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Introduction to Survival

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Title: Introduction to Survival


1
Introduction to Survival Stochastic Models -
Stochastic Models (STAT 2442)
  • Dr Shane Whelan, FFA
  • L527

2
Introductory Remarks
3
Course Notes
  • I give an Introduction to Stochastic Models in 12
    lectures at 10 on Tuesday in Q010.
  • Mr Gareth Colgan, MA, FIA, gives the other half
    of course, treating Survival Models.
  • My full website is at
  • http//www.ucd.ie/statdept/staff/swhelan.html
  • Following each lecture, the slides are put up on
    web as are problem sheets, etc. I am also writing
    up the lectures into chapters so full text for
    some chapters will also be up on the web.
  • My boardwork (including all proofs and
    supplementary examples) are part of the course
    (i.e., examinable)
  • Copy these down
  • Exam is in Summer (repeat Autumn) and is 1.75
    hours long (50 on Survival and 50 on
    Stochastic). No choice of questions do all.
  • This is introductory to 3rd Year Course I will
    cover 20 of the 3rd Year Course, filling out
    details and giving more examples.
  • Learn it nowyou must eventually!

4
Course Outline
  • Chapter 1 Introduction to (Actuarial) Modelling
    33
  • Chapter 2 Foundational concepts in Stochastic
    Processes 33
  • Chapter 3 Worked Examples 33

5
Chapter 1
  • Introduction to (Actuarial) Modelling

6
The Modelling Problem
  • I have yet to see any problem, however
    complicated, which, when you looked atit in the
    right way, did not become still more
    complicated.
  •  
  •  
  •  
  • Poul Anderson, science fiction writer, in New
    Scientist. (London, September 25, 1969).

7
Modelling
  • Model a simple, stylised imitation of a real
    world system or process.
  • Used to predict how process might respond to
    given changes enabling results of possible
    actions to be assessed or simply to understand
    how system will evolve in the future.
  • Other methods being too slow, too risky, or too
    expensive.
  • Objective of Model is paramount
  • we need to know what is best model and this is
    generally not the most accurate model need to
    balance cost with benefits.
  • e.g., macroeconometric model of economy
  • Price of share at each future date
  • Model life office

8
Model of Salary Progession
Salary Level
Years since Graduation
9
Classifying Models
  • Deterministic Model Unique output for given set
    of inputs. The output is not a random variable.
  • Stochastic Model Output is a random variable.
    Perhaps some inputs are also random variables.
  • A deterministic model can be seen as a special
    case of a stochastic model.

10
Stochastic Model of Salary Progession
Salary Level Some Possible Paths
Years since Graduation
11
Stochastic Analysis
Stochastic Model of a System
Future Period
10
9
8
7
6
5
4
3
2
1
0
12
Stochastic Analysis
Stochastic Model of a System
Future Period
10
9
8
7
6
5
4
3
2
1
0
13
Components of Model
  • Structural Part sets out the relationship
    between the variables modelled (inputs) so as to
    determine the functioning of the system
    (outputs).
  • Relationships are generally expressed in logical
    or mathematical terms.
  • Complexity of model is determined by the number
    of variables modelled and the form of
    relationship posited between them.
  • Parameters the value of the inputs.
  • Often estimated from past data, using statistical
    techniques.
  • Also current observation, subjective assessment,
    etc.

14
Building a Model
  • Anything Goes

15
Introduction to Real Modelling
  • Perspective we attempt to get, well captured in
  • Real Life Mathematics, Bernard Beauzamy, Irish
    Math. Soc. Bulletin 48 (Summer 2002), 4346.
  • Available on Web from
  • http//www.maths.tcd.ie/pub/ims/bull48/M4801.pdf
  • Repays the 20 minute read!

16
Quotes from Real Life Mathematics
  • It is always our duty to put the problem in
    mathematical terms, and this part of the work
    represents often one half of the total work
  • My concern is, primarily, to find people who are
    able and willing to discuss with our clients,
    trying to understand what they mean and what they
    want. This requires diplomacy, persistence, sense
    of contact, and many other human qualities.
  • Since our problem is real life, it never fits
    with the existing academic tools, so we have to
    create our own tools. The primary concern for
    these new tools is the robustness.

17
Building a Model 10 Helpful Steps
  • Set well-defined objectives for model.
  • Plan how model is to be validated
  • i.e., the diagnostic tests to ensure it meets
    objectives
  • Define the essence of the structural model the
    1st order approximation. Refinement and details
    can come later.
  • Collect analyse data for model (and any other
    parameters)
  • Involve experts on the real world system to get
    feedback on conceptual model.

