The Gompertz-Makeham distribution. Fredrik Norström. Supervisor: Yuri Belyaev

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1 The Gompertz-Makeham dstrbuto by Fredrk Norström Master s thess Mathematcal Statstcs, Umeå Uversty, 997 Supervsor: Yur Belyaev

2 Abstract Ths work s about the Gompertz-Makeham dstrbuto. The dstrbuto has bee appled plety of tmes. The propertes of the Gompertz-Makeham dstrbuto vestgated ths work are umodalty of the Gompertz-Makeham dstrbuto ad relatoshp betwee the meda value ad the mea resdual lfe tme of the Gompertz-Makeham dstrbuto. For most of the realstc set of parameter values the fucto s umodal but ot for all. The example used ths work gves a set of parameter values for whch the fucto s umodal. The relatoshp betwee the meda value ad the mea lfe tme, left after some gve tme, has also bee vestgated wth trucato at fxed age S. A program for smulatos of lfe tme wth the Gompertz-Makeham dstrbuto has bee wrtte Pascal. For the possblty to have estmators of the ukow parameters of the Gompertz-Makeham dstrbuto the least square estmato has bee appled by use of a program wrtte Pascal. Descrpto of the least square estmato ad also the method of Maxmum Lkelhood ad the EMalgorthm are also cluded ths work. The estmators of parameters ca also be obtaed by use of the last two methods. For testg the hypothess that extreme old ages follows the Gompertz-Makeham dstrbuto a goodess of ft test has bee appled to real demographc data. For ths example the hypothess that extreme old ages follows the Gompertz-Makeham dstrbuto, wth parameters estmated by use of the least square estmato, s rejected.

3 Cotets Itroducto. Propertes ad characterstcs of the Gompertz-Makeham dstrbuto. 5 3 Programs for smulato ad estmato of parameters Smulato of Gompertz-Makeham dstrbuted data Estmato of the parameters the Gompertz-Makeham dstrbuto wth use of the least square estmato Estmato of the parameters the Gompertz-Makeham dstrbuto wth use of the method of Maxmum Lkelhood Estmato of the parameters the Gompertz-Makeham dstrbuto wth use of the EM-algorthm. 8 4 Testg of some hypotheses. 4. The Goodess of ft test 4. Kolmogorov test 4.3 Lkelhood rato test 5 Refereces. 4 Appedx Itroducto

4 A terest of fdg a specfc fucto (dstrbuto) that well approxmates real lfe table data has exsted through may years. To fd a specfc dstrbuto fucto that approxmates lfe table data a reasoable way has always bee a major problem ad attempts wth dfferet fuctos have t solved the problem. There are several dfferet dstrbuto fuctos that have bee tested for ths purpose. The ma problem s f those fuctos descrbes the real stuato good eough. There s always a possblty to estmate parameters usg less complcated fuctos as lear fuctos ad quadratc fuctos. But these kd of fuctos are t of terest, because they wo t expla lfe table data good eough. It s smple to prove ths by usg dfferet tests. The tests wll make t possble to reject the hypothess that lfe table data are from oe of those fuctos. Whle may commo fuctos, fuctos that are ofte used other applcatos accordg to statstcal theory, are t possble to use. Aother fucto, whch has the possblty to expla lfe table data satsfactory s ecessary to fd. Of course, ths fucto must be a dstrbuto fucto (a dstrbuto has the characterstc that all of the possble outcomes of the dstrbuto has fucto values betwee ad ad the sum of all possble outcomes s ) otherwse t ca t expla lfe table data a atural way. If the fucto s t a dstrbuto fucto urealstc stuatos occurs. The fucto that explas lfe table data must have the characterstc that the error (whe the ukow parameters of the fucto are estmated accordg to lfe table data) seems to be a small radom error, or ay sese the error has the characterstc that t does t dsturb the real model more tha moderate. Several attempts have bee doe to fd fuctos that approxmates lfe table data a satsfactory way. Researches have bee doe for may dfferet types of fuctos. All of the fuctos wth dfferet type of theores of mortalty behd, as the Brody-Falla theory [Brody, 93; Falla, 958], the Smms-Joes theory [Smms, 94] formed by Smms 94 ad expouded by Joes later o, the Smms-Sacher theory [Smms, 94; Sacher, 956] expaded by Sacher from Smms orgal theory 94. All of these theores have bee aalysed the book Tme, cells, ad agg [Strehler, 96] ad the theores are show that they ca t be useful by Strehler. All of these theores have the outft that they use the Gompertz fucto, frst troduced by Bejam Gompertz 85, mathematcally expressed as R m R e at, Rm s rate of mortalty, a ad R are costats ad t s the tme parameter [Gompertz, 85]. Aother theory also usg the Gompertz fucto s the Strehler- Mldva theory [Strehler, 96; Lehoff, 959]. The theores explaed above are all related to a decrease, wth agg, the healthy state. However, the frst three of the theores above make certa predctos whch are keepg wth observatos, although they are ot completely cosstet wth certa other prmary observatos relatg tme, physologcal fucto ad mortalty. The Strehler-Mldva theory fts the Gompertza mortalty ketcs ad assumes a lear decay of physologcal fucto at a rate cosstet wth observato. Fally, t predcts the quattatvely verse relatoshp betwee Gompertz slope ad tercept whch has bee observed. The theory though has t bee used later years, because of the fact that there are fuctos that seems to gve much better approaches. I the last decades almost all of the researches that are related to approxmatos of lfe tables are made wth other fuctos tha the fuctos explaed above. Probably the most mportat fucto to apply s the Gompertz-Makeham dstrbuto. Ths dstrbuto fucto gves much better approxmatos for lfe table data tha the approaches descrbed above. The Gompertz-Makeham dstrbuto s also a fucto that uses the Gompertz fucto.

