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y(densities.)154 4423 y(B.)49 b(Bullo)s(c)m(k)32 b(Da)m(vis)g(\(1977\))f(prop)s(osed)j (letting)d Fj(h)d Fn(=)f(1)p Fj(=t)p Fn(,)33 b(where)h Fj(t)f Fn(is)f(the)h(smallest)e Fj(t)i Fn(for)f(whic)m(h)299 4556 y(the)23 b(estimate)f(of)g(the)h(optimal)d Ff(j)p Fj(')p Ff(j)1565 4519 y Fi(2)1627 4556 y Fn(equals)j(1)p Fj(=)p Fn(\()p Fj(n)r Fn(+)r(1\).)40 b(F)-8 b(or)21 b(the)j(F)-8 b(ourier)21 b(k)m(ernel)j(sin)o(\()p Fj(x)p Fn(\))p Fj(=)p Fn(\()p Fj(\031)t(x)p Fn(\),)299 4688 y(this)32 b(estimate)g(is)549 4945 y Ff(j)t Fg(b)-59 b Fj(')p Fn(\()p Fj(t)p Fn(\))p Ff(j)780 4904 y Fi(2)846 4945 y Fn(=)1045 4877 y Fj(n)p 960 4922 229 4 v 960 5013 a(n)22 b Ff(\000)h Fn(1)1215 4745 y Fg(0)1215 4924 y(@)1303 4774 y( )1396 4877 y Fn(1)p 1391 4922 59 4 v 1391 5013 a Fj(n)1527 4820 y Fh(n)1476 4850 y Fg(X)1491 5060 y Fh(i)p Fi(=1)1637 4945 y Fn(cos\()p Fj(tX)1921 4960 y Fh(i)1950 4945 y Fn(\))1988 4774 y Fg(!)2066 4797 y Fi(2)2128 4945 y Fn(+)2226 4774 y Fg( )2320 4877 y Fn(1)p 2315 4922 V 2315 5013 a Fj(n)2450 4820 y Fh(n)2400 4850 y Fg(X)2414 5060 y Fh(i)p Fi(=1)2560 4945 y Fn(sin\()p Fj(tX)2834 4960 y Fh(i)2862 4945 y Fn(\))2900 4774 y Fg(!)2979 4797 y Fi(2)3018 4745 y Fg(1)3018 4924 y(A)3128 4945 y Ff(\000)3327 4877 y Fn(1)p 3237 4922 229 4 v 3237 5013 a Fj(n)g Ff(\000)f Fn(1)3509 4945 y Fj(:)1872 5280 y Fl(2)p eop %%Page: 3 3 3 2 bop 299 83 a Fn(Blo)m(xom)27 b(\(1979\))h(pro)m(vides)h(some)g (encouraging)f(exp)s(erimen)m(tal)g(results)h(with)f(this)g(estimate.) 153 282 y(C.)49 b(W)-8 b(ah)m(ba)33 b(\(1981\))e(and)h(Hall)f(and)i (Marron)f(\(1988\))f(adaptiv)m(ely)h(estimate)g(parameters)g(of)g(the) 299 415 y(optimal)f(F)-8 b(ourier)33 b(transform)g(of)h(the)g(k)m (ernel)h(under)g(certain)f(tail)e(conditions)h(on)h(the)h(c)m(har-)299 548 y(acteristic)i(function)g(of)g Fj(f)11 b Fn(.)58 b(This)37 b(w)m(ould)h(require)f(kno)m(wledge)i(of)e(the)h(underlying)f (class)g(of)299 681 y(densities.)149 880 y(D.)48 b(Cline)g(\(1988\))g (uses)i(the)g(W)-8 b(atson-Leadb)s(etter)49 b(result)g(to)g(p)s(oin)m (t)f(out)g(that)h(the)h(optimal)299 1013 y(F)-8 b(ourier)45 b(transform)g(alw)m(a)m(ys)i(is)f(symmetric)f(and)h(p)s(ositiv)m(e.)84 b(As)47 b(the)g(F)-8 b(ourier)45 b(transform)299 1146 y(\(sa)m(y)-8 b(,)49 b Fj(\030)5 b Fn(\))44 b(of)h(the)g(Epanec)m(hnik) m(o)m(v)j(k)m(ernel)d(tak)m(es)h(negativ)m(e)g(v)-5 b(alues,)48 b(it)c(can)h(alw)m(a)m(ys)h(b)s(e)f(re-)299 1279 