A nonparametric visual test of mixed hazard models

S a is ics & Ope a ions Resea ch T ansac ions SORT 37 (2) July-Decembe 2013, 153-174 S a is ics & Ope a ions Resea ch T ansac ions c Ins i u d’Es ad´ ıs ica de Ca alunya [email p o ec ed] ISSN: 1696-2281 eISSN: 2013-8830 www.idesca .ca /so / A nonpa ame ic isual es o mixed haza d models Jaap Sp eeuw1, Jens Pe ch Nielsen2and Sø en Fiig Ja ne 3 Abs ac We conside mixed haza d models and in oduce a new isual inspec ion echnique capable o de ec ing he c edibili y o ou model assump ions. Ou echnique is based on a ans o med da a app oach, whe e he densi y o he ans o med da a should be close o he uni o m dis ibu ion when ou model assump ions a e co ec . To es ima e he densi y on he ans o med axis we ake ad an age o a ecen ly de ined local linea densi y es ima o based on il e ed da a. We apply he me hod o na ional mo ali y da a and show ha i is capable o de ec ing signs o he e ogenei y e en in small da a se s wi h subs an ial a iabili y in obse ed dea h a es. MSC: 62F10, 62N01, 62N02, 62P05. Keywo ds: Mo ali y da a, ail y models, isual inspec ion. 1. In oduc ion The e is an inc easing use o mo ali y models o answe a numbe o pension ela ed ques ions. Mo ali y ables and hei es ima ion ha e always been o impo ance while calcula ing app op ia e p ices o isk p oduc s depending on indi iduals’ su i al. Mo e ecen ly, mo ali y models a e being used in mo e complex models assessing he alue o inancial p oduc s inco po a ing su i al in a a ie y o ways. Financial use s o mo ali y models a e he e o e no only ac ua ies nowadays, bu also in es o s looking o oppo uni ies in su i al bonds and o he packages o su i al isks. Di e en pu poses o mo ali y models lead o di e en measu es o quali y. 1Facul y o Ac ua ial Science and Insu ance, Cass Business School, Ci y Uni e si y London, 106 Bunhill Row, London, EC1Y 8TZ, UK. E-mail: j.sp eeuw@ci y.ac.uk 2Facul y o Ac ua ial Science and Insu ance, Cass Business School, Ci y Uni e si y London, 106 Bunhill Row, London, EC1Y 8TZ, UK. 3Danish Labou Ma ke Supplemen a y Pension Fund, Kongens Vænge 8, 3400 Hille ød, Denma k. Recei ed: Oc obe 2012 Accep ed: July 2013 154 A nonpa ame ic isual es o mixed haza d models In his pape we de elop a isualiza ion echnique ha seems use ul o he indi idual assessmen o he quali y o a mo ali y model. One applica ion we a e hinking o is o ecas ing o mo ali ies ha is a basic building block o he inancial p icing o su i al, bu also a use ul ool in asse liabili y managemen o pension po olios. Typically, ela i ely simple pa ame ic mo ali y models including calenda e ec s a e used as s a ing poin o mo ali y o ecas s. The calenda e ec is he explici ool o he o ecas and is o en isola ed and es ima ed h ough s anda d ime se ies me hodology. A pe ec his o ical i o he pas is he e o e no always wha he mo ali y modelle is looking o . O en i is mo e impo an o ha e an o e all good i , wi hou oo sys ema ic de ia ions gi ing eliable and meaning ul o ecas s. These la e objec i es a e no easy o gene alize o some quan i a i e model ha can be es ed. O en simple mo ali y models a e ejec ed, simply because mo ali y da a o en is na ionwide and su icien ly abundan o in o m ela i ely complex unde lying pa ame ic s uc u es. The e o e, a es ejec ing ou simple model is o en no wha we wan . We do know ha ou simple model is no accu a e, we do no wan an excessi e i , wha we wan is a good, in ui i e and eliable o ecas . When modelling mo ali y o a popula ion, he e is a a ie y o po en ially sui able li e ime da a models a ailable. Po en ial models