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Fp(-)23 b Fs(\026)2082 1687 y Fy(n)2133 1672 y Fp(\()p Fs(A)p Fp(\))2288 1522 y Fh(\014)2288 1572 y(\014)2288 1622 y(\014)2288 1672 y(\014)2288 1721 y(\014)2337 1672 y Fp(-)g Fw(max)2448 1731 y Fy(A)p Fl(2A)2637 1522 y Fh(\014)2637 1572 y(\014)2637 1622 y(\014)2637 1672 y(\014)2637 1721 y(\014)2664 1536 y Fk(Z)2709 1761 y Fy(A)2785 1672 y Fs(f)f Fp(-)2946 1536 y Fk(Z)2990 1761 y Fy(A)3066 1672 y Fs(f)3108 1630 y Fr(\()p Fy(1)p Fr(\))3202 1522 y Fh(\014)3202 1572 y(\014)3202 1622 y(\014)3202 1672 y(\014)3202 1721 y(\014)3229 1531 y Fk(\015)743 1954 y Fo(\024)83 b Fp(\()p Fs(k)22 b Fp(-)h Fs(1)p Fp(\))17 b Fw(max)1258 2014 y Fy(j>1)1424 1954 y Fq(P)1499 1813 y Fk(\014)1576 1954 y Fw(max)1589 2014 y Fy(A)p Fl(2A)1778 1804 y Fh(\014)1778 1854 y(\014)1778 1904 y(\014)1778 1954 y(\014)1778 2004 y(\014)1806 1818 y Fk(Z)1850 2044 y Fy(A)1926 1954 y Fs(f)1968 1913 y Fr(\()p Fy(j)p Fr(\))2072 1954 y Fp(-)22 b Fs(\026)2231 1969 y Fy(n)2283 1954 y Fp(\()p Fs(A)p Fp(\))2438 1804 y Fh(\014)2438 1854 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3148 y(\017)p 2355 3192 V 2358 3284 a(2)2419 3075 y Fk(\015)1102 3423 y Fw(\(b)m(y)34 b(a)f(double)f(application)e(of)j(the)g(triangle)e (inequalit)m(y\))743 3631 y Fo(\024)83 b Fs(2)p Fp(\()p Fs(k)22 b Fp(-)h Fs(1)p Fp(\))p Fn(j)p Fo(A)p Fn(j)17 b Fw(max)1407 3691 y Fy(A)p Fl(2A)1596 3631 y Fq(P)1671 3490 y Fk(\014)1748 3481 y Fh(\014)1748 3531 y(\014)1748 3581 y(\014)1748 3631 y(\014)1748 3681 y(\014)1776 3495 y Fk(Z)1820 3721 y Fy(A)1896 3631 y Fs(f)22 b Fp(-)g Fs(\026)2119 3646 y Fy(n)2171 3631 y Fp(\()p Fs(A)p Fp(\))2326 3481 y Fh(\014)2326 3531 y(\014)2326 3581 y(\014)2326 3631 y(\014)2326 3681 y(\014)2380 3631 y Fs(>)2493 3563 y(\017)p 2493 3608 V 2496 3699 a(2)2558 3490 y Fk(\015)743 3863 y Fo(\024)83 b Fs(2k)p Fp(\()p Fs(k)23 b Fp(-)f Fs(1)p Fp(\))1311 3822 y Fy(2)1350 3863 y Fs(e)1400 3822 y Fr(-)p Fy(n\017)1539 3798 y Fi(2)1577 3822 y Fy(=2)1686 3863 y Fs(;)324 4095 y Fw(where)33 b(in)f(the)h(last)g(step)g(w)m(e)g (used)h(Ho)s(e\033ding's)d(inequalit)m(y)g([8].)656 b Fd(\003)324 4487 y Ft(2)155 b(Discussion)324 4725 y Fx(methodology.)68 b Fw(The)41 b(metho)s(dology)d(of)k(the)f(prop)s(osed)g(test)g(is)g (close)g(in)f(spirit)324 4858 y(to)c(Y)-8 b(atracos')36 b(minim)m(um)31 b(distance)37 b(parametric)d(densit)m(y)j(estimate,)f (see)i(Y)-8 b(atracos)324 4991 y([10)o(],)33 b(Devro)m(y)m(e)h(and)e (Lugosi)g([5,)h(6)o(,)g(7].)1918 5251 y(4)p eop %%Page: 5 5 5 4 bop 324 548 a Fx(comput)-9 b(a)g(tion.)83 b Fw(The)46 b(h)m(yp)s(othesis)g(testing)f(metho)s(d)f(prop)s(osed)h(ab)s(o)m(v)m (e)h(is)g(com-)324 681 y(putationally)38 b(quite)j(simple.)68 b(The)42 b(sets)g Fs(A)2030 696 y Fy(i;j)2146 681 y Fw(and)f(the)h(in)m (tegrals)2940 601 y Fk(R)2993 715 y Fy(A)3069 681 y Fs(f)3111 645 y Fr(\()p Fy(j)p Fr(\))3234 681 y Fw(ma)m(y)f(b)s(e)324 814 y(computed)g(and)h(stored)g(b)s(efore)h(seeing)f(the)h(data.)70 b(Then)43 b(one)f(merely)f(needs)i(to)324 946 y(calculate)f Fs(\026)811 961 y Fy(n)863 946 y Fp(\()p Fs(A)p Fp(\))h Fw(for)h(all)e Fs(A)k Fo(2)g(A)d Fw(and)h(compute)e(the)i(test)h (statistics)d(requiring)324 1079 y Fs(O)p Fp(\()p Fs(nk)564 1043 y Fy(2)621 1079 y Fp(+)17 b Fs(k)768 1043 y Fy(2)825 1079 y Fw(log)g Fs(k)p Fp(\))30 b Fw(time.)41 b(In)30 b(man)m(y)f(applications)f Fs(k)g Fp(=)g Fs(2)p Fw(.)43 b(In)30 b(these)h(cases)g(the)g(test)324 1212 y(b)s(ecomes)h(esp)s (ecially)g(simple)e(as)k(the)f(class)g Fo(A)g Fw(con)m(tains)f(just)h (one)g(set.)324 1395 y Fx(r)m(obustness.)49 b Fw(Note)35 b(that)f(the)h(theorem)e(do)s(es)j(not)e(require)h(an)m(y)g(assumption) d(for)324 1528 y(the)k(nominal)d(densities.)53 b(\(In)36 b(fact,)i(the)e(result)g(ma)m(y)f(b)s(e)i(form)m(ulated)d(in)i(a)g (similar)324 1660 y(fashion)43 b(without)e(ev)m(en)k(assuming)d(the)h (existence)h(of)g(the)g(densities.\))75 b(The)44 b(test)324 1793 y(is)e(robust)h(in)g(a)g(v)m(ery)g(strong)g(sense:)65 b(w)m(e)43 b(obtain)f(uniform)f(exp)s(onen)m(tial)g(b)s(ounds)324 1926 y(for)35 b(the)g(probabilit)m(y)e(of)i(failure)f(under)i(the)f 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b(densities)k(corrupted)g(b)m(y)g(an)g(additiv)m(e)e (noise.)59 b(Th)m(us,)39 b(assume)e(that)h(the)324 3171 y Fs(X)390 3186 y Fy(i)420 3171 y Fw('s)k(are)f(distributed)f (according)g(to)h(densit)m(y)h Fs(f)f Fp(=)h Fs(f)2408 3135 y Fr(\()p Fy(1)p Fr(\))2529 3171 y Fc(?)27 b Fs(g)p Fw(,)43 b(where)f(the)f(nominal)324 3304 