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gs 1 -1 sc (d) col0 sh gr
/Times-Italic ff 150.00 scf sf
4500 900 m
gs 1 -1 sc (3) col0 sh gr
/Times-Italic ff 150.00 scf sf
4500 1125 m
gs 1 -1 sc (3) col0 sh gr
/Times-Italic ff 150.00 scf sf
1200 900 m
gs 1 -1 sc (1) col0 sh gr
/Times-Italic ff 180.00 scf sf
1950 1575 m
gs 1 -1 sc (h) col0 sh gr
$F2psEnd
rs
%%EndDocument
@endspecial 270 3303 a Fm(Figure)c(3:)36 b(F)-7 b(ollo)n(w-up)34
b(of)g(an)g(individual)g Fe(i)g Fm(in)g(state)g Fe(j)40
b Fm(aged)33 b Fe(x)2338 3315 y Fd(1)2410 3303 y Fm(at)h(\034rst)g(in)n
(terview)g(\(time)g Fe(t)3320 3315 y Fd(1)3358 3303 y
Fm(\))633 3402 y(rein)n(terview)n(ed)26 b(at)i(times)g
Fe(t)1497 3414 y Fd(2)1561 3402 y Fm(and)g Fe(t)1753
3414 y Fd(3)1790 3402 y Fm(.)411 3609 y Fn(Let)41 b(us)f(supp)s(ose)g
(that)h(individual)h Fk(i)e Fn(observ)m(ed)h(at)g(state)g
Fk(j)k Fn(at)c(\034rst)f(examination)270 3722 y(\(time)f
Fk(t)552 3736 y Ff(1)628 3722 y Fn(and)g(age)f Fk(x)1032
3736 y Ff(1)1071 3722 y Fn(\),)i(is)d(classi\034ed)i(in)f(state)g
Fk(k)i Fn(and)f(age)f Fk(x)2488 3736 y Ff(2)2565 3722
y Fj(=)f Fk(x)2725 3736 y Ff(1)2790 3722 y Fj(+)25 b
Fk(d)2933 3736 y Ff(1)3010 3722 y Fn(at)38 b(second)270
3835 y(in)m(terview)32 b(\()p Fk(t)730 3849 y Ff(2)794
3835 y Fj(=)25 b Fk(t)923 3849 y Ff(1)983 3835 y Fj(+)20
b Fk(d)1121 3849 y Ff(1)1161 3835 y Fn(\))15 b(.)411
3948 y(It)34 b(is)h(assumed)g(that)g(the)g(lag)g(b)s(et)m(w)m(een)h(t)m
(w)m(o)h(in)m(terviews)e(is)g(expressed)f(in)h(terms)g(of)270
4061 y(elemen)m(tary)d(time)f(units)f(\()p Fk(h)p Fn(\),)h(i.e.)41
b Fk(d)1545 4075 y Ff(1)1610 4061 y Fj(=)25 b Fk(n)1761
4075 y Ff(1)1800 4061 y Fk(h)p Fn(.)411 4174 y(Then)31
b(the)g(con)m(tribution)i(so)d(far)g(of)g(individual)h
Fk(i)g Fn(to)f(the)h(lik)m(eliho)s(o)s(d)g(is:)1537 4378
y Fk(L)1599 4330 y Ff(\()p Fi(i)p Ff(\))1599 4402 y Fi(t)1624
4411 y Fc(1)1659 4402 y Fb(\000)p Fi(t)1739 4411 y Fc(2)1803
4378 y Fj(=)1874 4393 y Fi(d)1910 4402 y Fc(1)1974 4378
y Fk(p)2020 4341 y Fi(j)t(k)2020 4401 y(x)2060 4410 y
Fc(1)2098 4378 y Fk(;)327 4583 y Fn(where)43 b(the)f(probabilit)m(y)i
(on)f(the)f(righ)m(t)i(hand)e(side)g(is)g(the)h Fj(\()p
Fk(j;)15 b(k)s Fj(\))p Fn(th)44 b(elemen)m(t)g(of)d(the)270
4695 y(matrix)31 b(pro)s(duct:)1310 4946 y Fi(n)1353
4955 y Fc(1)1290 4974 y Fg(Y)1283 5168 y Fi(u)p Ff(=1)1429
5075 y Fi(h)1474 5060 y Fk(P)1532 5079 y Fi(x)1572 5088
y Fc(1)1607 5079 y Ff(+\()p Fi(u)p Fb(\000)p Ff(1\))p
Fi(h)1918 5060 y Fj(=)2014 4960 y Fg(\020)2068 5075 y
Fi(d)2104 5084 y Fc(1)2143 5060 y Fk(p)2189 5023 y Fi(j)t(k)2189
5083 y(x)2229 5092 y Fc(1)2267 4960 y Fg(\021)2352 5060
y Fk(:)897 b Fn(\(3\))411 5345 y(If)30 b(this)h(individual)h(is)e
(observ)m(ed)i(only)f(once)g(more)h(at)f(sa)m(y)g(time)g
Fk(t)2707 5359 y Ff(3)2773 5345 y Fj(=)26 b Fk(t)2903
5359 y Ff(2)2962 5345 y Fj(+)21 b Fk(d)3101 5359 y Ff(2)3171
