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2408: 5345 y Fn(\()p Fk(d)3253 5359 y Ff(2)3319 5345 y Fj(=)270
2409: 5458 y Fk(n)325 5472 y Ff(2)364 5458 y Fk(:h)p Fn(\))32
2410: b(and)g(noted)g(to)f(b)s(e)g(in)g(state)h Fk(l)r Fn(,)f(then)h(its)f
2411: (further)h(con)m(tribution)h(to)f(the)f(lik)m(eliho)s(o)s(d)1807
2412: 5841 y(3)p eop
2413: %%Page: 4 4
2414: 4 3 bop 270 399 a Fn(is)363 414 y Fi(d)399 423 y Fc(2)438
2415: 399 y Fk(p)484 366 y Fi(k)r(l)484 421 y(x)524 430 y Fc(2)594
2416: 399 y Fn(\()p Fk(x)681 413 y Ff(2)749 399 y Fj(=)29 b
2417: Fk(x)901 413 y Ff(1)962 399 y Fj(+)21 b Fk(d)1101 413
2418: y Ff(1)1141 399 y Fn(\).)46 b(In)32 b(this)g(case)g(the)h(comp)s(onen)m
2419: (t)h(of)e(the)h(total)g(lik)m(eliho)s(o)s(d)f(due)270
2420: 511 y(to)e(individual)i Fk(i)e Fn(is)1313 624 y Fk(L)1375
2421: 587 y Ff(\()p Fi(i)p Ff(\))1484 624 y Fj(=)1579 523 y
2422: Fg(\020)1634 639 y Fi(d)1670 648 y Fc(1)1709 624 y Fk(p)1755
2423: 587 y Fi(j)t(k)1755 647 y(x)1795 656 y Fc(1)1833 523
2424: y Fg(\021)1908 624 y Fh(\002)1998 523 y Fg(\020)2053
2425: 639 y Fi(d)2089 648 y Fc(2)2128 624 y Fk(p)2174 587 y
2426: Fi(k)r(l)2174 647 y(x)2214 656 y Fc(2)2252 523 y Fg(\021)2321
2427: 624 y Fk(:)411 808 y Fn(Similarly)-8 b(,)47 b(con)m(tributions)d(will)f
2428: (come)g(from)g(all)f(mem)m(b)s(ers)i(of)e(the)g(sample.)77
2429: b(One)270 921 y(observ)m(es)41 b(that)g(the)g(formation)g(of)f(the)h
2430: (lik)m(eliho)s(o)s(d)g(is)f(no)h(trivial)g(matter)g(since)g(there)270
2431: 1034 y(is)c(no)g(simple)g(analytical)h(expression)f(for)g(the)g(higher)
2432: h(order)f(transition)h(probabilities)270 1147 y(in)30
2433: b(\(3\).)411 1260 y(If)j(an)g(individual)i(is)e(not)h(aliv)m(e)g(at)g
2434: (the)f(time)h(of)f(the)h(in)m(terview,)i(the)e(date)g(of)f(death)270
2435: 1373 y(has)41 b(to)h(b)s(e)f(kno)m(wn)h(in)g(order)g(to)g(calculate)g
2436: (the)g(con)m(tribution)i(to)e(the)f(lik)m(eliho)s(o)s(d.)75
2437: b(If)270 1486 y(this)31 b(date)g(is)f(unkno)m(wn,)i(w)m(e)g(assumed)f
2438: (that)g(the)g(individual)g(died)h(at)e(the)h(mid)h(in)m(terv)-5
2439: b(al.)270 1599 y(If)34 b(it)h(is)g(aliv)m(e)g(but)h(its)e(status)h(is)g
2440: (unkno)m(wn,)i(there)f(is)f(no)g(con)m(tribution)i(un)m(til)f(the)g
2441: (next)270 1712 y(in)m(terview.)411 1824 y(In)30 b(this)g(w)m(a)m(y)-8
2442: b(,)32 b(the)f(log-lik)m(eliho)s(o)s(d)h(of)d(a)i(sample)g(of)f
2443: Fk(N)40 b Fn(individuals)31 b(can)g(b)s(e)f(formed:)1440
2444: 2110 y Fk(l)d Fj(=)1624 1996 y Fi(N)1590 2024 y Fg(X)1598
2445: 2219 y Fi(i)p Ff(=1)1736 2110 y Fj(log)17 b Fk(L)1931
2446: 2072 y Ff(\()p Fi(i)p Ff(\))2029 2110 y Fk(:)1220 b Fn(\(4\))411
2447: 2396 y(The)35 b(parameters)h Fk(\013)1131 2410 y Fi(ij)1192
2448: 2396 y Fj(\()p Fk(h)p Fj(\))g Fn(and)f Fk(\014)1581 2410
2449: y Fi(ij)1642 2396 y Fj(\()p Fk(h)p Fj(\))h Fn(of)e(\(2\))h(are)g
2450: (estimated)h(b)m(y)f(maximising)h(this)270 2509 y(quan)m(tit)m(y)-8
2451: b(.)411 2622 y(If)20 b Fk(\022)497 2681 y Ff(~)557 2622
2452: y Fn(denotes)h(the)g(v)m(ector)g(of)f(parameters)i(and)1997
2453: 2598 y Fj(^)1997 2622 y Fk(\022)2002 2681 y Ff(~)2063
2454: 2622 y Fn(its)e(maxim)m(um)i(lik)m(eliho)s(o)s(d)f(estimator,)270
2455: 2786 y(then)30 b(it)g(follo)m(ws)g(from)g(standard)g(theory)g(that)g
2456: (for)f(large)i(sample)f(size)f Fk(N)40 b Fn(the)29 b(MLE)3255
2457: 2762 y Fj(^)3255 2786 y Fk(\022)3260 2846 y Ff(~)3329
2458: 2786 y Fn(is)270 2926 y(appro)m(ximativ)m(ely)j(normally)g(distributed)
2459: f(with)g(mean)g Fk(\022)2229 2986 y Ff(~)2299 2926 y
2460: Fn(and)g(co)m(v)-5 b(ariance)32 b(matrix)1402 3182 y
2461: Fj(lim)1363 3242 y Fi(N)7 b Fb(!1)1597 3182 y Fj(E\()1694
2462: 3158 y(^)1694 3182 y Fk(\022)1699 3241 y Ff(~)1740 3182
2463: y Fj(\))25 b(=)g Fk(\022)1901 3241 y Ff(~)1957 3182 y
2464: Fk(;)1586 3405 y(V)20 b Fj(\()p Fk(\022)1699 3464 y Ff(~)1740
2465: 3405 y Fj(\))25 b(=)1925 3343 y(1)p 1906 3384 83 4 v
2466: 1906 3467 a Fk(N)1999 3405 y(I)2046 3367 y Fb(\000)p
2467: Ff(1)2141 3405 y Fj(\()p Fk(\022)2181 3464 y Ff(~)2221
2468: 3405 y Fj(\))15 b Fk(;)994 b Fn(\(5\))270 3641 y(where)25
2469: b Fk(I)7 b Fj(\()p Fk(\022)612 3701 y Ff(~)652 3641 y
2470: Fj(\))24 b Fn(is)g(the)g(information)h(matrix)f(computed)h(at)f(the)h
2471: (true)f(v)-5 b(alue)24 b Fk(\022)2795 3701 y Ff(~)2850
2472: 3641 y Fn(.)38 b(This)24 b(implies)270 3781 y(the)f(asymptotic)g
2473: (normalit)m(y)h(of)e(the)h(transition)g(probabilities)h(and)f(health)g
2474: (exp)s(ectancies,)270 3894 y(as)30 b(no)m(w)h(discussed.)270
2475: 4137 y Fl(2.2)112 b(Estimation)35 b(of)i(health)h(exp)s(ectancies)f
2476: (and)h(their)f(v)-6 b(ariances)411 4309 y Fn(F)e(or)43
2477: b(con)m(v)m(enience)h(tak)m(e)e(the)h(time/age)g(unit)g
2478: Fk(h)p Fn(=1)f(so)g(that)g(in)g(what)h(follo)m(ws)f(the)270
2479: 4422 y(temp)s(oral)h(unit)g(is)f(suppressed)h(in)f(the)h(notation.)78
2480: b(Recall,)46 b(ho)m(w)m(ev)m(er,)i(that)43 b(di\033eren)m(t)270
2481: 4535 y(v)-5 b(alues)42 b(of)f Fk(h)h Fn(will)g(giv)m(e)g(di\033eren)m
2482: (t)h(n)m(umerical)g(estimates)f(of)g(the)g(parameters.)75