di e in le els o complexi y and hey y o cap u e di e en ea u es o da a. Speci ic pa ame ic li e ables combined wi h ime se ies o ecas s a e omnip esen in he ac ua ial and demog aphic li e a u e. The li e a u e abou pa ame ic mo ali y p ojec ion has been de eloping apidly in he las ew yea s. Recen e iews o mains eam mo ali y o ecas ing models can be ound in Cai ns e al. (2009), Cai ns e al. (2011), Dowd e al. (2010a,b) and Habe man and Renshaw (2011). Cai ns e al. (2009) compa e eigh models on he basis o se e al desi able ex pos quali a i e p ope ies (like model pa simony, anspa ency, possibili y o gene a e sample pa hs, p esence (o absence) o coho e ec s and abili y o achie e a non i ial co ela ion s uc u e) and quan i a i e c i e ia (consis ency wi h his o ical da a and obus ness o pa ame e es ima es). Six o hese models a e subjec o subsequen in es iga ion by Dowd e al. (2010a,b) and Cai ns e al. (2011). These include he o iginal Lee-Ca e model (Lee and Ca e , 1992), he basic age-pe iod-coho model by Renshaw and Habe man (2006), an al e na i e age-pe iod-coho model by Cu ie (2006), he o iginal Cai ns-Blake-Dowd model as launched in Cai ns e al. (2006), and wo ex ensions he eo . The six models a e he subjec o o mal goodness-o - i es s in Dowd e al. (2010a) and back es ing in Dowd e al. (2010b). Cai ns e al. (2011) judges hese models on he basis o ex an e quali a i e aspec s like biological easonableness, plausibili y o o ecas le els o unce ain y in p ojec ions a se e al ages, and obus ness o o ecas s. In all hese pape s, he mo ali y da a applied was con ined o hose o indi iduals aged 60 o abo e. Habe man and Renshaw (2011), concen a ing on he key ac o s o li e expec ancy and annui y alues, i s conduc a de ailed compa ison o he se e al models a pensione ages. Apa om he models in he abo e pape s, hey also conside special cases o he Renshaw and Habe man (2006) model in hei s udy. La e on, hey ex end he age ange and in ol e he model by Pla (2009) and se e al a ian s he eo . Jaap Sp eeuw, Jens Pe ch Nielsen and Sø en Fiig Ja ne 155 The s abili y o he o ecas depends c ucially on he choice o he pa ame ic o m. Gene ally, a complex model wi h many pa ame e s is no a good choice e en hough such models migh be selec ed om classical ma hema ical s a is ical model selec ion designed o in-sample p edic ion. Models wi h many pa ame e s gene ally i da a be e han models wi h ewe pa ame e s. On he o he hand, a la ge numbe o pa ame e s a e ha de o o ecas han ewe pa ame e s. Fo ecas ing unce ain y inc eases d ama ically wi h he numbe o pa ame e s. Thus, o ob ain eliable o ecas s we wan models which desc ibe he key ea u es o da a wi h as ew pa ame e s as possible. The pu pose o his pape is o in oduce a isual diagnos ic ool which can be used o guide us when choosing a pa ame ic model. A good pa ame ic model is a simple model wi hou ob ious sys ema ic e o s. Tha model could be chosen by he well in o med s a is ician wo king wi h he pa icula mo ali y o ecas applica ion in mind. Ou isual diagnos ic ool will be jus one help ul ool in he o e all ma hema ical s a is ical oolbox. Ou me hod is inspi ed om ecen de elopmen s in ex eme alue es ima ion, whe e ans o ma ions o da a gi e isual in o ma ion on he quali y o he dis ibu ional i in he ail. This ecen me hodology has ound i s way in o insu ance p icing and also he ela ed ield o ope a ional isk. Fo a comp ehensi e o e iew o his new ans o ma ion me hodology in he la e con ex , see Bolanc´ e e al. (2012a). The ans o ma ion based me hod can o example compa e he pe o mance o se e al candida