Inference for Type-I and Type-II Hybrid Censored Minimal Repair and Record Data

In this paper, hybrid censoring mechanisms are applied to minimal repair and record data. Based on the derivation of the joint distribution of such data under hybrid censoring, likelihood inference is discussed. For illustration, Type-I and Type-II hybrid censoring schemes are considered for exponential distributions. In particular, the exact (conditional) distribution of the maximum likelihood estimator is obtained for an exponential distribution. This result is used to construct exact (conditional) confidence intervals using the method of pivoting the cumulative distribution function. Finally, the results are illustrated using two data sets taken from the literature on minimal repair models. Although the discussion of the results is in terms of minimal repair models, the results can be applied directly to record value data. By utilizing a connection of minimal repair times to occurrence times of non-homogeneous Poisson processes, a nonparametric estimate for the intensity rate of the process and the underlying lifetime distribution under hybrid censoring is also proposed. The paper is supplemented by simulational results.


Introduction
The notion of minimal repair has been considered in the literature as a concept where a repairable item is repaired with minimal effort so that the item continues to function at the same level of wear and tear as before the repair.Repairs are usually assumed to be performed without time loss, since the repair time is usually supposed relatively short compared to the expected operating time of the item.A simple example of such a situation is the replacement of a broken V-belt in a car engine since the age of the engine can be considered the same after the repair.Minimal repair models have been discussed in various ways in the literature and many extensions have been provided.For instance, Barlow and Hunter [13] considered minimal repair in connection with maintenance strategies.In Ascher [6], the model has been discussed in terms of a badas-old concept.Ascher [4] commented on models for reliability of repaired equipment whereas Ascher and Feingold [5] focused on repairable systems reliability.Further results, extensions, and connections are presented in Block et al. [17] and Gupta and Kirmani [39].Aven and Jensen [7] discussed a general minimal repair model in terms of point processes.A survey of results and models can be found in Tadj et al. [51].
Gupta and Kirmani [39] established a connection of the standard minimal repair model to record values and non-homogeneous Poisson processes (NHPP) (see also Pfeifer [50], Cramer [23]).This connection is quite useful in the statistical analysis of such data since it points out that record values from a common cumulative distribution function (cdf) F, minimal repair times with lifetime cdf F, as well as jump times of a NHPP with cumulative intensity rate = − log F can be treated identically in a probabilistic sense.In this respect, the results presented in the following can also be applied to the hybrid censored data that are generated by these models.
In this paper, we combine minimal repair models (or record data) with hybrid censoring strategies.A recent survey of hybrid censoring models and respective inferential results has been provided by Balakrishnan et al. [10] who mainly focus on models based on (progressively Type-II censored) order statistics (see also Balakrishnan and Kundu [12], Balakrishnan and Cramer [8]).These models have been entitled as nonreplacement cases of truncated life tests.In addition to the above-mentioned reviews, the following works, among others, are of importance for the case of an exponential distribution mainly dealt with in this article: Chen and Bhattacharyya [21], Gupta and Kundu [40], Childs et al. [22], Chandrasekar et al. [20], Balakrishnan and Iliopoulos [11], Cramer and Balakrishnan [25].
However, early publications like Epstein [34] considered also (Type-I) hybrid censored in a replacement scenario.This means that as soon as the item fails, it will be replaced with an identical item of the same virtual age.Clearly, this can be interpreted as a minimal repair.This model has also been addressed in Ebrahimi [31] (two-parameter exponential distributed lifetimes), Fairbanks et al. [35] (confidence intervals), Draper and Guttman [29] (Bayesian inference), and Ebrahimi [32] (prediction), but does not appear to have received further attention in the literature.
In this paper, we will take this idea and combine it with the censoring mechanisms of hybrid censoring.In particular, we will use the structural approach proposed in Cramer [24] to present a general approach to the statistical analysis of such data under hybrid censoring.The approach is based on ordered data X (1) ≤ • • • ≤ X (m) which is subject to hybrid censoring.In the following, these data will be generated from the minimal repairs in a sample of exactly n items (for details see eq. 2).We will illustrate the power of the approach using the basic models of Type-I and Type-II hybrid censoring.
Thus, given minimal repair data X (1) ≤ X (2) ≤ X (3) ≤ • • • , a Type-I hybrid censored sample with a desired number r of measurements and time threshold τ is obtained by terminating the life test at The corresponding sample is given the measurements X (1) ≤ • • • ≤ X (M I ) with random sample size M I = r j=1 1 (−∞,τ ] (X ( j) ).Here, 1 A : R → {0, 1} denotes the indicator function, that is, 1 A (x) = 1 ⇐⇒ x ∈ A. Clearly, the resulting data is bounded by the threshold τ .The sample size M I is at most r but may be zero.
For Type-II hybrid censoring, the test duration is given by which ensures a minimum sample size of r .In particular, we get the measurements with random sample size M II = r + ∞ j=r +1 1 (−∞,τ ] (X ( j) ) However, both the observed sample size M II and the test duration W II are unbounded.Note that, in non-replacement models as discussed in detail in Balakrishnan et al. [10], the sample size is always bounded.
In Sect.2, we introduce the data and the model in detail.We derive the joint distribution of the ordered repair time data and of the random number of measurements assuming a general life time distribution with cdf F. For subsequent use, we present also the respective results for exponentially distributed lifetimes.In particular, we establish the respective results for both Type-I and Type-II hybrid censoring.Sect. 3 addresses likelihood inference particularly in the exponential case for such data as well as nonparametric estimation of the intensity rate when the data is understood as occurrence times of a non-homogeneous Poisson process [see Sect.(2.2)].After establishing the maximum likelihood estimators for the mean θ of an exponential distribution in terms of the total time on test, we derive their exact (conditional) cdf using the modularization approach in the form presented in Balakrishnan et al. [10] and Cramer [24].Using the method of pivoting the cdf (see Balakrishnan et al. [9]) and illustrating its applicability, we show that it leads to exact (conditional) confidence intervals.In Sect.4, the results are illustrated by two data sets taken from Whitaker and Samaniego [54] and Kumar and Klefsjö [44], respectively.For comparison, we provide also confidence intervals due to Fairbanks et al. [35] under Type-I hybrid censoring.The discussion is supplemented by some simulations illustrating the concepts and results.Further directions of research are sketched in Sect. 5 (conclusions and outlook).
Finally, we would like to emphasize that, although we consider the data to be repair times, it is possible to interpret them as record values, too (see also Gupta and Kirmani [39] and comments given above).This is not important for the derivation of the presented statistical and probabilistic results.It is all a matter of interpretation.corresponding pdf is given by f (x) = θ −1 e −x/θ , x ≥ 0. Thus, θ > 0 denotes the mean lifetime of the items under study.
In the following, we study different minimal repair models with both a maximum or minimum number of r observations as well as a time threshold τ .The design of the hybrid censored life test is based on the time threshold τ which may be interpreted as either a maximum test duration or a desired test duration.In order to avoid trivialities, we assume that τ is included in the support of F, that is, 0 < F(τ ) < 1.In this regard, the models can be considered as a (Type-I/Type-II) hybrid censoring of minimal repair times or as Type-I/Type-II hybrid censored data with replacement.
As mentioned above the considered models are designed by performing Type-I and Type-II censoring on a sequence of minimal repair times.Therefore, we introduce first the underlying basic minimal repair model (see Model 2.1).