5 The major dfferece betwee the Gompertz-Makeham dstrbuto ad the fuctos explaed above s that the Gompertz-Makeham fucto uses more parameters tha the smple Gompertz fucto. The Gompertz-Makeham dstrbuto has the survval fucto: F s ( ) exp[-s- e s ], (,,), ad cosequetly the (cumulatve) hazard fucto: H (s) s+ s e. The cumulatve hazard fucto s descrbed secto. The Gompertz fucto has the dstrbuto fucto: F (s) B exp[as], (a,b), a <, whle the Gompertz-Makeham fucto has the dstrbuto fucto: F (s) -exp[-s- s e ]. The Gompertz-Makeham fucto has three ukow costats whle the Gompertz fucto has oly two costats. That s oe of the ma reasos why the Gompertz-Makeham fucto s to prefer for descrptos of real data stead of the Gompertz fucto. The Gompertz-Makeham dstrbuto has bee vestgated may ways. Stll there are thgs that have t bee doe. I ths work dfferet propertes wll be vestgated. Oe of the vestgated propertes s, f the probablty desty fucto of the Gompertz-Makeham dstrbuto s umodal,.e. f there exst oly oe local maxmum of the fucto (the local maxmum ca be at the boudary of the fucto). Aother property that wll be vestgated s the relatoshp betwee the meda ad the mea value of the Gompertz-Makeham dstrbuto. The results of these two vestgatos are preseted secto of ths work. The best way to have the possblty to vestgate some of the propertes of the Gompertz- Makeham dstrbuto may tmes s to wrte a computer program, a program that maages to obta the wated results. I ths works there has bee wrtte programs for dfferet purposes. The programs that has bee wrtte are: () a program that smulates lfe tmes assumg the parameters the model are kow, the program s descrbed secto 3., (smulated lfe tmes are useful to have for results of dfferet propertes), ad () a program that makes t possble to have the least square estmato of the Gompertz- Makeham dstrbuto related to lfe table data, a program that s descrbed more detals secto 3.. (3) a program that are testg hypotheses wth the goodess of ft test. Descrbed secto 4.. Other estmators tha the least square estmator are also possble to have for approxmatos of lfe table data. Some of these estmators gves most cases better approxmatos tha the estmator of the least square estmato. It s ecessary to have a good estmato of the ukow parameters,, ad, for the opportuty to recogse several dfferet propertes of the Gompertz-Makeham dstrbuto. Examples of good estmators are gve secto 3.3 ad 3.4. I these two sectos there s a descrpto of the method of Maxmum-Lkelhood ad also a descrpto of the EM-algorthm. The least square estmato s a method that mmses the square of the dfferece betwee the real value ad the value of the fucto. The method of Maxmum-Lkelhood s a

6 method whch searches the most lkely parameters of a fucto, the parameters that could have produced kow data. The EM-algorthm s almost smlar to the method of Maxmum- Lkelhood but most cases the EM-algorthm gves smpler expressos ad therefore t s a easer way to solve the problems. Aother property that wll be vestgated s, f the values of the parameters are cosstet for every ages. Most of the vestgatos of dfferet propertes of the Gompertz-Makeham dstrbuto uses oly estmatos of observatos that are the Gompertza perod, the Gompertza perod exteds from about the age 35 to the age 9. Stll there s a problem, whch s f the observatos that the Gompertz-Makeham dstrbuto s makg use of are estmated for ot all ages but oly for a perod, as the Gompertza perod. There s a possblty that the estmato reached ca t be used for extreme old ages. The observatos for extreme old ages mght ot follow the Gompertz-Makeham dstrbuto wth parameters, ad, whch gves good approxmatos for ages betwee 35 ad 9. I secto 4., a goodess of ft test has bee doe for testg the hypothess that extreme old ages follows the Gompertz-Makeham dstrbuto wth the estmated parameters, ad (the parameters are estmated by use of the least square estmato ad the perod for the estmato s 3-8 years). If the hypothess that lfe table data for extreme old ages follows the Gompertz-Makeham dstrbuto, wth estmated parameters, ad, does t hold, the hypothess wll be rejected. I case the hypothess explaed above wll be rejected t s ecessary to fd aother fucto whch gve rase to more realstc approxmatos of lfe table data for extreme old ages (extreme old ages ofte s deoted as ages over 9 or 95 years) tha the Gompertz-Makeham dstrbuto does. It mght be ecessary to approxmate extreme old ages wth a dfferet fucto tha the fucto that approxmates lfe table data for other ages. A alteratve s a desty fucto that has a dscotuty at a extreme old age. Ths fucto maybe gves a better approxmato for all of the ages tha the Gompertz-Makeham dstrbuto does. Attempts to get better approxmatos by use of a breakpot the mddle of the perod has bee doe, e.g. Pak ad Hrsaov [984] has used ths kd of approxmato. There are also other tests tha the goodess of ft test whch are possble to use for testg the hypothess that extreme old ages follows the Gompertz-Makeham dstrbuto wth estmated parameters. For example f the data are explaed a more exact form tha years the Kolmogorov test s possble to use. Ths test s explaed secto 4.. The Goodess of ft test ad the Kolmogorov test ca also be used for testg whether the Gompertz-Makeham dstrbuto s possble to use for approxmatos of real lfe table data at all. The thrd ad last of the test methods explaed ths work s the lkelhood rato test. Based o data x ad the lkelhood fucto p(x,),. The lkelhood rato test cossts of testg f x, ): ( x) sup{ p( } sup{ p( x, ): } s bgger tha some test value (kow values from tables). The test value depeds o the model assumpto ad the hypothess. sup{p( x, ): } ad are estmated wth the momet of Maxmum-Lkelhood, the estmators are ad. The methodology of the lkelhood rato test s possble to fd more detals secto 4.3. I that secto there s also descrbed how to test H : versus H :, gve ad. sup{p( x, ): }

7 For the model used ths work t wo t be ecessary to test whether the Gompertz-Makeham dstrbuto, wth estmated parameters, ad, gves a good approxmato of real lfe table data for ages betwee 3 ad 8 or ot. The reaso for ths s the fact that whe comparsos betwee the observatos for the lfe table data used ad the values of the Gompertz-Makeham dstrbuto for ages the perod 3 to 8 years has bee doe wth help of a plot. The plot shows that the Gompertz-Makeham dstrbuto gves very good approxmato ad o essetal argumet for a test exsts. May vestgatos ad dfferet attempts have bee doe to decde f the Gompertz- Makeham dstrbuto has all of the characterstcs that are ecessary,.e. f t s possble to use the Gompertz-Makeham dstrbuto accordg to real data. For example Pak ad Hrsaov [984] has doe a vestgato for checkg f the parameters have the same characterstcs the complete terval betwee 35 ad 75 years. Ivestgatos of the Gompertz-Makeham dstrbuto have also bee doe by several others. Most of them gves a good mpresso of the dstrbuto. Ths gves us a reaso to make the cocluso that the dstrbuto teds to gve us good approxmatos whe t s used accordg to lfe table data. A example where real lfe table data are used s llustrated secto 3.. The real demographc data are collected from Statstcal Abstract [983] ad the data are for Swedsh wome uder the perod A pece of the table Statstcal Abstract, the table used to have the llustrato metoed above, are show below. Age Survvors of bor alve Mea expectato of rested lfe, years Me Wome Me Wome Table.. Table.. are collected data that shows the umbers of bor alve that are stll alve whe they are x years of age (the age are couted whe a ew year beg) ad the umber are arraged tervals of legth oe year. The approxmato of real demographc data wth the Gompertz-Makeham dstrbuto (the parameters are estmated wth help of the least square estmato wth estmated parameters obtaed secto 3.) gves very good results. The relato betwee the Gompertz-Makeham dstrbuto ad real demographc data for Swedsh wome s show below Fgure..