e models o a da a se a hand. Assume we we e old by an o acle wha he exac ue dis ibu ion is, hen we would ans o m ou da a using his o acle in o ma ion such ha ou ans o med da a would exac ly o igina e om a uni o m dis ibu ion. Now we do no ha e access o any o acles. Howe e , i we ake some es ima ed pa ame ically i ed su i al dis ibu ion as de ining ou ans o ma ion, hen any de ec able de iance on he ans o med scale om he uni o m dis ibu ion implies de iances o he pa ame ic dis ibu ion used in he ans o ma ion s ep om he unde lying ue dis ibu ion. Ou me hodology uses a nonpa ame ic smoo h ke nel es ima o on he ans o med scale. One di icul y we mee he e is ha ou da a is classical su i al da a ha is no independen iden ically dis ibu ed. We he e o e use a ecen local linea ke nel densi y es ima o – speci ically he one o Nielsen e al. (2009) – ha is adjus ed o he unca ion and censo ing pa e n we mee in ou da a. Compa ison be ween di e en unde lying sugges ed pa ame ic models a e ca ied ou by i s es ima ing hese pa ame ic models and hen o in es iga e h ough isual inspec ion, whe he he densi y o he ans o med da a indeed looks uni o m. I he unde lying pa ame ic model unde in es iga ion would be ue, he es ima ed densi y should be close o one o e he uni in e al. The e o e di e en unde lying pa a- me ic models can be isualized and compa ed on he ans o med scale. In p inciple, he densi ies could also be es ima ed and compa ed on he o iginal scale. Howe e , he e a e se e al isual and es ima ional ad an ages o wo king on he ans o med scale. One o hese is ha ou me hod makes maximal use o spa se and ola ile da a and is hus pa icula ly well sui ed o explo e how po en ial models desc ibe he mo ali y a ad- 156 A nonpa ame ic isual es o mixed haza d models anced ages whe e exposu e is in a iably limi ed. We es ou me hod using da a om na ions o di e en size: USA, Uni ed Kingdom, Denma k and Iceland. Al hough he main ocus o ou pape is o model human mo ali y, i is wo h- while men ioning ha ou me hodology is applicable o any p obabili y densi y model, whe he i conce ns human su i al o no . 1.1. Mixed haza d models F ail y heo y o e s a possible explana ion o he p esence o an old-age mo ali y pla eau. Acco ding o his heo y popula ions a e he e ogeneous wi h some people being mo e ail, i.e. ha ing a highe haza d a e, han o he people. Since pe sons wi h high haza d a es end o die soone han pe sons wi h low haza d a es old age g oups will be domina ed by low ail y pe sons and his e ec educes he a e o inc ease a he popula ion le el. F ail y models we e in oduced in he demog aphic li e a u e by Vaupel e al. (1979). In a mul iplica i e ail y model, an indi idual’s haza d a e consis s o wo pa s, namely a ce ain s anda d in ensi y and a ce ain nonnega i e andom a iable, he ail y, ac ing mul iplica i ely on he s anda d in ensi y. A Gompe z o Makeham speci ica ion is usually aken o he s anda d in ensi y, al hough some imes a Weibull model can be seen. F ail y is usually assumed o ollow a Gamma dis ibu ion, which is known o be ma hema ically e y ac able. A ew publica ions abou ail y modelling appea ed in he ac ua ial li e a u e. Wang and B own (1998) use he Gompe z-Gamma o Pe ks model o g adua e mo ali y imp o emen ac o s in a Socie y o Ac ua ies’ Li e Table. Bu and Habe man (2004) employ Gene alized Linea Models o g adua e mo ali y o insu ed li es. They conside h ee mix u e models, namely i) Pe ks; ii) modi ied Pe ks, and iii) Gompe z-In e se Gaussian. The au ho s conclude ha he Pe ks model i s he da a bes . An o e iew o he e ogenei y models in li