5345 y Fn(\()p Fk(d)3253 5359 y Ff(2)3319 5345 y Fj(=)270
5458 y Fk(n)325 5472 y Ff(2)364 5458 y Fk(:h)p Fn(\))32
b(and)g(noted)g(to)f(b)s(e)g(in)g(state)h Fk(l)r Fn(,)f(then)h(its)f
(further)h(con)m(tribution)h(to)f(the)f(lik)m(eliho)s(o)s(d)1807
5841 y(3)p eop
%%Page: 4 4
4 3 bop 270 399 a Fn(is)363 414 y Fi(d)399 423 y Fc(2)438
399 y Fk(p)484 366 y Fi(k)r(l)484 421 y(x)524 430 y Fc(2)594
399 y Fn(\()p Fk(x)681 413 y Ff(2)749 399 y Fj(=)29 b
Fk(x)901 413 y Ff(1)962 399 y Fj(+)21 b Fk(d)1101 413
y Ff(1)1141 399 y Fn(\).)46 b(In)32 b(this)g(case)g(the)h(comp)s(onen)m
(t)h(of)e(the)h(total)g(lik)m(eliho)s(o)s(d)f(due)270
511 y(to)e(individual)i Fk(i)e Fn(is)1313 624 y Fk(L)1375
587 y Ff(\()p Fi(i)p Ff(\))1484 624 y Fj(=)1579 523 y
Fg(\020)1634 639 y Fi(d)1670 648 y Fc(1)1709 624 y Fk(p)1755
587 y Fi(j)t(k)1755 647 y(x)1795 656 y Fc(1)1833 523
y Fg(\021)1908 624 y Fh(\002)1998 523 y Fg(\020)2053
639 y Fi(d)2089 648 y Fc(2)2128 624 y Fk(p)2174 587 y
Fi(k)r(l)2174 647 y(x)2214 656 y Fc(2)2252 523 y Fg(\021)2321
624 y Fk(:)411 808 y Fn(Similarly)-8 b(,)47 b(con)m(tributions)d(will)f
(come)g(from)g(all)f(mem)m(b)s(ers)i(of)e(the)g(sample.)77
b(One)270 921 y(observ)m(es)41 b(that)g(the)g(formation)g(of)f(the)h
(lik)m(eliho)s(o)s(d)g(is)f(no)h(trivial)g(matter)g(since)g(there)270
1034 y(is)c(no)g(simple)g(analytical)h(expression)f(for)g(the)g(higher)
h(order)f(transition)h(probabilities)270 1147 y(in)30
b(\(3\).)411 1260 y(If)j(an)g(individual)i(is)e(not)h(aliv)m(e)g(at)g
(the)f(time)h(of)f(the)h(in)m(terview,)i(the)e(date)g(of)f(death)270
1373 y(has)41 b(to)h(b)s(e)f(kno)m(wn)h(in)g(order)g(to)g(calculate)g
(the)g(con)m(tribution)i(to)e(the)f(lik)m(eliho)s(o)s(d.)75
b(If)270 1486 y(this)31 b(date)g(is)f(unkno)m(wn,)i(w)m(e)g(assumed)f
(that)g(the)g(individual)g(died)h(at)e(the)h(mid)h(in)m(terv)-5
b(al.)270 1599 y(If)34 b(it)h(is)g(aliv)m(e)g(but)h(its)e(status)h(is)g
(unkno)m(wn,)i(there)f(is)f(no)g(con)m(tribution)i(un)m(til)f(the)g
(next)270 1712 y(in)m(terview.)411 1824 y(In)30 b(this)g(w)m(a)m(y)-8
b(,)32 b(the)f(log-lik)m(eliho)s(o)s(d)h(of)d(a)i(sample)g(of)f
Fk(N)40 b Fn(individuals)31 b(can)g(b)s(e)f(formed:)1440
2110 y Fk(l)d Fj(=)1624 1996 y Fi(N)1590 2024 y Fg(X)1598
2219 y Fi(i)p Ff(=1)1736 2110 y Fj(log)17 b Fk(L)1931
2072 y Ff(\()p Fi(i)p Ff(\))2029 2110 y Fk(:)1220 b Fn(\(4\))411
2396 y(The)35 b(parameters)h Fk(\013)1131 2410 y Fi(ij)1192
2396 y Fj(\()p Fk(h)p Fj(\))g Fn(and)f Fk(\014)1581 2410
y Fi(ij)1642 2396 y Fj(\()p Fk(h)p Fj(\))h Fn(of)e(\(2\))h(are)g
(estimated)h(b)m(y)f(maximising)h(this)270 2509 y(quan)m(tit)m(y)-8
b(.)411 2622 y(If)20 b Fk(\022)497 2681 y Ff(~)557 2622
y Fn(denotes)h(the)g(v)m(ector)g(of)f(parameters)i(and)1997
2598 y Fj(^)1997 2622 y Fk(\022)2002 2681 y Ff(~)2063
2622 y Fn(its)e(maxim)m(um)i(lik)m(eliho)s(o)s(d)f(estimator,)270
