Statistical Inference for Multi State Systems under the Generalized Modified Weibull Class

  • Andreas Makrides Laboratory of Statistics and Data Analysis, Department of Statistics and Actuarial-Financial Mathematics, University of the Aegean, Greece
Keywords: Multi-state system, semi-Markov processes, H-class of distributions, Modified Weibull distribution, parameter estimation


Multi state systems can be seen as semi-Markov processes by considering an arbitrary distribution function for sojourn times. Especially, in this work, the Modified Weibull distribution is employed to be the distribution of sojourn times with a shape parameter λλ such that is member of a distributions family that is closed under minima. Parameters estimators are provided and the proposed methodology is evaluated using a detailed simulation procedure.


Download data is not yet available.

Author Biography

Andreas Makrides, Laboratory of Statistics and Data Analysis, Department of Statistics and Actuarial-Financial Mathematics, University of the Aegean, Greece

Andreas Makrides is a Post Doc researcher at the Department of Statistics and Actuarial – Financial Mathematics, University of the Aegean, Samos, Greece (advisor: Prof. Alex Karagrigoriou).

His research interests lie among others in Stochastic Modeling, Stochastic Processes, Applied Probability, Mathematical Statistics, Semi-Markov Processes, Reliability Theory, Multi-State Systems, Entropy and Divergence, Goodness of Fit Tests and Control Charts – Statistical Quality Control.

He has worked as a Post Doc researcher (2018–2019) at the Laboratoire de Mathématiques Raphaël Salem, University of Rouen, France.

He received his Ph.D in Statistics (2016) from the Department of Mathematics and Statistics, University of Cyprus, his M.Sc. in Statistics and Modeling (with excellence, 2010) from the Department of Mathematics of Aristotle University of Thessaloniki, Greece and his B.Sc. (Hons, with excellence, 2008) from the Department of Mathematics of Aristotle University of Thessaloniki, Greece.

Andreas has been the representative of both Cyprus and France to the 21st EYSM, July 29–Aug 2, 2019, Belgrade, Serbia (selected by the Regional Committee of the Bernoulli Society). He is the recipient of two excellence awards: (1) IKY (State Scholarships Foundation in Greece) awards, for academic excellence in undergraduate studies during the academic years 2005–2006, 2006–2007, 2007–2008. (2) 1st ranked postgraduate candidate, M.Sc “Statistics and Modeling, Department of Mathematics, Aristotle University of Thessaloniki, Greece, 2008. IKYK (State Scholarships Foundation in Cyprus) Fellowship, for the graduate studies.

He is also a reviewer for several scientific journals.

Furthermore, he has teaching experience since he has been working from 2017 as an Associated Lecturer at the University of Cyprus, the Cyprus University of Technology and the Uclan University, Cyprus on various courses both undergraduate and postgraduate.


Azzalini, A. (1985). A Class of Distributions Which Includes the Normal Ones. Scandinavian Journal of Statistics, 12, 171–178.

Ahuja, J., C. and Nash, Stanley W. (1967). The generalized Gompertz-Verhulst family of distributions. Sankhya, Series A, 29, 141–156.

Balasubramanian, K., Beg, M. I. and Bapat, R. B. (1991). On families of distributions closed under extrema, Sankhya: The Indian Journal of Statistics A, 53, 375–388.

Barbu, V., S., Karagrigoriou, A., Makrides, A. (2017). Semi-Markov Modelling for Multi-State Systems, Meth. & Comput. Appl. Prob., 19, 1011–1028.

Barbu, V., S., Karagrigoriou, A., Makrides, A. (2019). Estimation and reliability for a special type of semi-Markov process, J. Math. Stat., 15(1), 259–272.

Barbu, V., S., Karagrigoriou, A., Makrides, A. (2020). Statistical inference for a general class of distributions with time-varying parameters, J. Appl. Stat., 1–20.

Barbu, V., S., Limnios N. (2004). Discrete time semi-Markov processes for reliability and survival analysis – a nonparametric estimation approach, Parametric and Semiparametric Models with Applications to Reliability, Survival Analysis and Quality of Life, eds. Narayanaswamy Balakrishnan, Mikhail Nikulin, Mounir Mesbah, Nikolaos Limnios, Birkhäuser, Collection Statistics for Industry and Technology, Boston, 487–502.

Delgarm, L. and Zadkarami, M., R. (2015). A new generalization of lifetime distributions, Comput. Stat. 30, 1185–1198.

Eugene N., Lee C., and Famoye F. (2002). Beta-normal distribution and its applications, Commun. Stat. Theory Methods 31, 497–512.

Lai, C., D., Xie, M. and Murthy, D. (2003). A modified Weibull distribution. IEEE Trans. Reliab., 52, 33–37.

Limnios, N. and Oprişan, G. (2001). Semi-Markov Processes and Reliability, Birkhäuser, Boston.

Lisnianski, A., Frenkel, Karagrigoriou A. (2017). Recent advances in multi-state systems reliability: Theory and applications, Springer.

Marshall A., W., Olkin I. (1967). A multivariate exponential distribution. J. Am. Stat. Assoc., 62, 30–44.

Ouhbi, B. and Limnios, N. (1996). Non-parametric estimation for semi-Markov kernels with application to reliability analysis, Appl. Stoch. Models Data Anal., 12, 209–220.

Ristic M., M. and Nadarajah S. (2014). A new lifetime distribution. J. Stat. Comput. Simul. 84, 135–150.

Zografos, K. and Balakrishnan, N. (2009). On families of beta- and generalized gamma-generated distributions and associated inference, Stat. Methodol., 6, 344–362.

Reliability and Stochastic Processes