Model 2.1 (Minimal repair model in a sample of n items)
Consider a life test with n items whose lifetimes are monitored.If the first failure occurs at the failure time X (n,1) , a minimal repair will be performed and the monitoring process will be continued.The failure times of different items are supposed independent.This procedure will be repeated unless the experiment is terminated and, thus, results in an increasing sequence of failure/repair times that is, X (n,k) k∈N .
In order to analyse the above data, one has to discuss the distribution of the observed failure/repair times X (n,k) k∈N .Clearly, X (n,1) = min{X 1 , . . ., X n }.But now, the failed component is repaired which affects the distribution of its lifetimes so that the resulting random variables are no longer identically distributed after the first repair.However, the independence is preserved.Therefore, we will discuss first the joint distribution of the first k repair times X (n,1) , . . ., X (n,k) generated according to Model 2.1.To obtain a formal construction of the minimal repair model, we introduce the standard minimal repair model (SMR) and illustrate its relation to a NHPP.For brevity, we introduce the notation t m = (t 1 , . . ., t m ) ∈ R m .The standard minimal repair model corresponds to the case n = 1 of Model 2.1, that is, we consider a life test with a single object whose lifetime is monitored.If a failure occurs then it will be instantly repaired in the sense of minimal repair so that always one object is under study.The corresponding sequence of failure/repair times is denoted by Block et al. [17] studied the standard minimal repair model under more general assumptions and provided a formal construction.Furthermore, Gupta and Kirmani [39] identified the failure/repair times of the standard minimal repair model with the record values of a sequence of i.i.d.r .v.s with common cdf F. In order to establish this result, they considered the counting process generated by the corresponding failure/repair times, which turns out to be a NHPP.The corresponding counting process (N (t)) t≥0 is defined by Hence, the extensively studied topic of record values (see, for example, Arnold et al. [3], Nevzorov [49], David and Nagaraja [28]) can be applied to minimal repair data.In particular, the failure times {X (1,n) | n ∈ N} from the SMR Model and the (upper) records in a sequence of i.i.d.r.v.s (Y n ) n∈N are identically distributed.This follows immediately from Theorem 1 in Gupta and Kirmani s[39, Section 2].Thus, we have the following proposition.

