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Hybrid Dynamic and Fuzzy Models of Morality - ebook/pdf
Hybrid Dynamic and Fuzzy Models of Morality - ebook/pdf
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Wydawca: Wydawnictwo Uniwersytetu Łódzkiego Język publikacji: polski
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The mortality modeling and forecasting is of fundamental importance in many areas, such as funding of public and private pensions, life insurance, the care of the elderly or the provision of health services. The book is an attempt to approach this subject from a new theoretical point of view, using theory of stochastic differential equations, theory of fuzzy numbers and complex numbers. These notes are addressed to tertiary students, doctoral students and specialists in the fields of demography, life insurance, statistics and economics.

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Agnieszka Rossa, Andrzej Szymański – University of Łódź, Faculty of Economics and Sociology Department of Demography and Social Gerontology, 90-214 Łódź, 41/43 Rewolucji 1905 r. St. Lesław Socha – Cardinal Stefan Wyszyński University in Warsaw Faculty of Mathematics and Natural Sciences. School of Exact Sciences, Institute of Informatics 01-938 Warszawa, 1/3 Wóycickiego St. EDITORIAL BOARD OF GERONTOLOGY SERIES Professor Elżbieta Kowalska-Dubas (Faculty of Educational Sciences, University of Łódź) – Chair of the Editorial Board; Members: Professor Bogusława Urbaniak (Faculty of Economics and Sociology, University of Łódź); Professor Grzegorz Bartosz (Faculty of Biology and Environmental Protection, University of Łódź); Professor Piotr Szukalski (Faculty of Economics and Sociology, University of Łódź); Dr. Natalia Piórczyńska (Łódź University Press), M.A Monika Kamieńska (Academic Centre for Artistic Initiatives) TRANSLATOR AND PROOFREADING Janusz Kwitecki REVIEWER Jan Paradysz INITIATING EDITOR Iwona Gos TYPESETTING Agnieszka Rossa TECHNICAL EDITOR Leonora Wojciechowska COVER DESIGN Katarzyna Turkowska Cover Image: © Depositphotos.com/hobbitart Printed directly from camera-ready materials provided to the Łódź University Press The research was supported by a grant from the National Science Centre, Poland, under contract UMO-2015/17/B/HS4/00927 © Copyright by Authors, Łódź 2018 © Copyright for this edition by Uniwersytet Łódzki, Łódź 2018 Published by Łódź University Press First edition. W.07707.16.0.K Printing sheets 18.875 ISBN 978-83-8088-926-2 e-ISBN 978-83-8088-927-9 Łódź University Press 90-131 Łódź, 8 Lindleya St. www.wydawnictwo.uni.lodz.pl e-mail: ksiegarnia@uni.lodz.pl phone. (42) 665 58 63 Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Abbreviation and notation . . . . . . . . . . . . . . . . . . . . . . 13 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Chapter 1. Basic mortality characteristics and models . . . . . 19 1.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.2. Discrete-time mortality frameworks . . . . . . . . . . . . . . . 19 1.2.1. Age-specic rates and probabilities of death . . . . . . 19 1.2.2. The relationship between mortality rates and death probabilities . . . . . . . . . . . . . . . . . . . . . . . . 21 1.2.3. Interpolation models . . . . . . . . . . . . . . . . . . . 22 1.2.4. Other life-table measures . . . . . . . . . . . . . . . . . 26 . . . . . . . . . . . . . 27 1.3.1. Survival distributions . . . . . . . . . . . . . . . . . . . 27 1.3.2. The relationship between the mortality rate and the 1.3. Continuous-time mortality frameworks 1.6. The dynamic LeeCarter model force of mortality . . . . . . . . . . . . . . . . . . . . . 30 1.4. Laws of mortality . . . . . . . . . . . . . . . . . . . . . . . . . 32 1.5. The LeeCarter model and its extensions . . . . . . . . . . . . 36 1.5.1. The LeeCarter model . . . . . . . . . . . . . . . . . . 37 1.5.2. Age-period-cohort modications . . . . . . . . . . . . . 43 1.5.3. The fuzzy LeeCarter model . . . . . . . . . . . . . . . 47 . . . . . . . . . . . . . . . . . 49 1.6.1. Dynamic LC model . . . . . . . . . . . . . . . . . . . . 49 1.6.2. Discrete dynamic LC model . . . . . . . . . . . . . . . 