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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 Order-q process References
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
Luis E. Nieto-Barajas Dependence structures
Contents Order-1 process Order-q process References
Order−1 process
η1 η2 η3 η4 η5
θ1 θ2 θ3 θ4 θ5
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Dependence among {θk} is induced through latents {ηk}Close form expressions when use conjugate distributions
Want to ensure a given marginal distribution
Luis E. Nieto-Barajas Dependence structures
Contents Order-1 process Order-q process References
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)
Luis E. Nieto-Barajas Dependence structures
Contents Order-1 process Order-q process References
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)
Luis E. Nieto-Barajas Dependence structures
Contents Order-1 process Order-q process References
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
Luis E. Nieto-Barajas Dependence structures
Contents Order-1 process Order-q process References
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
Luis E. Nieto-Barajas Dependence structures
Contents Order-1 process Order-q process References
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.
Luis E. Nieto-Barajas Dependence structures
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Survival Analysis: Order-1 Beta process
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Hazard functionConfidence band (95%)Nelson−Aalen based estimate
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Survival Analysis: Order-1 Beta process
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Estimate of Survival Function
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Model estimateConfidence bound (95%)Kaplan−MeierKM Confidence bound (95%)
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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.
Luis E. Nieto-Barajas Dependence structures
Contents Order-1 process Order-q process References
Survival Analysis: Order-1 Gamma process
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Estimate of Survival Function
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Model estimateConfidence bound (95%)Kaplan−MeierKM Confidence bound (95%)
Luis E. Nieto-Barajas Dependence structures
Contents Order-1 process Order-q process References
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
Luis E. Nieto-Barajas Dependence structures
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Claims Reserving (INBR)
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)
Luis E. Nieto-Barajas Dependence structures
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Claims Reserving (INBR)
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)
Luis E. Nieto-Barajas Dependence structures
Contents Order-1 process Order-q process References
Claims Reserving (INBR): Independence (cj = 0)
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Luis E. Nieto-Barajas Dependence structures
Contents Order-1 process Order-q process References
Claims Reserving (INBR): Dependence
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Luis E. Nieto-Barajas Dependence structures
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Claims Reserving (INBR): Dependence
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Luis E. Nieto-Barajas Dependence structures
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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
Luis E. Nieto-Barajas Dependence structures
Contents Order-1 process Order-q process References
Order−2 process
η1 η2 η3 η4 η5
θ1 θ2 θ3 θ4 θ5
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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)
Luis E. Nieto-Barajas Dependence structures
Contents Order-1 process Order-q process References
Order−2 process
η1 η2 η3 η4 η5
θ1 θ2 θ3 θ4 θ5
?
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HHHHH
HHHHHHHj
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HHHHHHHj
?
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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)
Luis E. Nieto-Barajas Dependence structures
Contents Order-1 process Order-q process References
Order−2 process
η1 η2 η3 η4 η5
θ1 θ2 θ3 θ4 θ5
?
@@@@@@R
HHHHH
HHHHHHHj
?
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HHHHHHHj
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HHHHH
HHHHHHHj
?
@@@@@@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)
Luis E. Nieto-Barajas Dependence structures
Contents Order-1 process Order-q process References
Order−2 process
ω
η1 η2 η3 η4 η5
θ1 θ2 θ3 θ4 θ5
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With this common ancestor ω we can through more arrowsand still ensure a given marginal
Luis E. Nieto-Barajas Dependence structures
Contents Order-1 process Order-q process References
Space and time process
This idea can be use to induce time and/or spatial dependence
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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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Luis E. Nieto-Barajas Dependence structures
Contents Order-1 process Order-q process References
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
Luis E. Nieto-Barajas Dependence structures
Contents Order-1 process Order-q process References
Order-q beta process
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.
Luis E. Nieto-Barajas Dependence structures
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Autocorrelation in {θt}
5 10 15
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0lag
Luis E. Nieto-Barajas Dependence structures
Contents Order-1 process Order-q process References
Order-q beta process
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)
Luis E. Nieto-Barajas Dependence structures
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Time series: Yt = Unemployement in Chile
1980 1985 1990 1995 2000 2005 2010
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Luis E. Nieto-Barajas Dependence structures
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Time series: Yt = Unemployement in Chile
1980 1990 2000 2010 2020
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BePBDM
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Spatial process
Luis E. Nieto-Barajas Dependence structures
Contents Order-1 process Order-q process References
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
Luis E. Nieto-Barajas Dependence structures
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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)
Luis E. Nieto-Barajas Dependence structures
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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
Luis E. Nieto-Barajas Dependence structures
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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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Luis E. Nieto-Barajas Dependence structures
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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.
Luis E. Nieto-Barajas Dependence structures