Proposition 2.1 The repair times (X (1,n) ) n∈N in the SMR model and the (upper) record values (Y
where F denotes the survival function of F.
We will see that a slight modification of this approach can be utilized to gain a similar identification for the Type-I/Type-II hybrid censored minimal repair models.
In a next step, we consider the minimal repair model 2.1.Note that if an object fails then a minimal repair is conducted, so that immediately after the failure again n items are under test.For illustration, the scenario including the repair times is depicted in Fig. 1 for n = 4 items on test.
Remark 2.3 1. Corollary 2.2 shows that the situation can also be interpreted as follows.Instead of considering an (initial) sample of size n of i.i.d.r.v.s and performing minimal repairs to each item upon failure, one may consider a series system with n i.i.d.components and conduct a minimal repair to the series system.The resulting samples in both scenarios have the same distribution.2. Notice that minimal repair times can be considered as particular sequential order statistics.Therefore, Type-I hybrid censored minimal repair times can be seen as a particular case of Type-I censored sequential order statistics which have been discussed in Burkschat et al. [18].3.In the preceding construction, the information about the failed (and repaired) item has been ignored.If this information is available then we can handle the situation as n independent samples with the same baseline distribution.Note that the stopping rule for the observation process can now be set in two ways.
(a) The stopping is performed per sub-sample, that is, we require for each subsample a specific hybrid censoring scheme.For instance, we can consider Type-I hybrid censoring with a threshold τ i and a maximum number m i for the i-th sub-sample.Of course, one can choose τ i = τ or m i = m if appropriate.This construction leads to n independent samples of hybrid censored minimal repair times subject to possibly different hybrid censoring schemes.
In particular, the sample sizes M Ii for each sub-sample are independent.Note that, technically, these hybrid censoring schemes can be different for each subsample.In case of progressively censored data, such multi-sample situations have been considered in Górny and Cramer [38] (Type-I hybrid) and Jansen et al. [42] (Type-II hybrid) which have led to explicit but messy results.It would be interesting to see whether such results can also be established for hybrid censored minimal repair data.(b) A hybrid censoring scheme is applied jointly to all sub-samples.In case of Type-I hybrid censoring, a threshold τ and a total number of observations m is fixed in advance leading to sample sizes M Ii for each sub-sample.However, the M Ii 's are dependent now since they depend on all sub-samples!This situation can also be analysed in terms of a marked NHPP (resulting from the superposition as above) which is subject to hybrid censoring.The marks are given by the sub-sample number.For information on marked Poisson processes, see, e.g., Møller and Waagepetersen [48], Last and Penrose [45].
It will be interesting to analyse these models which will be subject of future research.

Corollary 2.4 ( Exponential distribution) Assuming exponentially distributed lifetimes, the joint pdf in Corollary 2.2 for fixed k
From Proposition 2.1, we find the marginal cdf of X (n,i) (see Arnold et al.[3,p.10]).
Theorem 2. 5 The cdf of X (n,i) , for i ∈ N, in Model 2.1 with common cdf F is given by (2.6) Fig. 2 Possible data scenarios for the minimal repair model with time censoring at τ > 0 and maximum number r = 5 of repairs (with different choices of X (n,5) )

Type-I Hybrid Censored Data
For a given sequence of increasingly ordered r.v.s (Z k ) k∈N as well as parameters r ∈ N and τ > 0, the corresponding Type-I hybrid censored sample (Z This procedure is applied to minimal repair data generated from Model 2.1, that is, the life test is terminated when either the r -th repair occurs or the pre-determined maximum test duration τ > 0 expires.This situation is depicted in Fig. 2 for r > 4 and some τ > 0. The experimental time of such a life testing experiment is given by Note that this is obtained from Model 2.1 by bounding the test duration by τ .Denoting the number of repairs until τ by the random counter it is clear that M maybe zero (generally with positive probability).To be more precise, this happens with probability which is positive since 0 < F(τ ) < 1 has been assumed.Therefore, the experimenter is faced with the problem that the life test may terminate without observing a repair.As a result, inference will be of conditional nature since we have to ensure to observe at least one failure or repair.Thus, we will assume subsequently that M > 0. In the Type-I hybrid censoring case, the sample size is given by that is, the maximum number of observed failures is bounded by r .Notice that the same problem is present for Type-I censoring and Type-I hybrid censoring without replacement, respectively.The probability mass function (pmf) of M results from the equivalence and the marginal cdf of X (n,m) which is given in Theorem 2.5.

Lemma 2.6
The pmf of the r.v.
Further, the expectation of M I is given by Proof The pmf follows directly from Lemma 2.5.The expectation of M I is obtained by writing In the next step, we present an expression for the joint cdf of the Type-I hybrid censored minimal repair times (see Fig. 3).It can be easily deduced from the general results presented in Cramer [24] (see also Burkschat et al. [18]) and Lemma 2.6 provided that are available.However, proceeding by analogy with the case of order statistics discussed in Cramer and Balakrishnan [25], we find similar expressions so that we omit the technical details.In particular, for m ∈ {1, . . ., r }, one has to replace the joint cdf of the first m order statistics by that one of the first m minimal repair times, that is, by with f (1,...,m;n) given in (2.5).Furthermore, one has to take into account that the following identity is true for 1 ≤ m ≤ r − 1: where This follows from (2.9) by evaluating the most inner integral.Thus, we find the following expressions for the cdf and pdf, respectively.
Introducing the test duration w = t r ∧ τ , the pdf can be written as so that the case-by-case definition is not needed.For exponential distributions, the situation simplifies considerably.