2786 y(then)30 b(it)g(follo)m(ws)g(from)g(standard)g(theory)g(that)g
(for)f(large)i(sample)f(size)f Fk(N)40 b Fn(the)29 b(MLE)3255
2762 y Fj(^)3255 2786 y Fk(\022)3260 2846 y Ff(~)3329
2786 y Fn(is)270 2926 y(appro)m(ximativ)m(ely)j(normally)g(distributed)
f(with)g(mean)g Fk(\022)2229 2986 y Ff(~)2299 2926 y
Fn(and)g(co)m(v)-5 b(ariance)32 b(matrix)1402 3182 y
Fj(lim)1363 3242 y Fi(N)7 b Fb(!1)1597 3182 y Fj(E\()1694
3158 y(^)1694 3182 y Fk(\022)1699 3241 y Ff(~)1740 3182
y Fj(\))25 b(=)g Fk(\022)1901 3241 y Ff(~)1957 3182 y
Fk(;)1586 3405 y(V)20 b Fj(\()p Fk(\022)1699 3464 y Ff(~)1740
3405 y Fj(\))25 b(=)1925 3343 y(1)p 1906 3384 83 4 v
1906 3467 a Fk(N)1999 3405 y(I)2046 3367 y Fb(\000)p
Ff(1)2141 3405 y Fj(\()p Fk(\022)2181 3464 y Ff(~)2221
3405 y Fj(\))15 b Fk(;)994 b Fn(\(5\))270 3641 y(where)25
b Fk(I)7 b Fj(\()p Fk(\022)612 3701 y Ff(~)652 3641 y
Fj(\))24 b Fn(is)g(the)g(information)h(matrix)f(computed)h(at)f(the)h
(true)f(v)-5 b(alue)24 b Fk(\022)2795 3701 y Ff(~)2850
3641 y Fn(.)38 b(This)24 b(implies)270 3781 y(the)f(asymptotic)g
(normalit)m(y)h(of)e(the)h(transition)g(probabilities)h(and)f(health)g
(exp)s(ectancies,)270 3894 y(as)30 b(no)m(w)h(discussed.)270
4137 y Fl(2.2)112 b(Estimation)35 b(of)i(health)h(exp)s(ectancies)f
(and)h(their)f(v)-6 b(ariances)411 4309 y Fn(F)e(or)43
b(con)m(v)m(enience)h(tak)m(e)e(the)h(time/age)g(unit)g
Fk(h)p Fn(=1)f(so)g(that)g(in)g(what)h(follo)m(ws)f(the)270
4422 y(temp)s(oral)h(unit)g(is)f(suppressed)h(in)f(the)h(notation.)78
b(Recall,)46 b(ho)m(w)m(ev)m(er,)i(that)43 b(di\033eren)m(t)270
4535 y(v)-5 b(alues)42 b(of)f Fk(h)h Fn(will)g(giv)m(e)g(di\033eren)m
(t)h(n)m(umerical)g(estimates)f(of)g(the)g(parameters.)75
b(Giv)m(en)270 4648 y(implemen)m(tation)28 b(of)d(the)h(maxim)m(um)h
(lik)m(eho)s(o)s(d)f(pro)s(cedure)g(describ)s(ed)g(ab)s(o)m(v)m(e)g
(the)g(health)270 4760 y(exp)s(ectancies)31 b(are)f(estimated)i(as)e
(follo)m(ws.)411 4873 y(Let)37 b Fk(e)621 4829 y Fi(ij)621
4885 y(x)717 4873 y Fn(b)s(e)f(the)g(exp)s(ected)h(subsequen)m(t)f
(time)h(sp)s(en)m(t)f(in)g(state)g Fk(j)41 b Fn(b)m(y)c(an)f
(individual)270 4986 y(who)d(w)m(as)f(in)h(state)f Fk(i)g
Fn(at)h(age)f Fk(x)p Fn(.)46 b(F)-8 b(or)33 b(suc)m(h)g(individuals,)g
Fk(e)2329 4942 y Fi(ij)2329 4998 y(x)2422 4986 y Fn(is)f(the)g(health)i
(exp)s(ectancy)270 5099 y(of)c(state)g Fk(j)5 b Fn(.)41
b(If)940 5485 y Fi(y)982 5471 y Fk(I)1029 5433 y Fi(ij)1022
5493 y(x)1115 5471 y Fj(=)1211 5315 y Fg(\()1284 5407
y Fj(1)91 b Fn(if)30 b Fk(X)7 b Fj(\()p Fk(x)21 b Fj(+)e
Fk(y)s Fj(\))26 b(=)f Fk(j)36 b Fn(giv)m(en)31 b Fk(X)7
b Fj(\()p Fk(x)p Fj(\))26 b(=)f Fk(i)15 b Fn(,)1284 5543
y Fj(0)91 b Fn(otherwise.)1807 5841 y(4)p eop
%%Page: 5 5
5 4 bop 270 399 a Fn(then)1555 643 y Fk(O)1627 606 y
Fi(ij)1624 666 y(x)1713 643 y Fj(=)1839 529 y Fb(1)1809
557 y Fg(X)1811 751 y Fi(y)r Ff(=1)1956 657 y Fi(y)1997