y(placed)34 b(b)m(y)h Fj(\030)784 1294 y Fi(+)843 1279 y Fn(,)f(its)g(p)s(ositiv)m (e)f(truncation,)h(for)g(a)f(strict)h(impro)m(v)m(emen)m(t)g(in)f(the)i (exp)s(ected)h Fj(L)3746 1294 y Fi(2)299 1411 y Fn(error.)157 1611 y(E.)49 b(Cline)28 b(\(1990\))f(giv)m(es)i(precise)h(asymptotic)e (analysis)f(of)i(the)g(exp)s(ected)h Fj(L)3001 1626 y Fi(2)3070 1611 y Fn(error)e(based)i(up)s(on)299 1743 y(the)j(F)-8 b(ourier)31 b(transform)h(approac)m(h.)299 1943 y(Finally)-8 b(,)22 b(based)i(up)s(on)f(recen)m(t)h(dev)m(elopmen) m(ts)g(in)e(data-based)h(bandwidth)h(selection,)g(sev)m(eral)0 2076 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3254 y Fj(h)27 b Fn(=)1551 3114 y Fg(\022)1787 3105 y(R)1870 3185 y Fj(K)1960 3149 y Fi(2)p 1634 3231 519 4 v 1634 3324 a Fj(C)7 b(n)p Fn(\()1807 3243 y Fg(R)1890 3324 y Fj(x)1945 3295 y Fi(2)1985 3324 y Fj(K)g Fn(\))2113 3295 y Fi(2)2163 3114 y Fg(\023)2236 3131 y Fi(1)p Fh(=)p Fi(5)2395 3254 y Fj(;)0 3484 y Fn(where)1203 3650 y Fj(C)35 b Fn(=)1426 3582 y(1)p 1421 3627 59 4 v 1421 3718 a Fj(\031)1507 3514 y Fg(Z)1606 3541 y Fi(\003)1562 3740 y(0)1676 3650 y Fj(t)1711 3609 y Fi(4)1751 3650 y Fn(\()p Ff(j)p Fj(')1881 3665 y Fh(n)1927 3650 y Fn(\()p Fj(t)p Fn(\))p Ff(j)2066 3609 y Fi(2)2127 3650 y Ff(\000)23 b Fn(1)p Fj(=n)p Fn(\))17 b Fj(dt)32 b(:)0 3859 y Fn(This)h(is)g (related)f(to)h(a)f(plug-in)f(metho)s(d)i(suggested)h(b)m(y)g(P)m(ark)g (and)f(Marron)g(\(1990\),)f(in)g(whic)m(h)h Fj(C)40 b Fn(is)0 3992 y(tak)m(en)34 b(as)1044 4141 y Fj(C)g Fn(=)1267 4073 y(1)p 1262 4118 V 1262 4209 a Fj(\031)1347 4005 y Fg(Z)1447 4031 y Fd(1)1403 4231 y Fi(0)1538 4141 y Fj(t)1573 4099 y Fi(2)1613 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Fi(1)840 3350 y Fj(<)g(t)978 3365 y Fi(2)1046 3350 y Fj(<)g Ff(\001)17 b(\001)g(\001)26 b Fj(<)i(t)1432 3365 y Fh(k)1501 3350 y Fn(with)d(the)i(prop)s(ert)m(y)g(that)f Fj(t)2510 3365 y Fi(1)2577 3350 y Fn(=)i Ff(\000)p Fj(\013)q Fn(,)g Fj(t)2911 3365 y Fh(k)2981 3350 y Fn(=)g Fj(\013)q Fn(,)f Ff(j)p Fj(t)3265 3365 y Fh(i)3302 3350 y Ff(\000)9 b Fj(t)3423 3365 y Fh(i)p Fi(+1)3542 3350 y Ff(j)28 b(\024)g Fj(\015)5 b Fn(.)0 3483 y(Clearly)-8 b(,)32 b(w)m(e)h(can)g(assure)h (this)e(with)g Fj(k)f Ff(\024)d Fn(1)22 b(+)g(2)p Fj(\013)q(=\015)5 b Fn(.)43 b(W)-8 b(e)33 b(b)s(egin)f(with)479 3748 y Fe(P)557 3577 y Fg(\()647 3748 y Fn(sup)638 3833 y Fd(j)p Fh(t)p Fd(j)p Fh(<\013)819 3748 y Ff(j)p Fj(')p Fn(\()p Fj(t)p Fn(\))22 b Ff(\000)g