50 . . . 51 1.6.3. Parameters estimation of the dynamic LC model 1.7. The Vasi£ek and CoxIngersollRoss models . . . . . . . . . . 53 1.7.1. V and CIR models . . . . . . . . . . . . . . . . . . . . 53 1.7.2. Discrete V and CIR models . . . . . . . . . . . . . . . 55 1.7.3. Modied V and CIR models . . . . . . . . . . . . . . . 55 1.7.4. Discrete modied V and CIR models . . . . . . . . . . 57 1.7.5. Parameters estimation of the V and CIR models . . . 57 6 1.8. The MilevskyPromislow model . . . . . . . . . . . . . . . . . 59 1.8.1. MP model . . . . . . . . . . . . . . . . . . . . . . . . . 59 1.8.2. Discrete MP model . . . . . . . . . . . . . . . . . . . . 60 1.8.3. Parameters estimation of the MP model . . . . . . . . 62 1.9. The GiacomettiOrtobelliBertocchi model . . . . . . . . . . . 63 . . . . . . . . . . . . . . . . . . . . . . . . 63 1.9.1. GOB model 1.9.2. Discrete GOB model . . . . . . . . . . . . . . . . . . . 65 1.9.3. Parameters estimation of the GOB model . . . . . . . 68 1.10. The modied MilevskyPromislow model . . . . . . . . . . . . 68 1.10.1. Modied MP model . . . . . . . . . . . . . . . . . . . 68 1.10.2. Discrete modied MP model . . . . . . . . . . . . . . . 70 1.10.3. Parameters estimation of the modied MP model . . . 72 1.11. The MilevskyPromislow models with two or more linear 3.1. 3.2. scalar lters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 1.11.1. MP model with two dependent lters . . . . . . . . . . 72 1.11.2. MP model with two independent lters . . . . . . . . . 73 1.11.3. MP model with a vector lter . . . . . . . . . . . . . . 74 1.12. Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 Chapter 2. Static and dynamic hybrid models . . . . . . . . . . 77 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 2.1. 2.2. Static hybrid models . . . . . . . . . . . . . . . . . . . . . . . 78 2.3. Dynamic hybrid models . . . . . . . . . . . . . . . . . . . . . . 80 2.4. Moment equations for the hybrid models . . . . . . . . . . . . 86 2.5. Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Chapter 3. Dynamic hybrid mortality models . . . . . . . . . . 93 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Identication of switchings . . . . . . . . . . . . . . . . . . . . 95 3.2.1. An introductory example . . . . . . . . . . . . . . . . . 95 3.2.2. Theoretical backgrounds of the JL test . . . . . . . . . 98 3.2.3. Determining switching points from mortality rates . . 100 . . . . . . . . . . . . . 108 3.3.1. Dynamic LCH model . . . . . . . . . . . . . . . . . . . 108 . . . . . . . . . . . . . . . . . . . 110 3.3.2. Discrete LCH model 3.3.3. Parameters estimation of the dynamic LCH model . . 110 3.4. The Vasi£ek and CoxIngersollRoss hybrid models . . . . . . 111 . . . . . . . . . . . . . . . . . . 111 3.4.1. VH and CIRH models . . . . . . . . . . . . . 112 3.4.2. VH and CIRH moment models . . . . . . . . . . . . . 113 3.4.3. Discrete VH and CIRH models 3.4.4. Discrete VH and CIRH moment models . . . . . . . . 114 3.4.5. Modied VH and CIRH models . . . . . . . . . . . . . 115 3.4.6. Modied VH and CIRH moment models . . . . . . . . 116 3.3. The dynamic LeeCarter hybrid model 7 3.4.7. Discrete modied VH and CIRH models . . . . . . . . 116 3.4.8. Discrete modied VH and CIRH moment models . . . 117 3.4.9. Parameters estimation of the VH and CIRH models . 118 3.5. The MilevskyPromislow hybrid models with one linear scalar lter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 3.5.1. MPH model . . . . . . . . . . . . . . . . . . . . . . . . 121 3.5.2. MPH moment model . . . . . . . . . . . . . . . . . . . 122 3.5.3. Discrete MPH model . . . . . . . . . . . . . . . . . . . 123 3.5.4. Discrete MPH moment model . . . . . . . . . . . . . . 124 . . . . . . 125 3.5.5. Parameters estimation of the MPH models 3.6. The GiacomettiOrtobelliBertocchi hybrid models . . . . . . 126 . . . . . . . . . . . . . . . . . . . . . . . 126 . . . . . . . . . . . . . . . . . . 127 . . . . . . . . . . . . . . . . . . 128 . . . . . . . . . . . . . 129 . . . . . 130 3.7. Modied MilevskyPromislow hybrid models . . . . . . . . . . 