2483: b(Giv)m(en)270 4648 y(implemen)m(tation)28 b(of)d(the)h(maxim)m(um)h
2484: (lik)m(eho)s(o)s(d)f(pro)s(cedure)g(describ)s(ed)g(ab)s(o)m(v)m(e)g
2485: (the)g(health)270 4760 y(exp)s(ectancies)31 b(are)f(estimated)i(as)e
2486: (follo)m(ws.)411 4873 y(Let)37 b Fk(e)621 4829 y Fi(ij)621
2487: 4885 y(x)717 4873 y Fn(b)s(e)f(the)g(exp)s(ected)h(subsequen)m(t)f
2488: (time)h(sp)s(en)m(t)f(in)g(state)g Fk(j)41 b Fn(b)m(y)c(an)f
2489: (individual)270 4986 y(who)d(w)m(as)f(in)h(state)f Fk(i)g
2490: Fn(at)h(age)f Fk(x)p Fn(.)46 b(F)-8 b(or)33 b(suc)m(h)g(individuals,)g
2491: Fk(e)2329 4942 y Fi(ij)2329 4998 y(x)2422 4986 y Fn(is)f(the)g(health)i
2492: (exp)s(ectancy)270 5099 y(of)c(state)g Fk(j)5 b Fn(.)41
2493: b(If)940 5485 y Fi(y)982 5471 y Fk(I)1029 5433 y Fi(ij)1022
2494: 5493 y(x)1115 5471 y Fj(=)1211 5315 y Fg(\()1284 5407
2495: y Fj(1)91 b Fn(if)30 b Fk(X)7 b Fj(\()p Fk(x)21 b Fj(+)e
2496: Fk(y)s Fj(\))26 b(=)f Fk(j)36 b Fn(giv)m(en)31 b Fk(X)7
2497: b Fj(\()p Fk(x)p Fj(\))26 b(=)f Fk(i)15 b Fn(,)1284 5543
2498: y Fj(0)91 b Fn(otherwise.)1807 5841 y(4)p eop
2499: %%Page: 5 5
2500: 5 4 bop 270 399 a Fn(then)1555 643 y Fk(O)1627 606 y
2501: Fi(ij)1624 666 y(x)1713 643 y Fj(=)1839 529 y Fb(1)1809
2502: 557 y Fg(X)1811 751 y Fi(y)r Ff(=1)1956 657 y Fi(y)1997
2503: 643 y Fk(I)2044 606 y Fi(ij)2037 666 y(x)270 929 y Fn(is)31
2504: b(the)h(subsequen)m(t)h(time)f(sp)s(en)m(t)g(in)g(state)g
2505: Fk(j)37 b Fn(b)m(y)32 b(individuals)g(in)g(state)g Fk(i)g
2506: Fn(at)g(age)g Fk(x)p Fn(.)44 b(No)m(w)270 1042 y(the)31
2507: b(mean)g(of)770 1056 y Fi(y)811 1042 y Fk(I)858 998 y
2508: Fi(ij)851 1054 y(x)949 1042 y Fn(is)955 1353 y Fj(E)o(\()1051
2509: 1367 y Fi(y)1093 1353 y Fk(I)1140 1315 y Fi(ij)1133 1375
2510: y(x)1201 1353 y Fj(\))26 b(=)e(Pr\()p Fk(X)7 b Fj(\()p
2511: Fk(x)21 b Fj(+)f Fk(y)s Fj(\))26 b(=)f Fk(j)5 b Fh(j)p
2512: Fk(X)i Fj(\()p Fk(x)p Fj(\))27 b(=)e Fk(i)p Fj(\))h(=)2573
2513: 1367 y Fi(y)2614 1353 y Fk(p)2660 1315 y Fi(ij)2660 1375
2514: y(x)270 1551 y Fn(and)31 b(hence)1370 1795 y Fk(e)1412
2515: 1758 y Fi(ij)1412 1818 y(x)1498 1795 y Fj(=)40 b(E\()p
2516: Fk(O)1778 1758 y Fi(ij)1775 1818 y(x)1839 1795 y Fj(\))26
2517: b(=)2026 1682 y Fb(1)1996 1709 y Fg(X)1998 1903 y Fi(y)r
2518: Ff(=1)2142 1809 y Fi(y)2184 1795 y Fk(p)2230 1758 y Fi(ij)2230
2519: 1818 y(x)3274 1795 y Fn(\(6\))411 2107 y(Th)m(us)41 b(kno)m(wledge)g
2520: (of)f(the)g(transition)h(probabilities)2324 2121 y Fi(y)2365
2521: 2107 y Fk(p)2411 2062 y Fi(ij)2411 2118 y(x)2511 2107
2522: y Fn(p)s(ermits)f(calculation)i(of)270 2230 y(the)31
2523: b(health)g(exp)s(ectancies)g Fk(e)1267 2185 y Fi(ij)1267
2524: 2241 y(x)1343 2230 y Fn(.)411 2438 y(In)c(fact)g(the)g(probabilities)
2525: 1363 2452 y Fi(y)1404 2438 y Fk(p)1450 2394 y Fi(ij)1450
2526: 2450 y(x)1510 2438 y Fj(\()p Fk(\022)s Fj(\))g Fn(are)g(estimated)h(b)m
2527: (y)2332 2452 y Fi(y)2373 2438 y Fk(p)2419 2394 y Fi(ij)2419
2528: 2450 y(x)2479 2438 y Fj(\()2521 2414 y(^)2514 2438 y
2529: Fk(\022)s Fj(\))f Fn(where)2887 2414 y Fj(^)2880 2438
2530: y Fk(\022)i Fn(is)d(the)i(max-)270 2561 y(im)m(um)k(lik)m(eliho)s(o)s
2531: (d)e(estimator)i(of)e(the)g(previous)h(section.)41 b(Hence)31
2532: b Fk(e)2604 2517 y Fi(ij)2604 2573 y(x)2695 2561 y Fn(is)f(estimated)h
2533: (b)m(y)1266 2870 y Fg(c)1260 2919 y Fk(e)1302 2874 y
2534: Fi(ij)1302 2930 y(x)1388 2919 y Fj(=)25 b Fk(e)1526 2881
2535: y Fi(ij)1526 2941 y(x)1587 2919 y Fj(\()1629 2895 y(^)1622
2536: 2919 y Fk(\022)r Fj(\))h(=)1854 2805 y Fb(1)1824 2832
2537: y Fg(X)1826 3026 y Fi(y)r Ff(=1)1970 2933 y Fi(y)2012
2538: 2919 y Fk(p)2058 2881 y Fi(ij)2058 2941 y(x)2118 2919
2539: y Fj(\()2160 2895 y(^)2153 2919 y Fk(\022)s Fj(\))1040
2540: b Fn(\(7\))411 3230 y(Clearly)31 b Fk(e)767 3186 y Fi(ij)767
2541: 3242 y(x)828 3230 y Fj(\()870 3206 y(^)863 3230 y Fk(\022)r
2542: Fj(\))g Fn(is)f(a)g(consisten)m(t)i(normally)f(distributed)g(estimator)
2543: h(of)e Fk(e)2950 3186 y Fi(ij)2950 3242 y(x)3010 3230
2544: y Fn(.)411 3343 y(The)h(T)-8 b(a)m(ylor)32 b(appro)m(ximation)1031
2545: 3618 y Fi(y)1072 3604 y Fk(p)1118 3566 y Fi(ij)1118 3626
2546: y(x)1179 3604 y Fj(\()1221 3580 y(^)1214 3604 y Fk(\022)r
2547: Fj(\))26 b Fh(')1416 3618 y Fi(y)1457 3604 y Fk(p)1503
2548: 3566 y Fi(ij)1503 3626 y(x)1563 3604 y Fj(\()p Fk(\022)s
2549: Fj(\))20 b(+)g(\()1832 3580 y(^)1825 3604 y Fk(\022)j
2550: Fh(\000)d Fk(\022)s Fj(\))2063 3566 y Fb(0)2096 3542
2551: y Fk(@)2149 3556 y Fi(y)2191 3542 y Fk(p)2237 3498 y
2552: Fi(ij)2237 3554 y(x)2297 3542 y Fj(\()p Fk(\022)s Fj(\))p
2553: 2096 3583 317 4 v 2205 3666 a Fk(@)5 b(\022)2438 3604
2554: y(:)270 3822 y Fn(implies)31 b(the)g(large)g(sample)f(v)-5
2555: b(ariances)31 b(and)g(co)m(v)-5 b(ariances)995 4153 y
2556: Fj(V)d(ar)q(\()p Fk(e)1214 4116 y Fi(ij)1214 4176 y(x)1275
2557: 4153 y Fj(\()1317 4129 y(^)1310 4153 y Fk(\022)r Fj(\)\))26
2558: b(=)1547 4067 y Fg(X)1594 4258 y Fi(y)1694 4067 y Fg(X)1739
2559: 4258 y Fi(u)1825 4153 y Fj(\()1893 4092 y Fk(@)p 1870
