Table Of ContentStudies in Systems, Decision and Control 196
Magdalena Szymkowiak
Lifetime
Analysis by
Aging Intensity
Functions
Studies in Systems, Decision and Control
Volume 196
Series editor
Janusz Kacprzyk, Polish Academy of Sciences, Warsaw, Poland
e-mail: [email protected]
The series “Studies in Systems, Decision and Control” (SSDC) covers both new
developments and advances, as well as the state of the art, in the various areas of
broadly perceived systems, decision making and control–quickly, up to date and
withahighquality.Theintentistocoverthetheory,applications,andperspectives
on the state of the art and future developments relevant to systems, decision
making,control,complexprocessesandrelatedareas, asembeddedinthefieldsof
engineering,computerscience,physics,economics,socialandlifesciences,aswell
astheparadigmsandmethodologiesbehindthem.Theseriescontainsmonographs,
textbooks, lecture notes and edited volumes in systems, decision making and
control spanning the areas of Cyber-Physical Systems, Autonomous Systems,
Sensor Networks, Control Systems, Energy Systems, Automotive Systems,
Biological Systems, Vehicular Networking and Connected Vehicles, Aerospace
Systems, Automation, Manufacturing, Smart Grids, Nonlinear Systems, Power
Systems, Robotics, Social Systems, Economic Systems and other. Of particular
valuetoboththecontributorsandthereadershiparetheshortpublicationtimeframe
and the world-wide distribution and exposure which enable both a wide and rapid
dissemination of research output.
More information about this series at http://www.springer.com/series/13304
Magdalena Szymkowiak
Lifetime Analysis
by Aging Intensity Functions
123
MagdalenaSzymkowiak
Institute of Automation andRobotics
Poznań University of Technology
Poznań,Poland
ISSN 2198-4182 ISSN 2198-4190 (electronic)
Studies in Systems,DecisionandControl
ISBN978-3-030-12106-8 ISBN978-3-030-12107-5 (eBook)
https://doi.org/10.1007/978-3-030-12107-5
LibraryofCongressControlNumber:2018968098
©SpringerNatureSwitzerlandAG2020
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart
of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,
recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission
orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar
methodologynowknownorhereafterdeveloped.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this
publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom
therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse.
The publisher, the authors and the editors are safe to assume that the advice and information in this
book are believed to be true and accurate at the date of publication. Neither the publisher nor the
authorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinor
for any errors or omissions that may have been made. The publisher remains neutral with regard to
jurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations.
ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG
Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland
To my husband
for his support and forbearance,
to my parents, children, the whole family,
to my teachers and friends
for their help and continuous assistance.
Preface
Agingtendencyofitemsandcompoundstructuresisanimportantandchallenging
subject of the lifetime analysis. This phenomenon attracts the attention of an
increasing number of reliability researches. They usually present properties of
random lifetimes by means of the respective nonnegative univariate absolutely
continuous distributions using classic tools of reliability theory as distribution
function, survival function, and failure rate. The aim of this book is to define and
studyvariousagingintensityfunctionswhichareusedforgaugingvariedaspectsof
aging tendency. The classic version of the aging intensity was recently introduced
by Jiang, Ji, and Xiao [31].
In Part I, we introduce and analyze the different instances of classic aging
intensity determined for uniform and bivariate distributions, considering their
absolutely continuous and discrete cases. In Chap. 1, the classic aging intensity
functions are provided. For univariate and bivariate absolutely continuous random
variables,theirstudiesbymeansofagingintensitiesarepresentedinChaps.2and4,
respectively.Forunivariateandbivariatediscretedistributions(seeChaps.3and5,
respectively),weproposetwodifferentagingintensities,basicandalternativeones.
Bothofthemcanbeusedinthediscretelifetimeanalysis,butinsomesituations,the
analysisthroughone of them seemsto beeasier than through theother one.
Moreover, in Part II motivated by the concept of generalized failure rate pro-
posed by Barlow and van Zwet [7, 8], we introduce the G-generalized aging
intensity functions which allow us to measure and compare aging tendencies of
lifetimerandomvariablesinvarioustimescaling.InChap.6,wefocusourstudyto
thecasewhen Gisageneralized Paretodistribution.InChaps.7and8,weextend
our analysis for any strictly increasing absolutely continuous lifetime distribution
with possibly bounded support.
Some of the introduced aging intensities characterize families of distributions
dependent on a single parameter, and the others determine distributions uniquely.
Using aging intensities, we can define and study partial orders based on them.
Moreover, the recognition of the shape of a properly chosen aging intensity esti-
mate admits a simple identification of the data lifetime distributions.
vii
viii Preface
The readers should have some basic knowledge in probability and statistics to
understandthesubjectpresentedinthismanuscript.Bothreliabilityresearchersand
practitioners find the monograph useful for reference to put forward some new
ideas. To make the reading more clear, in Appendix A, we present some basic
continuousanddiscrete,univariateandbivariatelifetimedistributionsstudiedinthe
book. Moreover, in Appendix B, some mentioned stochastic orders are listed.