Corollary 2.8 (Exponential distribution)
Assuming exponentially distributed lifetimes, the joint pdf in Theorem 2.7 reads It follows from (2.12) that the joint pdf depends only on the test duration w = t r ∧τ so that we can write Note that TTT r = nw = n(t r ∧ τ ) denotes the total time on test of the life test.This observation will be very useful considering likelihood inference.

Type-II Hybrid Censored Data
For a given sequence of increasingly ordered r.v.s (Z k ) k∈N as well as parameters r ∈ N and τ > 0, the Type-II hybrid censored sample is defined by with M as in (2.7).Thus, M II = M ∨ r denotes the random sample size of the Type-II hybrid censored sample.Clearly, we have M II ≥ r so that at least r measurements are available.Moreover, since (Z k ) k is a sequence of r.v.s , it may be possible that M is infinite.In order to have a proper model, one may assume that Since, for any ∈ N, this holds, when P(Z ≤ τ ) →∞ − −− → 0. As above, we consider minimal repairs in a sample of n items.The life test is terminated when both the r -th repair has occurred and the pre-determined maximum test duration τ > 0 is expired.
Representing the repair times by X (n,1) , X (n,2) , X (n,3) , . . .as in Fig. 3, the experimental time of the life test is given by Therefore, the test duration is bounded from below by τ which can be considered as a minimal test duration in this setting.The number of observed failures is given by M II = M ∨ r with M as in (2.8).By similarity with Lemma 2.6, we get the pmf of M II .Note that we can write Proof The representation of the pmf follows directly from the definition of M II and the pmf of M. Since 0 < F(τ ) < 1 has been assumed throughout this paper, we get using the marginal distribution of the -th minimal repair time (or record value) (see Theorem 2.5) As mentioned above, this implies The expectation of M II is calculated by analogy with that of M I in Lemma 2.6.It should be noted that it converges to ∞ when τ → ∞ since this implies F(τ ) → 1.
Since M II is unbounded, we present in Theorem 2.10 the joint cdf of the data and the counter M (which is more convenient than M II in that case).Notice that where the cases {M < r }, {M = r } can be distinguished based on the observed data, that is, the sample (X (n,i) ) 1≤i≤M II , M II can be constructed from the sample (X (n,i) ) 1≤i≤M∨r , M and vice versa.
Theorem 2.10 Let (X (n,i) ) i∈N be the minimal repair times in a sample of size n.Then, for m ≥ r , the cdf F (I I ),M (1,...,r ;n) of the Type-II hybrid censored repair times (X (n,i) ) 1≤i≤M∨r and M is given by where F (1,...,m;n) denotes the joint cdf of X (n,1) , . . ., X (n,m) , m ∈ N, and H (1,...,m) is defined in (2.11).Furthermore, In the case M < r , we obtain that Thus, we arrive at the given expression for the joint cdf.
Since F is supposed absolutely continuous with pdf f , we get the following pdf from Theorem 2.10.

Corollary 2.11 Let (X (n,i)
) i∈N be the minimal repair times in a sample of size n.Then, for m ∈ {r , r + 1, . . .}, the pdf f X (n,1) ,...,X (n,M II ) ,M II of the Type-II hybrid censored repair times X (n,1) , . . ., X (n,M II ) and the sample size M II is given by where f (1,...,k;n) denotes the joint pdf of X (n,1) , . . ., By analogy with (2.12), we can write the pdf in terms of the test duration w = t r ∨τ as The expression in (2.14) simplifies considerably in case of exponentially distributed life times.In particular, the pdf depends only on the data via the total time on test Corollary 2.12 (Exponential distribution) Assuming exponentially distributed lifetimes, the joint pdf in Corollary 2.11 reads 13 Clearly, the preceding discussion can also be applied to other hybrid censoring schemes following the ideas presented in Cramer [24].Further details will be provided in future research.

Likelihood Inference
In this section, we assume that the baseline cdf F θ depends on some unknown parameter θ ∈ ⊆ R p .First, we consider likelihood inference for Type-I and Type-II hybrid censored data.From (2.12) and (2.14), we find that the likelihood function is given by for realizations x 1 , . . ., x m , m of X (n,1) , . . ., X (n,M HCS ) , M HCS , where M HCS ∈ {M I , M II } and w denotes the observed test duration (that is, the realization of W HCS ∈ {W I , W II } as mentioned above).
Remark 3.1 It follows from the results presented in Cramer [24] that the likelihood function under any hybrid censoring scheme has the form given in (3.1) with appropriately chosen values of m (observed sample size) and w (test duration) for the hybrid censoring scheme under consideration.