e insu ance is gi en in Oli ie i (2006), while Jones (1998) de elops a mul iple s a e model o measu e he impac o ail y on he p opensi y o lapse a policy. Finally, Li e al. (2009) ex end he Lee-Ca e model by allowing o unobse ed he e ogenei y wi hin a cell, de e mined by age and ime. In his pape we illus a e ou me hodology in he one dimensional case. Mos o ecas ing models ope a e wi h a mul iplica i e ela ionship be ween age e ec and ime e ec . To isualize he i o he age e ec , one would hen ha e o di ide ou he es ima ed ime e ec and ice e sa o isualize he ime e ec only. We a e happy o say ha ou pape – di used in p elimina y e sions – al eady has inspi ed a numbe o o he wo ks in ma hema ical and compu a ional s a is ics. I has o example been ci ed in he h ee ecen pape s G´ amiz-P´ e ez e al. (2013a,b,c). Jaap Sp eeuw, Jens Pe ch Nielsen and Sø en Fiig Ja ne 157 1.2. Ou line The se -up o his pape is as ollows. In Sec ion 2 we p esen he isual inspec ion echnique in de ail. Bo h he con inuous- ime amewo k wi h ans o med coun ing p ocesses and he implemen a ion wi h disc e e da a is discussed. Sec ion 3 discusses ail y models in gene al and in oduces he class o models we will be using. Sec ion 4 p esen s he nume ical applica ion. Fo ou coun ies a ying signi ican ly in size (Uni ed S a es, Uni ed Kingdom, Denma k and Iceland), one da a se pe coun y ( emale pe iod 2006 om he Human Mo ali y Da abase) and h ee di e en ail y speci ica ions, namely Gamma, In e se Gaussian, and degene a e (no ail y), we show he es ima es as well as he isual inspec ion echnique. In pa icula , we gi e a ho ough analysis o he mo ali y a ad anced ages ha can be ex ac ed om he con inuous g aphs. Sec ion 5 se s ou a conclusion. 2. Visual inspec ion echnique 2.1. Sampling scheme o he su i al da a Conside a da a se wi h mo ali y s a is ics o nli es. Le o each o hese nindi iduals Yibe an exposu e p ocess wi h alue one when he i’ h indi idual is ali e and unde obse a ion and le Nibe a coun ing p ocess aking he alue one i he i’ h indi idual has died while unde obse a ion. Bo h Yiand Nia e unc ions o he age x. Fo mally, we assume ha Niis a one-dimensional coun ing p ocess wi h espec o an inc easing igh con inuous comple e il a ion Fx,x∈R+,i.e. one ha obeys les condi ions habi uelles, see Ande sen e al. (1993, p. 60). We model he in ensi y as λc i(x) = µθ(x)Yi(x), whe e θbelongs o he pa ame e space Θo he pa ame e s de e mining he exac mo ali y and ail y. The es ima o b θo θis de i ed om minimizing he log likelihood o Bo gan (1984): l(θ) = n ∑ i=1Zlog{µθ(x)}dNi(x)− n ∑ i=1Zµθ(x)Yi(x)dx, ha is maximized o e he pa ame e space Θ. 2.2. Visual inspec ion by ans o ma ions Assume ha some o acle has gi en us he ue unde lying c.d. . Fθ. Then conside he ans o med coun ing p ocesses Ni=Ni◦F−1 θde ined on [0,1].I ou o acle eally had 158 A nonpa ame ic isual es o mixed haza d models old us he u h, hen Niwould ha e s ochas ic in ensi y λi(y) = α(y)Yi(y), whe e Yi(y) = YiF−1 θ(y)wi h α(y) = 1/(1−y)co esponding o he haza d o he uni o m dis ibu ion wi h densi y (y) = α(y)expZy 0−α(s)ds=1, o y∈[0,1]. Ano he mo e s a is ical e m o o acle in o ma ion is p io in o ma ion. I is ha ype o in o ma ion ha is ex e nal o he da a se a hand. In ou applica ion below ou p io in o ma ion will always be some pa ame ic speci ica ion o he model and ou o acle candida e o he ue c.d. will be Fb θ,whe e b θis he es ima ed pa ame e in he speci ied pa ame ic model. I Fb θ eally is a good desc ip ion o he ue c.d. . F, hen ou da a should be uni o mly dis ibu ed a e a ans o ma ion by Fb θ. To