# Marco Burkschat's research while affiliated with RWTH Aachen University and other places

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## Publications (50)

Two definitions of Birnbaum’s importance measure for coherent systems are studied in the case of exchangeable components. Representations of these measures in terms of distribution functions of the ordered component lifetimes are given. As an example, coherent systems with failure-dependent component lifetimes based on the notion of sequential orde...

In the failure dependent proportional hazard model, it is assumed that identical components work jointly in a system. At the moments of consecutive component failures the hazard rates of still operating components can change abruptly due to a change of the load acting on each component. The modification of the hazard rate consists in multiplying th...

Necessary and sufficient conditions for multivariate total positivity of order 2 (MTP2) for density functions of some class of exchangeable random variables are obtained. The considered densities occur via symmetrization of particular ordered random variables. As an example, a characterization of the MTP2 property for the Freund–Weinman multivariat...

In this note, the distribution of the lifetime of a coherent system with independent and identically distributed component lifetimes is considered. Conditions yielding unimodality or logconcavity of the density function of the system lifetime are obtained. In the conditions, only assumptions on the density function of the components and on the sign...

A conditional IFR property for a coherent system with i failed components, and a uniform IFR property, are introduced and studied. The class of mixed systems with the uniform IFR property is characterized. A pair of examples illustrate our findings.

We study the conditions for unimodality of the lifetime distribution of a coherent system when the ordered component lifetimes in the system are described by generalized order statistics. Results for systems with independent and identically distributed lifetimes of components are included in this setting. The findings are illustrated with some exam...

We consider coherent systems with components whose exchangeable lifetime distributions come from the failure-dependent proportional hazard model, i.e., the consecutive failures satisfy the assumptions of the generalized order statistics model. For a fixed system and given failure rate proportion jumps, we provide sharp bounds on the deviations of s...

We consider systems based on sequential order statistics (SOS) with underlying distributions possessing proportional hazard rates (PHRs). In that case, the lifetime distribution of the system can be expressed as a distorted distribution. Motivated by the distribution structure in the case of pairwise different model parameters, a particular class o...

Type-I censored sequential order statistics are introduced and their distribution is examined. In the case of underlying exponential distributions, the conditional joint and marginal distributions of the normalized spacings given the number of observed failure times are obtained. Furthermore, the conditional distribution of the maximum likelihood e...

Distributions of the maximum likelihood estimators (MLEs) in Type-II (progressive) hybrid censoring based on two-parameter exponential distributions have been obtained using a moment generating function approach. Although resulting in explicit expressions, the representations are complicated alternating sums. Using the spacings-based approach of Cr...

For successive failure times of components in a technical system, a flexible model based on sequential order statistics is proposed. Beyond the common assumption of proportionality, this model allows for structural adjustments of the hazard rates of the underlying lifetime distributions in situations, where failures have an impact on the entire sha...

As a
generalization of order statistics from independent and identically distributed random variables, sequential order statistics
(SOSs) may be applied as a model for ordered data, when assuming changes of underlying distributions immediately after the occurrences of ordered observations. For example, in the case of a model for k-out-of-
n-systems...

Sequential order statistics can be used to describe the lifetime of a system with n components which works as long as k components function assuming that failures possibly affect the lifetimes of remaining units. In this work, the reversed hazard rates of sequential order statistics are examined. Conditions for the reversed hazard rate ordering and...

The limiting behavior of the hazard rate of coherent systems based on sequential order statistics is examined. Related results for the survival function of the system lifetime are also considered. For deriving the results, properties of limits involving a relevation transform are studied in detail. Then, limits of characteristics in sequential
$k...

At first, I would like to thank the authors for giving the detailed survey on the current state of stochastic orderings of order statistics from heterogeneous populations. Their review illustrates the fast development of research in this area during the recent years, in particular including many important contributions by the authors themselves. Be...

In a scheme of independent, possibly Type-II censored samples, joint estimation of scale parameters is considered under an order statistics prior leading to strictly ascendingly ordered Bayes estimators. The focus is on deriving handy closed-form expressions for the estimators.

Sequential order statistics can be used to describe the ordered lifetimes of components in a system, where the failure of a component may affect the performance of remaining components. In this paper mixture representations of the residual lifetime and the inactivity time of systems with such failure-dependent components are considered. Stochastic...