131 3.7.1. Modied MPH model . . . . . . . . . . . . . . . . . . 131 3.7.2. Modied MPH moment model . . . . . . . . . . . . . . 131 3.7.3. Discrete modied MPH model . . . . . . . . . . . . . . 133 3.7.4. Discrete modied MPH moment model . . . . . . . . . 134 3.7.5. Parameters estimation of the modied MPH models . 134 3.6.1. GOBH model 3.6.2. GOBH moment model 3.6.3. Discrete GOBH model 3.6.4. Discrete GOBH moment model 3.6.5. Parameters estimation of the GOBH models 3.8. The MilevskyPromislow hybrid models with two or more linear lters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 3.8.1. MPH model with two dependent lters . . . . . . . . . 136 3.8.2. MPH moment model with two dependent lters . . . . 137 3.8.3. MPH model with two independent lters . . . . . . . . 140 3.8.4. MPH moment model with two independent lters . . . 141 . . . . . . . . . 143 3.8.5. MPH model with a vector linear lter 3.8.6. MPH moment model with a vector linear lter . . . . 144 3.8.7. Discrete MPH moment model with two dependent lters . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 3.8.8. Discrete MPH moment model with two independent lters . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 . . . . 154 3.8.9. Discrete MPH model with a vector linear lter 3.8.10. Parameters estimation of the DMPH moment models with two lters . . . . . . . . . . . . . . . . . . . . . . 155 3.9. Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 Chapter 4. Mortality model based on oriented fuzzy numbers 161 4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 4.2. Algebra of oriented fuzzy numbers . . . . . . . . . . . . . . . . 162 8 4.3. The extended KoissiShapiro mortality model . . . . . . . . . 173 4.4. Data fuzzication with switchings . . . . . . . . . . . . . . . . 176 4.5. Parameters estimation of the EFLC model . . . . . . . . . . . 180 4.6. Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 Chapter 5. Mortality models based on modied fuzzy numbers and complex functions . . . . . . . . . . . . . . . . . . 183 5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 5.2. Mortality model based on the algebra of modied fuzzy numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 5.3. Parameters estimation of the MFLC model . . . . . . . . . . 186 5.4. Mortality model based on complex functions . . . . . . . . . . 190 5.5. Parameters estimation of the CFLC model . . . . . . . . . . . 193 . . . . . . . . . . . . . . . 194 5.6. Quaternion-valued mortality model 5.7. Parameters estimation of the QVLC model . . . . . . . . . . 199 5.8. Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 Chapter 6. Models estimation and evaluation based on the real data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 6.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 6.2. Results of switching points identication for the mortality data of Poland . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 6.3. Estimation results . . . . . . . . . . . . . . . . . . . . . . . . . 205 6.3.1. The DLCH model . . . . . . . . . . . . . . . . . . . . . 205 6.3.2. The DGOBHM model . . . . . . . . . . . . . . . . . . 211 6.3.3. The MFLC model . . . . . . . . . . . . . . . . . . . . 219 6.3.4. The QVLC model . . . . . . . . . . . . . . . . . . . . . 225 6.4. Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 Appendix A. Elements of the analysis of stochastic processes . . . . . . . . . . . . . . . . . . . . . . 231 and stochastic equations A.1. Basic denitions of stochastic processes . . . . . . . . . . . . . 231 A.1.1. Second-order processes . . . . . . . . . . . . . . . . . . 233 A.1.2. Stationary processes . . . . . . . . . . . . . . . . . . . 235 A.1.3. Gaussian processes . . . . . . . . . . . . . . . . . . . . 236 A.1.4. Markov processes . . . . . . . . . . . . . . . . . . . . . 237 A.1.5. Processes with independent increments . . . . . . . . . 238 A.1.6. White noise . . . . . . . . . . . . . . . . . . . . . . . . 240 A.2. Dierential and integral calculus of stochastic processes . . . . 242 A.2.1. Integrating and dierentiating in the mean square sense242 A.2.2. Stochastic integrals with respect to diusion processes 244 A.2.3. Itô s formula for diusion processes . . . . . . . . . . . 