Fj(')1207 3763 y Fh(n)1254 3748 y Fn(\()p Fj(t)p Fn(\))p Ff(j)28 b Fj(>)f(\014)1585 3577 y Fg(\))1693 3748 y Ff(\024)h Fe(P)1877 3577 y Fg(\()2007 3748 y Fn(sup)1957 3833 y Fd(j)p Fh(t)p Fd(\000)p Fh(s)p Fd(j)p Fh(<\015)2221 3748 y Ff(j)p Fj(')p Fn(\()p Fj(t)p 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Fj(e)1624 4990 y Fh(i)p Fi(\()p Fh(t)p Fd(\000)p Fh(s)p Fi(\))p Fh(X)1884 5031 y Ff(j)k(\024)h Fe(E)p Ff(j)p Fn(\()p Fj(t)22 b Ff(\000)h Fj(s)p Fn(\))p Fj(X)8 b Ff(j)27 b(\024)h Fj(\015)5 b Fe(E)p Ff(j)p Fj(X)j Ff(j)27 b(\024)3067 4963 y Fj(\014)p 3067 5008 61 4 v 3073 5099 a Fn(3)1872 5280 y Fl(4)p eop %%Page: 5 5 5 4 bop 0 83 a Fn(when)40 b Ff(j)p Fj(t)26 b Ff(\000)h Fj(s)p Ff(j)38 b(\024)h Fj(\015)5 b Fn(.)62 b(Therefore,)42 b Fj(I)k Ff(\021)39 b Fn(0.)62 b(Next,)41 b(w)m(e)f(let)e Fj(Y)60 b Fn(b)s(e)39 b(the)g(random)f(v)-5 b(ariable)37 b(that)i(puts)0 216 y(mass)33 b(1)p Fj(=n)f Fn(at)g(eac)m(h)i(of)e(the) h Fj(X)1127 231 y Fh(i)1155 216 y Fn('s)g(in)f(the)h(sample)e(dra)m(wn) j(from)d Fj(X)8 b Fn(.)43 b(Then)1120 404 y Ff(j)p Fj(')1212 419 y Fh(n)1259 404 y Fn(\()p Fj(t)p Fn(\))22 b Ff(\000)h Fj(')1556 419 y Fh(n)1603 404 y Fn(\()p Fj(s)p Fn(\))p Ff(j)k(\024)h Fe(E)p Ff(j)p Fn(1)22 b Ff(\000)g Fj(e)2190 363 y Fh(i)p Fi(\()p Fh(t)p Fd(\000)p Fh(s)p Fi(\))p Fh(Y)2443 404 y Ff(j)1780 561 y(\024)28 b Fe(E)p Ff(j)p Fn(\()p Fj(t)22 b Ff(\000)h Fj(s)p Fn(\))p Fj(Y)e Ff(j)1780 794 y Fn(=)27 b Ff(j)p Fj(t)22 b Ff(\000)h Fj(s)p Ff(j)2159 620 y Fg(\014)2159 679 y(\014)2159 739 y(\014)2159 799 y(\014)2159 859 y(\014)2206 726 y Fn(1)p 2201 771 59 4 v 2201 862 a Fj(n)2337 669 y Fh(n)2286 699 y Fg(X)2301 909 y Fh(i)p Fi(=1)2447 794 y Fj(X)2528 809 y Fh(i)2556 620 y Fg(\014)2556 679 y(\014)2556 739 y(\014)2556 799 y(\014)2556 859 y(\014)2638 794 y Fj(:)0 1052 y Fn(Therefore,)1237 1309 y Fj(I)8 b(I)35 b Ff(\024)28 b Fe(P)1550 1138 y Fg(\()1630 1309 y Fj(\015)1703 1134 y Fg(\014)1703 1194 y(\014)1703 1254 y(\014)1703 1314 y(\014)1703 1373 y(\014)1751 1241 y Fn(1)p 1746 1286 V 1746 1377 a Fj(n)1881 1184 y Fh(n)1831 1214 y Fg(X)1846 1424 y Fh(i)p Fi(=1)1991 1309 y Fj(X)2072 1324 y Fh(i)2100 1134 y Fg(\014)2100 1194 y(\014)2100 1254 y(\014)2100 1314 y(\014)2100 1373 y(\014)2161 1309 y Ff(\025)2276 1241 y Fj(\014)p 2276 1286 61 4 v 2282 1377 a Fn(3)2347 1138 y Fg(\))1366 1641 y Ff(\024)g Fe(P)1550 1471 y Fg(\()1630 1467 y(\014)1630 1527 y(\014)1630 1587 y(\014)1630 1646 y(\014)1630 1706 y(\014)1678 1574 