y(densit)m(y)30 b Fs(f)709 3268 y Fr(\()p Fy(1)p Fr(\))832 3304 y Fw(is)g(no)m(w)f (assumed)g(to)h(b)s(e)g(Lipsc)m(hitz)f(\(i.e.,)g Fn(j)p Fs(f)2496 3268 y Fr(\()p Fy(1)p Fr(\))2589 3304 y Fp(\()p Fs(x)p Fp(\))16 b(-)g Fs(f)2868 3268 y Fr(\()p Fy(1)p Fr(\))2961 3304 y Fp(\()p Fs(y)p Fp(\))p Fn(j)28 b Fo(\024)g Fs(c)p Fn(j)p Fs(x)16 b Fp(-)g Fs(y)p Fn(j)324 3437 y Fw(for)37 b(some)g Fs(c)d(>)h(0)i Fw(for)h(all)e Fs(x;)17 b(y)34 b Fo(2)h Fq(R)6 b Fw(\),)44 b(supp)s(orted)37 b(on)g(the)g(b)s(ounded)g(set)h Fp([-)p Fs(M;)17 b(M)p Fp(])p Fw(,)324 3570 y(and)49 b(the)g(densit)m(y)g Fs(g)g Fw(of)g(the)h(additiv)m(e)d(noise)i(is)g(assumed)f(to)h(ha)m(v)m(e)h (supp)s(ort)f(in)324 3703 y(the)41 b(in)m(terv)-6 b(al)41 b Fp([-)p Fs(r;)17 b(r)p Fp(])p Fw(,)42 b(where)f Fs(r)h Fw(is)f(though)m(t)g(of)g(as)h(a)f(small)e(n)m(um)m(b)s(er.)67 b(The)42 b(other)324 3836 y Fs(k)20 b Fp(-)f Fs(1)32 b Fw(nominal)c(densities)j(are)h(arbitrary)-8 b(.)42 b(Then,)31 b(according)g(to)h(the)f(theorem,)g(the)324 3968 y(prop)s(osed)j(test)g(is)f(correct)i(with)d(probabilit)m(y)f (larger)i(than)h Fs(1)22 b Fp(-)h Fs(2k)p Fp(\()p Fs(k)g Fp(-)g Fs(1)p Fp(\))3219 3932 y Fy(2)3259 3968 y Fs(e)3309 3932 y Fr(-)p Fy(n\017)3448 3909 y Fi(2)3486 3932 y Fy(=2)324 4101 y Fw(as)33 b(long)f(as)h Fo(k)p Fs(f)22 b Fp(-)g Fs(f)1036 4065 y Fr(\()p Fy(1)p Fr(\))1130 4101 y Fo(k)27 b(\024)i Fs(\001)1384 4116 y Fy(1)1446 4101 y Fp(-)22 b Fs(\017)p Fw(.)44 b(But)695 4384 y Fo(k)p Fs(f)22 b Fp(-)h Fs(f)949 4342 y Fr(\()p Fy(1)p Fr(\))1042 4384 y Fo(k)84 b Fp(=)1335 4248 y Fk(Z)1407 4234 y Fh(\014)1407 4284 y(\014)1407 4334 y(\014)1407 4384 y(\014)1407 4433 y(\014)1435 4248 y Fk(Z)1507 4384 y Fs(f)1549 4342 y Fr(\()p Fy(1)p Fr(\))1642 4384 y Fp(\()p Fs(x)23 b Fp(-)f Fs(y)p Fp(\))p Fs(g)p Fp(\()p Fs(y)p Fp(\))p Fs(dy)g Fp(-)2383 4248 y Fk(Z)2455 4384 y Fs(f)2497 4342 y Fr(\()p Fy(1)p Fr(\))2591 4384 y Fp(\()p Fs(x)p Fp(\))p Fs(g)p Fp(\()p Fs(y)p Fp(\))p Fs(dy)3034 4234 y Fh(\014)3034 4284 y(\014)3034 4334 y(\014)3034 4384 y(\014)3034 4433 y(\014)3077 4384 y Fs(dx)1175 4652 y Fo(\024)1335 4516 y Fk(Z)28 b(Z)1479 4652 y Fn(j)p Fs(f)1542 4611 y Fr(\()p Fy(1)p Fr(\))1636 4652 y Fp(\()p Fs(x)22 b Fp(-)h Fs(y)p Fp(\))f(-)g Fs(f)2106 4611 y