Darmowy fragment publikacji:
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-speci c 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 modi cations . . . . . . . . . . . . . 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. Modi ed V and CIR models . . . . . . . . . . . . . . . 55
1.7.4. Discrete modi ed 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 modi ed MilevskyPromislow model . . . . . . . . . . . . 68
1.10.1. Modi ed MP model
. . . . . . . . . . . . . . . . . . . 68
1.10.2. Discrete modi ed MP model . . . . . . . . . . . . . . . 70
1.10.3. Parameters estimation of the modi ed 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
Identi cation 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. Modi ed VH and CIRH models . . . . . . . . . . . . . 115
3.4.6. Modi ed VH and CIRH moment models . . . . . . . . 116
3.3. The dynamic LeeCarter hybrid model
7
3.4.7. Discrete modi ed VH and CIRH models . . . . . . . . 116
3.4.8. Discrete modi ed 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. Modi ed MilevskyPromislow hybrid models . . . . . . . . . . 131
3.7.1. Modi ed MPH model
. . . . . . . . . . . . . . . . . . 131
3.7.2. Modi ed MPH moment model . . . . . . . . . . . . . . 131
3.7.3. Discrete modi ed MPH model . . . . . . . . . . . . . . 133
3.7.4. Discrete modi ed MPH moment model . . . . . . . . . 134
3.7.5. Parameters estimation of the modi ed 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 fuzzi cation with switchings . . . . . . . . . . . . . . . . 176
4.5. Parameters estimation of the EFLC model . . . . . . . . . . . 180
4.6. Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
Chapter 5. Mortality models based on modi ed fuzzy
numbers and complex functions . . . . . . . . . . . . . . . . . . 183
5.1.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
5.2. Mortality model based on the algebra of modi ed 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 identi cation 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 de nitions 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 modi ed fuzzy and
complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
B.1. Modi ed 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 bene ts, 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
Modi ed 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
Modi ed Vasi£ek
Discrete Modi ed Vasi£ek
Modi ed Vasi£ek hybrid
Discrete Modi ed Vasi£ek hybrid
Modi ed Vasi£ek hybrid moment
Discrete Modi ed Vasi£ek hybrid moment
CoxIngersollRoss
Discrete CoxIngersollRoss
CoxIngersollRoss hybrid
Discrete CoxIngersollRoss hybrid
CoxIngersollRoss hybrid moment
Discrete CoxIngersollRoss hybrid moment
Modi ed CoxIngersollRoss
Discrete Modi ed CoxIngersollRoss
Modi ed CoxIngersollRoss hybrid
Discrete Modi ed CoxIngersollRoss hybrid
Modi ed 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 Modi ed 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
Modi ed MilevskyPromislow
Discrete Modi ed MilevskyPromislow
Discrete MilevskyPromislow hybrid
MilevskyPromislow hybrid moment
Discrete MilevskyPromislow hybrid moment
Modi ed MilevskyPromislow hybrid
Discrete Modi ed MilevskyPromislow hybrid
Modi ed MilevskyPromislow hybrid moment
Discrete Modi ed 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-speci c mortality rates, a range
of mortality models have been proposed with modeling the probability
of death, the age-speci c mortality rate or the force of mortality.
Among mortality models three main approaches can be identi ed:
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, Bi s 2005,
Bi s, 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, Bi s 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 [Bi s, Denuit 2006, Bi s 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 di culty 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 con rmed 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 di culties,
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 modi cation 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 modi ed 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-speci c rates and probabilities of death
The de nition of a mortality rate used in this book draws on the
general de nition of a cohort (or period) demographic rate de ned as
a ratio of the number of demographic events occurring in some de ned
cohort (or in a real population within some de ned 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 de nitions of both a co-
hort and a period annualized age-speci c mortality rates are provided
[Rossa et al. 2011, pp. 229231].
20
An important notion used in the De nition 1.1 is a cohort, de ned
as a real or hypothetical aggregate of individuals that experience a spe-
ci c demographic event, e.g. births, during a speci c time interval.
The cohort is identi ed 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.
De nition 1.1. A cohort age-speci c 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)
De nition 1.2. A period age-speci c 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-speci c mor-
tality rates (1.2.1) are then denoted as nm(s)
x and the period age-speci c
mortality rates (1.2.2) as nmx,t.
Pobierz darmowy fragment (pdf)