MLEs for Exponentially Distributed Lifetime Distributions
Clearly, the expression in (3.1) has the same mathematical form as the likelihood function for the first n record values so that the computation of the maximum likelihood estimates uses the same methods and leads to the same expressions (see general comments in Cramer [24]).
For illustration, we discuss the case of exponential distributions.In particular, the MLE for exponentially distributed lifetimes with mean θ is given by provided it exists.Under Type-II hybrid censoring, this is always true.In case of Type-I hybrid censored data, one has to assume that M I > 0.
Remark 3.2 According to the comments provided in Cramer [24], results on Bayesian inference can also be easily established.

Nonparametric Estimates for the Intensity Rate and the Cumulative Distribution Function
Since the model can also be considered in terms of superposed NHPPs, one can consider estimation of the intensity rate λ(•).Using the parametric approach for exponentially distributed lifetimes, we get as parametric estimates for the intensity and the underlying distribution function, respectively.First note that inferential results for a NHPP under hybrid censoring are not available so far.However, utilizing the previous results we can construct a nonparametric estimator restricting the observation window of the test to the interval [0, W HCS ).We consider the superposed NHPP.
For illustration, we restrict the discussion to Type-I hybrid censoring.Other hybrid censoring schemes, can be handled similar by suitable adaptions.A more detailed analysis will be subject of future research.Let w I = x m ∧ τ and m I be the realizations of W I and M I , respectively.Then, a piecewise-linear Nelson-Aalen type estimator of the cumulative intensity function = − ln(1 − F) under Type-I hybrid censoring is constructed by 1. Type-I case, that is, where x 0 = 0, and, as above, and x 1 , . . ., x m are realizations of the ordered occurrence times X (n,1) , . . ., X (n,M I ) .As mentioned in Leemis [46], the factor m I /(m I + 1) takes into account that stopping at the threshold τ leads to m I + 1 gaps.2. Type-II case, that is, w I = x m < τ and m I = m: The corresponding estimate of the cumulative distribution function is given by t) , t ≥ 0. Note that a Type-I censored version of a NHPP has also been discussed in Henderson [41] (see also Lewis and Shedler [47], Leemis [46]).Furthermore, the intensity rate λ(•) of the NHPP is estimated by (i) Type-I case: Applications of these estimates to Type-I hybrid censored data are presented in Sect. 4.

Distribution of the MLE for Exponentially Distributed Lifetimes
As pointed out in (3.2), the MLE of θ can be written in terms of the total time on test statistic.Thus, we get a quite similar expression as for hybrid censored order statistics' data.Moreover, the distribution of the MLE can be found using the modularization approach (see, e.g., Górny and Cramer [37], Balakrishnan et al. [10], Cramer [24]).
In case of Type-I hybrid censoring, the conditional cdf can be taken directly from Burkschat et al. [18].However, since the derivation is quite forward, we present a direct proof which also yields a simpler representation of the conditional cdf.For convenience, we denote by g m,θ and G m,θ the pdf and cdf of a gamma distribution with scale parameter θ > 0 and shape parameter m ∈ N, that is, g m,θ is defined by of the MLE under Type-I hybrid censoring is given by, Proof We get by the law of total probability (w.r.t. the random counter M) that From the pdf we conclude that, for x ≥ 0, Hence, we get Furthermore, by Theorem 2.5, we get Depending on the value of nτ , we can split this expression into the following cases Consequently one gets the expression in (3.8) by dividing this formula by the probability P θ (M ≥ 1) = 1 − P θ (M = 0).
Note that the corresponding distribution of the MLE θ can be split into a continuous part with support on [0, nτ/r ] and a discrete part with point masses at the points nτ, nτ/2, . . ., nτ/(r − 1).
In order to construct exact conditional confidence intervals, we make use of the method of pivoting the cdf.Therefore, we have to ensure that the (conditional) cdf of the MLE is monotonic in θ > 0 and that the associated limits satisfy certain conditions (see Balakrishnan et al. [9]).Fortunately, these conditions can be concluded again from the results of Burkschat et al. [18] or by straightforward derivations using the representations of the conditional cdf in Theorem 3.3.The stochastic monotonicity of the MLE can alternatively be deduced from van Bentum and Cramer [14].Theorem 3.4 For the conditional MLE θ under Type-I hybrid censoring and arbitrary nτ >x ≥ 0, P θ ( θ > x | M ≥ 1) is a monotone increasing function in θ > 0 with Therefore, we get the following result.
for θ is obtained by solving the following two non-linear equations.
In particular, if r θ < nτ then one has to solve Theorem 3.5 follows directly from the method of pivoting the cdf (see, for example, Casella and Berger [19]) and Theorem 3.3.