2560: 4132 99 4 v 1870 4215 a(@)5 b(\022)1979 4167 y Fi(y)2020
2561: 4153 y Fk(p)2066 4116 y Fi(ij)2066 4176 y(x)2127 4153
2562: y Fj(\()p Fk(\022)s Fj(\)\))2278 4116 y Fb(0)2301 4153
2563: y Fk(V)20 b Fj(\()p Fk(\022)s Fj(\))2523 4092 y Fk(@)p
2564: 2500 4132 V 2500 4215 a(@)5 b(\022)2609 4167 y Fi(u)2654
2565: 4153 y Fk(p)2700 4116 y Fi(ij)2700 4176 y(x)2760 4153
2566: y Fj(\()p Fk(\022)s Fj(\))p Fk(;)744 4437 y(C)i(ov)s
2567: Fj(\()p Fk(e)984 4400 y Fi(ij)984 4460 y(x)1045 4437
2568: y Fj(\()1087 4413 y(^)1080 4437 y Fk(\022)r Fj(\))p Fk(;)15
2569: b(e)1242 4400 y Fi(ik)1242 4460 y(x)1310 4437 y Fj(\()1352
2570: 4413 y(^)1345 4437 y Fk(\022)s Fj(\))25 b(=)1547 4351
2571: y Fg(X)1594 4542 y Fi(y)1694 4351 y Fg(X)1739 4542 y
2572: Fi(u)1825 4437 y Fj(\()1893 4376 y Fk(@)p 1870 4417 V
2573: 1870 4500 a(@)5 b(\022)1979 4451 y Fi(y)2020 4437 y Fk(p)2066
2574: 4400 y Fi(ij)2066 4460 y(x)2127 4437 y Fj(\()p Fk(\022)s
2575: Fj(\)\))2278 4400 y Fb(0)2301 4437 y Fk(V)20 b Fj(\()p
2576: Fk(\022)s Fj(\))2523 4376 y Fk(@)p 2500 4417 V 2500 4500
2577: a(@)5 b(\022)2609 4451 y Fi(u)2654 4437 y Fk(p)2700 4400
2578: y Fi(ik)2700 4460 y(x)2766 4437 y Fj(\()p Fk(\022)s Fj(\))p
2579: Fk(;)411 4725 y Fn(where)31 b Fk(V)21 b Fj(\()p Fk(\022)s
2580: Fj(\))30 b Fn(is)f(the)i(co)m(v)-5 b(ariance)32 b(matrix)f(\(5\).)411
2581: 4934 y(If)f(one)g(w)m(ere)i(in)m(terested)g(in)e(the)h(health)g(exp)s
2582: (ectancy)1548 5215 y Fi(y)1590 5201 y Fk(e)1632 5163
2583: y Fi(ij)1632 5223 y(x)1718 5201 y Fj(=)1861 5082 y Fi(y)1814
2584: 5115 y Fg(X)1814 5309 y Fi(u)p Ff(=1)1960 5215 y Fi(u)2005
2585: 5201 y Fk(p)2051 5163 y Fi(ij)2051 5223 y(x)270 5479
2586: y Fn(o)m(v)m(er)39 b(the)e(in)m(terv)-5 b(al)39 b Fj(\()p
2587: Fk(x;)15 b(x)26 b Fj(+)e Fk(y)s Fj(\))p Fn(,)40 b(giv)m(en)e(the)g
2588: (initial)g(state)g Fk(i)f Fn(at)h(age)g Fk(x)p Fn(,)h(then)f(the)g(ab)s
2589: (o)m(v)m(e)270 5592 y(form)m(ulae)32 b(hold)f(mo)s(di\034ed)g(b)m(y)f
2590: (the)h(upp)s(er)g(limit)g Fk(y)h Fn(in)f(the)g(summations.)1807
2591: 5841 y(5)p eop
2592: %%Page: 6 6
2593: 6 5 bop 411 399 a Fn(Health)29 b(exp)s(ectancies)g(irresp)s(ectiv)m(e)g
2594: (of)f(the)h(initial)g(state)g(are)g(w)m(eigh)m(ted)h(a)m(v)m(erages)g
2595: (of)270 511 y Fk(e)312 467 y Ff(1)p Fi(j)312 523 y(x)418
2596: 511 y Fn(and)35 b Fk(e)640 467 y Ff(2)p Fi(j)640 523
2597: y(x)713 511 y Fn(.)53 b(The)35 b(w)m(eigh)m(ts)h(are)f(the)f(prop)s
2598: (ortions)i Fk(\031)2166 478 y Ff(1)2163 534 y Fi(x)2207
2599: 511 y Fn(,)f Fk(\031)2322 478 y Ff(2)2319 534 y Fi(x)2397
2600: 511 y Fn(of)f(the)h(p)s(opulation)h(aged)f Fk(x)270 624
2601: y Fn(that)c(are)g(in)f(states)g(1)h(and)f(2)h(resp)s(ectiv)m(ely)-8
2602: b(.)41 b(Th)m(us)1461 783 y Fk(e)1503 745 y Fi(j)1503
2603: 805 y(x)1572 783 y Fj(=)25 b Fk(\031)1723 745 y Ff(1)1720
2604: 805 y Fi(x)1764 783 y Fk(e)1806 745 y Ff(1)p Fi(j)1806
2605: 805 y(x)1898 783 y Fj(+)20 b Fk(\031)2044 745 y Ff(2)2041
2606: 805 y Fi(x)2085 783 y Fk(e)2127 745 y Ff(2)p Fi(j)2127
2607: 805 y(x)3274 783 y Fn(\(8\))270 958 y(is)34 b(the)h(\(marginal\))i
2608: (health)e(exp)s(ectancy)g(of)g(state)g Fk(j)40 b Fn(at)35
2609: b(age)g(x.)53 b(Its)34 b(estimator)i Fk(e)3135 914 y
2610: Fi(j)3135 970 y(x)3179 958 y Fj(\()3221 934 y(^)3214
2611: 958 y Fk(\022)r Fj(\))f Fn(is)270 1071 y(the)k(same)f(linear)h(com)m
2612: (bination)i(of)d Fk(e)1603 1027 y Fi(ij)1603 1083 y(x)1664
2613: 1071 y Fj(\()1706 1047 y(^)1699 1071 y Fk(\022)s Fj(\))p
2614: Fn(,)i Fk(i)p Fn(=\(1,2\),)i(and)d(as)f(suc)m(h)h(is)f(asymptotically)
2615: 270 1184 y(normally)31 b(distributed)h(with)f(mean)g
2616: Fk(e)1601 1140 y Fi(j)1601 1196 y(x)1675 1184 y Fn(and)g(v)-5
2617: b(ariance)660 1516 y Fj(V)d(ar\()p Fk(e)878 1479 y Fi(j)878
2618: 1539 y(x)923 1516 y Fj(\()965 1492 y(^)958 1516 y Fk(\022)r
2619: Fj(\)\))26 b(=)1243 1403 y Ff(2)1195 1430 y Fg(X)1204
2620: 1626 y Fi(i)p Ff(=1)1327 1516 y Fj(\()p Fk(\031)1417
2621: 1479 y Fi(i)1414 1539 y(x)1458 1516 y Fj(\))1493 1479
2622: y Ff(2)1548 1516 y Fj(V)-8 b(ar)p Fk(e)1731 1479 y Fi(ij)1731
2623: 1539 y(x)1792 1516 y Fj(\()1834 1492 y(^)1827 1516 y
2624: Fk(\022)s Fj(\))20 b(+)g(2)p Fk(\031)2119 1479 y Ff(1)2116
2625: 1539 y Fi(x)2160 1516 y Fk(\031)2215 1479 y Ff(2)2212
2626: 1539 y Fi(x)2256 1516 y Fk(C)7 b(ov)s Fj(\()p Fk(e)2496
2627: 1479 y Ff(1)p Fi(j)2496 1539 y(x)2568 1516 y Fj(\()2610
2628: 1492 y(^)2603 1516 y Fk(\022)s Fj(\))p Fk(;)15 b(e)2766
2629: 1479 y Ff(2)p Fi(j)2766 1539 y(x)2839 1516 y Fj(\()2881
2630: 1492 y(^)2874 1516 y Fk(\022)r Fj(\)\))p Fk(:)411 1776
2631: y Fn(In)28 b(fact)g Fk(e)740 1743 y Ff(1)740 1798 y Fi(x)785
2632: 1776 y Fj(\()827 1752 y(^)820 1776 y Fk(\022)r Fj(\))h
2633: Fn(and)f Fk(e)1144 1743 y Ff(2)1144 1798 y Fi(x)1188
2634: 1776 y Fj(\()1230 1752 y(^)1223 1776 y Fk(\022)s Fj(\))h
2635: Fn(are)f(asymptotically)h(normally)g(distributed)h(with)f(co)m(v)-5
2636: b(ari-)270 1888 y(ance)401 2160 y Fk(C)7 b(ov)s Fj(\()p
2637: Fk(e)641 2122 y Ff(1)641 2182 y Fi(x)685 2160 y Fj(\()727
2638: 2136 y(^)720 2160 y Fk(\022)s Fj(\))p Fk(;)15 b(e)883