8 f(s) age Fgure. the Gompertz-Makeham dstrbuto (wth parameters 5.* -3, 7.786* -6 ad.6) real demographc data. As you ca see of Fgure. the Gompertz-Makeham dstrbuto gves a very good approxmato of real demographc data. Ths explas why t s of terest to use the Gompertz-Makeham dstrbuto for dfferet approxmatos relatg to lfe legth theory. The basc reaso for makg approxmatos of real demographc data wth use of the Gompertz-Makeham dstrbuto s that there are may dfferet professos that have great use of these kd of approxmatos of lfe table data. Oe of the professos that have great use of the Gompertz-Makeham dstrbuto are surace compaes. The Gompertz- Makeham dstrbuto would gve them better possbltes to determe surace s that better explas the mortalty amog people for both accdets ad atural deaths ad t would be very helpful whe the fees of the surace s are decded. The Gompertz-Makeham dstrbuto s t oly useful for approxmatg lfe legths for huma populatos. It mght eve be of mportace to use t may dfferet bologcal ways, for example plat bology has great use of the Gompertz-Makeham dstrbuto. Also lfe legths for dfferet crops are a applcato where the Gompertz-Makeham dstrbuto ca be of mportace to use. The possblty to study lfe legths for dfferet crops mght eve gve the possblty to choose a treatmet that some sese rase the qualty of these crops. Propertes ad characterstcs of the Gompertz-Makeham dstrbuto The Gompertz-Makeham dstrbuto s a dstrbuto that gves very good approxmatos to emprcal dstrbutos of lfe legth ot oly for huma populatos but also for dfferet bologcal arts. There wll be vestgatos of some propertes of the Gompertz-Makeham dstrbuto ths secto. The Gompertz-Makeham dstrbuto has the survval fucto: F s ( ) exp[-s- s e ]. (.) Where s s lfe tme always o-egatve ( s ) ad, ad are kow o-egatve parameters ad + > ad hece F( s) > for s > ad F( ). (.)

9 How dfferet values of ths three parameters fluece to the model wll be researched. I geeral the hazard fucto of the Gompertz-Makeham dstrbuto ofte ca be used to express the behavour of the dstrbuto. The (cumulatve) hazard fucto of a dstrbuto ca be defed by the relato H (s): l( F ( s ) ), (,,), (.3) ad the testy hazard fucto s the dervatve of the (cumulatve) hazard fucto h (s): dh( s) ds. (.4) The Gompertz-Makeham dstrbuto has the cumulatve hazard fucto H (s) s+ e s ad the testy hazard fucto s (.5) h (s) +e s. (.6) Formulas (.5) ad (.6) ca easly be derved from t s survval fucto (.) ad the deftos of the hazard fuctos (.3) ad (.4). Note that H (s) > f s > due to (.). Aother property that ca be derved s the probablty desty fucto of the Gompertz- Makeham fucto f (s) df( s) ds d( - e ds -H(s) ) h (s) e -H(s) (+e s )exp(-(s+ e s )). (.7) It wll be of terest f there exsts ay pots of extremum ot at the boudary of the probablty desty fucto of the Gompertz-Makeham dstrbuto. Ths kowledge s ecessary for the possblty to coclude f the probablty desty fucto of the Gompertz- Makeham dstrbuto s umodal or ot. The probablty desty fucto s umodal f the dervatve s for at most oe value of the dervatve. The dervatve of the probablty desty fucto (.7) of the Gompertz-Makeham dstrbuto s df( s) ( h' ( s) - h (s)) e -H(s) -H(s) d( h (s)e ) ds ds ((e s ) - ( +e s ) ) e -H(s), (.8) Because as stated expresso (.6) the testy hazard fucto of the Gompertz-Makeham dstrbuto s h (s) + e s ad therefore the dervatve of t s h' (s) e s. Rght had sde of equato (.8) s oly whe the statemets (e s ) ( + e s ) s true, because < H(s) < ad e -H(s) >. There exsts oe or two pots of extremum oly whe (e s ) ( + e s ) ad the values of the lfe legth that gves ths equalty are postve. Set z e s, the (.8) ca be rewrtte as z ( +z) + z + z or z (-)z -. The value z ca be solved by a secod grade equato so the roots are z (- - 4 / )-, ad z s l ( ( (+ - 4 / )-, ad tme scale (- - 4 / )-)), (.8)

10 s l ( ( (+ - 4 / )-)). (.9) As earler defed, ad. We have s s, otherwse the soluto s a atural logarthm of a egatve value, whch s a complex value ad for that case s ad s wll have complex tme. Ths fact makes sure that the pots of extremum occur at o-egatve tme f they exst. If s s postve (s exsts) there are two possbltes. Ether there exst both a local mmum ad a local maxmum of the probablty desty fucto or there exsts a flecto pot ad there does t exst ay other pot of extremum separated from the boudary of the fucto, occurs f s s. If s s t postve ad s s postve the there exst oly a local maxmum of the desty probablty fucto. If s s t postve there are o pots of extremum for the probablty desty fucto of the Gompertz-Makeham dstrbuto. The fucto s umodal f there are o local maxmum ot at the boudary of the fucto. Corollary (.) There exsts a flecto pot of the probablty desty fucto of the Gompertz-Makeham dstrbuto f ad oly f 4 ad >. Proof of corollary (.) There exsts a flecto pot for the probablty desty fucto f ad oly f s s ad the value of s s postve. s s f expressos (.9) ad (.) are equal expressos. The expressos are equal f -4/ 4. Set 4 ad t follows that s s, s l( ( (- - 4 / )-))) s o-egatve mples that l( l( ( 4 -)) l( ( (+ - 4 / )-))) s (-)) > or ( -)/ > or >. There exsts a flecto pot f ad oly f 4 ad >. Whch was to be proved. Corollary (.) There exst a local mmum ot at the boudary of the probablty desty fucto of the Gompertz-Makeham dstrbuto f 4 <, (+) < ad < (+) are true statemets. Proof of corollary (.) Local mmum ot at the boudary of the probablty desty fucto of the Gompertz- Makeham dstrbuto exsts f s s postve or l( ( (- -(+)/ > (- - 4 / )-)) > ( - 4 / (- - 4 / )-) > - 4 / > (+)/ ) > (+) -, o complex solutos allowed - 4 / - 4 / > 4 < (.3) (f 4 the there s a flecto pot, stead of a local mmum ot at the boudary) -(+)/ > or (+) < (.4) ( - 4 / ) < ( - (+)/) f codtos (.3) ad (.4) are true the -4/ < +4(+) / -4(+)/ < (+) (.5)

