Dependence structures with applications to actuarial science

Transcription

1 with applications to actuarial science Department of Statistics, ITAM, Mexico Recent Advances in Actuarial Mathematics, OAX, MEX October 26, 2015

2 Contents Order-1 process Application in survival analysis Application in claims reserving (INBR) Application in solvency analysis (see Mendoza and Nieto-Barajas, 2006) Order-q process Application in time series modeling Application in disease mapping

3 Order1 process η 1 η 2 η 3 η 4 η 5 θ 1 θ 2 θ 3 θ 4 θ 5 Dependence among {θ k } is induced through latents {η k } Close form expressions when use conjugate distributions Want to ensure a given marginal distribution

4 Order1 process Nieto-Barajas and Walker (2002): Beta process: {θ k } BeP 1 (a, b, c) θ 1 Be(a, b), η k θ k Bin(c k, θ k ), θ k+1 η k Be(a + η k, b + c k η k ) θ k Be(a, b) marginally & Corr(θ k, θ k+1 ) = c k /(a + b + c k )

5 Order1 process Nieto-Barajas and Walker (2002): Beta process: {θ k } BeP 1 (a, b, c) θ 1 Be(a, b), η k θ k Bin(c k, θ k ), θ k+1 η k Be(a + η k, b + c k η k ) θ k Be(a, b) marginally & Corr(θ k, θ k+1 ) = c k /(a + b + c k ) Gamma process: {θ k } GaP 1 (a, b, c) θ 1 Ga(a, b), η k θ k Po(c k θ k ), θ k+1 η k Ga(a + c k, b + η k ) θ k Ga(a, b) marginally & Corr(θ k, θ k+1 ) = c k /(b + c k )

6 Survival Analysis Hazard rate modelling If T is a discrete r.v. with support on τ k then h(t) = θ k I (t = τ k ) with {θ k } BeP 1 (a, b, c) If T is a continuous r.v. and {τ k } are a partition of IR + then h(t) = θ k I (τ k1 < t τ k ) with {θ k } GaP 1 (a, b, c)

7 Survival Analysis Hazard rate modelling If T is a discrete r.v. with support on τ k then with {θ k } BeP 1 (a, b, c) h(t) = θ k I (t = τ k ) If T is a continuous r.v. and {τ k } are a partition of IR + then with {θ k } GaP 1 (a, b, c) h(t) = θ k I (τ k1 < t τ k ) This is old stuff!, but what it is new is that there is an R-package called BGPhazard that implements these models

8 Survival Analysis Example: Discrete survival model We analyse the 6-MP clinical trial data which consists of remission duration times (in months) for children with acute leukemia. 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 no censored observations. To define the prior we took a = b = and c t = 50 for all t. We use command BeMRes to fit the model and the command BePloth to produce graphs.

9 Survival Analysis: Order-1 Beta process Estimate of hazard rates time Hazard rate Hazard function Confidence band (95%) NelsonAalen based estimate

10 Survival Analysis: Order-1 Beta process Estimate of Survival Function Model estimate Confidence bound (95%) KaplanMeier KM Confidence bound (95%) times

11 Survival Analysis Example: Continuous survival model We define a piecewise hazard function The data are survival times of 33 leukemia patients. Times are measured in weeks from diagnosis. Three of the observations were censored. The prior was defined by taking a = b = and c k ξ iid Ga(1, ξ) for k = 1,..., K and ξ Ga(0.01, 0.01). We took K = 10 intervals and chose the partition τ k such that each interval contains approximately the same number of exact (not censored) observations. We used the command GaMRes to fit the model and command GaPloth to produce graphs.

12 Survival Analysis: Order-1 Gamma process Estimate of Survival Function Model estimate Confidence bound (95%) KaplanMeier KM Confidence bound (95%) times

13 Claims Reserving (INBR) Consider the run-off triangle Year of Development year origin 1 2 j n 1 n 1 X 11 X 12 X 1j X 1,n1 X 1n 2 X 21 X 22 X 2j X 2,n1.... i X i1 X i2 X i,n+1i... n 1 X n1,1 X n1,2 n X n1 X ij = Incremental claim amounts originated in year i and paid in development year j

14 Claims Reserving (INBR) de Alba and Nieto-Barajas (2008): Use a non-stationary GaP 1 (a, b, c) to introduce dependence across development years. Claims originated in different years remain independent X i1 Ga(a i1, b i1 ), η ik X ik Po(c ik X ik ), X i,k+1 η ik Ga(a i,k+1 + c ik, b i,k+1 + η ik ) {X i1,..., X i,n+1i } GaP 1 (a, b, c), where a ij = a i, b ij = b j and c ij = c j, with n i=1 (1/b j) = 1. This implies with λ j = c j /(b j + c j ) E(X ij X i,j1 ) = (1 λ j ) α i β j + λ j X i,j1,

15 Claims Reserving (INBR) Example: Taylor and Ashe s dataset Data consists of incremental claims in a n n triangle with n = 10 Transformed that data to millions to avoid numerical problems Took priors for (a, b, c): a i Ga(0.001, 0.001), (1/b 1,..., 1/b n ) Dir(1,..., 1) and c j Ga(0.01, 0.01)