be able o inspec he c edibili y o ou o acle in o ma ion o p io in o ma ion o pa ame ic assump ions, we es ima e he densi y based on he il e ed su i al da a N1,Y1,...,Nn,Ynon [0,1]and see whe he i looks la . This densi y es ima o should ha e good bounda y co ec ion because i is de ined on he ans o med axis [0,1].We sugges o use he na u al weigh ed local linea densi y es ima o o Nielsen e al. (2009): b (y) = n ∑ i=1ZKy,b(y−s)Yi(s)b S(s)dNi(s), whe e Ky,b(y−s) = a2(y)−a1(y)(y−s) a0(y)a2(y)−{a1(y)}2Kb(y−s), wi h Kb(y−s) = 1 bK(y−s b),(1) and aj(y) = n ∑ i=1ZKb(y−s)(y−s)jYi(s)ds, Jaap Sp eeuw, Jens Pe ch Nielsen and Sø en Fiig Ja ne 159 and b S(s) = ∏ ≤sn1−db Λ( )o, being he Kaplan-Meie es ima e o he su i al unc ion, wi h b Λ(s) = n ∑ i=1 s Z0nY(n)( )o−1 dNi( ), whe e Y(n)( ) = ∑n i=1Yi(s). 2.3. Implemen ing wi h disc e e da a In mos eal li e applica ions we only ha e disc e ized e sions o he s ochas ic p ocesses Yiand Nia ailable. Fi s we need o de ine he ele an disc e ized ime poin s H1,...,HKand he co esponding di e ences hk=Hk−Hk−1 o k∈{1,...,K}, wi h H0=0. We de ine HK=in  ;Fb θ( ) = 1 o any plausible su i al unc ion Fθ. Disc e ized da a a e o en de ined as occu ences and exposu es. Le espec i ely Ok= n ∑ i=1ZHk Hk−1 dNi(x) and Ek= n ∑ i=1ZHk Hk−1 Yi(x)dx. Now assume ha we only obse e hese disc e e occu ences – he Ok’s – and expo- su es – he Ek’s. Then a na u al app oxima ion o he log likelihood unc ion l(θ)abo e o ou disc e e obse a ions would be ld(θ) = ∑ k{logµθ(H∗ k)}Ok−∑ k µθ(H∗ k)Ek, whe e H∗ k= (Hk−1+Hk)/2 . Now conside disc e ized ime poin s on he axis ans o med by Fb θ.Le Hk= F∗ b θ(Hk).hk=Hk−Hk−1and H∗ k=Hk−1+Hk/2 o k∈{1,...,K}.No e ha HK=1. Also no e ha o en he disc e e ime poin s a e equidis an be o e he ime ans o ma ion bu no he ea e . 160 A nonpa ame ic isual es o mixed haza d models On he ans o med axis wi h ime, he se ies H∗ 1,...,H∗ Kis ans o med in o H∗ 1,...,H∗ K. We will ha e occu ences Ok=Ok and exposu es Ek=Ek∗hk/hk. Assume ha we we e gi en he ue c.d. . wi h e y la ge isk exposu es Ek. Then on he o iginal axis Ok∼µb θ(H∗ k)Ekhkwhile on he ans o med axis Ok=Ok∼αb θH∗ kEkhk. I he model we e he ue one, he haza d a es OkEkon he ans o med axis would be equal o 11−H∗ k, and hence he densi y unc ions would be cons an a 1. The local linea densi y es ima o on he ans o med axis will in he disc e e case be de ined as b d(y) = ∑ k Kd,y,b(y−H∗ k)b S d(H∗ k)Ok,(2) whe e Kd,y,b(y−s) = a2,d(y)−a1,d(y)(y−s) a0,d(y)a2,d(y)−{a1,d(y)}2Kb(y−s), aj,d(y) = K ∑ k=1 Kb(y−H∗ k)(y−H∗ k)jEk and b S d(H∗ k) = 0.5nb S d(Hk−1)+ b S d(Hk)o=0.5"exp(− k−1 ∑ i=1 hi Oi Ei)+exp(− k ∑ i=1 hi Oi Ei)#. The choice o he bandwid h bdepends on he a ailabili y o da a. La ge coun ies ha e a la ge isk exposu e; hen mos o he de ia ion be ween he densi y es ima e and 1 can be a ibu ed o model unce ain y. In such cases, no o ha dly any smoo hing is equi ed and bcan be small. Fo no so densely popula ed coun ies wi h small isk exposu e, on he o he hand, p ope smoo hing – wi h a la ge bandwid h – is needed o compensa e o pa ame e unce ain y. Jaap Sp eeuw, Jens Pe ch Nielsen and Sø en Fiig Ja ne 161 3. Mixed haza d models In an indi idual ail y model he indi idual e ec o a li e’s mo ali y ac s mul iplica- i ely. Assume ha a coho consis s o nindi iduals. Then o he i h pe son o he coho , he indi idual e ec is ep esen ed by he andom a iable Ziand he condi ional o ce o mo ali y a age x, gi en Zi=zi, is gi en by µ(x,zi) = ziµ(x),i∈{1,...,n}, wi h µ(x)deno ing he s anda d o ce o mo ali y a age x– which is he o ce o mo ali y o a li e wi h ail y le el 1 – and all Ziindependen and iden ically dis ibu ed, wi h a mean equal o 1. In his pape we will assume ha he indi idual haza d is o he o m µ(x) = exp(a0+a1x+a2x2).