11 Due to codtos (.3), (.4) ad (.5) there exsts a local mmum of the probablty desty fucto of the Gompertz-Makeham dstrbuto f 4 <, (+) < ad < (+) are true statemets. Whch has to be proved. If there s a local mmum ot at the boudary of the probablty desty fucto of the Gompertz-Makeham dstrbuto the >, because of the fact that (by use of codto (.4) ad (.5)) 4 < < (+) / (+) (+) /(+) (+). Imples that ever bgger tha (+) f local mmum exsts or that must be bgger or equal to f there exsts a local mmum ot at the boudary of the probablty desty fucto of the Gompertz-Makeham dstrbuto. If the /(+) (+) / 4. If the equalty 4 holds, the there s a flecto pot at the pot of extremum ot at the boudary ad ot a local mmum, due to corollary (.). Therefore ca t be equal to f there exsts a local mmum ot at the boudary, hece must be bgger tha. Corollary (.6) There exsts a local maxmum ot at the boudary of the probablty desty fucto of the Gompertz-Makeham dstrbuto f 4 < ad (+) < are true statemets or f 4 <, (+) ad > (+) are true statemets. Proof of corollary (.6) Local maxmum exsts ad s ot at the boudary f s s postve l ( (+ + ( (+ - 4 / - 4 / - 4 / ) > (+) > (+)/ )-)) > ( (+ - 4 / )-) > - 4 / > (+)/ - ad we have (.3). The followg two codtos are possble, codto (.4) ad (+). (.7) If codtos (.3) ad (.4) are true the there exsts a local maxmum ot at the boudary. The case whe the codto (.7) s true leads to followg ( ) > ( (+)/-) -4/ > +4(+) / - 4(+)/ > (+). (.8) There exsts a local maxmum ot at the boudary of the probablty desty fucto of the Gompertz-Makeham dstrbuto f codtos (.3), (.7) ad (.8) are true statemets. Due to codtos (.3), (.4), (.7) ad (.8) there exsts a local maxmum ot at the boudary f 4 < ad (+) < are true statemets or 4 <, (+) ad > (+) are true statemets. Whch was to be proved. - 4 / Defto A fucto s umodal f there exsts oly oe local maxmum of the fucto. Theorem (.9) (.)

12 The probablty desty fucto of the Gompertz-Makeham dstrbuto fucto s umodal f ad oly f 4 s a true statemet, or f (+) ad (+) are true statemets or f 4 < ad > (+) are true statemets. Proof of theorem (.) By defto (.9) the probablty desty fucto of the Gompertz-Makeham dstrbuto s umodal f the oly local maxmum that exsts of the fucto s at the boudary or f the oly local maxmum of t s ot at the boudary of the boudary. The oly local maxmum s at the boudary (s) of the fucto whe corollary (.6) s t true. If corollary (.6) s t true, the followg statemets are correct, 4 or both (+) ad (+) are true statemets. The oly local maxmum s t at the boudary f there exsts a local maxmum ot at the boudary ad there does t exst a local mmum separated from the boudary of the fucto. Ths s the case whe corollary (.6) s true ad corollary (.) s t true. Corollary (.6) s true f 4 < ad (+) < are true statemets or f 4 <, (+) ad > (+) are true statemets ad corollary (.) s t true whe 4 ad/or (+) ad/or (+) are true statemets. Corollary (.6) s true ad corollary (.) s t true oly f 4 <, (+) ad > (+) are true statemets or f 4 <, (+) < ad > (+) are true statemets,.e. corollary (.6) s true ad corollary (.) s t true whe 4 < ad > (+) are true statemets. The probablty desty fucto of the Gompertz-Makeham dstrbuto s umodal f 4 s a true statemet or (+) ad (+) are true statemets or f 4 < ad > (+) are true statemets. Whch has bee proved. There are four dfferet possbltes of the shape of the probablty desty fucto for the Gompertz-Makeham dstrbuto. The dfferet shapes that are possble are llustrated below Fgure F g u r e f ( s ) age Fgure. shows the fucto for the values of, ad that are estmated secto 4, amely 5.44* -3, 7.* -6 ad.3 ad s a example of the probablty desty fucto of the Gompertz-Makeham dstrbuto havg a local maxmum ot at the boudary of the fucto.

13 . 6 F g u r e F g u r e F g u r e f ( s ) age Fgure. shows a example of the probablty desty fucto for the Gompertz-Makeham dstrbuto havg a local maxmum both at the boudary ad ot at the boudary of the fucto where the parameter values are.,.5 ad.85. Fgure.3 shows a example of the probablty desty fucto of the Gompertz-Makeham dstrbuto where the fucto has a global maxmum at the boudary of the fucto. The values of the parameters are.,.4 ad.9. Fgure.4 shows a example of the probablty desty fucto of the Gompertz-Makeham dstrbuto where the fucto has a global maxmum at the boudary of the Gompertz-Makeham dstrbuto ad a flecto pot ot at the boudary of the fucto. The values of the three parameters are.,.5 ad.8. Aother property that s of terest to research are the relatoshps betwee the meda value ad the mea value of the Gompertz-Makeham dstrbuto. Is the meda smaller tha the mea value for kow, ad for the fucto or ot? Notable, the meda s the value where F (s) ½. F (s) -e -H(s) ½ e -H(s) ½ H (s) l() s + s + s e s e l() - l() (.) s ca t be solved explctly as a expresso of, ad. Therefore the oly possblty to have a soluto of the meda s to have a soluto wth help of a umercal method. A umercal method that ofte s used for ths kd of approxmatos s the Newto-Raphso method. Wth help of the Newto-Raphso the meda value wll be foud wth good approxmato. The meda value wll be foud as the root of fucto (.) (the root s where the equalty of (.) holds). The Newto-Raphso s a method that terates utl a approxmato has bee foud wth as may dgts as desrable of the meda value. The ma problem wth the Newto-Raphso method s that a start value must be chose for havg the possblty to start the teratos that ths method s buld upo. The start value that s used ca t be chose too far away from the root of the fucto. Otherwse the fucto wo t coverge to a value ad t wo t gve ay good soluto of the exstg problem. To fd a useful start value t s ofte ecessary to use a dfferet method. Ths method must have the possblty to gve a value that s close eough to the searched root, so that the Newto- Raphso method ca be used. The advatage of the method s that the method coverges very fast to the searched root. The Newto-Raphso method s faster tha most of the other methods. That s the justfcato of the method.