2639: 2122 y Ff(2)883 2182 y Fi(x)928 2160 y Fj(\()970 2136
2640: y(^)963 2160 y Fk(\022)r Fj(\)\))84 b(=)f(\()p Fk(\031)1406
2641: 2122 y Fi(i)1403 2182 y(x)1447 2160 y Fj(\))1482 2122
2642: y Ff(2)1521 2160 y Fk(C)7 b(ov)s Fj(\()p Fk(e)1761 2122
2643: y Ff(11)1761 2182 y Fi(x)1837 2160 y Fj(\()1879 2136
2644: y(^)1872 2160 y Fk(\022)r Fj(\))p Fk(;)15 b(e)2034 2122
2645: y Ff(12)2034 2182 y Fi(x)2110 2160 y Fj(\()2152 2136
2646: y(^)2145 2160 y Fk(\022)s Fj(\)\))1162 2320 y(+)83 b(2)p
2647: Fk(\031)1416 2283 y Ff(1)1413 2343 y Fi(x)1457 2320 y
2648: Fk(\031)1512 2283 y Ff(2)1509 2343 y Fi(x)1568 2219 y
2649: Fg(h)1610 2320 y Fk(C)7 b(ov)s Fj(\()p Fk(e)1850 2283
2650: y Ff(11)1850 2343 y Fi(x)1926 2320 y Fj(\()1968 2296
2651: y(^)1961 2320 y Fk(\022)r Fj(\))p Fk(;)15 b(e)2123 2283
2652: y Ff(22)2123 2343 y Fi(x)2199 2320 y Fj(\()2241 2296
2653: y(^)2234 2320 y Fk(\022)s Fj(\)\))21 b(+)e Fk(C)7 b(ov)s
2654: Fj(\()p Fk(e)2701 2283 y Ff(12)2701 2343 y Fi(x)2777
2655: 2320 y Fj(\()2819 2296 y(^)2812 2320 y Fk(\022)r Fj(\))p
2656: Fk(;)15 b(e)2974 2283 y Ff(21)2974 2343 y Fi(x)3050 2320
2657: y Fj(\()3092 2296 y(^)3085 2320 y Fk(\022)s Fj(\)\))3201
2658: 2219 y Fg(i)1162 2500 y Fj(+)83 b(\()p Fk(\031)1406 2462
2659: y Ff(2)1403 2522 y Fi(x)1447 2500 y Fj(\))1482 2462 y
2660: Ff(2)1521 2500 y Fk(C)7 b(ov)s Fj(\()p Fk(e)1761 2462
2661: y Ff(21)1761 2522 y Fi(x)1837 2500 y Fj(\()1879 2476
2662: y(^)1872 2500 y Fk(\022)r Fj(\))p Fk(;)15 b(e)2034 2462
2663: y Ff(22)2034 2522 y Fi(x)2110 2500 y Fj(\()2152 2476
2664: y(^)2145 2500 y Fk(\022)s Fj(\)\))p Fk(:)411 2658 y Fn(These)32
2665: b(form)m(ulae)i(apply)e(if)g Fk(\031)1438 2625 y Ff(1)1435
2666: 2680 y Fi(x)1510 2658 y Fn(and)h Fk(\031)1743 2625 y
2667: Ff(2)1740 2680 y Fi(x)1815 2658 y Fn(are)g(kno)m(wn.)46
2668: b(Ho)m(w)m(ev)m(er,)34 b(this)e(will)h(generally)270
2669: 2771 y(not)c(b)s(e)g(the)h(case)f(and)g(the)g(prop)s(ortions)h(will)g
2670: (ha)m(v)m(e)g(to)f(b)s(e)g(estimated)h(from)f(the)g(sample.)270
2671: 2884 y(One)i(w)m(a)m(y)h(of)f(pro)s(ceeding)h(is)f(to)g(use)g(the)g
2672: (age)h(sp)s(eci\034c)f(prop)s(ortions.)43 b(Recall)31
2673: b(that)h(these)270 2997 y(are)24 b(obtained)g(from)g(p)s(opulation)h
2674: (surv)m(eys)e(and)g(are)h(commonly)h(used)e(to)h(estimate)g(health)270
2675: 3110 y(exp)s(ectancies)41 b(from)g(cross-sectional)h(data.)72
2676: b(W)-8 b(e)41 b(ha)m(v)m(e)h(c)m(hosen)g(not)f(to)g(use)g(observ)m(ed)
2677: 270 3223 y(prev)-5 b(alences)33 b(but)f(rather)h(what)g(w)m(e)g(called)
2678: g(quasi-stationary)g(prev)-5 b(alences.)46 b(They)32
2679: b(ma)m(y)270 3335 y(b)s(e)e(of)g(indep)s(enden)m(t)i(in)m(terest)g(and)
2680: e(are)h(discussed)f(in)h(the)g(next)f(section.)270 3572
2681: y Fl(2.3)112 b(Stationary)37 b(prev)-6 b(alences)411
2682: 3743 y Fn(Consider)40 b(inhomogeneous)g(pro)s(ducts)f(of)f(non-negativ)
2683: m(e)j(sto)s(c)m(hastic)e(matrices)g(ob-)270 3856 y(tained)g(from)g(a)f
2684: (mark)m(o)m(v)i(pro)s(cess)e(de\034ned)i(on)e(2)h(non-absorbing)h
2685: (states.)65 b(W)-8 b(e)39 b(are)g(in-)270 3969 y(terested)33
2686: b(in)g(the)f(b)s(eha)m(viour)i(b)s(efore)e(absorption)h(of)f(the)h
2687: (transition)h(matrices)3065 3983 y Fi(t)3094 3969 y Fk(P)3152
2688: 3983 y Fi(x)p Fb(\000)p Fi(t)3309 3969 y Fn(as)270 4082
2689: y Fk(t)25 b Fh(!)g(1)p Fn(.)411 4264 y(Let)1342 4467
2690: y Fi(t)1372 4453 y Fk(w)r Fj(\()p Fk(x)p Fj(\))h(=)1683
2691: 4325 y Fg(\022)1750 4411 y Fi(t)1780 4397 y Fk(w)1847
2692: 4364 y Fi(:)p Ff(1)1906 4397 y Fj(\()p Fk(x)p Fj(\))1750
2693: 4524 y Fi(t)1780 4510 y Fk(w)1847 4477 y Fi(:)p Ff(2)1906
2694: 4510 y Fj(\()p Fk(x)p Fj(\))2029 4325 y Fg(\023)2126
2695: 4453 y Fk(:)270 4673 y Fn(b)s(e)h(the)i(conditional)g(distribution)g
2696: (at)f(age)g Fk(x)p Fn(.)40 b(The)28 b(quan)m(tit)m(y)2423
2697: 4687 y Fi(t)2452 4673 y Fk(w)2519 4640 y Fi(:)p Ff(1)2579
2698: 4673 y Fj(\()p Fk(x)p Fj(\))g Fn(is)g(the)g(prop)s(ortion)270
2699: 4786 y(of)36 b(individuals)h(in)g(state)f(1)g(at)h(age)g
2700: Fk(x)f Fn(and)g(can)h(b)s(e)f(expressed)h(in)f(terms)h(of)3009
2701: 4800 y Fi(t)3038 4786 y Fk(p)3084 4753 y Fi(i)p Ff(1)3084
2702: 4808 y Fi(x)p Fb(\000)p Fi(t)3244 4786 y Fn(and)270 4913
2703: y Fi(t)300 4899 y Fk(p)346 4866 y Fi(i)p Ff(2)346 4921
2704: y Fi(x)p Fb(\000)p Fi(t)499 4899 y Fn(\()p Fk(i)25 b
2705: Fj(=)g(1)p Fk(;)15 b Fj(2)p Fn(\))31 b(that)e(are)h(the)f
2706: (probabilities)i(to)e(b)s(e,)g(resp)s(ectiv)m(ely)-8
2707: b(,)30 b(in)g(state)f(1)g(and)h(2)f(at)270 5012 y(age)i
2708: Fk(x)p Fn(.)40 b(It)30 b(follo)m(ws)h(that)1345 5239
2709: y Fi(t)1374 5225 y Fk(w)1441 5187 y Fi(:)p Ff(1)1501
2710: 5225 y Fj(\()p Fk(x)p Fj(\))26 b(=)1910 5174 y Fi(t)1940
2711: 5160 y Fk(p)1986 5127 y Fi(i)p Ff(1)1986 5183 y Fi(x)p
2712: Fb(\000)p Fi(t)p 1755 5204 511 4 v 1755 5304 a(t)1784
2713: 5290 y Fk(p)1830 5259 y Fi(i)p Ff(1)1830 5314 y Fi(x)p
2714: Fb(\000)p Fi(t)1974 5290 y Fj(+)2045 5304 y Fi(t)2095
2715: 5290 y Fk(p)2141 5259 y Fi(i)p Ff(2)2141 5314 y Fi(x)p
2716: Fb(\000)p Fi(t)1345 5510 y(t)1374 5496 y Fk(w)1441 5459