y Fn(1)p 1673 1619 59 4 v 1673 1710 a Fj(n)1809 1517 y Fh(n)1758 1547 y Fg(X)1773 1757 y Fh(i)p Fi(=1)1919 1641 y Fj(X)2000 1656 y Fh(i)2028 1467 y Fg(\014)2028 1527 y(\014)2028 1587 y(\014)2028 1646 y(\014)2028 1706 y(\014)2089 1641 y Ff(\025)2204 1574 y Fn(4)p Fe(E)p Ff(j)p Fj(X)8 b Ff(j)p 2204 1619 256 4 v 2307 1710 a Fn(3)2469 1471 y Fg(\))1366 1864 y Ff(!)27 b Fn(0)0 2052 y(b)m(y)37 b(the)f(la)m(w)g(of)f(large)g(n)m(um)m(b)s (ers.)54 b(Finally)-8 b(,)34 b(w)m(e)j(let)e Fj(\020)43 b Fn(and)36 b Fj(\021)k Fn(b)s(e)c(the)g(real)f(and)h(imaginary)d (parts)j(of)0 2185 y(the)41 b(F)-8 b(ourier)39 b(transform)h Fj(')p Fn(.)66 b(Let)41 b Fj(\020)1360 2200 y Fh(n)1447 2185 y Fn(and)g Fj(\021)1693 2200 y Fh(n)1780 2185 y Fn(b)s(e)g(the)g(corresp)s(onding)f(empirical)d(functions.)68 b(F)-8 b(or)0 2318 y(example,)32 b Fj(\020)452 2333 y Fh(n)499 2318 y Fn(\()p Fj(t)p Fn(\))c(=)f(\(1)p Fj(=n)p Fn(\))990 2243 y Fg(P)1094 2269 y Fh(n)1094 2347 y(i)p Fi(=1)1229 2318 y Fn(cos)q(\()p Fj(tX)1514 2333 y Fh(i)1542 2318 y Fn(\).)44 b(Then,)34 b(for)e(\014xed)h Fj(t)2352 2333 y Fh(i)2381 2318 y Fn(,)151 2505 y Fe(P)16 b Ff(fj)p Fj(')p Fn(\()p Fj(t)444 2520 y Fh(i)472 2505 y Fn(\))22 b Ff(\000)h Fj(')696 2520 y Fh(n)743 2505 y Fn(\()p Fj(t)816 2520 y Fh(i)844 2505 y Fn(\))p Ff(j)k Fj(>)h(\014)6 b(=)p Fn(3)p Ff(g)27 b(\024)h Fe(P)16 b Ff(fj)p Fj(\020)8 b Fn(\()p Fj(t)1662 2520 y Fh(i)1689 2505 y Fn(\))22 b Ff(\000)h Fj(\020)1892 2520 y Fh(n)1939 2505 y Fn(\()p Fj(t)2012 2520 y Fh(i)2040 2505 y Fn(\))p Ff(j)k Fj(>)h(\014)6 b(=)p Fn(6)p Ff(g)21 b Fn(+)h Fe(P)17 b Ff(fj)p Fj(\021)t Fn(\()p Fj(t)2847 2520 y Fh(i)2874 2505 y Fn(\))22 b Ff(\000)h Fj(\021)3082 2520 y Fh(n)3129 2505 y Fn(\()p Fj(t)3202 2520 y Fh(i)3230 2505 y Fn(\))p Ff(j)28 b Fj(>)f(\014)6 b(=)p Fn(6)p Ff(g)1277 2663 y(\024)28 b Fn(4)p Fj(e)1476 2622 y Fd(\000)p Fh(n\014)1617 2599 y Fa(2)1651 2622 y Fh(=)p Fi(72)0 2851 y Fn(b)m(y)f(Ho)s(e\013ding's)f(inequalit)m(y)f (for)h(b)s(ounded)g(random)g(v)-5 b(ariables)25 b(\(Ho)s(e\013ding,)h (1963\).)40 b(This)27 b(concludes)0 2984 y(the)33 b(pro)s(of)f(of)g (Theorem)h(1.)p 1065 2930 65 4 v 1065 2988 4 59 v 1126 2988 V 1065 2991 65 4 v 0 3383 a Fk(The)38 b(main)e(result.)299 3635 y Fn(In)43 b(the)h(Theorem)f(b)s(elo)m(w,)j(w)m(e)e(describ)s(e)g (a)f(simple)e(class)i(of)g(densities)g(for)g(whic)m(h)g(Chiu's)0 3768 y(metho)s(d)32 b(is)g(non-consisten)m(t.)44 b(No)33 b(attempt)f(w)m(as)h(made)f(to)h(obtain)e(a)h(general)g(result.)0 