14 Defto (.) The Newto-Raphso method has the costructo that t uses a start value of x, x, whch wll be chose ot too far away from the searched root x. The the Newto-Raphso method s a method that terates utl the value of x s close eough to the value of x -, so that the dfferece betwee x ad x - s as small as wated,.e. the level of the tolerace s desrable. The Newto-Raphso method has the followg desg : x + x - f (x ) f '(x ), where f(x) s a fucto that s at the root of x. More formato about the theory of the Newto-Raphso method ca be foud dfferet books, e.g. Elde ad Wttmeyer-Koch [987], Flaery et al. [986] ad Adams [99]. Fucto (.) gves the meda value of the Gompertz-Makeham dstrbuto, f(s) s+(exp(s)-)/ - l(). The dervatve of fucto (.) s f (s) +exp(s). The meda value of the Gompertz-Makeham dstrbuto has bee approxmated usg the values of, ad estmated wth use of the least square estmato secto 3. (the ukow parameters, ad are estmated from lfe tables for Swedsh wome uder the perod [Statstcal Abstract, 983]). The meda value s 8.76 years (t s possble to have a approxmato wth as may dgts as wated). The mea value of the Gompertz-Makeham dstrbuto s m F, (s) E[S] sdf ( s) F ( s)ds -H(s) e ds e -( s+ (exp( s)-)/ ) ds (.3) Ths tegral s t a kow tegral ths maer, so the tegral must be smplfed to a kow tegral, otherwse the tegral must be solved a umercal way. Ths tegral s possble to trasform to a tegral that has a kow umercal soluto. The result after several trasformatos shows that t s possble to rewrte the tegral to a well-kow tegral as show below. Set u s, du ds e -( s+ (exp( s)-)/ ) e u/ ds e / du e ds. -(u+ exp( u/ )/ ) v du e / Set v,dv v du du dv e -u ( ) v e / -(u+ exp( u/ )/ ) e du e / -v ( ) e v v dv e/ ( ) 7 -v Itegrate e v 7 -v e v dv dv. 7 e -(u+ exp( u/ )/ ) so by parts tmes so that > / ad < + / the we have the recurret relato as below (f < the formula (.4) s the soluto): du

15 -v e v / dv (e -/ ( ) + m F, (s) e/ ( ) ( + - If < the ( e -/( ) - ( j - -v e v / ) j- / (e -/ ( ) ( ( ) j- / ( ) j )- ( ) e -/ dv ( ) )... (.4) - - ( ( ) j- / j ( ) - / e / - / ( ) e -/( ) -v e v / )- ( ) - dv ). ( ) - / -v e v / ( ) dv) (-/,/)) (.5) m F, (s) (- e/ (/) / (-/,/)) (.6) Where (-/,/) deotes a complete gamma-fucto. The complete gamma-fucto has the advatage that t s a kow fucto that has bee evaluated umercal. Wth help of tables ad dfferet packages for computers (for example Mathematca ) t s possble to have ay well-approxmated value of the fucto for ay combatos of, ad. By help of Mathematca the mea lfe tme for Swedsh wome (usg lfe table data for the perod ) [Statstcal abstract, 983], based o (.6), s calculated wth use of the values of, ad estmated secto 3. (The values of the parameters are 5.44* -3, 7.* -6 ad.3). The mea lfe tme ths case ca be derved from fucto (.6), whle <. It s very lkely that formula (.6) ca be used all cases whe mea lfe tme for huma populatos are calculated. The mea lfe tme for Swedsh wome the gve perod s 78.7 years. The table value for the mea lfe tme of Swedsh wome calculated Statstcal Abstract [983] s 78.5, ths value s close to the mea value for the Gompertz- Makeham dstrbuto wth estmated parameters. Ths fact dcates that the mea value for the Gompertz-Makeham dstrbuto s useful. The relatoshp betwee the meda ad the mea value s researched wth the quotato of the meda ad the mea value for dfferet values of, ad. The quotato s me F ad m F, from (.) ad (.5) we ca calculate the quotato for gve parameters. By use of the estmated values for, ad the quotet, estmated for the case of Swedsh wome descrbed above, the quotato s calculated ad the quotato s 8.76/ That meas that f the assumpto that the lfe legth follows the Gompertz-Makeham dstrbuto s correct more tha half of the Swedsh wome lves at least a lttle bt loger tha the mea lfe legth for Swedsh wome. The trucated dstrbutos for the meda value ad eve the mea value, wth the trucato at the pot t T ca be of some terest to estmate. The estmatos ca be useful for dfferet braches, for example for the surace compaes t ca be of great use whe the lfe surace fee s set for dfferet people. The trucated dstrbuto of the meda value s : me F (s t T) F s + t ( ) e H(s+t) -H(t) e -(s+t)-(exp((s+t)-)/) /e -t-(exp(t)-)/) F ( t) e

16 e -(s+exp(t)(exp(s)-)/) ½ s+e t (e s -)/ - l(). (.7) The trucated dstrbuto of the meda value ca the, as the case wthout havg a trucato at t T (just set T ad t wll be exactly the case as wthout trucato, because f T the equato (.7) wll be equal to equato (.),.e. s+e (e s -)/ s+ (e s -)/ s+ (e s -)/ ), be used ad t wll gve a approxmato of the meda value wth very good exactess. The approxmatos of the meda value wll be doe wth use of the Newto-Raphso defto (.) as descrbed earler ths secto. The trucated mea value s: m F, (s t T) E[S t T ] e ((st)+ (exp( (s+t))-)/ ) e -( t + (exp( t)-)/ ds sdf ( s) e F ( s t) ds F ( t) -( s+ exp( t)(exp( s)-)/ ds e e H(st) H(t) ds (.8) Set e t, ad e t are costats so that wth use of equato (.4) ad chagg wth equato (.8) wll have the same outft as equato (.4), but wth the dfferece that wll be chaged wth equato (.8). If the trucato s at T (o trucato at all), the mea lfe tme for the rest of the lfe tme s the same as for ew-bor babes, amely the equalty e. The dfferece betwee the mea lfe tme left from a trucato at the age t ad the mea lfe tme from day of brth s the costat. The costat s e t tmes bgger tha the costat for mea lfe tme, the costat, used for the calculato of mea lfe tme from day of brth. The trucated dstrbuto for the meda ad the codtoal dstrbuto for the mea lfe tme are for the ages 5, 55, 6,..., 85, 9 as show table (.) below From the table below the patter seems to be that wth growg age the mea value wll beg to decrease less tha the meda value of lfe tme left at the age t. Somewhere betwee 65 ad 7 mea lfe tme starts to be bgger tha the meda age left. t me F m F, quotet table value of mea lfe tme left Table (.), the meda of the lfe tme left ad the mea lfe tme left, at age t. From the table for Swedsh wome earler metoed ths secto[statstcal Abstract, 983].