Remark 3.6 (i)
To utilize Theorem 3.5, one has to ensure the solvability of the above equations.For r > 1, this is guaranteed by Theorem 3.4, while the given equations do not have to have a solution in the case r = 1.Therefore, one has to modify the bounds of the confidence interval in this case slightly (see Balakrishnan et al. [9]).However, if possible, the problematic case r = 1 should be excluded by the design of the experiment.(ii) In order to apply the pivoting method, it is important to take into account that the (conditional) distribution of the maximum likelihood estimator has point masses under hybrid censoring.As pointed out in Casella and Berger [19, p. 434], these point masses have to be considered in the sense that we do not consider the survival function but the probability P θ ( θ ≥ x | M ≥ 1) to determine the lower bound θ L when the value θ obs of the maximum likelihood estimator satisfies θ obs > nτ/r .This ensures the desired level which, due to the point masses, may be somewhat higher than required.
Rewriting the gamma cdf in Theorem 3.5 as a cdf of a χ 2 -distribution, we get G ,θ (x) = F χ 2 (2 ) (2x/θ ) illustrating that the conditions are connected to χ 2 (2 )-distributions.This shows the similarity to Fairbank's construction of confidence intervals (see Fairbanks et al. [35] and Remark 3.7).(iii) In general, one has to calculate the interval bounds θ L and θ U numerically.
Examples are presented in Sect. 4.
Remark 3.7 Fairbanks et al. [35] proposed (1 − α)-confidence intervals for Type-I hybrid censored data given the observed sample size M I .In particular, they gave the following confidence intervals (written in our notation) A comparison to the confidence intervals constructed by the method of pivoting the cdf is given in the data examples in Sect. 4.
In case of Type-II hybrid censoring, we get similar (but unconditional) results.First, we present an explicit expression of the cdf of the MLE.Theorem 3.8 For x ≥ 0, the cdf G II r ,θ of the MLE θ under Type-II hybrid censoring is given by Arguing as in the proof of Theorem 3.3, we obtain with the cdf of the gamma distribution the following expression for the cdf of the MLE: Furthermore, using (3.9), we arrive at the desired representation of the cdf G II r ,θ .Depending on the value of nτ , we can split the above expression again into the following cases Consequently one gets the expression in (3.10).

Remark 3.9
The representation of the cdf in (3.10) shows that the MLE has both a continuous part with support (nτ/r , ∞) and a discrete part with point masses at nτ/k, k = r , r +1, . . . .Notice that the cdf has infinitely many jumps in the interval [0, nτ/r ].
A plot of a density function (blue curve) is depicted in Fig. 4 with n = 2, τ = 500, r = 10, θ = 100.The jumps are located at 100, 1000/11, 1000/12, 1000/13, . . . .It is worth mentioning that the situation is in some sense reflected to the Type-I hybrid censoring scheme where one has a continuous part first followed by a finite number of jumps.
The stochastic monotonicity of the MLE follows directly from Bentum and Cramer [14, Example 1] so that it remains to show that the limits of the cdf w.r.t.θ are 0 and 1, respectively.This follows directly from (3.10) by noticing that is, θ is a scale parameter.Hence, lim θ→0 G m,1 (z/θ ) = 1, lim θ→∞ G m,1 (z/θ ) = 0.Then, considering the expression in (3.10) for a given x, the result follows immediately.Furthermore, it should be noted that the monotonicity in θ can also be directly The resulting total time on tests, number of observed measurements (in brackets), and maximum likelihood estimates for θ are summarized in Table 3.Notice that the MLE for θ given the two (independent) samples is given by (see Cramer and Kamps [26], Beutner and Cramer [16]).For exponential distributions, an exact (1 − α)-confidence interval for θ is given by (32) α=.05Despite the rather small sample sizes present in the censored data, the results for both estimates and conditional confidence intervals are quite reasonable compared to the full available information without hybrid censoring, that is, the data taken from Tables 1 and 2. The aspect is also discussed using simulations in Sect.4.2.In particular, the results improve as the desired sample size r and/or the time thresholds increase.This is quite natural since this is associated with a larger observation window, so that more information is available in the estimation process.Note that the results are worse for small thresholds under Type-I hybrid censoring, i.e. for a short maximum test duration, which usually results in few or no measurements.Furthermore, the results are better under Type-II hybrid censoring which is also due to more information provided by the sampling process.Finally, it turns out that the (conditional) confidence intervals for Type-I hybrid censored data due to Fairbanks et al. [35] are close to our results.For a larger threshold τ and larger values of r , they are almost identical.
Estimators of the intensity rate and the cumulative distribution function based on Type-I hybrid censored data are depicted in Figs. 5 and 6.First, it should be noted that the nonparametric piecewise-linear estimate of the distribution function and the parametric estimate based on an exponential distribution assumption are quite close, despite the small sample sizes.The parametric estimate of the distribution function majorizes (in these examples) the nonparametric estimate so that early failures are somewhat more likely assuming an exponential distribution.The nonparametric intensity rate estimate is close to the constant estimate obtained from the exponential assumption.However, at the end of the observation window it indicates an increase in the rate