This monograph is a summary of the research carried out during the last years,
describedinthepapersthatcanbefoundinthereferencelistattheendofthebook.
I wish to thank, in particular, Maria Iwińska who was the coauthor of my first
articles in this area, for her help and professional advice. I also greatly appreciate
the meticulousness of the anonymous papers’ reviewers, and I wish to record my
sincere thanks for their constructive comments and suggestions.
Poznań, Poland Magdalena Szymkowiak
October 2018
Contents
Part I Classic Aging Intensity Functions
1 Basic Reliability Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1 Univariate Absolutely Continuous Distributions . . . . . . . . . . . . . . 3
1.2 Univariate Discrete Distributions . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Bivariate Absolutely Continuous Distributions . . . . . . . . . . . . . . . 6
1.4 Bivariate Discrete Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Aging Intensity of Nonnegative Univariate Absolutely
Continuous Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1 Characterizations of Nonnegative Univariate Absolutely
Continuous Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.1 Characterization of Univariate Distribution . . . . . . . . . . . . 12
2.1.2 Characterization of Univariate Inverse Distribution . . . . . . 13
2.2 Weibull Related Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.1 Two-Parameter Weibull Distribution. . . . . . . . . . . . . . . . . 17
2.2.2 Exponential Distribution. . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2.3 Modified Weibull Distribution . . . . . . . . . . . . . . . . . . . . . 21
2.2.4 Inverse Two-Parameter Weibull Distribution . . . . . . . . . . . 21
2.2.5 Inverse Modified Weibull Distribution . . . . . . . . . . . . . . . 25
2.3 Some Others Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3.1 Linear Failure Rate Distribution . . . . . . . . . . . . . . . . . . . . 26
2.3.2 Gompertz Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.3.3 Makeham Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.4 Some Properties of Aging Intensity Order . . . . . . . . . . . . . . . . . . 28
2.4.1 AI Order for Weibull Related Distributions . . . . . . . . . . . . 28
2.4.2 RAI Order for Inverse Weibull Related Distributions . . . . . 29
ix
x Contents
2.5 Analysis of Aging Intensity Through Data. . . . . . . . . . . . . . . . . . 30
2.5.1 Analysis of Aging Intensity Through Generated Data . . . . 30
2.5.2 Analysis of Aging Intensity Through Real
Complete Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.5.3 Analysis of Aging Intensity Through Censored Data . . . . . 36
3 Aging Intensities of Discrete Distributions . . . . . . . . . . . . . . . . . . . . 39
3.1 Characterizations of Discrete Distribution. . . . . . . . . . . . . . . . . . . 39
3.2 Discrete Weibull Related Distribution . . . . . . . . . . . . . . . . . . . . . 43
3.2.1 Discrete Weibull (I) Distribution. . . . . . . . . . . . . . . . . . . . 43
3.2.2 Geometric Distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.2.3 Discrete Weibull (III) Distribution . . . . . . . . . . . . . . . . . . 45
3.2.4 Discrete Modified Weibull Distribution. . . . . . . . . . . . . . . 48
3.3 Some Properties of Discrete Aging Intensity Order. . . . . . . . . . . . 48
3.3.1 Discrete Aging Intensity Order. . . . . . . . . . . . . . . . . . . . . 49
3.3.2 Relationship Between DAI Order and Other
Stochastic Orders. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4 Aging Intensities Vector for Bivariate Absolutely Continuous
Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.1 Characterizations of Nonnegative Bivariate Absolutely
Continuous Distributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.2 Bivariate Absolutely Continuous Weibull Related
Distributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.2.1 Bivariate Exponential Distribution . . . . . . . . . . . . . . . . . . 56
4.2.2 Bivariate Weibull Distribution . . . . . . . . . . . . . . . . . . . . . 58
4.3 Bivariate Aging Intensity Order and Its Properties . . . . . . . . . . . . 59
4.3.1 Bivariate Aging Intensity Order . . . . . . . . . . . . . . . . . . . . 59
4.3.2 Relationships Between Bivariate AI Order and Other
Bivariate Stochastic Orders. . . . . . . . . . . . . . . . . . . . . . . . 60
4.4 Analysis of Aging Intensity Through Real Bivariate Data. . . . . . . 62
5 Aging Intensities Vectors for Bivariate Discrete Distributions . . . . . 65
5.1 Characterizations of Bivariate Discrete Distributions. . . . . . . . . . . 65
5.2 Bivariate Discrete Weibull Distribution . . . . . . . . . . . . . . . . . . . . 72
5.2.1 Introduction to Bivariate Discrete Weibull Distribution . . . 72
5.2.2 Analysis of Bivariate Discrete Weibull Distribution. . . . . . 73
5.2.3 Bivariate Geometric Distribution. . . . . . . . . . . . . . . . . . . . 74
5.3 Some Properties of Bivariate Discrete Aging Intensity Order. . . . . 75
5.3.1 Bivariate Discrete Aging Intensity Order. . . . . . . . . . . . . . 75
5.3.2 Relationship Between Bivariate DAI Order and Other
Bivariate Stochastic Orders. . . . . . . . . . . . . . . . . . . . . . . . 76