17 3 Programs for smulato ad estmato of parameters I ths secto of the work there has bee wrtte a program to produce smulated values of lfe tme of the Gompertz-Makeham dstrbuto. Ths program s explaed secto 3.. A program for the least square estmato has also bee wrtte, the program estmates the ukow parameters the Gompertz-Makeham dstrbuto. Explaato of the program ca be foud secto 3.. Descrptos of dfferet methods of estmato ca be foud secto 3., 3.3 ad 3.4. The least square estmato s explaed secto 3.. Two other methods of estmato are also descrbed, amely the method of Maxmum-Lkelhood secto 3.3 ad the EM(expectato-maxmsato)-algorthm secto Smulato of the Gompertz-Makeham dstrbuto I order to fd a way of determg dfferet propertes of the Gompertz-Makeham dstrbuto there ofte arses stuatos where smulato of lfe tmes are ecessary to do. That s because of the fact that t ca be very dffcult to solve some problems wthout usg smulato ad some cases there are t eve ay kow methods to solve the problems explctly wthout help of smulato. There s also the possblty that the solutos some cases ca be qute complcated to fd ad the solutos are t that much better tha the results obtaed by smulato. Therefore t s reasoable to smulate the soluto stead of solvg the problem explctly. Whe smulato s used t s most ofte very mportat that the umber of smulatos for lfe tmes wll be large, otherwse t s possble that too much formato wll be lost. The lfe tme wll be smulated by splttg the hazard fucto two parts. The two parts of the hazard fucto are s ad (e s -)/. Takg, ad as kow parameters, of course whe we have real data, ad are t kow parameters, to make t possble to smulate lfe tmes. The ukow parameters must some way be estmated. By splttg the hazard fucto two parts there wll be two depedet lfe tmes smulated. The mmum of ths two smulated lfe tmes (whe the hazard fucto s splt two parts) s used as the real lfe tme. The lfe tmes obtaed are the useful for vestgatos of dfferet propertes. Ths smulato study gves possbltes to compare dfferet methods of estmato. For example the expectato ad varace of dfferet estmators such as the estmators for the least square estmate, the method of Maxmum-Lkelhood (ML) ad the estmator of the method of momets ca be compared ad coclusos about the best estmator of those are possble to do. Whle the survval fucto are as stated (.) exp[-s- ]. The two parts to be separated are w : exp[-s ] ad (3..) w : exp[-exp((s )-)/]. (3..) H (s) -s ad H (s) - From expressos (3..) ad (3..), s ca be solved explctly so that s ad

18 s l(- l(w )). A ormal explaato of the two dfferet parts of the hazard fucto s that the frst part, wth the lfe legth s, descrbes the rsk for death a accdetal way ad the other part, wth the lfe legth s, descrbes the rsk for death caused by dseases ad other decremets the healthy state due to the ageg process. Ths explaato gves us a reaso why the hazard fucto s splt two parts. Therefore a mprovemet to categores the faces of the dfferet parts the hazard fucto s possble to do ad t s also possble to gve a logcal explaato how the deaths are dstrbuted real lfe (the percetage of deaths caused by accdets ad deaths caused by decremets the healthy state). Aother advatage of splttg the hazard fucto two parts s that ths smulato techque s much faster tha the techque wthout splttg the hazard fucto. The faster smulato techque s therefore atural to use whle also a logcal explaato of the techque exst. Theorem (3..3) If S m(s, S ), where S - ad S l(- l(w )),.the H(u) H (u) + H (u). S ad S are depedet stochastc varables ad W ad W are stochastc varables uformly dstrbuted the terval [,]. Note: W ad W ca be thought of as umbers beg produced from a radom umber geerator. Proof of theorem (3..3) P(S >u) P(m (S, S ) > u) P((S > u) (S > u)) (because the fact that f m(s, S ) > u the both S ad S must be bgger tha u) P(S >u) P(S >u) F u ( ) F u ( ) e l( F u H(u) H (u) + H (u) whch was to be proved. ( ) ) +l( F u ( ) ) - H (u)- H (u) e A program has bee wrtte Pascal for smulato of lfe tmes. Ths program s possble to use for dfferet applcatos accordg to the Gompertz-Makeham dstrbuto. The program uses the equalty S m (U r,v r ). The program has the followg costructo : () Fx values of, ad are take, all of them followg statemet (.),.e., ad are kow o-egatve parameters ad + > ad hece F s ( ) > for s >. For havg ay set of the data t s ecessary to have values of, ad that are chose a realstc way. For example f the value of s chose close to. or bgger, the values of the Gompertz-Makeham dstrbuto for that values wll, f eve the value of s chose usatsfactory, ted to be very large ad depedg o software the computer wo t have the ablty to do all of the calculatos that are ecessary for a soluto. () Set U r -l(w r )/ ad V r l(-/l(w r+ ))/. Set S r m(u r, V r ), r,...,. Where U r, V r ad S r deotes the r:th value of the smulated lfe tme ad W r s a stochastc uformly dstrbuted varable the terval [,]. S r wll be the smulated value of