643 y Fk(I)2044 606 y Fi(ij)2037 666 y(x)270 929 y Fn(is)31
b(the)h(subsequen)m(t)h(time)f(sp)s(en)m(t)g(in)g(state)g
Fk(j)37 b Fn(b)m(y)32 b(individuals)g(in)g(state)g Fk(i)g
Fn(at)g(age)g Fk(x)p Fn(.)44 b(No)m(w)270 1042 y(the)31
b(mean)g(of)770 1056 y Fi(y)811 1042 y Fk(I)858 998 y
Fi(ij)851 1054 y(x)949 1042 y Fn(is)955 1353 y Fj(E)o(\()1051
1367 y Fi(y)1093 1353 y Fk(I)1140 1315 y Fi(ij)1133 1375
y(x)1201 1353 y Fj(\))26 b(=)e(Pr\()p Fk(X)7 b Fj(\()p
Fk(x)21 b Fj(+)f Fk(y)s Fj(\))26 b(=)f Fk(j)5 b Fh(j)p
Fk(X)i Fj(\()p Fk(x)p Fj(\))27 b(=)e Fk(i)p Fj(\))h(=)2573
1367 y Fi(y)2614 1353 y Fk(p)2660 1315 y Fi(ij)2660 1375
y(x)270 1551 y Fn(and)31 b(hence)1370 1795 y Fk(e)1412
1758 y Fi(ij)1412 1818 y(x)1498 1795 y Fj(=)40 b(E\()p
Fk(O)1778 1758 y Fi(ij)1775 1818 y(x)1839 1795 y Fj(\))26
b(=)2026 1682 y Fb(1)1996 1709 y Fg(X)1998 1903 y Fi(y)r
Ff(=1)2142 1809 y Fi(y)2184 1795 y Fk(p)2230 1758 y Fi(ij)2230
1818 y(x)3274 1795 y Fn(\(6\))411 2107 y(Th)m(us)41 b(kno)m(wledge)g
(of)f(the)g(transition)h(probabilities)2324 2121 y Fi(y)2365
2107 y Fk(p)2411 2062 y Fi(ij)2411 2118 y(x)2511 2107
y Fn(p)s(ermits)f(calculation)i(of)270 2230 y(the)31
b(health)g(exp)s(ectancies)g Fk(e)1267 2185 y Fi(ij)1267
2241 y(x)1343 2230 y Fn(.)411 2438 y(In)c(fact)g(the)g(probabilities)
1363 2452 y Fi(y)1404 2438 y Fk(p)1450 2394 y Fi(ij)1450
2450 y(x)1510 2438 y Fj(\()p Fk(\022)s Fj(\))g Fn(are)g(estimated)h(b)m
(y)2332 2452 y Fi(y)2373 2438 y Fk(p)2419 2394 y Fi(ij)2419
2450 y(x)2479 2438 y Fj(\()2521 2414 y(^)2514 2438 y
Fk(\022)s Fj(\))f Fn(where)2887 2414 y Fj(^)2880 2438
y Fk(\022)i Fn(is)d(the)i(max-)270 2561 y(im)m(um)k(lik)m(eliho)s(o)s
(d)e(estimator)i(of)e(the)g(previous)h(section.)41 b(Hence)31
b Fk(e)2604 2517 y Fi(ij)2604 2573 y(x)2695 2561 y Fn(is)f(estimated)h
(b)m(y)1266 2870 y Fg(c)1260 2919 y Fk(e)1302 2874 y
Fi(ij)1302 2930 y(x)1388 2919 y Fj(=)25 b Fk(e)1526 2881
y Fi(ij)1526 2941 y(x)1587 2919 y Fj(\()1629 2895 y(^)1622
2919 y Fk(\022)r Fj(\))h(=)1854 2805 y Fb(1)1824 2832
y Fg(X)1826 3026 y Fi(y)r Ff(=1)1970 2933 y Fi(y)2012
2919 y Fk(p)2058 2881 y Fi(ij)2058 2941 y(x)2118 2919
y Fj(\()2160 2895 y(^)2153 2919 y Fk(\022)s Fj(\))1040
b Fn(\(7\))411 3230 y(Clearly)31 b Fk(e)767 3186 y Fi(ij)767
3242 y(x)828 3230 y Fj(\()870 3206 y(^)863 3230 y Fk(\022)r
Fj(\))g Fn(is)f(a)g(consisten)m(t)i(normally)f(distributed)g(estimator)
h(of)e Fk(e)2950 3186 y Fi(ij)2950 3242 y(x)3010 3230
y Fn(.)411 3343 y(The)h(T)-8 b(a)m(ylor)32 b(appro)m(ximation)1031
3618 y Fi(y)1072 3604 y Fk(p)1118 3566 y Fi(ij)1118 3626
y(x)1179 3604 y Fj(\()1221 3580 y(^)1214 3604 y Fk(\022)r
Fj(\))26 b Fh(')1416 3618 y Fi(y)1457 3604 y Fk(p)1503
3566 y Fi(ij)1503 3626 y(x)1563 3604 y Fj(\()p Fk(\022)s
Fj(\))20 b(+)g(\()1832 3580 y(^)1825 3604 y Fk(\022)j
Fh(\000)d Fk(\022)s Fj(\))2063 3566 y Fb(0)2096 3542
y Fk(@)2149 3556 y Fi(y)2191 3542 y Fk(p)2237 3498 y