Table 3
Test durations, number of observed measurements (in brackets), and maximum likelihood estimates for θ based on the data in Table 1 for some values of the threshold τ and the sample size r under Type-I and Type-II hybrid censoring Sample sizes r   The conclusions drawn from Table 4 are very similar to those obtained for the data analysed in Example 4.1 so that we do not provide further comments here.
Estimators of the intensity rate and the cumulative distribution function based on Type-I hybrid censored data are depicted in Figs.7 and 8.For both intensity rate and cumulative distribution function, the behaviour of the estimates is similar to that already described for the air conditioner data in Example 4.1.In particular, the impact on the observation window caused by the thresholds τ = 500 and τ = 750 is more evident in this example than in Example 4.1.

Simulations
In order to assess properties of the point estimator, we have conducted a simulation study.For brevity, we restrict ourselves to the Type-I hybrid censoring scheme but, of course, similar simulations can be performed for other hybrid censoring schemes.We consider a situation similar to the load-haul dump machine data, that is, we assume (i) maximum sample sizes m ∈ {1, 2, . . ., 25}, (ii) threshold τ ∈ {50, 60, . . ., 1200}, (iii) number of simulation runs N = 5000 for each combination of m and τ , (iv) a true parameter θ = 91.9048,and n = 2 (superposed) minimal repair samples.Similar results have been observed for other input parameters.The evaluations are based on the same seeds for any m and τ .This enables us to study the impact of both the sample size and the threshold on the estimates.A histogram and kernel density estimate of the estimates θ I is presented in Fig. 9 for m = 20 and selected values of τ .The plots clearly illustrate the point masses of the distribution.For comparison, the density function of a scaled χ 2 -distribution is additionally provided.The degrees of freedom df are taken as rounded values of 2m I where m I is computed as mean sample size from the simulation (see also Table 5).Comparing the continuous kernel density estimate and the plotted χ 2 -density function, it turns out that the curves are somewhat close (particularly on the right tail), that is, the χ 2 -density function looks like a smoothed version of the kernel density estimate.The difference gets smaller for increasing τ which corresponds to less cases corresponding to Type-I censoring (see Table 5, column 6).In fact, this might illustrate why Fairbanks' construction of confidence intervals works although it ignores the existence of point masses in the distribution of the maximum likelihood estimator.
For a better visualization, the results are presented in terms of heatmaps generated by the R package 'gplots' (see Warnes et al. [53]).In particular, we provide the proportion of non-existing MLEs (Fig. 10) as well as bias (Fig. 11), standard deviation (Fig. 12), and mean squared error (Fig. 13) of the maximum likelihood estimates, respectively.
It turns out that, as expected, the observed proportions of non-existing MLEs rapidly decrease with an increasing threshold τ .In particular, the proportion is less than 5% and 1% for τ ≥ 140 and τ ≥ 210, respectively.Its zero for τ ≥ 560.If τ = 100 which is close to the true value of θ then the MLE does not exist in 11.42% of the simulation runs.For early Type-I censoring (τ = 50), the MLE does not exist in 33.46% of the runs.The heatmap in Fig. 11 clearly illustrates that the threshold has a huge impact on the bias of the estimator.If τ is small (e.g., τ ≤ 140) then the MLE tends to underestimate the true parameter.Interestingly, this seems to be rather independent of the choice of m (at least if m ≥ 3).Then, the bias is increasing-decreasing (as a function of τ ).Furthermore, it should be mentioned that, for larger τ , the maximum likelihood estimates seem to have a lower bias for smaller m.The histogram provided in the upper left corner of Fig. 11 supports these observations.Although the bias has a wide range, most of the scenarios show a smaller bias.
For the deviation, the situation is slightly different (see Figs.   with increasing τ and m for the other cases.The heatmaps show also some 'column structure' which indicates that the deviation seems to be quite robust against the desired sample size m whereas the threshold seems to have a larger impact.Moreover, the heatmap shows that the deviations do not generally improve for an increasing τ .These patterns may be caused by the fact that the distribution of the MLE has point masses at certain values.Hence, the relative position of the point masses and the thresholds have a larger impact on the deviation.In case of the confidence intervals, we provide the simulation results given in Table 5.The results illustrate that the coverage probabilities are quite close to the desired level and usually are larger so that the confidence intervals can be considered as conservative (as expected).Furthermore, the differences for the confidence intervals based on the pivoting method and Fairbanks' proposal are-at least for larger thresholds-quite small.In particular, they yield (on average) the same bounds in many cases.For smaller thresholds, the pivoting methods yields smaller average length of the confidence interval with a comparable coverage probability.On the other hand, Fairbanks' approach yields one-sided unbounded confidence intervals when the experiment is terminated without observing a minimal repair.In this case, the pivoting method does not provide a confidence interval.Therefore, in this case, one should use Fairbanks' confidence interval.Finally, it should be mentioned that the lengths of the confidence intervals decrease with both increasing desired sample size r and increasing threshold τ .