(3) In he no a ion o Fo a e al. (1988) his model is labelled GM(0,3). No e ha he special case a2=0 leads o he Gompe z model (GM(0,2)). The s uc u e in (3) o ms he basis o na ional and in e na ional mo ali y modelling in Ja ne and K yge (2011). We ha e dµ(x)/dx =µ(x)(a1+2a2x). I is easonable o assume ha mo ali y is inc easing as a unc ion o age. This would imply a1≥0 and a2≥0. Nonnega i e es ima es o a1and a2a e also ob ained in Ja ne and K yge (2011). The ela i e change o mo ali y as a unc ion o age x– de ined in Ho iuchi and Coale (1990) as k(x) = dlnµ(x)dx – is a linea unc ion o age: k(x) = a1+2a2x. The coho mo ali y a age xis gi en as µθ(x) = E[Z|x]·µ(x), whe e E[Z|x] deno es he mean ail y o li es su i ing o age x. Le LZdeno e he Laplace ans o m o ail y a bi h, i.e. LZ(s) = E[exp(−sZ)]. I can hen be shown, see e.g. Hougaa d (1984), ha E[Z|x] = −L′ Z[M(x)] LZ[M(x)] wi h M(x) = x Z0 µ(s)ds. Hence he coho mo ali y can be easily calcula ed o all ail y speci ica ions wi h known Laplace ans o m. In he li e a u e, he Gamma dis ibu ion has been by a he mos popula speci ica ion in he ail y model. This is pa ly due o i s ma hema - ical ac abili y. Abb ing and Van den Be g (2007) show ha , unde mild condi ions 168 A nonpa ame ic isual es o mixed haza d models 0.0 0.2 0.4 0.6 0.8 1.0 0.96 0.98 1.00 1.02 1.04 T ans o medaxis Figu e 6: Denma k: Local linea densi y es ima o as in (2), wi h b =1/6, on he ans o med scale: Gamma ail y (solid) compa ed wi h In e se Gaussian (do ed) and no ail y (dashed). 0.0 0.2 0.4 0.6 0.8 1.0 0.80 0.85 0.90 0.95 1.00 1.05 1.10 T ans o medaxis Figu e 7: Iceland: Local linea densi y es ima o as in (2), wi h b =1/2, on he ans o med scale: Gamma ail y (solid) compa ed wi h In e se Gaussian (do ed) and no ail y (dashed). Jaap Sp eeuw, Jens Pe ch Nielsen and Sø en Fiig Ja ne 169 The e o e, he e is no eason o ea ha we ha e smoo hed oo much and ha should be he eason o he small de iance. This USA s udy gi es us some con idence ha he Gamma ail y su i al model is wo king well also o smalle da a se s, whe e bigge luc ua ions a e o be expec ed. Fo he Uni ed Kingdom he Gamma ail y su i al model also i s ela i ely well, bu now wi h de iances up o i e pe cen . Su p isingly he Danish Gamma ail y su i al densi y has e y small de iances wi h he bigges being less han h ee pe cen . Iceland is ano he case, de iances up o 20% a e ound and he wo ail y models do no seem o imp o e he i compa ed o ha ing no ail y a all. O e all he conclusion om he g aphs is ha he Gamma ail y makes he bes i , he In e se Gaussian less so, bu wi h bo h ail y models being supe io o ha ing no ail y a all. I we ake a close look a he ail o he h ee i ed Danish su i al models a he ans o med scale, we can ge some u he insigh in o he ques ion posed in he in oduc ion. I is indeed e y clea ha he la ening ou o he Gamma ail y densi y in he ail helps he i . The Gamma ail y e sion is much close o one a ound he ail wi h abou hal he de iance om one compa ed o he no- ail y densi y e sion. In gene al, he pe o mance o In e se Gaussian is somewha be ween ha o Gamma and no ail y. Fo Iceland, he cu es a e almos iden ical, due o he small es ima e o σ2 and e y simila es ima es o he o he pa ame e s. In o he wo ds, o Iceland, he cases o Gamma ail y, In e se Gaussian ail y and no ail y a e nea ly he same. Fo US and UK he Gamma speci ica ion clea ly p o ides he bes desc ip ion o da a o he candida es