Fi(ij)2237 3554 y(x)2297 3542 y Fj(\()p Fk(\022)s Fj(\))p
2096 3583 317 4 v 2205 3666 a Fk(@)5 b(\022)2438 3604
y(:)270 3822 y Fn(implies)31 b(the)g(large)g(sample)f(v)-5
b(ariances)31 b(and)g(co)m(v)-5 b(ariances)995 4153 y
Fj(V)d(ar)q(\()p Fk(e)1214 4116 y Fi(ij)1214 4176 y(x)1275
4153 y Fj(\()1317 4129 y(^)1310 4153 y Fk(\022)r Fj(\)\))26
b(=)1547 4067 y Fg(X)1594 4258 y Fi(y)1694 4067 y Fg(X)1739
4258 y Fi(u)1825 4153 y Fj(\()1893 4092 y Fk(@)p 1870
4132 99 4 v 1870 4215 a(@)5 b(\022)1979 4167 y Fi(y)2020
4153 y Fk(p)2066 4116 y Fi(ij)2066 4176 y(x)2127 4153
y Fj(\()p Fk(\022)s Fj(\)\))2278 4116 y Fb(0)2301 4153
y Fk(V)20 b Fj(\()p Fk(\022)s Fj(\))2523 4092 y Fk(@)p
2500 4132 V 2500 4215 a(@)5 b(\022)2609 4167 y Fi(u)2654
4153 y Fk(p)2700 4116 y Fi(ij)2700 4176 y(x)2760 4153
y Fj(\()p Fk(\022)s Fj(\))p Fk(;)744 4437 y(C)i(ov)s
Fj(\()p Fk(e)984 4400 y Fi(ij)984 4460 y(x)1045 4437
y Fj(\()1087 4413 y(^)1080 4437 y Fk(\022)r Fj(\))p Fk(;)15
b(e)1242 4400 y Fi(ik)1242 4460 y(x)1310 4437 y Fj(\()1352
4413 y(^)1345 4437 y Fk(\022)s Fj(\))25 b(=)1547 4351
y Fg(X)1594 4542 y Fi(y)1694 4351 y Fg(X)1739 4542 y
Fi(u)1825 4437 y Fj(\()1893 4376 y Fk(@)p 1870 4417 V
1870 4500 a(@)5 b(\022)1979 4451 y Fi(y)2020 4437 y Fk(p)2066
4400 y Fi(ij)2066 4460 y(x)2127 4437 y Fj(\()p Fk(\022)s
Fj(\)\))2278 4400 y Fb(0)2301 4437 y Fk(V)20 b Fj(\()p
Fk(\022)s Fj(\))2523 4376 y Fk(@)p 2500 4417 V 2500 4500
a(@)5 b(\022)2609 4451 y Fi(u)2654 4437 y Fk(p)2700 4400
y Fi(ik)2700 4460 y(x)2766 4437 y Fj(\()p Fk(\022)s Fj(\))p
Fk(;)411 4725 y Fn(where)31 b Fk(V)21 b Fj(\()p Fk(\022)s
Fj(\))30 b Fn(is)f(the)i(co)m(v)-5 b(ariance)32 b(matrix)f(\(5\).)411
4934 y(If)f(one)g(w)m(ere)i(in)m(terested)g(in)e(the)h(health)g(exp)s
(ectancy)1548 5215 y Fi(y)1590 5201 y Fk(e)1632 5163
y Fi(ij)1632 5223 y(x)1718 5201 y Fj(=)1861 5082 y Fi(y)1814
5115 y Fg(X)1814 5309 y Fi(u)p Ff(=1)1960 5215 y Fi(u)2005
5201 y Fk(p)2051 5163 y Fi(ij)2051 5223 y(x)270 5479
y Fn(o)m(v)m(er)39 b(the)e(in)m(terv)-5 b(al)39 b Fj(\()p
Fk(x;)15 b(x)26 b Fj(+)e Fk(y)s Fj(\))p Fn(,)40 b(giv)m(en)e(the)g
(initial)g(state)g Fk(i)f Fn(at)h(age)g Fk(x)p Fn(,)h(then)f(the)g(ab)s
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%%Page: 8 8
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856 y Fk(e)976 818 y Fi(j)976 878 y(x)1020 856 y Fj(\()p
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Fj(1)c Fh(\000)f Fk(w)1529 818 y Ff(2)1568 856 y Fj(\()p
Fk(x;)15 b(\022)1701 915 y Ff(~)1741 856 y Fj(\))1776
728 y Fg(\025)1840 856 y Fk(e)1882 818 y Ff(1)p Fi(j)1882
878 y(x)1954 856 y Fj(\()p Fk(\022)s Fj(\))20 b(+)g Fk(w)2248
818 y Ff(2)2288 856 y Fj(\()p Fk(x;)15 b(\022)2421 915
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1104 y(It)30 b(is)f(these)h(health)h(exp)s(ectancies)g(that)f(w)m(e)h