4140 y Fm(Theorem)39 b(2.)55 b Fb(Let)34 b Fj(f)45 b Fb(b)s(e)34 b(a)g(densit)m(y)g(with)g(\014nite)f(\014rst)i(momen)m(t,)e (and)h(with)g(real)f(unimo)s(dal)e(c)m(har-)0 4273 y(acteristic)36 b(function)f Fj(')h Fb(satisfying)g Fj(')p Fn(\()p Fj(t)p Fn(\))e Ff(\030)g Fj(t)1701 4237 y Fd(\000)p Fh(c)1827 4273 y Fb(as)j Fj(t)d Ff(!)g(1)p Fb(,)j(where)g Fj(c)d(<)g Fn(1)p Fj(=)p Fn(2)i Fb(is)f(a)h(\014xed)i(constan)m(t.)0 4406 y(Then,)c(if)d Fj(H)40 b Fb(is)32 b(the)h(bandwidth)g(c)m(hoice)g (for)f(Chiu's)h(metho)s(d,)f(w)m(e)i(ha)m(v)m(e)1335 4624 y Fn(lim)17 b(inf)1378 4684 y Fh(n)p Fd(!1)1622 4624 y Fe(E)1701 4489 y Fg(Z)1817 4624 y Ff(j)p Fj(f)1893 4639 y Fh(nH)2025 4624 y Ff(\000)23 b Fj(f)11 b Ff(j)27 b Fj(>)g Fn(0)32 b Fj(:)1872 5280 y Fl(5)p eop %%Page: 6 6 6 5 bop 0 83 a Fm(An)42 b(example.)110 b Fn(The)38 b(condition)d(of)h (the)h(theorem)g(is)f(satis\014ed)h(for)f(the)h(random)f(v)-5 b(ariable)35 b Fj(X)42 b Fn(=)0 216 y Fj(Z)25 b Ff(\000)18 b Fj(Z)261 180 y Fd(0)285 216 y Fn(,)31 b(where)g Fj(Z)38 b Fn(and)30 b Fj(Z)988 180 y Fd(0)1042 216 y Fn(are)h(i.i.d.)e(gamma)f (random)i(v)-5 b(ariables)29 b(with)h(parameter)g Fj(a)e(<)g Fn(1)p Fj(=)p Fn(4)i(\(note)0 349 y(that)i Fj(')p Fn(\()p Fj(t)p Fn(\))c(=)g(1)p Fj(=)p Fn(\(1)21 b(+)h Fj(t)857 313 y Fi(2)897 349 y Fn(\))935 313 y Fh(a)976 349 y Fn(\).)0 715 y Fm(Other)46 b(modes)h(of)g(conver)n(gence.)123 b Fn(W)-8 b(e)41 b(can't)g(giv)m(e)g(an)g Fj(L)2533 730 y Fi(2)2614 715 y Fn(v)m(ersion)g(of)g(the)g(Theorem,)j(as)0 767 y Fg(R)83 848 y Fj(f)142 812 y Fi(2)214 848 y Fn(=)33 b Ff(1)i Fn(for)g(the)h(densities)f(under)i(consideration.)51 b(It)36 b(should)g(come)f(as)h(no)f(surprise)h(that)g(it)e(is)0 981 y(precisely)29 b(for)g(these)h(densities)f(that)f(problems)h(o)s (ccur,)h(as)f(the)g(design)g(of)f(the)i(metho)s(d)e(is)g Fj(L)3447 996 y Fi(2)3487 981 y Fn(-based.)0 1113 y(This)40 b(raises)f(the)i(in)m(teresting)e(question)h(of)f(whether)i(w)m(e)g (should)e(test)h(for)g(the)g(\014niteness)g(of)3605 1033 y Fg(R)3688 1113 y Fj(f)3747 1077 y Fi(2)0 1246 y Fn(b)s(efore)32 b(applying)e(an)i Fj(L)890 1261 y Fi(2)929 1246 y Fn(-based)h (bandwidth)e(selector.)44 b(Nev)m(ertheless,)34 b Fj(f)2746 1261 y Fh(nH)2888 1246 y Fn(is)d(undesirable)h(b)m(y)g(an)m(y)0 1379 y(standard)25 b(as)f(w)m(e)h(will)d(sho)m(w)j(that)f Fj(nH)35 b Ff(!)28 b Fn(0)23 b(in)h(probabilit)m(y)-8 b(,)24 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