Conclusions and Outlook
We have obtained results for hybrid censored minimal repair data (or, equivalently, for hybrid censored record values or jump times of a NHPP).In particular, we have shown how results established for Type-I and Type-II censored minimal repair time data can be utilized under hybrid censoring.Specifically, we have illustrated the power of the present approach assuming exponentially distributed life times.In this case, we showed the stochastic monotonicity of the MLE.Using this property, we constructed exact (conditional) confidence intervals for the mean of the life time with the method of pivoting the cumulative distribution.
An obvious extension of the discussed model with exponentially distributed lifetimes is the consideration of Weibull distributions, which also lead to explicit representations for the maximum likelihood estimators of the distribution parameters using results of Wang and Ye [52] (or, in terms of the power law process, see, e.g., Crow [27], Finkelstein [36], Engelhardt and Bain [33]).To be more precise, for Weibull distributed data with parameters θ > 0 and β > 0, the likelihood function is given by L(θ, β; (5.1) From the presented results as well as the likelihood function in Wang and Ye [52, eq. ( 2)], we find immediately the respective maximum likelihood estimators, that is, i=1 ln(W HCS /X (n,i) ) , θ = W HCS n M HCS 1/ β (provided M HCS ≥ 1).Related results for minimal repair/record data under hybrid censoring for Weibull distributions will be studied in a future paper.Another interesting direction of future research is the multi-sample case sketched in Remark 2.3.Following the approaches presented in Górny and Cramer [38] and Jansen et al. [42], it seems to be possible to establish corresponding results for minimal repair/record data subject to hybrid censoring.Moreover, the case of joint hybrid censoring seems to be of interest but, at a first glance, more challenging than the multiple sample case with sample-wise hybrid censoring.
Furthermore, it would be interesting to study the nonparametric approach in more detail.In particular, the derivation of asymptotic results for hybrid censoring schemes will be worth to address.This will supplement studies already available for Type-I and Type-II censored jump times of a NHPP (see, e.g., Henderson [41] and references cited therein).
Finally, it would be of interest to study large sample results.This will be particularly useful in the case where a larger number of measurements is available since the construction of the exact confidence intervals is computationally expensive.

Fig. 1
Fig. 1 Illustration of generating of a minimal repair sample starting with n = 4 items

Fig. 3
Fig. 3 Illustration of generating of a minimal repair sample starting with n = 4 items under Type-I hybrid censoring

Fig. 5 Fig. 6
Fig. 5 Plots of intensity rate estimators (left) and corresponding estimated cumulative distribution function (right) for air conditioner data under Type-I hybrid censoring with τ = 500 (leading to the Type-I censoring case) and m = 10.The solid red line corresponds to the nonparametric estimation (see (3.4) and (3.6)) whereas the dashed blue line is obtained from the estimate given in (3.3).The corresponding distribution function is estimated under an exponential assumption

Example 4 . 2 (
Load-haul dump machine data) Proceeding as in Example 4.1, we start with the ordered pooled data generated from the two samples in

Fig. 7 Fig. 8
Fig. 7 Plots of intensity rate estimators (left) and corresponding estimated cumulative distribution function (right) for load haul dump machine data under Type-I hybrid censoring with τ = 500 (leading to the Type-I censoring case) and m = 10.The solid red line corresponds to the nonparametric estimation (see (3.4) and (3.6)) whereas the dashed blue line is obtained from the estimate given in (3.3).The corresponding distribution function is estimated under an exponential assumption

Fig. 12
Fig.12 Heatmap of standard deviation of maximum likelihood estimates (true value θ = 91.9048;see beginning of Sect.4.2).The threshold is fixed in each column, whereas the desired sample size m is given in the rows.Note that only cases are included where the MLE exists (see Fig.10)

Fig. 13
Fig.13 Heatmap of mean squared error of maximum likelihood estimates (true value θ = 91.9048;see beginning of Sect.4.2).The threshold is fixed in each column, whereas the desired sample size m is given in the rows.Note that only cases are included where the MLE exists (see Fig.10) x 1 , . . ., x m , m) = nβ θ β m e −nw β /θ β Type-II hybrid censoring

Table 5
Results of a simulation study and comparison of confidence intervals under Type-I hybrid censoring Heatmap of bias of maximum likelihood estimates (true value θ = 91.9048;seebeginning of Sect.4.2).The threshold is fixed in each column, whereas the desired sample size m is given in the rows.Note that only cases are included where the MLE exists (see Fig.10) Fig. 10 Proportion of non-existing maximum likelihood estimates as a function of the threshold τ Count Fig. 11