246 A.2.4. The Itô and Stratonovich stochastic dierential 9 A.3. Moment equations for linear stochastic dynamic systems equations for diusion processes . . . . . . . . . . . . . 248 . . . 253 A.3.1. Linear systems with additive excitation . . . . . . . . . 253 A.3.2. Linear systems with additive and parametric excitation 255 . 259 A.4. Methods of discretization of stochastic dierential equations Appendix B. Elements of the algebra of modied fuzzy and complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 B.1. Modied fuzzy numbers . . . . . . . . . . . . . . . . . . . . . . 261 B.2. Complex numbers and complex functions . . . . . . . . . . . . 268 B.2.1. The Banach C∗algebra . . . . . . . . . . . . . . . . . 268 B.2.2. The Banach C(T )algebra . . . . . . . . . . . . . . . . 269 B.2.3. The quaternion space . . . . . . . . . . . . . . . . . . . 275 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 Preface Mortality is generally considered relatively easy to forecast, particu- larly when the forecasting horizon is short. In longer periods however, its course may be aected by various changes brought about by all kinds of disturbances and events. A case in point is the health crisis in Poland of the 1970s and 1980s [Okólski 2003]. In such cases, it is of key importance that appropriate assumptions and an adequate model are selected. Mortality forecasting is usually supported by extrapolative models, making use of the regularity found in age patterns and trends of death rates or probabilities over time. There are several reasons why one should learn more about morta- lity models. Forecasting of mortality has a wide range of applications outside the eld of statistics and mathematics. It is of fundamen- tal importance in such areas as funding of public or private pensions and life insurance. Annuity providers and policy makers use mortality projections to determine appropriate pension benets, to assess retire- ment income or life insurance products, to hold additional reserving capital or to manage the long term demographic risk. Thus, one of the important question arises: What is the best way to forecast future mortality rates and to model the uncertainty of such forecasts? A key input to address this question is the development of advanced mortality modeling methodology. These notes are an attempt to capture the stochastic nature of mortality by approaching the subject of mortality modeling and fore- casting from a new theoretical point of view, using theory of stochastic dierential equations, theory of fuzzy numbers and complex numbers. The book is addressed to tertiary students, doctoral students and specialists in the elds of demography, life insurance, statistics and economics. This research project was funded by the National Science Center pursuant to its decision no. 2015/17/B/HS4/00927. 11 Abbreviation and notation Throughout this book, the following abbreviations for mortality models have been adopted: Standard LeeCarter LeeCarter hybrid Dynamic LeeCarter Discrete Dynamic LeeCarter model LeeCarter hybrid discrete LeeCarter hybrid Fuzzy LeeCarter Extended Fuzzy LeeCarter Modied Fuzzy LeeCarter Complex-Function LeeCarter Quaternion-Valued LeeCarter Vasi£ek Discrete Vasi£ek Vasi£ek hybrid Discrete Vasi£ek hybrid Vasi£ek hybrid moment Discrete Vasi£ek hybrid moment Modied Vasi£ek Discrete Modied Vasi£ek Modied Vasi£ek hybrid Discrete Modied Vasi£ek hybrid Modied Vasi£ek hybrid moment Discrete Modied Vasi£ek hybrid moment CoxIngersollRoss Discrete CoxIngersollRoss CoxIngersollRoss hybrid Discrete CoxIngersollRoss hybrid CoxIngersollRoss hybrid moment Discrete CoxIngersollRoss hybrid moment Modied CoxIngersollRoss Discrete Modied CoxIngersollRoss Modied CoxIngersollRoss hybrid Discrete Modied CoxIngersollRoss hybrid Modied CoxIngersollRoss hybrid moment SLC LCH DLC DDLC LCH DLCH FLC EFLC MFLC CFLC QVLC V DV VH DVH VHM DVHM MV DMV MVH DMVH MVHM DMVHM CIR DCIR CIRH DCIRH CIRHM DCIRHM MCIR DMCIR MCIRH DMCIRH MCIRHM DMCIRHM Discrete Modied CoxIngersollRoss hybrid moment GOB DGOB GOBH DGOBH GOBHM DGOBHM GiacomettiOrtobelliBertocchi Discrete