (2005). Frequentist prediction intervals and predictive distributions. Biometrika, 92(3):529–542.
Lawless, J. F. and Thiagarajah, K. (1996). A point-process model incorporating renewals
and time trends, with application to repairable systems. Technometrics, 38(2):131–138.
Lee, L. and Lee, S. K. (1978). Some results on inference for the weibull process. Technometrics, 20(1):41–45.
Meeker, W. Q., Escobar, L. A., and Lu, C. J. (1998). Accelerated degradation tests:
Modeling and analysis. Technometrics, 40(2):89–99.
Meerschaert, M. M., Nane, E., and Xiao, Y. (2009). Correlated continuous time random
walks. Stat. Probab. Lett., 79(9):1194–1202.
Mizera, I. (2002). On depth and deep points: A calculus. Ann. Stat., 30(6):1681–1736.
Mood, A. M., Graybill, F. A., and Boes, D. C. (1974). Introduction to the theory of
statistics. McGraw-Hill New York, 3rd ed. edition.
Müller, C. H. (2005). Depth estimators and tests based on the likelihood principle with
application to regression. J. Multivariate Anal., 95(1):153–181.
Müller, C. H. (2013). D-optimal designs for lifetime experiments with exponential distribution and censoring. In Ucinski, D., Atkinson, A. C., and Patan, M., editors, mODa 10
– Advances in Model-Oriented Design and Analysis, Contributions to Statistics, pages
179–186. Springer International Publishing.
Nordman, D. J. and Meeker, W. Q. (2002). Weibull prediction intervals for a future number
of failures. Technometrics, 44(1):15–23.
Patel, J. K. (1989). Prediction intervals - a review. Communications in Statistics - Theory
and Methods, 18(7):2393–2465.
Rambaldi, M., Pennesi, P., and Lillo, F. (2015). Modeling foreign exchange market activity
around macroeconomic news: Hawkes-process approach. Phys. Rev. E, 91:012819.
30
Rao, C. R. (1965). Linear statistical inference and its applications. New York-LondonSydney: John Wiley and Sons, Inc. XVIII, 522 p. (1965).
Ross, S. M. (2014).
Introduction to probability models. 11th ed.
Amsterdam: Else-
vier/Academic Press, 11th ed. edition.
Sánchez-Silva, M. and Klutke, G.-A. (2016). Reliability and Life-Cycle Analysis of Deteriorating Systems. Heidelberg: Springer.
Schervish, M. J. (1995). Theory of statistics. New York, NY: Springer-Verlag.
Schnurr, A. (2012). On the semimartingale nature of Feller processes with killing. Stochastic
Processes Appl., 122(7):2758–2780.
Snyder, D. L. and Miller, M. I. (1991). Random point processes in time and space. 2nd ed.
New York etc.: Springer-Verlag, 2nd ed. edition.
Sobczyk, K. and Spencer, B. F. (1992). Random Fatigue. From Data to Theory. Boston:
Academic Press.
Wang, X. and Xu, D. (2010). An inverse gaussian process model for degradation data.
Technometrics, 52(2):188–197.
Wheatley, S., Filimonov, V., and Sornette, D. (2016). The hawkes process with renewal
immigration & its estimation with an EM algorithm. Computational Statistics & Data
Analysis, 94:120 – 135.
Xiong, C. and Milliken, G. A. (2002). Prediction for exponential lifetimes based on stepstress testing. Commun. Stat., Simulation Comput., 31(4):539–556.
31