Sequential order statistics can be used to describe the ordered
lifetimes of components in a system, where the failure of a component
may affect the performance of remaining components. In this paper
mixture representations of the residual lifetime and the inactivity
time of systems with such failure-dependent components are considered.
Stochastic...

Beispiel Temperaturdaten In einer Stadt wurden in ausgewählten Monaten folgende monatlichen Durchschnittstemperaturen im Zeitraum 1996-2002 gemessen.

Beispiel (Fortsetzung 3▸Beispiel Befragung der Mitarbeiterinnen) In der Personalabteilung des Unternehmens nahmen 15 Beschäftigte an der Fragebogenerhebung teil. Einen Ausschnitt des entstandenen Datensatzes für die Merkmale Pl Geschlecht (Kodierung: 0 männlich, 1 weiblich), P4 Dauer der Betriebszugehörigkeit (in Monaten), Bl Zufriedenheit mit dem...

Beispiel Produktion In einem Unternehmen sollen die Beschaffungsmengen und Beschaffungskosten der eingesetzten Rohstoffe sowie deren zeitliche Entwicklungen analysiert werden. Die beschafften Mengen der Rohstoffe A, B, C (in Tonnen) in den Jahren 1985 bis 2002 und die Rohstoffpreise (in €/Tonne) in den Jahren 1995 bis 2002 sind folgender Tabelle zu...

Beispiel Drogerieketten Die Drogeriekette A hat neben einem Stammhaus weitere 14 Filialen in anderen Städten. Die Geschäftsleitung möchte — zu ihrer Information und zum Vergleich mit Daten der Konkurrenz — im Rahmen einer statistischen Analyse der Umsätze (in Mio. €) des vergangenen Jahres in den Niederlassungen einige statistische Kenngrößen ermit...

Beispiel Stadtfeste Für die Planungen der jährlichen Stadtfeste möchte eine Stadtverwaltung klären, wie stark der Zusammenhang zwischen der jeweiligen Besucherzahl und den Kosten für die Stadtreinigung ausgeprägt ist. Die Beziehung zwischen dem Merkmal X Besucherzahl und dem Merkmal X Reinigungskosten soll möglichst durch eine Funktion quantitativ...

Im ersten Anwendungsbeispiel 3▸Befragung der Mitarbeiterinnen eines Unternehmens werden einige grundsätzliche Gedanken zur Systematisierung und Auswertung einer statistischen Erhebung formuliert. Die resultierenden statistischen Grundbegriffe zur Beschreibung und Einordnung der interessierenden Größen einer Erhebung werden im ersten Kapitel erläute...

Fisher information about multiple parameters in a progressively Type-II censored sample is discussed. A representation of the Fisher information matrix in terms of the hazard rate of the baseline distribution is established which can be used for efficient computation of the Fisher information. This expression generalizes a result of Zheng and Park...

Beispiel Geschwindigkeitsrnessung Eine Stadtverwaltung lässt das Verhalten der Verkehrsteilnehmerinnen gezielt und mit quantitativen
Methoden untersuchen, um eine fundierte Grundlage zur Unterstützung von Entscheidungen zu erhalten. Dazu wurden beispielsweise
am späten Abend Radarmessungen an einem Kontrollpunkt innerhalb der Stadt vorgenommen (zul...

Beispiel Reiseveranstalter In einem Marktsegment der Tourismusbranche konkurrierten im Jahr 2001 zehn Anbieter um die Gunst
der reiselustigen Kunden. Die Umsätze (in Mio. €) des Jahres 2001 sind in der folgenden Tabelle zusammengestellt.

Sequential order statistics describe the ordered failure times in a k-out-of-n system, where the failures of components might affect the performance of remaining working components. In this article aging properties of sequential order statistics are examined and conditions are given such that the distribution of a sequential order statistic is ILR,...

Based on multiply Type-II censored samples of sequential order statistics, Bayesian estimators are derived for the parameters of one- and two-parameter exponential distributions. In the one-parameter set-up, the posterior density is obtained under the assumption that the prior distribution is given by an inverse Gamma distribution, and the Bayes es...

A representation of the Fisher information in generalized order statistics in terms of the hazard rate of the underlying distribution function is derived under mild regularity conditions. This expression supplements results for complete, Type-II censored, and progressively Type-II censored data. As a byproduct, we find a hazard rate based represent...

