Annotation of imach096d/doc/docmath.ps, revision 1.1.1.1

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                   2366:  @endspecial 270 3303 a Fm(Figure)c(3:)36 b(F)-7 b(ollo)n(w-up)34
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                   2398: 4955 y Fc(1)1290 4974 y Fg(Y)1283 5168 y Fi(u)p Ff(=1)1429
                   2399: 5075 y Fi(h)1474 5060 y Fk(P)1532 5079 y Fi(x)1572 5088
                   2400: y Fc(1)1607 5079 y Ff(+\()p Fi(u)p Fb(\000)p Ff(1\))p
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                   2416: 399 y Fn(\()p Fk(x)681 413 y Ff(2)749 399 y Fj(=)29 b
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                   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
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                   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
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                   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
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                   2458: 2786 y Fn(is)270 2926 y(appro)m(ximativ)m(ely)j(normally)g(distributed)
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                   2465: 3405 y Fj(\))25 b(=)1925 3343 y(1)p 1906 3384 83 4 v
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                   2468: 3405 y Fj(\))15 b Fk(;)994 b Fn(\(5\))270 3641 y(where)25
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                   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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