conside ed. Bo h he In e se Gaussian and no ail y al e na i e de ia e subs an ially mo e in he igh ail han Gamma ail y. These wo speci ica ions bo h o e es ima e old age mo ali y subs an ially, while he Gamma ail y seems o cap u e he old age mo ali y pla eau e iden in da a. Mo eo e , he igh ail de ia ions o Gamma ail y is o he same magni ude as de ia ions o younge age segmen s, while he igh ail de ia ions o In e se Gaussian and no ail y seems o di e ge. While he pic u e is less clea o Denma k, he ail y densi ies also he e imp o e he desc ip ion o old age mo ali y. I also seems ha wi hou ail y he de ia ion di e ges in he igh ail, bu he magni ude o de ia ion is much smalle han o US and UK. In con as o US and UK, he Gamma and In e se Gaussian essen ially pe o m equally well. Thus we conclude ha he e is enough in o ma ion in da a o indica e he p esence o he e ogenei y, bu no enough in o ma ion o dis inguish be ween he di e en kinds o he e ogenei y. Las ly, Iceland has so li le exposu e and so much unce ain y in da a ha e en wi h he me hod de i ed in his pape we canno dis inguish be ween he models. A c i ical pa o he s udy conce ns he pe o mance o he es ima o o ad anced ages. To his end, o each coun y we calcula e he second la ges and la ges poin s o in e sec ion o he es ima o wi h he ho izon al line (i.e. he wo la ges oo s o he equa ion b d(y) = 1) and ansla e his back in o he co esponding ages. Fo compa abili y, we ha e le ou in his in es iga ion he la e spike o no ail y and he In e se Gaussian in he USA case. The esul s a e gi en in Table 2 below. I is qui e clea ha Gamma ail y densi ies in all case a e ha ing he las c ossing poin . This indica es 170 A nonpa ame ic isual es o mixed haza d models ha he Gamma ail y densi y p o ides he bes desc ip ion o old age mo ali y among he conside ed models. In he USA case, we ge almos o he age o 100 be o e ou ans o med densi y d ops below one. The lowes las c ossing o he gamma ail y is s ill qui e high, namely 93 yea s. Abo e his las c ossing poin on he ans o med scale, all he i ed pa ame ic models seem o ha e oo low densi ies. Thus, abo e he las c ossing poin ou pa ame ic models a e o e s a ing he possibili y o dying. In o he wo ds, abo e he las c ossing poin all models seem o be on he sa e side. In pa icula he Gamma ail y seems well beha ed o annui y pu poses. The densi y o e y old a e a bi oo high, bu a ely mo e han wo pe cen , and hese wo pe cen a e on he sa e side when calcula ing o example annui ies. Mos o he ex a old age mass is aken om he in e al be ween he nex las c ossing poin and he las c ossing poin , whe e he unde lying pa ame ic densi ies a e o e es ima ed in all cases. The e o e, while none o he densi ies a e making a pe ec i , he Gamma ail y densi y is e y close and wi h good p ope ies o he annui y o ecas e . I is on he sa e side o he e y old ages, wi h an o e all annui y ha seems o be close o he u h, o e es ima ing he densi y in he e y old ages, bu compensa ing o ha o e es ima ion in an in e al leading up o hose old ages. Wi hou ail y he de ia ions o old ages a e subs an ially la ge han wi h (gamma) ail y. The ans o med densi y is below 1 which indica es ha he p obabili y o dying old is o e es ima ed. A i s glance his appea s o be a odds wi h he ac ha wi hou ail y he old-age haza d is o e es ima ed, c . Figu e 1. The explana ion is ha while he old-age haza d is o e es ima ed he haza d is unde es ima ed in he age g oups below and he e o e oo many a ain he (high) age o 90, say, a e which hey die oo quickly. The model wi hou ail y is on he sa e when se ing