The sequential order statistics (SOS) are a good way to model the lifetimes of the components in a system when the failure of a component at time t affects the performance of the working components at this age t. In this article, we study properties of the lifetimes of the coherent systems obtained using SOS. Specifically, we obtain a mixture repre...

In the model of sequential order statistics, prior distributions are considered for the model parameters, which, for example, describe increasing load put on remaining components. Gamma priors are examined as well as priors out of a class of extended truncated Erlang distributions (ETED), which is introduced along with some properties. The choice o...

Linear estimation and prediction based on several samples of generalized order statistics from generalized Pareto distributions is considered. Representations of best linear unbiased estimators (BLUEs) and best linear equivariant estimators in location-scale families are derived, as well as corresponding optimal linear predictors. Moreover, we stud...

A model for describing the lifetimes of coherent systems, where the failures of components may have an impact on the lifetimes of the remaining components, is proposed. The model is motivated by the definition of sequential order statistics (cf. Kamps (1995)). Sequential order statistics describe the successive failure times in a sequential k -out-...

A model for describing the lifetimes of coherent systems, where
the failures of components may have an impact on the lifetimes of
the remaining components, is proposed. The model is motivated by
the definition of sequential order statistics (cf. Kamps (1995)).
Sequential order statistics describe the successive failure times
in a sequential
k-out-o...

Multivariate dependence of spacings of generalized order statistics is studied. It is shown that spacings of generalized order statistics from DFR (IFR) distributions have the CIS (CDS) property. By restricting the choice of the model parameters and strengthening the assumptions on the underlying distribution, stronger dependence relations are esta...

By using results from the distribution theory of generalized order statistics, the finite dimensional marginal distributions of the associated counting process are derived in terms of Meijer's G-functions. Moreover, some properties of the corresponding transition probabilities are given. The results are obtained by using a new representation of the...

In life tests, the progressive Type-II censoring methodology allows for the possibility of censoring a number of units each time a failure is observed. This results in a large number of possible censoring plans, depending on the number of both censoring times and censoring numbers. Employing maximum Fisher Information as an optimality criterion, op...

We consider best linear equivariant estimation in a particular location-scale family based on several progressively type II censored samples. The censoring schemes that minimize the mean squared error matrix of the estimators with respect to the Löwner ordering are obtained. Uniqueness of the schemes, which minimize the smallest and the largest eig...

Let X1:n ≤ X2:n ≤···≤ Xn:n denote the order statistics of a sample of n independent random variables X1, X2,…, Xn, all identically distributed as some X. It is shown that if X has a log-convex [log-concave] density function, then the general spacing vector (Xk1:n, Xk2:n − Xk1:n,…, Xkr:n − Xkr−1:n) is MTP2 [S-MRR2] whenever 1 ≤ k1 < k2 <···< kr ≤ n...

In a progressively type II censored life-testing experiment intact units may be removed from the experiment after every failure. If the initial number of units in the experiment and the total number of failures are fixed, the experimenter may choose between different censoring schemes. By specifying an optimality criterion, one may improve the outc...

In this paper, we derive representations of the best (affine) linear equivariant estimators (BLEEs) of the location and scale parameters in the location, scale, and location-scale set-ups. Furthermore, we show that the best linear equivariant predictor (BLEP) with respect to standardized mean squared error has a representation similar to the well k...

Best linear unbiased estimation for parameters of a particular location-scale family based on progressively Type-II censored order statistics is considered and optimal censoring schemes are determined. As optimality criteria serve the φ p - criteria from experimental designs which are applied to the covariance matrix of the BLUEs. The results are s...

Optimal censoring schemes in the model of progressive type II censoring are obtained for a location-scale family of distributions which includes exponential, uniform and Pareto distributions. In the one-parameter set-up the variance of the respective BLUE is used as an optimality criterion. In the two-parameter situation two criteria are applied wh...

Beispiel (Fortsetzung 3▸Beispiel Befragung der Mitarbeiterinnen) Für den Teilbereich Produktion des Unternehmens (120 Beschäftigte)
wurden die monatlichen Bruttogehälter des Monats Juli den Personalunterlagen entnommen. Die Urliste der Daten x1,…, x120 des Merkmals Bruttogehalt wurde bereits aufbereitet und liegt als Rangwertreihe x1,…, x120 vor.