2717: y Fi(:)p Ff(2)1501 5496 y Fj(\()p Fk(x)p Fj(\))g(=)1910
2718: 5445 y Fi(t)1940 5431 y Fk(p)1986 5398 y Fi(i)p Ff(2)1986
2719: 5454 y Fi(x)p Fb(\000)p Fi(t)p 1755 5475 V 1755 5575
2720: a(t)1784 5561 y Fk(p)1830 5530 y Fi(i)p Ff(1)1830 5585
2721: y Fi(x)p Fb(\000)p Fi(t)1974 5561 y Fj(+)2045 5575 y
2722: Fi(t)2095 5561 y Fk(p)2141 5530 y Fi(i)p Ff(2)2141 5585
2723: y Fi(x)p Fb(\000)p Fi(t)2290 5496 y Fk(:)1807 5841 y
2724: Fn(6)p eop
2725: %%Page: 7 7
2726: 7 6 bop 411 399 a Fn(W)-8 b(e)37 b(call)745 413 y Fi(t)775
2727: 399 y Fk(Q)847 413 y Fi(x)p Fb(\000)p Fi(t)1007 399 y
2728: Fn(a)g(submatrix)g(\(2)g(x)f(2\))h(of)f(the)h(transition)g(matrix)g
2729: (pro)s(duct)3178 413 y Fi(t)3207 399 y Fk(P)3265 413
2730: y Fi(x)p Fb(\000)p Fi(t)270 511 y Fn(de\034ned)31 b(in)g(\(3\).)1238
2731: 783 y Fi(t)1267 769 y Fk(Q)1339 783 y Fi(x)p Fb(\000)p
2732: Fi(t)1489 769 y Fj(=)1585 641 y Fg(\022)1652 727 y Fi(t)1681
2733: 713 y Fk(p)1727 680 y Ff(11)1727 735 y Fi(x)p Fb(\000)p
2734: Fi(t)1934 727 y(t)1964 713 y Fk(p)2010 680 y Ff(12)2010
2735: 735 y Fi(x)p Fb(\000)p Fi(t)1652 840 y(t)1681 826 y Fk(p)1727
2736: 793 y Ff(21)1727 848 y Fi(x)p Fb(\000)p Fi(t)1934 840
2737: y(t)1964 826 y Fk(p)2010 793 y Ff(22)2010 848 y Fi(x)p
2738: Fb(\000)p Fi(t)2134 641 y Fg(\023)2231 769 y Fk(:)411
2739: 1027 y Fn(If)501 1041 y Fi(t)531 1027 y Fk(w)r Fj(\()p
2740: Fk(x)p Fj(\))g Fn(=)f Fk(w)r Fj(\()p Fk(x)p Fj(\))i Fn(for)e(all)g
2741: Fk(t)25 b Fh(\025)g Fj(0)30 b Fn(then)h Fk(w)r Fj(\()p
2742: Fk(x)p Fj(\))g Fn(is)f(a)g(stationary)h(conditional)h(distribu-)270
2743: 1140 y(tion.)41 b(Giv)m(en)31 b(times)f Fk(t)1027 1154
2744: y Ff(1)1097 1140 y Fn(and)h Fk(t)1306 1154 y Ff(2)1345
2745: 1140 y Fn(,)f(w)m(e)h(can)g(deduce)g(from)g(the)f(pro)s(cess)g(that)
2746: 1032 1359 y Fi(t)1057 1368 y Fc(1)1093 1359 y Ff(+)p
2747: Fi(t)1173 1368 y Fc(2)1212 1345 y Fk(p)1258 1300 y Fi(ij)1258
2748: 1369 y(x)p Fb(\000)p Fi(t)1378 1378 y Fc(1)1412 1369
2749: y Fb(\000)p Fi(t)1492 1378 y Fc(2)1556 1345 y Fj(=)1668
2750: 1359 y Fi(t)1693 1368 y Fc(1)1732 1345 y Fk(p)1778 1300
2751: y Fi(ij)1778 1369 y(x)p Fb(\000)p Fi(t)1898 1378 y Fc(1)1972
2752: 1345 y Fh(\002)2078 1359 y Fi(t)2103 1368 y Fc(2)2142
2753: 1345 y Fk(p)2188 1300 y Fi(ij)2188 1369 y(x)p Fb(\000)p
2754: Fi(t)2308 1378 y Fc(1)2342 1369 y Fb(\000)p Fi(t)2422
2755: 1378 y Fc(2)270 1549 y Fn(where)h Fk(t)564 1563 y Ff(1)603
2756: 1549 y Fn(,)g Fk(t)692 1563 y Ff(2)756 1549 y Fh(\025)25
2757: b Fj(0)31 b Fn(and)g Fk(j)f Fj(=)25 b(1)p Fk(;)15 b Fj(2)p
2758: Fn(.)411 1761 y(F)-8 b(rom)25 b(this)f(form)m(ula,)j(w)m(e)e(can)g(mak)
2759: m(e)f(the)h(assumption)g(that)g(a)f(function)h Fk(\025)f
2760: Fn(dep)s(ending)270 1874 y(of)30 b Fk(t)406 1888 y Ff(1)445
2761: 1874 y Fn(,)p Fk(t)503 1888 y Ff(2)573 1874 y Fn(and)h
2762: Fk(x)f Fn(exists)f(and)i(satisfy)804 2092 y Fi(t)829
2763: 2101 y Fc(1)869 2078 y Fk(w)936 2041 y Fb(0)959 2078
2764: y Fj(\()p Fk(x)p Fj(\))36 b Fh(\002)1223 2092 y Fi(t)1248
2765: 2101 y Fc(2)1288 2078 y Fk(Q)1360 2092 y Fi(x)p Fb(\000)p
2766: Fi(t)1480 2101 y Fc(1)1514 2092 y Fb(\000)p Fi(t)1594
2767: 2101 y Fc(2)1659 2078 y Fj(=)25 b Fk(\025)p Fj(\()p Fk(t)1876
2768: 2092 y Ff(1)1915 2078 y Fk(;)15 b(t)1988 2092 y Ff(2)2028
2769: 2078 y Fk(;)g(x)p Fj(\))36 b Fh(\002)2297 2092 y Fi(t)2322
2770: 2101 y Fc(1)2357 2092 y Ff(+)p Fi(t)2437 2101 y Fc(2)2476
2771: 2078 y Fk(w)2543 2041 y Fb(0)2567 2078 y Fj(\()p Fk(x)p
2772: Fj(\))411 2283 y Fn(If)30 b(suc)m(h)g(a)h Fk(w)r Fj(\()p
2773: Fk(x)p Fj(\))g Fn(exists)f(as)g Fk(t)1395 2297 y Ff(1)1434
2774: 2283 y Fn(,)g Fk(t)1522 2297 y Ff(2)1587 2283 y Fh(!)25
2775: b(1)p Fn(,)30 b Fk(w)r Fj(\()p Fk(x)p Fj(\))h Fn(should)g(v)m(erify)
2776: 1257 2487 y Fk(w)1324 2449 y Fb(0)1348 2487 y Fj(\()p
2777: Fk(x)p Fj(\))g Fk(Q)p Fj(\()p Fk(x)p Fj(\))26 b(=)f Fk(\025)p
2778: Fj(\()p Fk(x)p Fj(\))31 b Fk(w)2090 2449 y Fb(0)2114
2779: 2487 y Fj(\()p Fk(x)p Fj(\))411 2691 y Fn(Since)38 b
2780: Fk(Q)p Fj(\()p Fk(x)p Fj(\))g Fn(is)f(a)g(non-negativ)m(e)i(substo)s(c)
2781: m(hastic)f(matrix,)i(it)d(follo)m(ws)h(that)g Fk(Q)p
2782: Fj(\()p Fk(x)p Fj(\))f Fn(is)270 2804 y(irreductible.)82
2783: b(Because)44 b Fk(w)r Fj(\()p Fk(x)p Fj(\))g Fn(is)f(non-negativ)m(e,)
2784: 49 b(w)m(e)44 b(can)g(assert,)j(according)e(to)e(the)270
2785: 2917 y(theorem)28 b(of)f(P)m(erron-F)-8 b(rob)s(enius)30
2786: b(\(Theorem)f(3.4)e(in)h(Seneta)g([2]\),)g(that)g Fk(\025)p
2787: Fj(\()p Fk(x)p Fj(\))f Fn(is)g(the)h(max-)270 3030 y(im)m(um)38
2788: b(eigen)m(v)-5 b(alue)37 b(of)f Fk(Q)p Fj(\()p Fk(x)p
2789: Fj(\))p Fn(,)j Fk(\025)1387 3044 y Ff(1)1462 3030 y Fn(sa)m(y)-8
2790: b(,)39 b(and)d(that)h Fk(w)r Fj(\()p Fk(x)p Fj(\))p Fn(=)h
2791: Fk(v)s Fj(\()p Fk(x)p Fj(\))f Fn(where)g Fk(v)s Fj(\()p
2792: Fk(x)p Fj(\))g Fn(is)f(the)h(left)270 3143 y(eigen)m(v)m(ector)g
2793: (\(normalised)g(to)e(mak)m(e)h Fk(v)1645 3110 y Ff(1)1685
2794: 3143 y Fj(\()p Fk(x)p Fj(\))24 b(+)f Fk(v)1972 3110 y
2795: Ff(2)2012 3143 y Fj(\()p Fk(x)p Fj(\))34 b(=)f(1)p Fn(\))i(of)g
