Goodness-of-fit tests for type-II right censored data [Elektronische Ressource] : structure preserving transformations and power studies / Tim Fischer

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Goodness-of-Fit Tests for Type-II Right Censored Data:Structure Preserving TransformationsandPower StudiesVon der Fakultät für Mathematik, Informatik und Naturwissenschaften der RWTHAachen University zur Erlangung des akademischen Grades eines Doktors derNaturwissenschaften genehmigte Dissertationvorgelegt vonDiplom-MathematikerTim Fischeraus MönchengladbachBerichter: Universitätsprofessor Dr. Udo KampsProfessor Dr. Eric BeutnerTag der mündlichen Prüfung: 24. November 2010Diese Dissertation ist auf den Internetseiten der Hochschulbibliothek online verfügbar.AcknowledgementsI would like to show my gratitude to my supervisor, Professor Udo Kamps, for giving methe opportunity to do interesting research with continued support in a cooperative atmo-sphere. It has been a special pleasure for me to experience his commitment and kindnessduring my time at the ’Institut für Statistik und Wirtschaftsmathematik’.I am grateful to my co-supervisor, Professor Eric Beutner, for many helpful discussionsand giving me valuable suggestions.I also thank Professor Marco Burkschat for always being interested in my work and helpingme with his comments and encouragement.Professor Erhard Cramer introduced me to doing mathematical research on my own duringmy diploma thesis and I would like to thank him very much for his support.I am also thankful to Dr. Wolfgang Herff for several fruitful discussions and always havingtime for me.
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01 janvier 2011