Beispiel (Fortsetzung 3▸Beispiel Befragung der Mitarbeiterinnen) Im Teilbereich Produktion des Unternehmens (120 Beschäftigte)
wurde eine Fragebogenerhebung durchgeführt, deren Ergebnisse für einige der erhobenen Merkmale visualisiert werden sollen.
Die Altersstruktur des Produktionsbereichs wurde bereits tabellarisch aufbereitet, indem die Mitarbe...

In Kamps (1995a) generalized order statistics have been introduced as a unification of several models of ascendingly ordered random variables. In this paper a dual model is proposed that enables a common approach to descendingly ordered random variables like reversed ordered order statistics, lower k-records and lower Pfeifer records. Connections b...

## Citations

... Most of the studies on the reliability and stochastic properties of the coherent systems are investigated in the case that the component lifetimes are i.i.d. or exchangeable. In this case the concept of signature associated to the coherent systems plays an important role (see Bieniek et al. (2020), Gertsbakh and Shpungin (2011), Navarro et al. (2008), and Samaniego (1985), Samaniego (2007)). Gurler and Bairamov (2009) considered systems with nonidentically distributed components. ...

... [99] provide an interesting interpretation of ∞ -norm symmetric densities, which is briefly explained. Every non-negative function g d satisfying (3.10) is of the form 11 The authors even allow for p ∈ (0, 1), but in this case . p is no longer a norm. ...

... The assumption is not very restrictive, because coherent systems with non-unimodal signatures are very rare. An example of a system of size 5 with bimodal signature was presented in Jasiński et al. (2009), and the construction was extended to higher dimensions by Bieniek and Burkschat (2018). In the proof, we apply a characterization of unimodality based on sign change behavior (cf., e.g., Marshall and Olkin 2007, proof of Proposition B.2., p. 99). ...

... Balakrishnan et al. [15] investigated the maximum likelihood estimation method when the SOS samples were obtained from a step stress test. Burkschat and Navarro [16] examined mixture representations of the residual lifetime and the inactivity time of system with failure-dependent components. Burkschat and Torrado [17] examined the reversed hazard rate property of SOS and presented the conditions of reversed hazard rate ordering and the decreasing reversed hazard rate property of SOS. Park [18] proposed an expectation-maximization algorithm to obtain the MLE of the survival function for an equally load-sharing system. ...

... (2011) Burkschat and Torrado (2014) obtained Part (i) of Theorem 2.5 for SOS. Once again, as mentioned above, if we consider some conditions in our theorem as well as in their theorem, then these two results coincide; (c) Burkschat and Navarro (2018) obtained Corollary 2.6 under different conditions and also by assuming that γ (i,n, m n ,k) = γ ( j,n, m n ,k) and γ (i,n , m n ,k ) = γ ( j,n , m n ,k ) for all i = j. ...

... Burkschat [8] extended the SOS concept to study exchangeable random variables under a weaker condition than the independent and identically distributed assumption. Burkschat's idea is able to apply to general coherent systems, such as the coherent systems of which components have dependent and non-identically distributed failure time distributions. ...

... More results on inference with SOSs can be found, for instance, in [29] and [30]. For recent works, see, e.g., [31] and [32]. Finally, notice that SOSs with proportional hazard rates are closely related to generalized order statistics; see [26,27,33]. ...

... [14] considered the statistical inference of a two-parameter exponential distribution under the Type-II progressively hybrid censoring scheme. Distributions of the maximum likelihood estimators (MLEs) in Type-II (progressive) hybrid censoring based on two-parameter exponential distributions have been obtained using a moment generating function approach by [15]. The statistical inference for the Gompertz distribution based on Type-II progressively hybrid censored data is discussed by [16]. ...

... Burkschat and Torrado [17] examined the reversed hazard rate property of SOS and presented the conditions of reversed hazard rate ordering and the decreasing reversed hazard rate property of SOS. Park [18] proposed an expectation-maximization algorithm to obtain the MLE of the survival function for an equally load-sharing system. Burkschat et al. [19] studied statistical inference methods based on type-I censored SOS samples. ...

... Reliability of coherent systems based on sequential order statistics 847 We refer the reader to, for example, [4]- [6], [15], and [20] for more details on stochastic properties and applications of the SOS model in reliability. The model of component failures based on SOS is closely related to the model of component failures based on nonhomogeneous pure-birth (NHPB) process. ...