GiacomettiOrtobelliBertocchi GiacomettiOrtobelliBertocchi hybrid Discrete GiacomettiOrtobelliBertocchi hybrid GiacomettiOrtobelliBertocchi hybrid moment Discrete GiacomettiOrtobelliBertocchi hybrid moment 14 MP DMP MMP DMMP DMPH MPHM DMPHM MMPH DMMPH MMPHM DMMPHM MP-2DF MPH-2DF MPHM-2DF DMPHM-2DF Discrete MilevskyPromislow hybrid moment MilevskyPromislow Discrete MilevskyPromislow Modied MilevskyPromislow Discrete Modied MilevskyPromislow Discrete MilevskyPromislow hybrid MilevskyPromislow hybrid moment Discrete MilevskyPromislow hybrid moment Modied MilevskyPromislow hybrid Discrete Modied MilevskyPromislow hybrid Modied MilevskyPromislow hybrid moment Discrete Modied MilevskyPromislow hybrid moment MilevskyPromislow, with 2 dependent lters MilevskyPromislow hybrid, with 2 dependent lters MilevskyPromislow hybrid moment, with 2 dependent lters MP-2IF MPH-2IF MPHM-2IF DMPHM-2IF MP-VLF MPH MPH-VLF MPHM-VLF DMPH-VLF with 2 dependent lters MilevskyPromislow, with 2 independent lters MilevskyPromislow hybrid, with 2 independent lters MilevskyPromislow hybrid moment, with 2 independent lters Discrete MilevskyPromislow hybrid moment with 2 independent lters MilevskyPromislow with vector linear lter MilevskyPromislow hybrid MilevskyPromislow hybrid, with a vector linear lter MilevskyPromislow hybrid moment, with a vector linear lter Discrete MilevskyPromislow hybrid, with a vector linear lter Introduction The phenomenon of mortality has been studied for many centuries. In the early 3rd c., a Roman jurist, Domitius Ulpianus, created for scal purposes the so-called Ulpian table containing life expectancies for the citizens of the Roman Empire. As historical sources do not mention what calculation method and source materials he had used, the Ulpian table is mainly of historical value [Rosset 1979, pp. 102103]. It is recognized that the father of the mortality table methodology is John Graunt (16201674), since his work [Graunt, 1662] where mor- tality of generations of London residents was examined. Graunt based his analysis on the records of London parishes, but did not specify which periods they concerned. Graunt s research was continued by an English astronomer Edmond Halley (16561742), who proposed mor- tality tables for the Wrocªaw population [Halley 1693]. The modern methodology for constructing mortality tables, also known as life-tables, is credited to Chin L. Chiang (19142014) and his book [Chiang 1968]. The more works on life-tables and mortality models come from 19th c. [Gompertz 1825, Thiele, Sprague 1871], but it is only during the last decades that the mortality modeling methodo- logy started to develop, as evidenced by numerous books on this sub- ject [Rosset 1979, Keilman 1990, Okólski 1990, Benjamin, Pollard 1993, Kannisto 1994, Tabeau et al. 2001, Keilman 2005, Alho, Spencer 2005, Girosi, King 2006, K edelski, Paradysz 2006, Rossa et al. 2011]. Since the introduction of the LeeCarter model [Lee, Carter 1992] proposed to forecast the trend of age-specic mortality rates, a range of mortality models have been proposed with modeling the probability of death, the age-specic mortality rate or the force of mortality. Among mortality models three main approaches can be identied: extrapolation, expectation and explanation [Pitacco 2004, Booth 2006, Tabeau et al. 2001]. The most common one is an extrapolative approach 16 which uses a real or fuzzy variable functions of age and time to de- scribe patterns and trends in death probabilities, mortality rates (or their transformations) and other measures [Heligman, Pollard 1980, Brouhns et al. 2002, Lee, Miller 2001, Renshaw, Haberman 2003a, 2003b, 2003c, 2006, 2008, Koissi, Shapiro 2006, Cairns et al. 2006, 2008a, 2008b, 2009, 2011, Denuit 2007, Debon et al. 2008, Haberman, Renshaw 2008, 2009, 2011, Hatzopoulos, Haberman 2011, Fung et al. 2017]. Mortality models can be divided also into two main categories: static and dynamic models. Models in the rst group are based on some algebraic equations, while in dynamic models of the second group the force of mortality (the intensity process) is expressed as a solution of stochastic dierential