Non-Parametric Inference of Transition Probabilities Based on - PowerPoint PPT Presentation

Non-Parametric Inference of Transition Probabilities Based on Aalen-Johansen Integral Estimators for Semi-Competing Risks Data Application to LTC Insurance Quentin Guibert Autorité de Contrôle Prudentiel et de Résolution ISFA, Université de Lyon, Université Claude-Bernard Lyon-1 Email: quentin.guibert@acpr.banque-france.fr IAA Colloquia, Oslo, 7-10 June 2015. Joint work with F. Planchet (ISFA and Prim’Act). The views expressed in this presentation are those of the authors and do not necessarily reflect those of the Autorité de Contrôle Prudentiel et de Résolution (ACPR), neither those of the Banque de France.

Introduction Non-parametric Estimation Asymptotic Results Application Summary Outline Introduction 1 Non-parametric Estimation 2 Asymptotic Results 3 Application 4 Guibert and Planchet IAA Colloquia, 7-10 June 2015 2/25

Introduction Non-parametric Estimation Asymptotic Results Application Summary Insurance context � Multi-state models are the suitable framework for modeling health and life insurance contracts (Haberman and Pitacco, 1998; Christiansen, 2012). � For a LTC insurance model, transition probabilities are generally fitted assuming the Markov assumption holds. These quantities are the main inputs for pricing or reserving models. � Need for realistic (best estimate) assumptions for the Solvency II purpose. Academics and practitioners generally use parametric models with the Markov assumption. Markov assumption is too strong. � Goodness of fit checks are complicated to implement as non-parametric estimators are not available for multi-state models when this assumption does not hold. Guibert and Planchet IAA Colloquia, 7-10 June 2015 3/25

Introduction Non-parametric Estimation Asymptotic Results Application Summary Acyclic multi-state model Consider an acyclic multi-state model which refers to a situation where both terminal and non-terminal events can occur during the lifetime of an individual. d 1 Formally, two lifetimes are identified: � S , the lifetime in healthy state e 1 . . . S = inf { t : X t � = a 0 } , . . . � T , the overall lifetime a 0 d i . T = inf { t : X t ∈ { d 1 , . . . , d m 2 }} , . . . where ( X t ) t ≥ 0 is the current state of the in- e m 1 . . dividual. d j d m 2 Guibert and Planchet IAA Colloquia, 7-10 June 2015 4/25

Introduction Non-parametric Estimation Asymptotic Results Application Summary Motivation In the French insurance framework, we have longitudinal data with independent right-censoring (administrative censoring). Right-censoring data Let C the unique right censoring variable. The following variables are available � Y = min ( S , C ) and γ = ✶ { S ≤ C } , Z = min ( T , C ) and δ = ✶ { T ≤ C } . No Markov assumption. Main goals � Non-parametric estimation of transition probabilities for a such a right censoring acyclic multi-state model � Non-parametric association measure between the failure time in healthy state and the overall failure time when non-terminal event occurs Guibert and Planchet IAA Colloquia, 7-10 June 2015 5/25

Introduction Non-parametric Estimation Asymptotic Results Application Summary Existing Estimators for Competing Risks Data � Non-parametric estimation framework for Markov multi-state model (Andersen et al. , 1993) � Let V be the indicator of the type of failure. The Aalen-Johansen (AJ) estimator for the cumulative incidence function (CIF) which is the joint distribution of ( T , V ) is F ( v ) ( t ) = P ( T ≤ t , V = v ) . Non-parametric estimator for CIF � i.i.d. observations are composed of ( Z i , δ i , δ i V i , ) 1 ≤ i ≤ n � Estimator can be expressed as a sum considering the ordered Z -values � � δ [ j : n ] � n � i − 1 δ [ i : n ] n − j W in J ( v ) � F ( v ) � [ i : n ] ✶ { Z i : n ≤ z } , � ( z ) = W in = n n − i + 1 n − j + 1 i = 1 j = 1 W in is the Kaplan-Meier (KM) weights and J ( v ) � � = ✶ { V i = v } i ( · ) converges w.p.1 to F ( v ) ( · ) and is asymptotic normal F ( v ) � � n Guibert and Planchet IAA Colloquia, 7-10 June 2015 6/25

Introduction Non-parametric Estimation Asymptotic Results Application Summary Existing Estimators for other multi-state models � No general framework for non-parametric estimation of multi-state models when Markov assumption does not hold � Particular models for: � state occupation probabilities (Datta and Satten, 2002) � transition probabilities for illness-death model (Meira-Machado et al. , 2006) � Classical approaches for semi-competing risks data use competing risks techniques and focus on estimating the survival function from the latent failure time to the non-terminal event: � non-parametric estimation with left-truncation and right-censoring (Peng and Fine, 2006) � semi-parametric model using copula-graphic estimators (e.g. Lakhal et al. , 2008) Guibert and Planchet IAA Colloquia, 7-10 June 2015 7/25