2796: Fk(Q)p Fj(\()p Fk(x)p Fj(\))g Fn(corresp)s(onding)i(to)270
2797: 3256 y Fk(\025)323 3270 y Ff(1)362 3256 y Fn(.)411 3369
2798: y(Then)31 b(the)g(stationary)g(prev)-5 b(alence)31 b(of)f(disabilit)m
2799: (y)h(at)g(age)g Fk(x)f Fn(is)g(de\034ned)h(to)g(b)s(e)1409
2800: 3686 y Fk(w)1476 3648 y Ff(2)1516 3686 y Fj(\()p Fk(x)p
2801: Fj(\))26 b(=)45 b(lim)1760 3742 y Fi(t)p Fb(!1)1942 3700
2802: y Fi(t)1972 3686 y Fk(w)2039 3648 y Fi(i)p Ff(2)2103
2803: 3686 y Fj(\()p Fk(x)p Fj(\))p Fk(;)1040 b Fn(\(9\))411
2804: 3913 y(pro)m(vided)35 b(the)g(limit)f(exists)g(indep)s(enden)m(tly)h
2805: (of)f Fk(i)p Fn(.)52 b(F)-8 b(rom)35 b(a)e(demographic)j(p)s(oin)m(t)f
2806: (of)270 4026 y(view)c(the)g(limit)h(in)f(\(9\))g(is)g(a)g(useful)g
2807: (\034ction)h(\(h)m(uman)g(ages)f(cannot)h(b)s(e)f(negativ)m(e\))h(and)g
2808: (it)270 4139 y(should)f(b)s(e)f(in)m(terpreted)i(as)1377
2809: 4470 y Fi(t)1407 4456 y Fk(w)1474 4419 y Fi(i)p Ff(2)1538
2810: 4456 y Fj(\()p Fk(x)p Fj(\))26 b(=)f Fk(w)1849 4419 y
2811: Ff(1)1889 4456 y Fj(\()p Fk(x)p Fj(\))c(+)f Fk(o)p Fj(\(1\))411
2812: 4660 y Fn(where)37 b(the)g Fk(o)p Fj(\(1\))g Fn(term)g(b)s(ecomes)f
2813: (negligible)i(ev)m(en)f(for)f(mo)s(derate)h(v)-5 b(alues)36
2814: b(of)g Fk(t)f Fn(and)270 4773 y(p)s(ositiv)m(e)c Fk(x)20
2815: b Fh(\000)g Fk(t)p Fn(.)270 5017 y Fl(2.4)112 b(Health)45
2816: b(exp)s(ectancies)h(estimated)e(with)g(the)i(stationary)f(prev)-6
2817: b(a-)525 5133 y(lences)411 5304 y Fn(A)46 b(consequence)i(of)e(these)h
2818: (considerations)h(is)f(that)g(the)g(\(marginal\))h(health)g(ex-)270
2819: 5417 y(p)s(ectancy)41 b(of)f(state)h Fk(j)47 b Fn(at)41
2820: b(age)g Fk(x)p Fn(,)i(giv)m(en)e(only)g(that)g(the)h(individual)f(is)g
2821: (aliv)m(e)g(at)g Fk(x)p Fn(,)i(is)270 5530 y(as)33 b(in)h(\(8\))g(with)
2822: h(the)f(prev)-5 b(alence)35 b Fk(\031)1509 5497 y Ff(2)1506
2823: 5553 y Fi(x)1583 5530 y Fn(replaced)g(b)m(y)f Fk(w)2139
2824: 5497 y Ff(2)2179 5530 y Fj(\()p Fk(x)p Fj(\))p Fn(.)51
2825: b(W)-8 b(riting)35 b Fk(w)2779 5497 y Ff(2)2819 5530
2826: y Fj(\()p Fk(x;)15 b(\022)2952 5590 y Ff(~)2992 5530
2827: y Fj(\))32 b(=)e Fk(w)3227 5497 y Ff(2)3267 5530 y Fj(\()p
2828: Fk(x)p Fj(\))1807 5841 y Fn(7)p eop
2829: %%Page: 8 8
2830: 8 7 bop 270 399 a Fn(to)37 b(indicate)h(the)f(parametric)i(dep)s
2831: (endence)f(of)e(this)h(quasi-stationary)g(prev)-5 b(alence,)40
2832: b(w)m(e)270 511 y(ha)m(v)m(e)31 b(for)g Fk(j)5 b Fn(=\(1,2\),)934
2833: 856 y Fk(e)976 818 y Fi(j)976 878 y(x)1020 856 y Fj(\()p
2834: Fk(\022)s Fj(\))25 b(=)1257 728 y Fg(\024)1305 856 y
2835: Fj(1)c Fh(\000)f Fk(w)1529 818 y Ff(2)1568 856 y Fj(\()p
2836: Fk(x;)15 b(\022)1701 915 y Ff(~)1741 856 y Fj(\))1776
2837: 728 y Fg(\025)1840 856 y Fk(e)1882 818 y Ff(1)p Fi(j)1882
2838: 878 y(x)1954 856 y Fj(\()p Fk(\022)s Fj(\))20 b(+)g Fk(w)2248
2839: 818 y Ff(2)2288 856 y Fj(\()p Fk(x;)15 b(\022)2421 915
2840: y Ff(~)2461 856 y Fj(\))p Fk(e)2538 818 y Ff(2)p Fi(j)2538
2841: 878 y(x)2610 856 y Fj(\()p Fk(\022)s Fj(\))503 b Fn(\(10\))411
2842: 1104 y(It)30 b(is)f(these)h(health)h(exp)s(ectancies)g(that)f(w)m(e)h
2843: (tak)m(e)f(as)g(the)g(marginal)i(exp)s(ectancies)e(of)270
2844: 1217 y(in)m(terest.)411 1430 y(The)24 b(estimate)g Fk(e)987
2845: 1386 y Fi(j)987 1442 y(x)1032 1430 y Fj(\()1074 1406
2846: y(^)1067 1430 y Fk(\022)r Fj(\))g Fn(of)f Fk(e)1309 1386
2847: y Fi(j)1309 1442 y(x)1353 1430 y Fj(\()p Fk(\022)s Fj(\))g
2848: Fn(is)g(obtained)i(b)m(y)f(replacing)h Fk(\022)2445 1490
2849: y Ff(~)2508 1430 y Fn(with)2707 1406 y Fj(^)2707 1430
2850: y Fk(\022)2712 1490 y Ff(~)2776 1430 y Fn(in)e(\(10\).)39
2851: b(A)23 b(reali-)270 1595 y(sation)h(of)f Fk(e)668 1551
2852: y Fi(j)668 1607 y(x)712 1595 y Fj(\()754 1571 y(^)747
2853: 1595 y Fk(\022)s Fj(\))g Fn(can)h(b)s(e)f(calculated)i(without)g
2854: (di\036cult)m(y)g(once)f(the)f(maxim)m(um)i(lik)m(eliho)s(o)s(d)270
2855: 1708 y(estimation)f(of)f Fk(\022)806 1768 y Ff(~)869
2856: 1708 y Fn(has)h(b)s(een)f(done)h(but)f(its)g(v)-5 b(ariance)24
2857: b(m)m(ust)g(b)s(e)f(estimated)h(indirectly)h(since)270
2858: 1861 y Fk(w)337 1828 y Ff(2)377 1861 y Fj(\()p Fk(x;)15
2859: b(\022)510 1921 y Ff(~)550 1861 y Fj(\))30 b Fn(is)g(not)h(kno)m(wn)g
2860: (explicitly)-8 b(.)42 b(Pro)s(ceed)31 b(as)f(follo)m(ws.)41
2861: b(Let)808 2161 y Fk(e)850 2124 y Fi(j)850 2184 y(x)894
2862: 2161 y Fj(\()p Fk(i;)15 b(t;)g(\022)s Fj(\))26 b(=)1276
2863: 2033 y Fg(\024)1324 2161 y Fj(1)20 b Fh(\000)1460 2175
2864: y Fi(t)1510 2161 y Fk(w)1577 2124 y Fi(i)p Ff(2)1641
2865: 2161 y Fj(\()p Fk(x;)15 b(\022)1774 2221 y Ff(~)1814
2866: 2161 y Fj(\))1849 2033 y Fg(\025)1912 2161 y Fk(e)1954
2867: 2124 y Ff(1)p Fi(j)1954 2184 y(x)2027 2161 y Fj(\()p
2868: Fk(\022)s Fj(\))20 b(+)2234 2175 y Fi(t)2283 2161 y Fk(w)2350
2869: 2124 y Fi(i)p Ff(2)2414 2161 y Fj(\()p Fk(x;)15 b(\022)2547
2870: 2221 y Ff(~)2587 2161 y Fj(\))p Fk(e)2664 2124 y Ff(2)p
2871: Fi(j)2664 2184 y(x)2737 2161 y Fj(\()p Fk(\022)s Fj(\))270
2872: 2435 y Fn(the)27 b(quan)m(tit)m(y)h(whose)e(limit)i(as)e