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y Fi(j)1309 1442 y(x)1353 1430 y Fj(\()p Fk(\022)s Fj(\))g
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1861 y Fk(w)337 1828 y Ff(2)377 1861 y Fj(\()p Fk(x;)15
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(explicitly)-8 b(.)42 b(Pro)s(ceed)31 b(as)f(follo)m(ws.)41
b(Let)808 2161 y Fk(e)850 2124 y Fi(j)850 2184 y(x)894
2161 y Fj(\()p Fk(i;)15 b(t;)g(\022)s Fj(\))26 b(=)1276
2033 y Fg(\024)1324 2161 y Fj(1)20 b Fh(\000)1460 2175
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2161 y Fj(\()p Fk(x;)15 b(\022)1774 2221 y Ff(~)1814
2161 y Fj(\))1849 2033 y Fg(\025)1912 2161 y Fk(e)1954
2124 y Ff(1)p Fi(j)1954 2184 y(x)2027 2161 y Fj(\()p
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2124 y Fi(i)p Ff(2)2414 2161 y Fj(\()p Fk(x;)15 b(\022)2547
2221 y Ff(~)2587 2161 y Fj(\))p Fk(e)2664 2124 y Ff(2)p
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2447 y(x)1826 2435 y Fj(\()p Fk(\022)s Fj(\))h Fn(of)f(\(10\).)40
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2548 y(w)m(e)31 b(ha)m(v)m(e,)h(using)e(the)h(de\034nitions)g(of)f(the)
h Fk(e)1767 2504 y Fi(ij)1767 2560 y(x)1828 2548 y Fj(\()p
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2874 y Fi(j)778 2934 y(x)822 2912 y Fj(\()p Fk(i;)15
b(t;)1010 2888 y Fj(^)1001 2912 y Fk(\022)t Fj(\))20
b Fh(\000)g Fk(e)1236 2874 y Fi(j)1236 2934 y(x)1280
2912 y Fj(\()p Fk(i;)15 b(t;)g(\022)s Fj(\))84 b(=)1808
2798 y Fb(1)1778 2826 y Fg(X)1780 3020 y Fi(y)r Ff(=0)1925
2926 y Fi(y)1966 2912 y Fk(p)2012 2874 y Ff(1)p Fi(j)2083
2943 y(x)2127 2912 y Fj(\()2169 2888 y(^)2162 2912 y
Fk(\022)s Fj(\))20 b Fh(\000)2354 2926 y Fi(y)2396 2912
y Fk(p)2442 2874 y Ff(1)p Fi(j)2513 2943 y(x)2557 2912
y Fj(\()p Fk(\022)s Fj(\))1624 3180 y(+)1778 3194 y Fi(t)1808
3180 y Fk(w)1875 3142 y Fi(i)p Ff(2)1939 3180 y Fj(\()p
Fk(x;)2074 3156 y Fj(^)2066 3180 y Fk(\022)s Fj(\))2162
3079 y Fg(h)2205 3194 y Fi(y)2246 3180 y Fk(p)2292 3142
y Ff(2)p Fi(j)2292 3202 y(x)2364 3180 y Fj(\()2406 3156
y(^)2399 3180 y Fk(\022)s Fj(\))g Fh(\000)2571 3194 y
Fi(y)2632 3180 y Fk(p)2678 3142 y Ff(1)p Fi(j)2678 3202
y(x)2750 3180 y Fj(\()2792 3156 y(^)2785 3180 y Fk(\022)s
Fj(\))2866 3079 y Fg(i)1624 3353 y Fh(\000)1778 3367
y Fi(t)1808 3353 y Fk(w)1875 3316 y Fi(i)p Ff(2)1939
3353 y Fj(\()p Fk(x;)15 b(\022)s Fj(\))2162 3280 y Fg(\002)2200
3367 y Fi(y)2241 3353 y Fk(p)2287 3316 y Ff(2)p Fi(j)2287
3376 y(x)2359 3353 y Fj(\()p Fk(\022)s Fj(\))20 b Fh(\000)2566
3367 y Fi(y)2627 3353 y Fk(p)2673 3316 y Ff(1)p Fi(j)2673
3376 y(x)2745 3353 y Fj(\()p Fk(\022)s Fj(\))2861 3280
y Fg(\003)411 3573 y Fn(No)m(w)39 b(use)g(the)g(T)-8
b(a)m(ylor)40 b(appro)m(ximation)h(for)2003 3587 y Fi(y)2045
3573 y Fk(p)2091 3529 y Fi(ij)2091 3585 y(x)2151 3573
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(and)h(also)g(that)g(for)270 3709 y Fi(t)300 3695 y Fk(w)367