18
Building a Model 10 Helpful Steps
  • Decide how to implement model
  • e.g. C, Excel, some statistical package. Often
    random number generator needed.
  • 7. Write and debug program.
  • Test the reasonableness of the output from the
    model and otherwise analyse output.
  • Does it replicate historic episodes reasonably
    well?
  • 9. Test sensitivity of output to input parameters
  • i.e., ensure small change to inputs has small
    affect on output.
  • We do not want a chaotic system in actuarial
    applications.
  • Communicate and document results and the model.
  • 10.a Review and update in the light of new data
    and other changes.

19
Advantages of Modelling
  • Modelling can claim all the advantages of the
    scientific programme over any other logical,
    critical ,and evidence-based study of phenomenon
    that builds, often incrementally, to a body of
    knowledge.
  • Complex systems, including stochastic systems,
    that are otherwise not tractable mathematically
    (in closed form) can be studied.
  • It is quicker (system studied in compressed
    time), and less expensive than alternatives.
  • Consequences of different policy actions can be
    assessed, so option can be selected that
    optimizes output.
  • We can reduce variance of model as we can better
    control experimental conditions.

20
Drawbacks of Modelling (that must be guarded
against)
  • Requires considerable investment of time and
    expertise..not free.
  • Often time-consuming to use many simulations
    needed and results analysed.
  • Not especially good at optimising outputs (better
    at comparing results of input variations)
  • Impressive-looking models (especially complex
    ones) can lead to overconfidence in model.
  • Model only as good as parameter inputs quality
    and credibility of data.
  • Must understand limitations of model (i.e., its
    proper use)
  • Must recognise that a model will become obsolete
    change in circumstances.
  • Sometimes difficult to interpret output.

21
Quotes from Real Life Mathematics
  • Most current mathematical research, since the
    1960s, is devoted to fancy situations it
    brings solutions which nobody understands to
    questions nobody asked. Nevertheless, those who
    bring these solutions are called distinguished
    by the academic community. This word by itself
    gives a measure of the social distance real life
    mathematics do not require distinguished
    mathematicians. On the contrary, it requires
    barbarians people willing to fight, to conquer,
    to build, to understand, with no predetermined
    idea about which tool should be used.

22
Computers Modelling
  • First generation of civiliation computers (say
    the UNIVAC computer) were used as calculators
    performing repetitive calculations
  • First was bought by the Census Bureau, second by
    A.C. Nielson Market Research and the third by the
    Prudential Insurance Company.
  • Second generation of computers (say the IBM 360
    series) were used as real time databases
  • airline reservations processing inventory
    control insurance industry semi-automated its
    back office
  • Subsequent generations have been used for, inter
    alia, design (CAD) or, put another way, modelling
  • Cars and airplanes designed without wind-tunnels
    the next generations of computer chips model
    life offices, etc.

23
Computers Modelling
  • And since the PC and, argubly, the invention of
    the spreadsheet (first Visicalc, then Lotus
    1-2-3, and finally Microsoft Excel), we all have
    access to a user-friendly aid to modelling.
  • The computer is revolutionising modelling.

24
Classifying Models Another Look
  • Deterministic Model Unique output for given set
    of inputs. The output is not a random variable.
  • Gives one scenario.
  • Simple systems can be solved for explicitly
    that is solution (output) is known in closed form
    a simple function, f(.).
  • But often need numerical methods to solve.
  • Stochastic Model Output is a random variable.
    Perhaps some inputs are also random variables.
  • Gives multiple scenarios, weighted by
    probability.
  • If possible, attempt at least a partial analytic
    solution it simplifies the modelling
    considerably.
  • In Monte Carlo simulation a single random drawing
    for each input random variable (and a realisation
    of each randomiser in model) is taken to give an
    input and the process repeated a large no. of
    times - equally likely deterministic models. This
    build up a picture of the output random variable.
    It is a very general and powerful technique but
    its precision depends, inter alia, on the no. of
    simulations.

25
Discrete Continuous Time and States
  • Consider the system (stochastic process) ltxigt,
    i?T
  • State of a model/process a set of variables
    describing the system at time t, i.e. xt
  • State space the set of all possible values for
    the process, xt, ?t
  • State space is either continous or discrete
    (finite or countable number of possibilities).
  • Time can also be considered continous or
    discrete.
  • Hence discrete time stochastic process
    continuous time stochastic process.
  • Note 1 Whether one employs discrete or
    continuous time or state space depends on the
    objectives of the modelling not solely on the
    underlying reality.
  • Note 2 Discrete systems lend themselves for
    easily to simulation. However, sometimes assuming
    continuity can lead to closed form solutions
    making them more tactable mathematically.