16 Claims Reserving (INBR): Independence (c j = 0) alpha Origin year 1/beta Development year

17 Claims Reserving (INBR): Dependence alpha Origin year 1/beta Development year

18 Claims Reserving (INBR): Dependence lambda Development year

19 Claims Reserving (INBR): Reserves comparison Overdispersed Poisson Gamma-GLM Indep. Gamma Dep. Gamma De Jong Reserve Reserve Reserve Reserve Year Estimate 95%-q Estimate 95%-q Estimate 95%-q Estimate 95%-q Estimate Total

20 Order2 process η 1 η 2 η 3 η 4 η 5 θ 1 θ 2 θ 3 θ 4 θ 5 Throw more arrows to induce higher order dependence

21 Order2 process η 1 η 2 η 3 η 4 η 5 θ 1 θ 2 θ 3 θ 4 θ 5 Throw more arrows to induce higher order dependence There is no way to obtain a given marginal distribution: say beta or gamma

22 Order2 process η 1 η 2 η 3 η 4 η 5 θ 1 θ 2 θ 3 θ 4 θ 5 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)

23 Order2 process ω η 1 η 2 η 3 η 4 η 5 θ 1 θ 2 θ 3 θ 4 θ 5 With this common ancestor ω we can through more arrows and still ensure a given marginal

24 Space and time process This idea can be use to induce time and/or spatial dependence t = 1 t = 2 t = 3 θ 1,1 (η 1,1 ) θ 1,3 (η 1,3 ) θ 1,2 (η 1,2 ) θ 1,4 (η 1,4 ) θ 2,1 (η 2,1 ) θ 2,3 (η 2,3 ) θ 2,2 (η 2,2 ) θ 2,4 (η 2,4 ) θ 3,1 θ 3,2 (η 3,1 ) (η 6 3,2 ) θ 3,3 (η 3,3 ) θ 3,4 (η 3,4 ) ω

25 Order-q beta process Jara and al. (2013): Orderq (AR) beta process: {θ t } BeP q (a, b, c) ω Be(a, b) η t ω ind Bin(c t, ω) q q θ t η Be a + η tj, b + (c tj η tj ) θ t Be(a, b) marginally j=0 j=0

26 Order-q beta process Properties: (a + b) Corr(θ t, θ t+s ) = for s 1. ( qs ) j=0 c tj ( a + b + q j=0 c tj ( q ( + j=0 tj) c q j=0 c t+sj ) ( a + b + ), q j=0 c t+sj If c t = c for all t then {θ t } becomes strictly stationary with Corr(θ t, θ t+s ) = (a + b) max{q s + 1, 0}c + (q + 1)2 c 2 {a + b + (q + 1)c} 2. )

27 Autocorrelation in {θ t } lag lag

28 Order-q beta process Example: Unemployment rate in Chile Annual data from 1980 to 2010 Use our BeP q as likelihood for the data {Y t } Took priors for (a, b, c): a Un(0, 1000), b Un(0, 1000) and c t λ iid Po(λ) and λ Un(0, 1000)

29 Time series: Y t = Unemployement in Chile Unemployment Rate in Chile q=3 q=4 q=5 q=6 q=7 q= Year

30 Time series: Y t = Unemployement in Chile BeP BDM Year

31 Spatial process

32 Spatial process Nieto-Barajas and Bandyopadhyay (2013): Spatial gamma process: {θ t } SGaP(a, b, c) ω Ga(a, b) η ij ω ind Ga(c ij, ω) θ i η Ga a + c ij, b + η ij j i j i i is the set of neighbours of region i θ t Ga(a, b) marginally

33 Disease mapping Study: Mortality in pregnant women due to hypertensive disorder in Mexico in Areas are the States Y i = Number of deaths in region i E i = At risk: Number of births (in thousands) λ i = Maternity mortality rate Zero-inflated model f (y i ) = π i I (y i = 0) + (1 π i )Po(y i λ i E i ) λ i = θ i exp(β x i ) π i = ξ ie δ z i 1 + ξ i e δ z i β is a vector of reg. coeff. s.t. β k N(0, σ 2 0 ) θ i SGaP(a, a, c) ξ i Ga(b, b)

34 Disease mapping Six explanatory variables: X 1 number of medical units (hospitals + clinics) X 2 proportion of pregnant women with soc. sec. X 3 prop. of pregnant women who were seen by a physician in the first trimester of pregnancy X 4 public expenditure in health per capita in thousands of MX Z 1 poverty index Z 2 proportion of births in clinics and hospitals

35 Estimated mortality rate λ i [3.05,6.33) [6.33,6.67) [6.67,7.38) [7.38,8.73) [8.73,21.07]

36 Estimated zero inflated prob. π i [0,0.01) [0.01,0.04) [0.04,0.06) [0.06,0.5) [0.5,0.6]

37 References de Alba, E. & Nieto-Barajas, L. E. (2008). Claims reserving: A correlated bayesian model. Insurance: Mathematics and Economics 43, Jara, A., Nieto-Barajas, L. E. & Quintana, F. (2013). A time series model for responses on the unit interval. Bayesian Analysis 8, Mendoza, M. & Nieto-Barajas, L. E. (2006). Bayesian solvency analysis with autocorrelated observations. Applied Stochastic Models in Business and Industry 22, Nieto-Barajas, L. E. & Bandyopadhyay, D. (2013). A zero-inflated spatial gamma process model with applications to disease mapping. Journal of Agricultural, Biological and Environmental Statistics 18, Nieto-Barajas, L. E. & Walker, S. G. (2002). Markov beta and gamma processes for modelling hazard rates. Scandinavian Journal of Statistics 29,