equations [Vasi£ek 1977, Cox et al. 1985a, 1985b, Janssen,Skiadas 1995, Milevsky, Promislow 2001, Dahl 2004, Bis 2005, Bis, Denuit 2006, Schrager 2006, Bravo, Braumann 2007, Yashin 2007, Hainaut, Devolder 2007, 2008, Luciano et al. 2008, Luciano, Vigna 2008, Plat 2009, Bayraktar et al. 2009, Bis et al. 2010, Coelho et al. 2010, Giacometti et al. 2011, Russoet al. 2011, Wanget al. 2011, Hainaut 2012, Rossa, Socha 2013]. Unfortunately, the simple dynamic models based on stochastic die- rential equations can be inadequate to describe demographic processes. In particular, they may fail to explain evolution of the phenomena, meaning that their behavior changes in continuous time or discrete time intervals. To make up for this disadvantage, researchers put for- ward a new type of models, called hybrid models, which account for interactions between continuous and discrete dynamics. Hybrid models, or switching models [Boukas 2005], are construc- ted as the generalizations of the models with switching points that have been already used for automatic control and for random struc- ture models [Kazakov, Artemiev 1980] describing phenomena within mechanics, biology, economics or empirical sciences. The authors of some studies have proposed complex mortality models sharing charac- teristics with the hybrid models [Bis, Denuit 2006, Bis et al. 2010, Hainaut 2012, Rossa, Socha 2013]. For the purposes of this study, a hybrid system will henceforth be understood as a family of static or dynamic models where the switch- ings take place according to some switching rule. The dynamic models will be described using stochastic dierential equations. There exists a class of equations for which analytical solutions of relatively complex structure can be found, therefore a new group of hybrid models will 17 be proposed called the moment hybrid models. The idea underlying their construction involves the replacement of the stochastic models by equivalent dierential equations for moments. Another promising approach to mortality modeling oers theory It is well-known that the main diculty in the of fuzzy numbers. applications of the LeeCarter model is due to the assumed homo- geneity of random terms. However, this property is not conrmed by the real-life data. The problem prompted search for solutions that could do without this assumption. One of the possible options is to set research in the framework of the fuzzy number theory. This line of thinking was adopted by [Koissi, Shapiro 2006], where empirical obser- vations and parameters of the LeeCarter model were converted into fuzzy symmetric triangular numbers. Unfortunately, the KoissiShapiro model involves some diculties, which arise from the necessity to nd the minimum of a criterion func- tion containing a max-type operator and cannot be solved using stan- dard optimization algorithms. One approach to such a problem can be applying the Banach algebra of oriented fuzzy numbers (OFN) deve- loped by [Kosi«ski et al. 2003]. The results of using this algebra to the KoissiShapiro model have been published in [Szyma«ski, Rossa 2014]. A more sophisticated modication of the KoissiShapiro model in- volves the replacement of the Banach OFN algebra by the Banach C∗algebra to allow the use of the GelfandMazur theorem about iso- metric isomorphism between the C∗algebra and the Banach algebra of complex functions and, consequently, to move the optimization prob- lem into the framework of complex analysis. To our best knowledge, this is an innovative approach to mortality modeling. This book has the following structure. In Chapter 1, basic mor- tality characteristics and some static and dynamic mortality models are discussed, especially the oldest historical mortality models (the so-called mortality laws), the well-known LeeCarter model with its extensions and generalizations, the Vasi£ek and CoxIngersoll-Ross models, the GiacomettiOrtobelliBertocchi model and some variants of the MilevskyPromislow model. Chapter 2 introduces theoretical backgrounds of hybrid modeling. In Chapter 3, hybrid counterparts of the dynamic models presented in Chapter 1 are provided and