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Goodness-of-Fit Tests for Type-II Right Censored Data:
Structure Preserving Transformations
and
Power Studies
Von der Fakultät für Mathematik, Informatik und Naturwissenschaften der RWTH
Aachen University zur Erlangung des akademischen Grades eines Doktors der
Naturwissenschaften genehmigte Dissertation
vorgelegt von
Diplom-Mathematiker
Tim Fischer
aus Mönchengladbach
Berichter: Universitätsprofessor Dr. Udo Kamps
Professor Dr. Eric Beutner
Tag der mündlichen Prüfung: 24. November 2010
Diese Dissertation ist auf den Internetseiten der Hochschulbibliothek online verfügbar.Acknowledgements
I would like to show my gratitude to my supervisor, Professor Udo Kamps, for giving me
the opportunity to do interesting research with continued support in a cooperative atmo-
sphere. It has been a special pleasure for me to experience his commitment and kindness
during my time at the ’Institut für Statistik und Wirtschaftsmathematik’.
I am grateful to my co-supervisor, Professor Eric Beutner, for many helpful discussions
and giving me valuable suggestions.
I also thank Professor Marco Burkschat for always being interested in my work and helping
me with his comments and encouragement.
Professor Erhard Cramer introduced me to doing mathematical research on my own during
my diploma thesis and I would like to thank him very much for his support.
I am also thankful to Dr. Wolfgang Herff for several fruitful discussions and always having
time for me. Both as a student and as a colleague, I learned a lot from him.
Furthermore, I am indebted to all of my colleagues at the ’Institut für Statistik und
Wirtschaftsmathematik’ for providing a stimulating and enjoyable environment. It was
a pleasure for me to work with such friendly and helpful people. In particular, I thank Mo-
hammed Abujarad, Johann Alexin, Ramona Au, Katinka Fischer, Simone Gerwert, Katrin
Herlé, Hassan Satvat, Bettina Schmiedt, Birgit Tegguer, Quan Nhon ’Ti’ Vuong, Xiaofang
Wang and Sabine Weidauer.
Special thanks go to my friend, colleague and former fellow student Stefan Bedbur for help-
ing me get through any hard times and putting up with me anytime when I was frustrated
by mathematics. I could not have done my study and work without him.
Lastly, and most importantly, I thank my family, my friends and my love Katja Fitzen. I
am indescribably grateful to my parents Hildegard Fischer and Friedbert Fischer and to
my brother Kai Fischer for the unconditional support they provided me through my entire
life. I owe my deepest gratitude to all of my friends I can always count on at times of need.
And to you, Katja, I am more grateful than words can say for giving me your love, your
trust and security in my life. I am very sorry that I had so little time in the last two years
and I thank you for understanding and always giving me encouragement. I love you.Contents
1 Introduction 1
1.1 Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Goodness-of-Fit Tests for Complete Samples . . . . . . . . . . . . . . . . . 2
1.2.1 Probability Plots and Correlation Type Goodness-of-Fit Statistics . 2
1.2.2 Statistics Based on Spacings . . . . . . . . . . . . . . . . . . . . . . 3
21.2.3 Neyman’s Smooth and Tests . . . . . . . . . . . . . . . . . . . . 4
1.2.4 EDF Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.5 Other Test Statistics in Literature . . . . . . . . . . . . . . . . . . . 8
1.2.6 Distributions of Test Statistics . . . . . . . . . . . . . . . . . . . . . 8
1.3 Goodness-of-Fit Tests for Type-II Right Censored Data . . . . . . . . . . . 11
1.3.1 Modified Test Statistics . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3.2 Distributions of Modified Test Statistics . . . . . . . . . . . . . . . 13
1.3.3 The Alternative Approach to Goodness-of-Fit Testing for Censored
Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.4 Aim of This Work and Outline . . . . . . . . . . . . . . . . . . . . . . . . . 14
2 Modifications of Samples from the Uniform Distribution 15
2.1 Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Random Dilation and Contraction . . . . . . . . . . . . . . . . . . . . . . . 23
3 Transformations of Samples from Arbitrary Distributions 25
3.1 The Transformation of O’Reilly and Stephens . . . . . . . . . . . . . . . . 28
3.2 The T of Michael and Schucany . . . . . . . . . . . . . . . . 36
3.2.1 On the Structure of the Vector of the Transformed Variables . . . . 36
3.2.2 On the Distribution of the Maximum of the Transformed Variables 44
3.3 More General Transformations . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.4 Transformations into r-1 Order Statistics . . . . . . . . . . . . . . . . . . . 80
3.5 T into i.i.d. Random Variables . . . . . . . . . . . . . . . . 83
4 Empirical Power Study 91
4.1 Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.2 Classical Goodness-of-Fit Tests . . . . . . . . . . . . . . . . . . . . . . . . 934.3 Modified Test Statistics for Type-II Right Censored Data . . . . . . . . . . 97
4.4 Tests Based on Transformed Data . . . . . . . . . . . . . . . . . . . . . . . 102
5 Outlook 115
Appendix 117
Bibliography 1191 Introduction 1
1 Introduction
1.1 Preliminary Remarks
When a statistician intends to examine a problem she or he always has to abstract from
reality and build a statistical model. This first step is very crucial because the relevance
of every result and conclusion based on the model for the real situation depends directly
on the adequate description of the natural circumstances.
For example, in the context of quality management or product development a company
is interested in the reliability of its products. Assuming these products are more or less
complex machines each consisting of elementary components that are frequently used for
such systems, it would be sensible to try making inferences from the lifetime distributions
of the components about the reliability of the more complex systems or possible new devel-
opments. In this approach, it is essential (among others) to suppose appropriate lifetime
distributions for the components because otherwise the results for the products or new
developments are not transferable to reality, although all the rest of the statistical com-
putations are correct. If, e.g., the lifetimes (in days) of the components of a 4-out-of-4
system (i.e., the system consists of 4 components and it fails if one or more components
fail) are assumed to be independent and identically exp(0:001) distributed (exponential
1
distribution with mean = 1000), then, consequently, the lifetime of the system is
0:001
1
supposed to be exponentially distributed with mean = 250. But, if the true lifetime
0:004
distribution of the components is wei(0:1; 0:4) then the lifetime distribution of the system
will be wei(0:4; 0:4) with mean 33, where, for ; > 0, wei(;) denotes a Weibull
1 with densitiy function x7! x exp( x ), x > 0. I.e., even though the
mean of the lifetimes of the components was estimated rather accurately (the mean of
wei(0:1; 0:4) is approximately 1051) the mean of the lifetime of the system in the model
differs dramatically from the true value.
Of course, the exponential distribution is a special case of the Weibull distribution, nev-
ertheless, this example also shows, that it might not suffice to restrict oneself to consid-
eration of a specific family of distributions, such that only some paramerters have to be
determined. I.e., if there is no additional information on the unknown distributions, non-
parametric models and methods must be applied.
After puttingn structurally identical components on a life-testing experiment and observ-
ing their failure times, one might get an idea of the true lifetime distribution by histograms,
kernel density estimation or the empirical distribution function, for instance. Then, a
goodness-of-fit test can be conducted to assess whether this idea may be kept or should be
rejected.
In this thesis, the term ’goodness-of-fit test’ always means a statistical test for the presence
of a certain distribution. More precisely, let X ;:::;X , n2N, be independent and iden-1 n
tically distributed (i.i.d.) random variables (rv’s) with absolutely continuous cumulative
distribution function (cdf) F. We wish to test the (null) hypothesis F = F against the0
alternative F = F , where F is a completely specified absolutely continuous cdf. Then,0 0
we call such a statistical test goodness-of-fit test.
6

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