dependence structures with applications to actuarial...

Contents Order-1 process Order-q process References

Dependence structures with applications toactuarial science

Luis E. Nieto-Barajas

Department of Statistics, ITAM, Mexico

Recent Advances in Actuarial Mathematics, OAX, MEX

October 26, 2015

Luis E. Nieto-Barajas Dependence structures


Contents

Order-1 process

Application in survival analysisApplication in claims reserving (INBR)Application in solvency analysis (see Mendoza andNieto-Barajas, 2006)

Order-q process

Application in time series modelingApplication in disease mapping



Order−1 process

η1 η2 η3 η4 η5

θ1 θ2 θ3 θ4 θ5

6@@@@@@R

6@@@@@@R

6@@@@@@R

6@@@@@@R

6

Dependence among {θk} is induced through latents {ηk}Close form expressions when use conjugate distributions

Want to ensure a given marginal distribution



Order−1 process

Nieto-Barajas and Walker (2002):

Beta process: {θk} ∼ BeP1(a, b, c)

θ1 ∼ Be(a, b), ηk | θk ∼ Bin(ck , θk),

θk+1 | ηk ∼ Be(a + ηk , b + ck − ηk)

⇒ θk ∼ Be(a, b) marginally & Corr(θk , θk+1) = ck/(a + b + ck)

Gamma process: {θk} ∼ GaP1(a, b, c)

θ1 ∼ Ga(a, b), ηk | θk ∼ Po(ckθk),

θk+1 | ηk ∼ Ga(a + ck , b + ηk)

⇒ θk ∼ Ga(a, b) marginally & Corr(θk , θk+1) = ck/(b + ck)



Survival Analysis

Hazard rate modelling

If T is a discrete r.v. with support on τk then

h(t) = θk I (t = τk)

with {θk} ∼ BeP1(a, b, c)

If T is a continuous r.v. and {τk} are a partition of IR+ then

h(t) = θk I (τk−1 < t ≤ τk)

with {θk} ∼ GaP1(a, b, c)

This is old stuff!, but what it is new is that there is anR-package called BGPhazard that implements these models



Survival Analysis

Example: Discrete survival model

We analyse the 6-MP clinical trial data which consists ofremission duration times (in months) for children with acuteleukemia.

The study consisted in comparing drug 6-MP versus placebo.We concentrate on the 21 patients who received placebo.

Observed time values range from 1 to 23 and there are nocensored observations.

To define the prior we took a = b = 0.0001 and ct = 50 forall t. We use command BeMRes to fit the model and thecommand BePloth to produce graphs.



Survival Analysis: Order-1 Beta process

0 5 10 15 20

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0.2

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Estimate of hazard rates

time

Haz

ard

rate

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Hazard functionConfidence band (95%)Nelson−Aalen based estimate



Survival Analysis: Order-1 Beta process

0 5 10 15 20

0.0

0.2

0.4

0.6

0.8

1.0

Estimate of Survival Function

times

Model estimateConfidence bound (95%)Kaplan−MeierKM Confidence bound (95%)



Survival Analysis

Example: Continuous survival model

We define a piecewise hazard function

The data are survival times of 33 leukemia patients. Timesare measured in weeks from diagnosis. Three of theobservations were censored.

The prior was defined by taking a = b = 0.0001 and

ck |ξiid∼ Ga(1, ξ) for k = 1, . . . ,K and ξ ∼ Ga(0.01, 0.01). We

took K = 10 intervals and chose the partition τk such thateach interval contains approximately the same number ofexact (not censored) observations.

We used the command GaMRes to fit the model andcommand GaPloth to produce graphs.



Survival Analysis: Order-1 Gamma process

0 20 40 60 80 100 120 140

0.0

0.2

0.4

0.6

0.8

1.0

Estimate of Survival Function

times

Model estimateConfidence bound (95%)Kaplan−MeierKM Confidence bound (95%)



Claims Reserving (INBR)

Consider the run-off triangle

Year of Development yearorigin 1 2 · · · j · · · n − 1 n

1 X11 X12 · · · X1j X1,n−1 X1n

2 X21 X22 · · · X2j X2,n−1

......

... · · ·...

i Xi1 Xi2 · · · Xi,n+1−i

......