26
Evaluation of Suitability of a Model
  • Evaluate in context of objectives and purpose to
    which it is put.
  • Consider data and techniques used to calibate
    model, especially estimation errors. Assess the
    credibility of the inputs.
  • Consider correlation structure between variables
    driving the model.
  • Consider correlation structure of model outputs.
  • Continued relevance of model (if past model).
  • Credibility of outputs.
  • Dangers of spurious accuracy.
  • Ease of use and how results can be communicated.

27
Further Considerations in Modelling
  • Short and long run properties of model
  • are the coded relationships stable over time?
  • should we factor in relationships that are second
    order in the short-term but manifest over
    long-term?
  • Analysing the output
  • generally by statistical sampling techniquesbut
    beware as observations are, in general
    correlated. IID assumption never, in general,
    valid.
  • Use failure in Turing-type (or Working) test to
    better model.
  • Sensitivity Testing
  • Check small changes to inputs produce small
    changes to outputs. Check results robust to
    statistical distribution of inputs.
  • Monitor and, perhaps expand on key sensitivities
    in model.
  • Use optimistic, best estimate, and pessimistic
    assumptions.

28
Further Considerations in Modelling
  • Communication documentation of results
  • Take account of knowledge and background of
    audience.
  • Build confidence in model so seen as useful tool.
  • Outline limitations of models.

29
Macro-Econometric Modelling A Case Study
30
Macro-Economics -V- Macro-Econometrics
31
Macro-Economics -V- Macro-Econometrics
32
Macro-Economic Management with Large Econometric
Models UK
33
Modelling in Early 1970s (UK)
  • Four Major Econometric Models
  • Bank of England
  • Treasury
  • NIESR (the National Institute of Economic and
    Social Research)
  • London Business School
  • Consisting of 500-1,000 equations
  • Modelling whole economy
  • So complex that, in effect, Black Boxes.

34
Macro-Economic Management with Large Econometric
Models UK
35
Macro-Economic Management with Large Econometric
Models UK
36
Modelling Errors
  • General Uncertainty error term in model.
  • Parameter misestimation the form of the model
    is right but the parameters are not.
  • Model misspecification the form of the model is
    wrong.

37
What went wrong in UK models
  • First thought to be parameter misestimation
  • Exchange rate floats in 1973 but no data to
    estimate what will happen so ignored it.
  • Oil shock nothing like it seen before so
    pushing models to extreme.
  • But the real problem was...
  • Model Misspecification the graph simply could
    not go in that way under Keynesian Theory

38
Opinion on Models, Early 1980s
  • Treasury forecasters in 1980 were predicting
    the worst economic downturn since the Great Slump
    of 1929-1931. Yet they expected no fall in
    inflation at all. This clearly was absurd and
    underlined the inadequacies of the model.
  •   Nigel Lawson, The View from No. 11.
  • Modelling was seen as a second-rate activity
    done by people who were not good enough to get
    proper academic jobs.
  • Earlier expectations of what models might
    achieve had evidently been set too high, with
    unrealistic claims about their reliability and
    scope.
  •  
  • Quoted from Economic Models and Policy-Making.
    Bank of England, Quarterly Bulletin, May 1997.

39
Modelling from Mid-1980s to Date
  • Must satisfy four criteria
  • Models and their outputs can be explained in a
    way consistent with basic economic analysis.
  • The judgement part of the process is made
    explicit.
  • The models produce results consistent with
    relevant historic episodes.
  • Results are consistent over time (e.g., the
    parameters are not sensitive to period studied)
  • Leads to small-scale models.
  • Simple, parsimonious in parameters, designed for
    purpose on hand.

40
Macro-Economic Management with Large Econometric
Models UK
41
Macro-Economic Management with Large Econometric
Models UK
42
Macro-Econometric Modelling Lessons Learned
  • Be limited in our expectations of what can
    reasonably be achieved.
  • Build disposal models.
  • Small, stylised, parsimonious models are
    beautiful.

43
Completes Case Study
44
Have Modest Ambitions when modellingModelling
Orders of Complexity
  • Level 1 - Two body problem
  • e.g., gravity, light through prism, etc.
  • Level 2 - N-identical body with local interaction
  • e.g., Maxwell-Boltzmanns thermodynamics
  • Ising model of ferromagnetism
  • Level 3 - N-identical body with long-range
    interaction
  • Level 4 - N-non-identical body with
    multi-interactions
  • Modelling Markets
  • Modelling economics systems generally
  • General actuarial modelling
  • The History of Science gives us no example of a
    complex problem of Level 3 or 4 being adequately
    modelled

From Roehner, B.M., Patterns of Speculation A
Study in Observational Econophysics, CUP 2002
45
Final Word
  • One thing I have learned in a long life that
    all our science, measured against reality, is
    primitive and childlike and yet it is the most
    precious thing we have.

Albert Einstein
46
Completes Chapter 1
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