3662 y Fi(i)p Ff(2)431 3695 y Fj(\()p Fk(x;)566 3671
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3942 y Fk(w)947 3904 y Fi(i)p Ff(2)1011 3942 y Fj(\()p
Fk(x;)1146 3918 y Fj(^)1138 3942 y Fk(\022)s Fj(\))25
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Fi(i)p Ff(2)1516 3942 y Fj(\()p Fk(x;)15 b(\022)s Fj(\))21
b(+)f(\()1878 3918 y(^)1871 3942 y Fk(\022)i Fh(\000)e
Fk(\022)s Fj(\))2108 3904 y Fb(0)2164 3880 y Fk(@)p 2141
3921 99 4 v 2141 4004 a(@)5 b(\022)2250 4008 y Fi(t)2279
3942 y Fk(w)2346 3904 y Fi(i)p Ff(2)2410 3942 y Fj(\()p
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4535 y Fk(e)750 4497 y Fi(j)750 4557 y(x)794 4535 y Fj(\()p
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4421 y Fb(1)1649 4448 y Fg(X)1651 4643 y Fi(y)r Ff(=0)1781
4535 y Fj(\()1823 4511 y(^)1816 4535 y Fk(\022)d Fh(\000)c
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4514 V 2086 4597 a(@)5 b(\022)2195 4535 y(g)s Fj(\()p
Fk(i;)15 b(t;)g(y)s(;)g(x;)g(j;)g(\022)s Fj(\))p Fk(;)411
4827 y Fn(where)630 5144 y Fk(g)s Fj(\()p Fk(i;)g(t;)g(y)s(;)g(x;)g(j;)
g(\022)s Fj(\))28 b(=)1292 5158 y Fi(y)1359 5144 y Fk(p)1405
5106 y Ff(1)p Fi(j)1405 5166 y(x)1476 5144 y Fj(\()p
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Fh(!)g(1)p Fn(,)30 b(giv)m(es)h(the)g(k)m(ey)f(result)1807
5841 y(8)p eop
%%Page: 9 9
9 8 bop 925 649 a Fk(e)967 612 y Fi(j)967 672 y(x)1012
649 y Fj(\()1054 625 y(^)1047 649 y Fk(\022)r Fj(\))21
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563 y Fg(X)1579 757 y Fi(y)r Ff(=0)1708 649 y Fj(\()1750
625 y(^)1743 649 y Fk(\022)e Fh(\000)d Fk(\022)s Fj(\))1981
612 y Fb(0)2036 588 y Fk(@)p 2014 629 99 4 v 2014 712
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Fi(j)1359 1168 y(x)1430 1146 y Fj(\()p Fk(\022)s Fj(\))21
b(+)e Fk(w)1724 1108 y Ff(2)1764 1146 y Fj(\()p Fk(x;)c(\022)s
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y Fk(p)2113 1108 y Ff(2)p Fi(j)2113 1168 y(x)2185 1146
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1146 y Fk(p)2499 1108 y Ff(1)p Fi(j)2499 1168 y(x)2571
1146 y Fj(\()p Fk(\022)s Fj(\))2687 1072 y Fg(\003)2740
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Fk(e)1505 1321 y Fi(j)1505 1377 y(x)1549 1365 y Fj(\()1591
1341 y(^)1584 1365 y Fk(\022)s Fj(\))g Fn(is)f(an)i(immediate)g
(consequence)g(of)f(\(11\).)54 b(The)270 1478 y(second)31
b(momen)m(ts)g(are)446 1775 y Fj(V)-8 b(ar\()p Fk(e)664
1789 y Fi(j)702 1775 y Fj(\()p Fk(x;)837 1751 y Fj(^)829
1775 y Fk(\022)s Fj(\)\))25 b Fh(')1082 1689 y Fg(X)1082
1883 y Fi(u)p Ff(=0)1228 1689 y Fg(X)1230 1883 y Fi(y)r
Ff(=0)1375 1592 y Fg(0)1375 1756 y(@)1487 1713 y Fk(@)p
1464 1754 V 1464 1837 a(@)5 b(\022)1522 1897 y Ff(~)1572
1775 y Fk(g)s Fj(\()p Fk(u;)15 b(x;)g(j;)g(\022)1922
1835 y Ff(~)1962 1775 y Fj(\))1997 1592 y Fg(1)1997 1756
y(A)2077 1615 y Fb(0)2116 1775 y Fk(V)20 b Fj(\()p Fk(\022)2229