19 lfe tme. The smulated lfe tmes are ow avalable to use for dfferet applcatos. More formato about the program for smulatos of lfe tme s possble to fd appedx. 3. Estmato of the parameters the Gompertz-Makeham dstrbuto wth use of the least square estmato The least square estmato s used ths secto to make t possble to estmate the ukow parameters the Gompertz-Makeham dstrbuto correspodg to real lfe table data. The least square estmato s a method of estmato that mmses the sum of all squares betwee kow observatos ad a gve fucto wth ukow parameters. The combato of the ukow parameters that has the least sum of squares wll the be used as estmated parameters. I ths work a program has bee wrtte the program laguage Pascal. The program makes t possble to obta the least square estmato for the ukow parameters (, ad ) of the Gompertz-Makeham dstrbuto. The structure of the program s descrbed the appedx of ths work wth for example the algorthm of the program. The appedx also cota a example where use of real demographc data has bee doe. The example makes use of lfe table data for Swedsh wome [Statstcal Abstract, 98] uder the perod The program that has bee wrtte for estmatg the least square estmator has got the followg overall algorthm (the fucto that the least square estmato s usg s the survval fucto of the Gompertz-Makeham dstrbuto wth ukow parameters, ad ): () Fd values of the three ukow parameters that approxmately correspods to the demographc data that are avalable, just by testg dfferet values of the ukow parameters. () Use the values of the ukow parameters that approxmately correspods to real data ad use them, the values are obtaed (). The values of the parameters wll the be tested for dfferet possbltes of the frst dgt of ecessty of the value of the three parameters. The combato of the frst umber for the three parameters wll the be used ext step. Next step wll be to test ext dgt of the parameters ad the combato of the parameters that gves the best estmato wll be used ext step ad ths wll cotue utl eough dgts of the parameters have bee calculated. (3) Repeat (), but chage the start value of the parameters to the value of the parameters that was obtaed () as the best estmator. Stop ths procedure whe the least square estmator wo t gve more tha small eough chages betwee the two steps. Note, the program there wll be a comparg procedure that chages the least square estmator ad the parameters of t f the value of the sum of squares are less tha the least value that has bee derved earler. More about the program s possble to fd the appedx. If the steps the algorthm s followed t s possble to have the estmated values of, ad wth as may correct dgts as wated. The parameters for the example metoed above are estmated wth help of the program ad the values of the estmated parameters are 5.* -3, 7.786* -6 ad.6. Better values of the parameters s possble to have but the values are good eough for havg a acceptable estmato. Fgure. shows the Gompertz-Makeham dstrbuto ad real lfe table data wth estmated parameters ad t s easy to uderstad that the parameters are useful. The plots of the two curves are subjoed the same plot by use of MatLab. The pots for real demographc are ths computer program coected to each other, though the fact that the real demographc data behaves lke a dscrete fucto. The ages for the demographc data are 3, 3,..., 79, 8 years. 3.3 Estmato of the parameters the Gompertz-Makeham dstrbuto wth use of the method of Maxmum-Lkelhood

20 The least square estmato s oe of the methods of estmato that ca be used to estmate the parameters the Gompertz-Makeham dstrbuto. Aother frequetly used method for estmato s the method of Maxmum- Lkelhood. The method of Maxmum-Lkelhood was frst proposed by Gauss 8 ad s a method that makes sese mostly parametrc models. Suppose f(x,) s the desty fucto of x f s true ad that s a subset of dmesoal space. Cosder f(x,) as a fucto of for fxed x. Ths fucto s ormally called the Lkelhood fucto wth the otato L(,x), where x s thought of as a set of observatos. The method of Maxmum- Lkelhood cossts of fdg the values (x) ( (,,..., k )) whch are "most lkely" to have produced kow data L( (x), x). It s also possble to have the Maxmum-Lkelhood estmator wthout assumg that the complete data are kow. If Xx, the (x) whch satsfes L( (x), x) f(x, (x)) max { f(x,) : } s ecessary to fd. If such a exsts, ths value estmates ay cotuous fucto q() by q( (x)). The estmate q( (x)) s called the Maxmum-Lkelhood estmato of q(). The method of Maxmum-Lkelhood s closer descrbed several books for example Bckel ad Doksum [977]. The estmato obtaed for the Gompertz-Makeham dstrbuto wth use of the method of Maxmum-Lkelhood s at the pot where the dervatve, wth respect to all parameters the model s,.e. the pot where f(x, ) for,,3 ad, ad 3. The survval fucto of the Gompertz-Makeham dstrbuto s, as stated (.), F s ( ) exp(-(s+ f ' e s e s )). The fucto ca be separated two parts, F s )), whose have the probablty desty fuctos (s) exp(-s) ad f '' (s) exp(s)exp(- e s ' ( ) exp(-s) ad F s '' ( ) exp(- )). The probablty space of the fucto are x o (S,, ' S,...,S, ), wth S,,...,, kow observatos ad I(S S ), most ofte there s o kowledge of. The probablty of the :th outcome s '' p (s, ) f ' (s )F s so the Lkelhood fucto of the Gompertz-Makeham dstrbuto s L(,x) (s )F s (f ' ( ) +f '' '' (s )F s ( ) +f '' ' ( )(- ) ' (s )F s ( )(- ) exp(-s )(( exp( s )) exp(-s )(( exp( From the expresso above the loglkelhood fucto ca easly be derved e s l(,s, ) l L(,s ) -s +l(+ ) - e s. s )) ). The Maxmum-Lkelhood ca be solved wth 3 o-lear systems, where,, 3 ad, ad 3. l(, x) for 3.4 Estmato of the parameters the Gompertz-Makeham dstrbuto wth use of the EM algorthm Aother method that s possble to use for the estmato of parameters of the Gompertz-Makeham dstrbuto s the EM(expectato-maxmsato)-algorthm. The EM-algorthm s almost lke the method of Maxmum- Lkelhood, but most cases there wll be a smpler way to solve the problems whe the EM-algorthm s used stead of the method of Maxmum-Lkelhood.

21 Cosder two statstc models, l (,x) ad l (,y), where x s t observed ad y s observed ad y s a fucto of x. Wth the use of the method of Maxmum-Lkelhood the systems of equatos eed to be solved. Cosder the fucto G(', ) E ( l (,X) Yy). Ths fucto s of course for fxed y ad y ca be observed. The loglkelhood fucto ca be wrtte the followg form : l ( X - (l ( X Form the expectato E ( l (,Y) Yy) l ( Y the l ( ', Y ) ', ) ', ) E ( l (,Y) Yy) l ( Y H(', ) E E ', ) G( L( ', X) (l( ) Y y) L ( ', Y) L( ', X Y) (l( ) Y y) L (, X Y) +l ( ', Y )) ', ) ', )- H( l ( ', X ) ', ) wth E L X Y) Y y + E ( l(l (,X Y) Yy). It s easy to proof that ths fucto has maxmum at ', so l ( Y ', ) - l (,Y) G( For every M, G( ', ) - G(,) - (H( ', )has the maxmum at ', ( ) > G(,). G( M,) maxg( k) The t s possble to state that l ( M,Y)-l (,Y) G( M,)-G(,) >. l (, y) -l( L ( ', X) / L ( ', X)). (l( ( ', ) ', ) - H(,)) G( ', ) - G(,). From the above expresso t s possble to state the two-step-approach s algorthm : (E): I a eghbourhood of (k) the fucto G( k) ', ( ) s determed.,,,...,m, (M) : The eghbourhood (k+) s foud where G((k+),(k)) max G( k) If the maxmum ca t be determed ths way the approach (k+) are the estmato of the parameters obtaed by the EM-algorthm. ', ( )> G((k),(k)). Estmators of the parameters for the Gompertz-Makeham dstrbuto ca be determed wth help of the EMalgorthm. G(', ) E x S s,,..., ( l ( ', ) ) ( e s ( s e s e ) l ' +' s ' s se s e - exp( ( ' s e - )) ' A l ' -B '+C l ' + D '- E (','), where A e s C s e e s, D s e s e s ad E (',') )l ' - ' s, B exp( ( ' +, whch ca be rewrtte as s s, ' s e - )) ' Now G(', ) ca be wrtte the form G(', ) G (';,,) + G (',';,,) wth G (';,,) A l ' - B ' ad G (',';,,) C l ' + D'- E (','). To fd the estmatos of the parameters wth help of the EM-algorthm the fuctos G ad G wll be maxmsed. The maxmum s at the pots where G ad G are maxmsed. The fuctos are maxmsed where G( ', ) A - B, ' s G( ', ) C e - ' - ' ',.

