Table Of ContentSPRINGER BRIEFS IN STATISTICS
JSS RESEARCH SERIES IN STATISTICS
Li-Hsien Sun · Xin-Wei Huang ·
Mohammed S. Alqawba ·
Jong-Min Kim · Takeshi Emura
Copula-Based
Markov Models
for Time Series
Parametric
Inference and
Process Control
SpringerBriefs in Statistics
JSS Research Series in Statistics
Editors-in-Chief
Naoto Kunitomo,Economics,MeijiUniversity, Chiyoda-ku,Tokyo,Tokyo,Japan
Akimichi Takemura, The Center for Data Science Education and Research, Shiga
University, Bunkyo-ku, Tokyo, Japan
Series Editors
Genshiro Kitagawa, Meiji Institute for Advanced Study of Mathematical Sciences,
Nakano-ku, Tokyo, Japan
Shigeyuki Matsui, Graduate School of Medicine, Nagoya University, Nagoya,
Aichi, Japan
Manabu Iwasaki, School of Data Science, Yokohama City University, Yokohama,
Tokyo, Japan
Yasuhiro Omori, Graduate School of Economics, The University of Tokyo,
Bunkyo-ku, Tokyo, Japan
Masafumi Akahira, Institute of Mathematics, University of Tsukuba, Tsukuba,
Ibaraki, Japan
Masanobu Taniguchi, Department of Mathematical Sciences/School, Waseda
University/Science & Engineering, Shinjuku-ku, Japan
Hiroe Tsubaki, The Institute of Statistical Mathematics, Tachikawa, Tokyo, Japan
Satoshi Hattori, Faculty of Medicine, Osaka University, Suita, Osaka, Japan
Kosuke Oya, School of Economics, Osaka University, Toyonaka, Osaka, Japan
ThecurrentresearchofstatisticsinJapanhasexpandedinseveraldirectionsinline
with recent trends in academic activities in the area of statistics and statistical
sciences over the globe. The core of these research activities in statistics in Japan
has been the Japan Statistical Society (JSS). This society, the oldest and largest
academicorganization for statistics inJapan, was founded in1931by ahandful of
pioneerstatisticiansandeconomistsandnowhasahistoryofabout80years.Many
distinguished scholars have been members, including the influential statistician
Hirotugu Akaike, who was a past president of JSS, and the notable mathematician
Kiyosi Itô, who was an earlier member of the Institute of Statistical Mathematics
(ISM), which has been a closely related organization since the establishment of
ISM. The society has two academic journals: the Journal of the Japan Statistical
Society (English Series) and the Journal of the Japan Statistical Society (Japanese
Series). The membership of JSS consists of researchers, teachers, and professional
statisticians in many different fields including mathematics, statistics, engineering,
medical sciences, government statistics, economics, business, psychology, educa-
tion, and many other natural, biological, and social sciences. The JSS Series of
Statisticsaimstopublishrecent results ofcurrentresearchactivities intheareas of
statistics and statistical sciences in Japan that otherwise would not be available in
English; they are complementary to the two JSS academic journals, both English
andJapanese.Becausethescopeofaresearchpaperinacademicjournalsinevitably
hasbecomenarrowlyfocusedandcondensedinrecentyears,thisseriesisintended
to fill the gap between academic research activities and the form of a single
academic paper. The series will be of great interest to a wide audience of
researchers, teachers, professional statisticians, and graduate students in many
countrieswhoareinterestedinstatisticsandstatisticalsciences,instatisticaltheory,
and in various areas of statistical applications.
More information about this subseries at http://www.springer.com/series/13497
Li-Hsien Sun Xin-Wei Huang
(cid:129) (cid:129)
Mohammed S. Alqawba
(cid:129)
Jong-Min Kim Takeshi Emura
(cid:129)
Copula-Based Markov
Models for Time Series
Parametric Inference and Process Control
123
Li-Hsien Sun Xin-Wei Huang
Graduate Institute of Statistics Institute of Statistics
National Central University National Chiao TungUniversity
Taoyuan,Taiwan Hsinchu, Taiwan
Mohammed S. Alqawba Jong-Min Kim
Department ofMathematics Division of ScienceandMathematics
Collegeof Sciences andArts at AlRass University of Minnesota at Morris
Qassim University Morris, MN, USA
Unayzah, SaudiArabia
Takeshi Emura
Department ofInformation Management
Chang GungUniversity
Taoyuan,Taiwan
ISSN 2191-544X ISSN 2191-5458 (electronic)
SpringerBriefs inStatistics
ISSN 2364-0057 ISSN 2364-0065 (electronic)
JSSResearch Series in Statistics
ISBN978-981-15-4997-7 ISBN978-981-15-4998-4 (eBook)
https://doi.org/10.1007/978-981-15-4998-4
©TheAuthor(s),underexclusivelicensetoSpringerNatureSingaporePteLtd.2020
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Preface
Thisbookprovidesstatisticalmethodologiesforfittingcopula-basedMarkovchain
models to a serially correlated time series. These methods are illustrated through a
varietyofillustrativeexamplesfromfinance,industry,sports,andotherfields.Itis
our hope that the book serves as an accessible textbook for learning statistical
analyses of time series data using copulas for researchers/students in the fields of
economics, management, mathematics, statistics, and others. The book can also
serve as a research monograph, where each chapter can be read independently.
As the subtitle “Parametric inference” suggests, we focus on parametric models
based on the normal distribution, t-distribution, normal mixture distribution,
Poisson distribution, and others. The book adopts likelihood-based methods as the
main statistical tools for fitting the models and develops computing techniques to
find the maximum likelihood estimator. Some chapters discuss statistical process
control, Bayesian methods, and regression methods. We provide computer codes
for most presented statistical methods to help readers analyze their data.