Introduction Non-parametric Estimation Asymptotic Results Application Summary Bivariate Competing Risks Data � Idea: there is recent literature on estimating bivariate competing risks models (Cheng et al. , 2007). Our acyclical model can be viewed as a particular case with a unique right censoring process. � Let ( S , V 1 ) and ( T , V ) be 2 competing risks processes where: � V 1 is indicator taking its values in the set of arrival states by direct transition from a 0 � V = ( V 1 , V 2 ) with is V 2 indicator taken its values in the set of arrival states from non-terminal events Bivariate CIF estimator � n � F ( v ) W in J ( v ) � 0 n ( y , z ) = [ i : n ] ✶ { Y [ i : n ] ≤ y , Z i : n ≤ z , } i = 1 � Simple form for the weights as ( S , V 1 ) is observed whether T is observed � � F 0 n is weakly convergent under independent censoring Guibert and Planchet IAA Colloquia, 7-10 June 2015 8/25

Introduction Non-parametric Estimation Asymptotic Results Application Summary Aalen-Johansen Integrals Estimators � � Consider an integral of the form S ( v ) ( ϕ ) = ϕ dF ( v ) with ϕ a generic function 0 � S can be considered as a covariate AJ integrals � � n � � S ( v ) F ( v ) W ( v ) � ϕ ( s , t ) � � ( ϕ ) = 0 n ( ds , dt ) = , 0 ≤ s ≤ t ≤ τ Z . in ϕ Y [ i : n ] , Z i : n n i = 1 � W ( v ) = W in J ( v ) [ i : n ] , AJ weights (Suzukawa, 2002) for competing risks data in � Possibility to take account for left-truncation L considering � � δ [ j : n ] δ [ i : n ] J ( v ) � i − 1 1 W ( v ) [ i : n ] � 1 − = , in nC n ( Z i : n ) nC n ( Z i : n ) j = 1 where C n ( x ) = n − 1 � n i = 1 ✶ L i ≤ x ≤ Z i Guibert and Planchet IAA Colloquia, 7-10 June 2015 9/25

Introduction Non-parametric Estimation Asymptotic Results Application Summary Transition Probabilities Estimators Application for estimating key probabilities in actuarial science i.e. p 0 e ( s , t , η ) = P ( s < S ≤ min ( t , t − η ) , T > t , V 1 = e ) , P ( S > s ) p ee ( s , t ) = P ( S ≤ s , T > t , V 1 = e ) P ( S ≤ s , T > s , V 1 = e ) , p ed ( s , t , η, ζ ) = P ( η < T − S ≤ ζ, s < S ≤ t , V = ( e , d )) . P ( T − S > η, s < S ≤ t , V 1 = e ) Remarking that { V 1 = e } = { V 1 = e , V 2 ∈ C e } where C e is the set of children (i.e transition states from e ) related to the state e , we can refer to our AJ integrals estimators Guibert and Planchet IAA Colloquia, 7-10 June 2015 10/25

Introduction Non-parametric Estimation Asymptotic Results Application Summary Transition Probabilities Estimators Our estimators enlarge those of Meira-Machado et al. (2006). � � � S ( e , C e ) ϕ ( 1 ) n s , t ,η , with ϕ ( 1 ) p 0 e ( s , t , η ) = � s , t ,η ( x , y ) = ✶ { s < x ≤ min ( t , t − η ) , y > t } , 1 − � H n ( s ) � � � S ( e , C e ) ϕ ( 2 ) n s , t � , with ϕ ( 2 ) p ee ( s , t ) = � s , t ( x , y ) = ✶ { x ≤ s , y > t } , � S ( e , C e ) ϕ ( 2 ) � n s , s � � S ( e , d ) ϕ ( 3 ) � n s ,ζ � , with ϕ ( 3 ) � p ed ( s , η, ζ ) = � s ,ζ ( x , y ) = ✶ { s < x ≤ t ,η< y − x ≤ ζ } , S ( e , C e ) ϕ ( 4 ) � n s ,η ϕ ( 4 ) s ,η ( x , y ) = ✶ { s < x ≤ t ,η< y − x } and � H n is the KM estimator of the distribution function of S . Guibert and Planchet IAA Colloquia, 7-10 June 2015 11/25

Introduction Non-parametric Estimation Asymptotic Results Application Summary Association measures As Scheike and Sun (2012) for multivariate competing risks model, we regard local association measures based on cross-odds ratio. ( s , t ) = odds ( T ≤ t , V 2 = d | S ≤ s , V 1 = e ) π ( e , d ) , 0 odds ( T ≤ t , V 2 = d | V 1 = e ) P ( A ) where odds ( A ) = 1 − P ( A ) . For a couple ( e , d ) , this non-parametric-estimator measures the effect of the duration spent in healthy state on the total lifetime. F ( e , d ) � ( s , t ) 0 n H ( e ) F ( e , d ) � 0 n ( s ) − � ( s , t ) π ( e , d ) 0 n � ( s , t ) = , 0 n F ( e , d ) � ( t ) n H ( e ) F ( e , d ) � 0 n ( ∞ ) − � ( t ) n H ( e ) F ( e , d ) where � 0 n is the estimator of the CIF of S for cause V 1 = e and � is that of T for n cause V = ( e , d ) . Guibert and Planchet IAA Colloquia, 7-10 June 2015 12/25