aside ese es o annui ies o 40 yea -olds, bu i we we e o use he same model o olde age g oups i would only be conse a i e up o a ce ain poin . This clea ly is no a desi able ea u e, and i illus a es he poin ha o e s a ing he p obabili y o dying old o one coho is no necessa ily a conse a i e assump ion o o he coho s. No ice ha i would be ha d o ge his kind o de ailed in o ma ion om es ing he unde lying densi ies o e en om g aphical isualiza ion echniques on he o iginal scale. The e o e, ou simple ans o ma ion echnique has enabled us o com o he s a is ician o ecas ing mo ali y models based on simple unde lying pa ame ic su i al dis ibu ions. Table 2: Second la ges and la ges c ossing poin o densi y es ima o wi h ho izon al line a 1. Coun y Gamma In e se Gaussian No ail y Second la ges La ges Second la ges La ges Second la ges La ges c ossing c ossing c ossing c ossing c ossing c ossing poin poin poin poin poin poin US 92.37 98.68 92.14 95.21 92.14 95.21 UK 89.51 96.85 89.15 95.34 87.61 92.10 Denma k 85.53 94.77 85.47 94.67 84.91 93.63 Iceland 84.38 93.02 84.34 93.01 84.34 93.01 Jaap Sp eeuw, Jens Pe ch Nielsen and Sø en Fiig Ja ne 171 5. Conclusions We ha e de eloped a new isual inspec ion echnique o su i al models. I gene alizes de elopmen s o ans o ma ion echniques o i.i.d. da a, see o example Bolanc´ e e al. (2008, 2012a, 2012b, 2013). The me hod seems use ul in many e sions o ollow-up s udies, see o example Guill´ en e al. (2012) and Pinque e al. (2011). We imagine i o be use ul when he applied s a is ician wan s he da a o guide his in ui ion. The wo king me hodology could be h ough unning he knowledge loop cycle: Da a→Visualiza ion→New Assump ion a numbe o imes un il he inal assump ions seem in ui i ely easonable and well beha ed also acco ding o mo e s anda d s a is ical echniques. All he mo ali y p ojec ion models discussed in he In oduc ion in ol e bo h an age and a ime dimension. As men ioned in he In oduc ion one can use ou one- dimensional isualiza ion echnique o he age e ec a e ha ing adjus ed o he ime e ec and ice e sa when isualizing he ime e ec . A ull mul idimensional e sion o ou me hodology is also possible. One could use mul idimensional densi y es ima ion o il e ed da a o in oduce a simila isual inspec ion echnique o assessing he quali y o mo ali y depending on bo h age and ime. See o example Buch-K omann and Nielsen (2012) o a ecen mul i a ia e densi y es ima o ha could be used in ou isual diagnos ic s ep a e ha ing ans o med ou da a wi h ou a ou i e o ecas ing mo ali y model. T ans o ma ions and isual i ing as de eloped in his pape would also seem ele an in o he a eas o ac ua ial science as, o example, ese ing, see he ecen pape s Ma ´ ınez-Mi anda e al. (2012) and Kuang e al. (2011). Acknowledgemen This p ojec was unded by a esea ch g an om The Ac ua ial Founda ion and he Socie y o Ac ua ies. Re e ences Abb ing, J. H. and Van den Be g, G. J. (2007). The unobse ed he e ogenei y dis ibu ion in du a ion anal- ysis. Biome ika, 94 (1), 87–99. Ande sen, P. K., Bo gan, O., Gill, R. D. and Keiding, N. (1993). S a is ical Models Based on Coun ing P o- cesses. Sp inge -Ve lag, New Yo k. Bolanc´ e, C., Guillen, M. and Nielsen, J. P. (2008). In e se be a ans o ma ion in ke nel densi y es ima ion. S a is ics and P obabili y Le e s, 78, 1757–1764. Bolanc´ e, C., Guill´ en, M., Nielsen, J. P. and Gus a sson, J. (2012a). Quan i a i e Ope a ional Risk Models. Chapman and Hall/CRC Finance Se ies, New Yo k. Bolanc´ e, C., Ayuso, M. and Guill´ en, M. (2012b). 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