2873: Fk(t)f Fh(!)g(1)h Fn(is)g Fk(e)1782 2391 y Fi(j)1782
2874: 2447 y(x)1826 2435 y Fj(\()p Fk(\022)s Fj(\))h Fn(of)f(\(10\).)40
2875: b(With)26 b Fk(e)2564 2391 y Fi(j)2564 2447 y(x)2608
2876: 2435 y Fj(\()p Fk(i;)15 b(t;)g(\022)s Fj(\))28 b Fn(its)e(estimator)270
2877: 2548 y(w)m(e)31 b(ha)m(v)m(e,)h(using)e(the)h(de\034nitions)g(of)f(the)
2878: h Fk(e)1767 2504 y Fi(ij)1767 2560 y(x)1828 2548 y Fj(\()p
2879: Fk(\022)s Fj(\))f Fn(and)h(rearranging,)736 2912 y Fk(e)778
2880: 2874 y Fi(j)778 2934 y(x)822 2912 y Fj(\()p Fk(i;)15
2881: b(t;)1010 2888 y Fj(^)1001 2912 y Fk(\022)t Fj(\))20
2882: b Fh(\000)g Fk(e)1236 2874 y Fi(j)1236 2934 y(x)1280
2883: 2912 y Fj(\()p Fk(i;)15 b(t;)g(\022)s Fj(\))84 b(=)1808
2884: 2798 y Fb(1)1778 2826 y Fg(X)1780 3020 y Fi(y)r Ff(=0)1925
2885: 2926 y Fi(y)1966 2912 y Fk(p)2012 2874 y Ff(1)p Fi(j)2083
2886: 2943 y(x)2127 2912 y Fj(\()2169 2888 y(^)2162 2912 y
2887: Fk(\022)s Fj(\))20 b Fh(\000)2354 2926 y Fi(y)2396 2912
2888: y Fk(p)2442 2874 y Ff(1)p Fi(j)2513 2943 y(x)2557 2912
2889: y Fj(\()p Fk(\022)s Fj(\))1624 3180 y(+)1778 3194 y Fi(t)1808
2890: 3180 y Fk(w)1875 3142 y Fi(i)p Ff(2)1939 3180 y Fj(\()p
2891: Fk(x;)2074 3156 y Fj(^)2066 3180 y Fk(\022)s Fj(\))2162
2892: 3079 y Fg(h)2205 3194 y Fi(y)2246 3180 y Fk(p)2292 3142
2893: y Ff(2)p Fi(j)2292 3202 y(x)2364 3180 y Fj(\()2406 3156
2894: y(^)2399 3180 y Fk(\022)s Fj(\))g Fh(\000)2571 3194 y
2895: Fi(y)2632 3180 y Fk(p)2678 3142 y Ff(1)p Fi(j)2678 3202
2896: y(x)2750 3180 y Fj(\()2792 3156 y(^)2785 3180 y Fk(\022)s
2897: Fj(\))2866 3079 y Fg(i)1624 3353 y Fh(\000)1778 3367
2898: y Fi(t)1808 3353 y Fk(w)1875 3316 y Fi(i)p Ff(2)1939
2899: 3353 y Fj(\()p Fk(x;)15 b(\022)s Fj(\))2162 3280 y Fg(\002)2200
2900: 3367 y Fi(y)2241 3353 y Fk(p)2287 3316 y Ff(2)p Fi(j)2287
2901: 3376 y(x)2359 3353 y Fj(\()p Fk(\022)s Fj(\))20 b Fh(\000)2566
2902: 3367 y Fi(y)2627 3353 y Fk(p)2673 3316 y Ff(1)p Fi(j)2673
2903: 3376 y(x)2745 3353 y Fj(\()p Fk(\022)s Fj(\))2861 3280
2904: y Fg(\003)411 3573 y Fn(No)m(w)39 b(use)g(the)g(T)-8
2905: b(a)m(ylor)40 b(appro)m(ximation)h(for)2003 3587 y Fi(y)2045
2906: 3573 y Fk(p)2091 3529 y Fi(ij)2091 3585 y(x)2151 3573
2907: y Fj(\()2193 3549 y(^)2186 3573 y Fk(\022)s Fj(\))d Fn(as)h(b)s(efore)f
2908: (and)h(also)g(that)g(for)270 3709 y Fi(t)300 3695 y Fk(w)367
2909: 3662 y Fi(i)p Ff(2)431 3695 y Fj(\()p Fk(x;)566 3671
2910: y Fj(^)558 3695 y Fk(\022)s Fj(\))p Fn(,)850 3956 y Fi(t)880
2911: 3942 y Fk(w)947 3904 y Fi(i)p Ff(2)1011 3942 y Fj(\()p
2912: Fk(x;)1146 3918 y Fj(^)1138 3942 y Fk(\022)s Fj(\))25
2913: b Fh(')1356 3956 y Fi(t)1385 3942 y Fk(w)1452 3904 y
2914: Fi(i)p Ff(2)1516 3942 y Fj(\()p Fk(x;)15 b(\022)s Fj(\))21
2915: b(+)f(\()1878 3918 y(^)1871 3942 y Fk(\022)i Fh(\000)e
2916: Fk(\022)s Fj(\))2108 3904 y Fb(0)2164 3880 y Fk(@)p 2141
2917: 3921 99 4 v 2141 4004 a(@)5 b(\022)2250 4008 y Fi(t)2279
2918: 3942 y Fk(w)2346 3904 y Fi(i)p Ff(2)2410 3942 y Fj(\()p
2919: Fk(x;)15 b(\022)s Fj(\))p Fk(:)411 4171 y Fn(Substitution)32
2920: b(of)e(these)g(appro)m(ximations)j(and)e(simpli\034cation)h(yields)708
2921: 4535 y Fk(e)750 4497 y Fi(j)750 4557 y(x)794 4535 y Fj(\()p
2922: Fk(i;)15 b(t;)982 4511 y Fj(^)973 4535 y Fk(\022)t Fj(\))20
2923: b Fh(\000)g Fk(e)1208 4497 y Fi(j)1208 4557 y(x)1252
2924: 4535 y Fj(\()p Fk(i;)15 b(t;)g(\022)s Fj(\))26 b Fh(')1680
2925: 4421 y Fb(1)1649 4448 y Fg(X)1651 4643 y Fi(y)r Ff(=0)1781
2926: 4535 y Fj(\()1823 4511 y(^)1816 4535 y Fk(\022)d Fh(\000)c
2927: Fk(\022)s Fj(\))2053 4497 y Fb(0)2109 4473 y Fk(@)p 2086
2928: 4514 V 2086 4597 a(@)5 b(\022)2195 4535 y(g)s Fj(\()p
2929: Fk(i;)15 b(t;)g(y)s(;)g(x;)g(j;)g(\022)s Fj(\))p Fk(;)411
2930: 4827 y Fn(where)630 5144 y Fk(g)s Fj(\()p Fk(i;)g(t;)g(y)s(;)g(x;)g(j;)
2931: g(\022)s Fj(\))28 b(=)1292 5158 y Fi(y)1359 5144 y Fk(p)1405
2932: 5106 y Ff(1)p Fi(j)1405 5166 y(x)1476 5144 y Fj(\()p
2933: Fk(\022)s Fj(\))20 b(+)1703 5158 y Fi(t)1733 5144 y Fk(w)1800
2934: 5106 y Fi(i)p Ff(2)1864 5144 y Fj(\()p Fk(x;)15 b(\022)s
2935: Fj(\))2087 5070 y Fg(\002)2125 5158 y Fi(y)2166 5144
2936: y Fk(p)2212 5106 y Ff(2)p Fi(j)2212 5166 y(x)2284 5144
2937: y Fj(\()p Fk(\022)s Fj(\))20 b Fh(\000)2491 5158 y Fi(y)2552
2938: 5144 y Fk(p)2598 5106 y Ff(1)p Fi(j)2598 5166 y(x)2670
2939: 5144 y Fj(\()p Fk(\022)s Fj(\))2786 5070 y Fg(\003)2839
2940: 5144 y Fk(:)411 5348 y Fn(Assuming)33 b(uniform)g(di\033eren)m
2941: (tiabilit)m(y)j(with)d(resp)s(ect)g(to)g Fk(\022)h Fn(and)g(pro)s
2942: (ceeding)g(to)e(the)270 5461 y(limit)f(as)f Fk(t)25 b
2943: Fh(!)g(1)p Fn(,)30 b(giv)m(es)h(the)g(k)m(ey)f(result)1807
2944: 5841 y(8)p eop
2945: %%Page: 9 9
2946: 9 8 bop 925 649 a Fk(e)967 612 y Fi(j)967 672 y(x)1012
2947: 649 y Fj(\()1054 625 y(^)1047 649 y Fk(\022)r Fj(\))21
2948: b Fh(\000)f Fk(e)1281 612 y Fi(j)1281 672 y(x)1325 649
2949: y Fj(\()p Fk(\022)s Fj(\))25 b Fh(')1607 536 y Fb(1)1577
2950: 563 y Fg(X)1579 757 y Fi(y)r Ff(=0)1708 649 y Fj(\()1750
2951: 625 y(^)1743 649 y Fk(\022)e Fh(\000)d Fk(\022)s Fj(\))1981