some es- timation procedures are proposed. Chapter 4 discusses the theoretical underpinnings of the fuzzy mortality modeling based on the algebra of Oriented Fuzzy Numbers (OFN), whereas Chapter 5 presents mortality 18 models from the perspective of the so-called modied fuzzy numbers (MFN) and complex functions. Chapter 6 illustrates results of estima- tions of some proposed models, the parameters of which were estimated using empirical mortality data sets. The comparative analysis of the models prediction accuracy is also performed. Chapter 1 Basic mortality characteristics and models 1.1. Introduction Demographic models are an attempt to generalize and simplify real demographic processes by means of mathematical functions or a set of mathematical relations in order to approximate possible variations observed in the real data and to support demographic forecasting. In this chapter basic notions, relations and some discrete-time as well as continuous-time extrapolative mortality models are introduced. The main attention is focused on the well-known LeeCarter model, its generalizations, the Vasi£ek and CoxIngersollRoss models as well as the MilevskyPromislow and GiacomettiOrtobelliBertocchi mo- dels. They will be converted to hybrid models in Chapter 3. 1.2. Discrete-time mortality frameworks 1.2.1. Age-specic rates and probabilities of death The denition of a mortality rate used in this book draws on the general denition of a cohort (or period) demographic rate dened as a ratio of the number of demographic events occurring in some dened cohort (or in a real population within some dened time period) to the time-to-exposure, understood as the number of time units lived by the cohort (or by the population during the given time period) [Preston et al. 2001, pp. 532]. If person-years are used in the denominator, a demographic rate is termed an annualized rate. Below the denitions of both a co- hort and a period annualized age-specic mortality rates are provided [Rossa et al. 2011, pp. 229231]. 20 An important notion used in the Denition 1.1 is a cohort, dened as a real or hypothetical aggregate of individuals that experience a spe- cic demographic event, e.g. births, during a specic time interval. The cohort is identied by the event itself and by its time frame. For the purposes of this discussion, let index t indicate a calendar year from the given set {1, 2, . . . , T}, and index x the attained age, meaning that it takes values from the set {0, 1, . . . , X}, where X is the xed upper age limit. Denition 1.1. A cohort age-specic mortality rate m(s) x cohort is a ratio of the number of deaths, D(s) x years last birthday to the number of person-years, K (s) age range [x, x + 1) in the s-th x , among individuals aged x , lived in the m(s) x = D(s) x K (s) x . (1.2.1) Denition 1.2. A period age-specic mortality rate mx,t is a ratio of the number of deaths, Dx,t, among individuals in the age range [x, x+1) years during the calendar year t to the number of person-years, Kx,t, lived in th age interval [x, x + 1) during this year mx,t = Dx,t Kx,t . (1.2.2) It is worth noting that the denominators K (s) x in (1.2.1) and Kx,t in (1.2.2) can be treated as the number of individuals exposed to the risk of death in the given age interval or in the age-time interval, re- spectively. In the case of (1.2.2) the denominator is usually replaced by the midyear population ¯Lx,t, lived in the age range [x, x + 1) during the given year t. Therefore, period mortality rates (1.2.2) are often described as central death rates because of a midyear population used in the denominator. For convenience (1.2.1), (1.2.2) are often expressed in thousands as m(s) x = D(s) x K (s) x · 1 000, mx,t = · 1 000. Dx,t Kx,t (1.2.3) In a more general discrete approach, it is possible to consider an age interval [x, x+n), where n ∈ N and n 1. The cohort age-specic mor- tality rates (1.2.1) are then denoted as nm(s) x and the period age-specic mortality rates (1.2.2) as nmx,t.
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Hybrid Dynamic and Fuzzy Models of Morality
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