...n − 1 Xn−1,1 Xn−1,2

n Xn1

Xij = Incremental claim amounts originated in year i and paidin development year j




de Alba and Nieto-Barajas (2008): Use a non-stationaryGaP1(a,b, c) to introduce dependence across developmentyears. Claims originated in different years remain independent

Xi1 ∼ Ga(ai1, bi1), ηik | Xik ∼ Po(cikXik),

Xi ,k+1 | ηik ∼ Ga(ai ,k+1 + cik , bi ,k+1 + ηik)

{Xi1, . . . ,Xi ,n+1−i} ∼ GaP1(a,b, c),where aij = ai , bij = bj and cij = cj , with

∑ni=1(1/bj) = 1.

This implies

E(Xij | Xi ,j−1) = (1− λj)αi

βj+ λjXi ,j−1,

with λj = cj/(bj + cj)




Example: Taylor and Ashe’s dataset

Data consists of incremental claims in a n × n triangle withn = 10

Transformed that data to millions to avoid numerical problems

Took priors for (a,b, c): ai ∼ Ga(0.001, 0.001),(1/b1, . . . , 1/bn) ∼ Dir(1, . . . , 1) and cj ∼ Ga(0.01, 0.01)



Claims Reserving (INBR): Independence (cj = 0)

1 2 3 4 5 6 7 8 9 10

45

67

89

Origin year

alph

a

1 2 3 4 5 6 7 8 9 10

0.0

0.10

0.20

0.30

Development year

1/be

ta



Claims Reserving (INBR): Dependence

1 2 3 4 5 6 7 8 9 10

45

67

89

Origin year

alph

a

1 2 3 4 5 6 7 8 9 10

0.0

0.10

0.20

0.30

Development year

1/be

ta



Claims Reserving (INBR): Dependence

2 3 4 5 6 7 8 9 10

0.0

0.2

0.4

0.6

Development year

lam

bda



Claims Reserving (INBR): Reserves comparison

Overdispersed Poisson Gamma-GLM Indep. Gamma Dep. Gamma De JongReserve Reserve Reserve Reserve

Year Estimate 95%-q Estimate 95%-q Estimate 95%-q Estimate 95%-q Estimate2 94634 275992 93316 169810 177813 470929 151028 376588 948363 469511 825235 446504 718423 603503 1184040 513163 9770530 4607244 709638 1139132 611145 911971 803354 1438850 655077 1164100 6957725 984889 1484632 992023 1422996 1319050 2162980 1279140 2054060 9638186 1419459 2036894 1453085 2048107 1823180 2864560 1802190 2853090 14275107 2177641 2993342 2186161 3077486 2680160 4063930 2599430 4035940 22203048 3920301 5221658 3665066 5259294 4235690 6464740 4278850 6235630 39381959 4278972 6003804 4122398 6113988 4761630 7634100 4670040 7076910 430090010 4625811 7890405 4516073 7339978 4948840 9344010 4669300 8261310 5967585

Total 18680856 23536181 18085772 22663092 21353200 28535000 20618200 27063500 20089343



Order−2 process

η1 η2 η3 η4 η5

θ1 θ2 θ3 θ4 θ5

?

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HHHHH

HHHHHHHj

?

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HHHHH

HHHHHHHj

?

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HHHHH

HHHHHHHj

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@@@@@@R ?

Throw more arrows to induce higher order dependence

There is no way to obtain a given marginal distribution:say beta or gamma

Unless we include an extra latent (layer)



Order−2 process

ω

η1 η2 η3 η4 η5

θ1 θ2 θ3 θ4 θ5

?

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HHHH

HHHHHH

HHj?

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HHHH

HHHHHH

HHj?

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HHj?

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With this common ancestor ω we can through more arrowsand still ensure a given marginal



Space and time process

This idea can be use to induce time and/or spatial dependence

'

&

$

%

'

&

$

%

'

&

$

%

t = 1 t = 2 t = 3

θ1,1(η1,1)

θ2,1(η2,1)

θ3,1(η3,1)

θ1,2(η1,2)

θ2,2(η2,2)

θ3,2(η3,2)

θ1,3(η1,3)

θ2,3(η2,3)

θ3,3(η3,3)

θ1,4(η1,4)

θ2,4(η2,4)

θ3,4(η3,4)

��>

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CCCCCCCCO ZZ}

ω



Order-q beta process

Jara and al. (2013):