Taoyuan, Taiwan Li-Hsien Sun
Hsinchu, Taiwan Xin-Wei Huang
Qassim, Saudi Arabia Mohammed S. Alqawba
Minnesota, USA Jong-Min Kim
Taoyuan, Taiwan Takeshi Emura
v
Acknowledgements
Wethanktheserieseditor,Dr.ShigeyukiMatsui,forhisvaluablecommentsonthis
book.
Li-Hsien Sun thanks his former graduate students, Chang-Shang Lee and
Wei-Cheng Lin, for their prior contribution to our published articles. He is finan-
cially supported by Ministry of Science and Technology, Taiwan (MOST
108-2118-M-008 -002 -MY2).
Xin-Wei Huang would like to thank the advisor of his master’s degree,
Dr.TakeshiEmura,whoisalsotheauthorofthisbook.Hewouldalsoliketothank
Dr. Jia-Han Shih for his kind help.
Mohammed Alqawba thanks his advisor Dr. Norou Diawara for his guidance
and valuable comments that lead to several published articles.
Takeshi Emura thanks his former graduate student, Ting-Hsuan Long, for his
priorcontributiontoourpublishedarticles.HeisfinanciallysupportedbyMinistry
of Science and Technology, Taiwan (MOST 107-2118-M-008-003-MY3).
vii
Contents
1 Overview of the Book with Data Examples. . . . . . . . . . . . . . . . . . . . 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Copulas and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Chemical Process Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.4 S&P 500 Stock Market Index Data . . . . . . . . . . . . . . . . . . . . . . . 3
1.5 Batting Average Data in MLB . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.6 Stock Price Data of Dow Jones Industrial Average. . . . . . . . . . . . 4
1.7 Data on the Count of Arsons. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.8 Concluding Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Copula and Markov Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Copulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Kendall’s Tau. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.4 Archimedean Copulas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.5 Random Number Generation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.6 Copula-Based Markov Chain. . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Appendix A: The Proof of Cðu;vÞ¼uþv(cid:2)1þCð1(cid:2)u;1(cid:2)vÞ
being a Copula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Appendix B: Proofs of Copulas Approaching to the Independence . . . . 24
Appendix C: Derivations of Kendall’s Tau. . . . . . . . . . . . . . . . . . . . . . 24
Appendix D: Derivation of Ca½1;1(cid:3)ðu;vÞ Under the Frank Copula. . . . . . . 26
Appendix E: Derivation of Cq½1;0(cid:3)ðu;vÞ Under the Gaussian Copula . . . . 26
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
ix
x Contents
3 Estimation, Model Diagnosis, and Process Control Under
the Normal Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.1 Serial Dependence, Statistical Process Control, and Copulas . . . . . 29
3.2 Model and Likelihood. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.3 Asymptotic Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.4 Goodness-of-Fit Tests. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.5 Model Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.6 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.7 Data Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.7.1 Chemical Process Data. . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.7.2 Financial Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.7.3 Baseball Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
Appendix: R Codes for Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 50
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4 Estimation Under Normal Mixture Models for Financial
Time Series Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.2 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.2.1 Copulas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.2.2 Copula-Based Markov Chain . . . . . . . . . . . . . . . . . . . . . . 58
4.3 Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.3.1 Maximum Likelihood Estimators . . . . . . . . . . . . . . . . . . . 60
4.3.2 Interval Estimation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.3.3 Initial Values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.4 Data Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.5 Data Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
Appendix: R codes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5 Bayesian Estimation Under the t-Distribution for Financial
Time Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.2 Models and Likelihood. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.2.1 Copula-Based Markov Models . . . . . . . . . . . . . . . . . . . . . 74
5.2.2 Non-standardized t-Distribution . . . . . . . . . . . . . . . . . . . . 75
5.2.3 Likelihood. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.3 Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.3.1 Estimation of Hyperparameters via Resampling. . . . . . . . . 77
5.3.2 Metropolis–Hastings Algorithm . . . . . . . . . . . . . . . . . . . . 79
Contents xi
5.4 Data Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
Appendix: Moment Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6 Control Charts of Mean by Using Copula Markov SPC
and Conditional Distribution by Copula. . . . . . . . . . . . . . . . . . . . . . 87
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6.2 Copula Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.2.1 Copula and Directional Dependence . . . . . . . . . . . . . . . . . 88
6.2.2 Copula Markov Statistical Process Control Chart. . . . . . . . 89
6.2.3 Control Charts of Mean by Using Copula Conditional
Distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6.3 Real Data Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.4 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
Appendix: R Codes for Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 96
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
7 Copula Markov Models for Count Series with Excess Zeros . . . . . . 101
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
7.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
7.2.1 Zero-Inflated Count Regression Models . . . . . . . . . . . . . . 103
7.3 Markov Chain Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
7.3.1 First-Order Markov Models . . . . . . . . . . . . . . . . . . . . . . . 107
7.3.2 Second-Order Markov Models . . . . . . . . . . . . . . . . . . . . . 108
7.3.3 Model Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
7.4 Statistical Inference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
7.4.1 Log-Likelihood Functions . . . . . . . . . . . . . . . . . . . . . . . . 111
7.4.2 Asymptotic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
7.5 Model Selection and Prediction. . . . . . . . . . . . . . . . . . . . . . . . . . 116
7.6 Arson Data Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
Appendix A: Trivariate Max-Id Copula Function with Positive
Stable LT and Bivariate Gumbel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
Appendix B: R Codes for Data Analysis . . . . . . . . . . . . . . . . . . . . . . . 122
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
Index .... .... .... .... .... ..... .... .... .... .... .... ..... .... 129