22 G( ', ) D -' ( se ' 's ' s e - ' ). Set (k), (k) ad (k) ad wth use of the maxmsato-step the algorthm the values of ', ' ad ' are set to ' (k+), ' (k+) ad ' (k+). The soluto wll the have the followg EM-algorthm (k+) (k+) ( k ) B(k) A(k), C( k ) ( (e k ) s ) (k +)s se ( k ) s D(k) C(k) (e, ). For the frst two, (k+) ad (k+), a soluto ca be decded explctly from the formulas whle (k+) has to be solved umercally. Now t s possble to estmate the parameters of the Gompertz-Makeham dstrbuto wth respect to the EM-algorthm. More about the EM-algorthm s possble to fd Cox ad Oakes [984] ad Belyaev ad Kahle [996]. 4. Testg of some hypotheses Of terest t s f the Gompertz-Makeham dstrbuto wth estmated parameters (parameters that are maly estmated for the ages betwee 3 ad 8) s acceptable eve for extreme old ages. Is there a possblty to use the estmated parameters of the Gompertz-Makeham dstrbuto eve for extreme old ages or ot? Wth dfferet well-kow tests t s possble to test f the estmated parameters of the Gompertz-Makeham dstrbuto are possble to use eve for extreme old ages. Three of the tests that are possble to use are the goodess of ft test, the Kolmogorov test ad the lkelhood rato test. There s a bg dsadvatage wth the Kolmogorov test, amely that most ofte there has to be more formato about the age tha whole years for havg the possblty to use the Kolmogorov test. The hypothess that the Gompertz-Makeham dstrbuto wth estmated parameters ca be tested also for extreme old ages s tested versus the hypothess that the estmated parameters ca t be used for these extreme old ages. If the hypothess that lfe table data for extreme old ages follows the Gompertz-Makeham dstrbuto wth estmated parameters s rejected, the those parameters ca't be used eve for extreme old ages. If that s the case, the best choce maybe s to use aother fucto to have approxmato of extreme old ages. If the Gompertz- Makeham dstrbuto descrbes all but the extreme old ages ad aother fucto descrbes the extreme old ages, there wll be a trucated fucto. The trucated fucto must have the characterstc of a dstrbuto fucto,.e. < f(s ) < ad k f(s ) for,...,k (t's possble that k s ). Note, two other possbltes are that stead of a trucated fucto aother fucto tha the Gompertz-Makeham dstrbuto s used for the whole age terval or the Gompertz-Makeham dstrbuto s used wth a more lkely set of the estmated parameters. Aother fucto tha the Gompertz-Makeham dstrbuto s t of terest ths work whle t seems lke the Gompertz-Makeham dstrbuto gves the best descrpto of lfe table data. 4. The Goodess of ft test Oe of the methods that are possble to use for the test of such a hypothess s the goodess of ft test. The (multomal) goodess of ft test s a test that makes use of a test statstc called Pearso's. Ths statstc s N k (f (s s ) f (s s )) j j j f (s s ) j. (4.)

23 Where N s the total populato sze of the researched ages. f (s s j ) s real demographc data for death at the ages s j,..., s k- ad death at ages older tha s k-. s k deotes the probablty to de after the age s k-, gve that alve at age s j (had had j:th brthday). f (s s j ) s the value of the codtoal probablty desty fucto of the Gompertz-Makeham dstrbuto at age s gve the codto that s s j. Note : k j f (s s j ). The test statstc wll be compared wth (k-j-) for gve sgfcace level. If the value s bgger tha (k-j-) the test wll reject the hypothess. A goodess of ft test has bee doe as descrbed formula (4.) testg that the demographc data explaed secto 3. follows the Gompertz-Makeham dstrbuto wth estmated parameters. The test s for ages over 95 years ad the hypothess s that ages over 95 years follows the Gompertz-Makeham dstrbuto wth the parameters estmated secto 3.. Fgure 4. below shows a plot wth both the survval fucto of the Gompertz-Makeham dstrbuto (wth the parameters estmated by use of the least square estmato secto 3.) ad values of lfe tme from lfe table data for wome [Statstcal Abstract, 983] for ages betwee 95 ad Fgure 4. x - lfe table data for Swedsh wome, - the Gompertz-Makeham dstrbuto wth parameters.5,.7786 ad.6. By the outft of Fgure 4. t s reasoable to assume that lfe table data for the ages betwee 95 ad does t follow the Gompertz-Makeham dstrbuto wth the estmated parameters Ths motvates the fact that there s a eed to use dfferet tests to prove ths statemet. A (multomal) goodess of ft test for lfe table data of Swedsh wome has bee doe wth help of a program wrtte MatLab. See appedx for more formato about the structure of the program. The goodess of ft test has bee doe assumg that the populato of the lfe table data are of sze 4673 wome (assumg that the total populato of wome are, the total populato s deftely much bgger tha ths) ad the values of f (s) are calculated wth help of the least square estmato,, of the Gompertz-Makeham dstrbuto, wth ( ) (5.* -4, 7.786* -6,.6). The ages that are used the goodess of ft test s 95, 96, 97, 98, 99 ad years. The result of the goodess of ft test s.. Ths value s compared wth the value for the percetage sgfcace level of the -dstrbuto. The value for the percetage level of the -dstrbuto s : 95. ( k j) 95. (5) 5.9. The hypothess that lfe table data for extreme old ages follows the Gompertz-Makeham dstrbuto wth the, parameters (, ) ca be rejected at the percetage level ad therefore at all relevat levels. The cocluso s that the Gompertz-Makeham dstrbuto wth estmated parameters ca ot be used for extreme old ages. There has also bee tested f the ages 9, 9, 9, 93, 94 ad 95 years correspods to ths estmated parameters ad the result of the goodess of ft test for those ages s The probablty to de at the ages 95, 96, 97, 98, 99 ad years, gve the codto that stll alve at 95 years of age are : f(d 95 (de at age 95), s 95 (stll alve at age 95)).96, f(d 96 s 95).3, f(d 97 s 95).7, f(d 98 s 95)., f(d 99 s 95).8 ad f(d s 95).5.

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