Order−q (AR) beta process: {θt} ∼ BePq(a, b, c)

ω ∼ Be(a, b) ηt | ωind∼ Bin(ct , ω)

θt | η ∼ Be

a +

q∑j=0

ηt−j , b +

q∑j=0

(ct−j − ηt−j)

θt ∼ Be(a, b) marginally




Properties:

Corr(θt , θt+s) =(a + b)

(∑q−sj=0 ct−j

)+(∑q

j=0 ct−j

)(∑qj=0 ct+s−j

)(a + b +

∑qj=0 ct−j

)(a + b +

∑qj=0 ct+s−j

) ,

for s ≥ 1.

If ct = c for all t then {θt} becomes strictly stationary with

Corr(θt , θt+s) =(a + b) max{q − s + 1, 0}c + (q + 1)2c2

{a + b + (q + 1)c}2.



Autocorrelation in {θt}

5 10 15

0.0

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lag

5 10 150.

00.

20.

40.

60.

81.

0lag




Example: Unemployment rate in Chile

Annual data from 1980 to 2010

Use our BePq as likelihood for the data {Yt}Took priors for (a, b, c): a ∼ Un(0, 1000), b ∼ Un(0, 1000)

and ct | λiid∼ Po(λ) and λ ∼ Un(0, 1000)



Time series: Yt = Unemployement in Chile

1980 1985 1990 1995 2000 2005 2010

0.05

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0.25

Year

Unem

ploym

ent Ra

te in C

hile

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q=3q=4q=5q=6q=7q=8



Time series: Yt = Unemployement in Chile

1980 1990 2000 2010 2020

0.00.1

0.20.3

0.4

Year

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BePBDM



Spatial process



Spatial process

Nieto-Barajas and Bandyopadhyay (2013):

Spatial gamma process: {θt} ∼ SGaP(a, b, c)

ω ∼ Ga(a, b) ηij | ωind∼ Ga(cij , ω)

θi | η ∼ Ga

a +∑j∈∂i

cij , b +∑j∈∂i

ηij

∂i is the set of neighbours of region i

θt ∼ Ga(a, b) marginally



Disease mapping

Study: Mortality in pregnant women due to hypertensive disorderin Mexico in 2009. Areas are the States

Yi = Number of deaths in region iEi = At risk: Number of births (in thousands)λi = Maternity mortality rate

Zero-inflated model

f (yi ) = πi I (yi = 0) + (1− πi )Po(yi | λiEi )

λi = θi exp(β′xi ) πi =ξie

δ′zi

1 + ξieδ′zi

β is a vector of reg. coeff. s.t. βk ∼ N(0, σ20)

θi ∼ SGaP(a, a, c)ξi ∼ Ga(b, b)



Disease mapping

Six explanatory variables:

X1 number of medical units (hospitals + clinics)

X2 proportion of pregnant women with soc. sec.

X3 prop. of pregnant women who were seen by a physician inthe first trimester of pregnancy

X4 public expenditure in health per capita in thousands of MX

Z1 poverty index

Z2 proportion of births in clinics and hospitals



Estimated mortality rate λi

[3.05,6.33)[6.33,6.67)[6.67,7.38)[7.38,8.73)[8.73,21.07]

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Estimated zero inflated prob. πi

[0,0.01)[0.01,0.04)[0.04,0.06)[0.06,0.5)[0.5,0.6]

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References

de Alba, E. & Nieto-Barajas, L. E. (2008). Claims reserving: A correlatedbayesian model. Insurance: Mathematics and Economics 43, 368–376.

Jara, A., Nieto-Barajas, L. E. & Quintana, F. (2013). A time series model forresponses on the unit interval. Bayesian Analysis 8, 723–740.

Mendoza, M. & Nieto-Barajas, L. E. (2006). Bayesian solvency analysis withautocorrelated observations. Applied Stochastic Models in Business andIndustry 22, 169–180.

Nieto-Barajas, L. E. & Bandyopadhyay, D. (2013). A zero-inflated spatialgamma process model with applications to disease mapping. Journal ofAgricultural, Biological and Environmental Statistics 18, 137–158.

Nieto-Barajas, L. E. & Walker, S. G. (2002). Markov beta and gammaprocesses for modelling hazard rates. Scandinavian Journal of Statistics 29,413–424.


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