1835 y Ff(~)2270 1775 y Fj(\))2320 1592 y Fg(0)2320 1756
y(@)2432 1713 y Fk(@)p 2410 1754 V 2410 1837 a(@)5 b(\022)2468
1897 y Ff(~)2518 1775 y Fk(g)s Fj(\()p Fk(u;)15 b(x;)g(j;)g(\022)2868
1835 y Ff(~)2908 1775 y Fj(\))2943 1592 y Fg(1)2943 1756
y(A)3038 1775 y Fk(;)166 b Fn(\(12\))270 2304 y Fk(C)7
b(ov)s Fj(\()p Fk(e)510 2318 y Fi(j)547 2304 y Fj(\()p
Fk(x;)682 2280 y Fj(^)674 2304 y Fk(\022)s Fj(\))p Fk(;)15
b(e)837 2319 y Fi(k)881 2304 y Fj(\()p Fk(x;)1016 2280
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y Fg(X)1261 2412 y Fi(u)p Ff(=0)1407 2217 y Fg(X)1409
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2242 y Fk(@)p 1643 2283 V 1643 2366 a(@)5 b(\022)1701
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2364 y Ff(~)2142 2304 y Fj(\))2177 2121 y Fg(1)2177 2285
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2364 y Ff(~)2449 2304 y Fj(\))2499 2121 y Fg(0)2499 2285
y(@)2611 2242 y Fk(@)p 2589 2283 V 2589 2366 a(@)5 b(\022)2647
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3060 2364 y Ff(~)3100 2304 y Fj(\))3135 2121 y Fg(1)3135
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(circuitous)i(route)f(to)g(\(12\))g(is)f(that)h(the)g(deriv)-5
b(ativ)m(e)3006 2590 y Fi(@)p 2989 2605 76 4 v 2989 2658
a(@)t(\022)3075 2626 y Fk(w)3142 2593 y Ff(2)3182 2626
y Fj(\()p Fk(x;)15 b(\022)s Fj(\))270 2748 y Fn(cannot)21
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g(calculating)2642 2712 y Fi(@)p 2625 2727 V 2625 2779
a(@)t(\022)2711 2762 y(t)2740 2748 y Fk(w)2807 2715 y
Fi(i)p Ff(2)2871 2748 y Fj(\()p Fk(x;)15 b(\022)s Fj(\))21
b Fn(presen)m(ts)270 2861 y(no)28 b(problems)g(once)g(the)f(deriv)-5
b(ativ)m(es)28 b(of)f(the)h(transition)g(probabilities)h(are)f(kno)m
(wn.)40 b(Cal-)270 2974 y(culation)34 b(of)f(these)h(latter)g(deriv)-5
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3199 y(estimator)686 3175 y Fj(^)679 3199 y Fk(\022)39
b Fn(had)e(b)s(een)g(found.)60 b(Th)m(us)38 b(the)f(ev)-5
b(aluation)38 b(of)f(\(12\))g(in)m(v)m(olv)m(es)h(calculating)270
3312 y(the)d(realised)g(v)-5 b(alue)34 b(of)g(the)h(deriv)-5
b(ativ)m(e)35 b(of)1790 3326 y Fi(t)1820 3312 y Fk(w)1887
3279 y Fi(i)p Ff(2)1985 3312 y Fn(for)f(large)h Fk(t)p
Fn(,)g(meaning)h(of)d(the)i(order)g(of)270 3425 y(200)c(mon)m(ths)g
(for)f(our)h(data)g(set.)270 3712 y Fp(References)270
3915 y Fn([1])46 b(Laditk)-5 b(a)35 b(S.)f(and)g(W)-8
b(olf)34 b(D.)52 b(New)34 b(metho)s(ds)g(for)g(analysing)g(activ)m(e)h
(life)f(exp)s(ectancy)-8 b(.)411 4028 y Fa(Journal)33
b(of)f(A)-5 b(ging)33 b(and)f(He)-5 b(alth)p Fn(,)32
b(1997.)270 4215 y([2])46 b(Seneta)31 b(E.)41 b Fa(Non-ne)-5
b(gative)34 b(matric)-5 b(es)p Fn(.)41 b(John)31 b(Wiley)f(and)h(Sons,)
g(New)f(Y)-8 b(ork,)31 b(1973.)1807 5841 y(9)p eop
%%Trailer
end
userdict /end-hook known{end-hook}if
%%EOF
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