2952: 612 y Fb(0)2036 588 y Fk(@)p 2014 629 99 4 v 2014 712
2953: a(@)5 b(\022)2122 649 y(g)s Fj(\()p Fk(y)s(;)15 b(x;)g(j;)g(\022)s
2954: Fj(\))p Fk(;)663 b Fn(\(11\))411 942 y(with)729 1146
2955: y Fk(g)s Fj(\()p Fk(y)s(;)15 b(x;)g(j;)g(\022)s Fj(\))27
2956: b(=)1246 1160 y Fi(y)1313 1146 y Fk(p)1359 1108 y Ff(1)p
2957: Fi(j)1359 1168 y(x)1430 1146 y Fj(\()p Fk(\022)s Fj(\))21
2958: b(+)e Fk(w)1724 1108 y Ff(2)1764 1146 y Fj(\()p Fk(x;)c(\022)s
2959: Fj(\))1987 1072 y Fg(\002)2026 1160 y Fi(y)2067 1146
2960: y Fk(p)2113 1108 y Ff(2)p Fi(j)2113 1168 y(x)2185 1146
2961: y Fj(\()p Fk(\022)s Fj(\))20 b Fh(\000)2392 1160 y Fi(y)2453
2962: 1146 y Fk(p)2499 1108 y Ff(1)p Fi(j)2499 1168 y(x)2571
2963: 1146 y Fj(\()p Fk(\022)s Fj(\))2687 1072 y Fg(\003)2740
2964: 1146 y Fk(:)411 1365 y Fn(Assymptotic)35 b(normalit)m(y)h(of)f
2965: Fk(e)1505 1321 y Fi(j)1505 1377 y(x)1549 1365 y Fj(\()1591
2966: 1341 y(^)1584 1365 y Fk(\022)s Fj(\))g Fn(is)f(an)i(immediate)g
2967: (consequence)g(of)f(\(11\).)54 b(The)270 1478 y(second)31
2968: b(momen)m(ts)g(are)446 1775 y Fj(V)-8 b(ar\()p Fk(e)664
2969: 1789 y Fi(j)702 1775 y Fj(\()p Fk(x;)837 1751 y Fj(^)829
2970: 1775 y Fk(\022)s Fj(\)\))25 b Fh(')1082 1689 y Fg(X)1082
2971: 1883 y Fi(u)p Ff(=0)1228 1689 y Fg(X)1230 1883 y Fi(y)r
2972: Ff(=0)1375 1592 y Fg(0)1375 1756 y(@)1487 1713 y Fk(@)p
2973: 1464 1754 V 1464 1837 a(@)5 b(\022)1522 1897 y Ff(~)1572
2974: 1775 y Fk(g)s Fj(\()p Fk(u;)15 b(x;)g(j;)g(\022)1922
2975: 1835 y Ff(~)1962 1775 y Fj(\))1997 1592 y Fg(1)1997 1756
2976: y(A)2077 1615 y Fb(0)2116 1775 y Fk(V)20 b Fj(\()p Fk(\022)2229
2977: 1835 y Ff(~)2270 1775 y Fj(\))2320 1592 y Fg(0)2320 1756
2978: y(@)2432 1713 y Fk(@)p 2410 1754 V 2410 1837 a(@)5 b(\022)2468
2979: 1897 y Ff(~)2518 1775 y Fk(g)s Fj(\()p Fk(u;)15 b(x;)g(j;)g(\022)2868
2980: 1835 y Ff(~)2908 1775 y Fj(\))2943 1592 y Fg(1)2943 1756
2981: y(A)3038 1775 y Fk(;)166 b Fn(\(12\))270 2304 y Fk(C)7
2982: b(ov)s Fj(\()p Fk(e)510 2318 y Fi(j)547 2304 y Fj(\()p
2983: Fk(x;)682 2280 y Fj(^)674 2304 y Fk(\022)s Fj(\))p Fk(;)15
2984: b(e)837 2319 y Fi(k)881 2304 y Fj(\()p Fk(x;)1016 2280
2985: y Fj(^)1008 2304 y Fk(\022)s Fj(\)\))26 b Fh(')1261 2217
2986: y Fg(X)1261 2412 y Fi(u)p Ff(=0)1407 2217 y Fg(X)1409
2987: 2412 y Fi(y)r Ff(=0)1554 2121 y Fg(0)1554 2285 y(@)1666
2988: 2242 y Fk(@)p 1643 2283 V 1643 2366 a(@)5 b(\022)1701
2989: 2426 y Ff(~)1752 2304 y Fk(g)s Fj(\()p Fk(u;)15 b(x;)g(j;)g(\022)2101
2990: 2364 y Ff(~)2142 2304 y Fj(\))2177 2121 y Fg(1)2177 2285
2991: y(A)2257 2144 y Fb(0)2295 2304 y Fk(V)20 b Fj(\()p Fk(\022)2408
2992: 2364 y Ff(~)2449 2304 y Fj(\))2499 2121 y Fg(0)2499 2285
2993: y(@)2611 2242 y Fk(@)p 2589 2283 V 2589 2366 a(@)5 b(\022)2647
2994: 2426 y Ff(~)2697 2304 y Fk(g)s Fj(\()p Fk(u;)15 b(x;)g(k)s(;)g(\022)
2995: 3060 2364 y Ff(~)3100 2304 y Fj(\))3135 2121 y Fg(1)3135
2996: 2285 y(A)3230 2304 y Fk(:)411 2626 y Fn(The)29 b(reason)g(for)g(this)f
2997: (circuitous)i(route)f(to)g(\(12\))g(is)f(that)h(the)g(deriv)-5
2998: b(ativ)m(e)3006 2590 y Fi(@)p 2989 2605 76 4 v 2989 2658
2999: a(@)t(\022)3075 2626 y Fk(w)3142 2593 y Ff(2)3182 2626
3000: y Fj(\()p Fk(x;)15 b(\022)s Fj(\))270 2748 y Fn(cannot)21
3001: b(b)s(e)f(calculated)i(directly)-8 b(.)38 b(On)21 b(the)g(other)g(hand)
3002: g(calculating)2642 2712 y Fi(@)p 2625 2727 V 2625 2779
3003: a(@)t(\022)2711 2762 y(t)2740 2748 y Fk(w)2807 2715 y
3004: Fi(i)p Ff(2)2871 2748 y Fj(\()p Fk(x;)15 b(\022)s Fj(\))21
3005: b Fn(presen)m(ts)270 2861 y(no)28 b(problems)g(once)g(the)f(deriv)-5
3006: b(ativ)m(es)28 b(of)f(the)h(transition)g(probabilities)h(are)f(kno)m
3007: (wn.)40 b(Cal-)270 2974 y(culation)34 b(of)f(these)h(latter)g(deriv)-5
3008: b(ativ)m(es)33 b(is)g(a)g(precondition)i(for)e(estimating)i(second)e
3009: (mo-)270 3087 y(men)m(ts)d(and)g(w)m(as)g(done)g(n)m(umerically)h(for)f
3010: (the)g(data)g(set)f(after)h(the)f(maxim)m(um)i(lik)m(eliho)s(o)s(d)270
3011: 3199 y(estimator)686 3175 y Fj(^)679 3199 y Fk(\022)39
3012: b Fn(had)e(b)s(een)g(found.)60 b(Th)m(us)38 b(the)f(ev)-5
3013: b(aluation)38 b(of)f(\(12\))g(in)m(v)m(olv)m(es)h(calculating)270
3014: 3312 y(the)d(realised)g(v)-5 b(alue)34 b(of)g(the)h(deriv)-5
3015: b(ativ)m(e)35 b(of)1790 3326 y Fi(t)1820 3312 y Fk(w)1887
3016: 3279 y Fi(i)p Ff(2)1985 3312 y Fn(for)f(large)h Fk(t)p
3017: Fn(,)g(meaning)h(of)d(the)i(order)g(of)270 3425 y(200)c(mon)m(ths)g
3018: (for)f(our)h(data)g(set.)270 3712 y Fp(References)270
3019: 3915 y Fn([1])46 b(Laditk)-5 b(a)35 b(S.)f(and)g(W)-8
3020: b(olf)34 b(D.)52 b(New)34 b(metho)s(ds)g(for)g(analysing)g(activ)m(e)h
3021: (life)f(exp)s(ectancy)-8 b(.)411 4028 y Fa(Journal)33
3022: b(of)f(A)-5 b(ging)33 b(and)f(He)-5 b(alth)p Fn(,)32
3023: b(1997.)270 4215 y([2])46 b(Seneta)31 b(E.)41 b Fa(Non-ne)-5
3024: b(gative)34 b(matric)-5 b(es)p Fn(.)41 b(John)31 b(Wiley)f(and)h(Sons,)
3025: g(New)f(Y)-8 b(ork,)31 b(1973.)1807 5841 y(9)p eop
3026: %%Trailer
3027: end
3028: userdict /end-hook known{end-hook}if
3029: %%EOF
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