Fr(\()p Fy(1)p Fr(\))2200 4652 y Fp(\()p Fs(x)p Fp(\))p Fn(j)p Fs(g)p Fp(\()p Fs(y)p Fp(\))p Fs(dydx)1918 5251 y Fw(5)p eop %%Page: 6 6 6 5 bop 1175 604 a Fo(\024)1335 468 y Fk(Z)1391 494 y Fy(M)p Fr(+)p Fy(r)1380 694 y Fr(-)p Fy(M)p Fr(-)p Fy(r)1611 468 y Fk(Z)1683 604 y Fs(c)p Fn(j)p Fs(y)p Fn(j)p Fs(g)p Fp(\()p Fs(y)p Fp(\))p Fs(dydx)1175 807 y Fo(\024)83 b Fs(2c)p Fp(\()p Fs(M)22 b Fp(+)g Fs(r)p Fp(\))p Fs(r)33 b(:)324 982 y Fw(Th)m(us,)d(the)g(condition)d(is)i(satis\034ed)g(if)h Fs(r)f Fw(is)h(so)f(small)e(that)i Fs(r)f Fo(\024)g Fp(\()p Fs(\001)2817 997 y Fy(1)2872 982 y Fp(-)15 b Fs(\017)p Fp(\))p Fs(=2c)p Fp(\()p Fs(M)g Fp(+)g Fs(r)p Fp(\))p Fw(.)324 1115 y(This)40 b(is)g(the)g(only)g(assumption)e(on)i(the)h (noise)f(densit)m(y)g Fs(g)p Fw(,)i(otherwise)e(it)f(ma)m(y)h(b)s(e)324 1248 y(completely)31 b(arbitrary!)43 b(\(Note)32 b(that)h(b)s (oundedness)g(of)h(the)g(supp)s(ort)e(of)i Fs(g)f Fw(is)g(not)g(a)324 1381 y(necessary)h(condition;)d(w)m(e)i(assumed)f(it)g(to)h(simplify)d (the)k(example.\))324 1547 y Fx(maximum)42 b(likelihood)f(does)g(not)g (w)m(ork.)56 b Fw(P)m(erhaps)38 b(the)e(most)f(standard)h(de-)324 1680 y(tection)47 b(metho)s(d)e(is)j(maxim)m(um)42 b(lik)m(eliho)s(o)s (d,)48 b(whic)m(h)f(accepts)h(the)g Fs(j)p Fw(-th)f(nominal)324 1813 y(densit)m(y)33 b Fs(f)712 1776 y Fr(\()p Fy(j)p Fr(\))826 1813 y Fw(if)861 1975 y Fs(1)p 853 2019 67 4 v 853 2111 a(n)993 1921 y Fy(n)945 1948 y Fk(X)959 2157 y Fy(`)p Fr(=)p Fy(1)1106 2043 y Fw(log)17 b Fs(f)1294 2002 y Fr(\()p Fy(j)p Fr(\))1375 2043 y Fp(\()p Fs(X)1480 2058 y Fy(`)1512 2043 y Fp(\))28 b Fs(>)1700 1975 y(1)p 1692 2019 V 1692 2111 a(n)1832 1921 y Fy(n)1784 1948 y Fk(X)1798 2157 y Fy(`)p Fr(=)p Fy(1)1945 2043 y Fw(log)17 b Fs(f)2133 2002 y Fr(\()p Fy(i)p Fr(\))2216 2043 y Fp(\()p Fs(X)2321 2058 y Fy(`)2353 2043 y Fp(\))100 b Fw(for)33 b(all)e Fs(i)d Fo(6)p Fp(=)g Fs(j)33 b(:)324 2285 y Fw(It)h(is)f(easy)i (to)f(sho)m(w)f(that)h(this)f(test)h(do)s(es)h(not)e(share)i(the)f(pro) m(v)m(ed)g(prop)s(ert)m(y)g(of)h(the)324 2418 y(prop)s(osed)i(test.)57 b(Indeed,)39 b(consider)e(the)h(simple)d(example)h(when)h Fs(k)e Fp(=)g Fs(2)p Fw(,)k(and)e(the)324 2551 y(t)m(w)m(o)29 b(nominal)e(densities)i(are)h(standard)f(normal)e(and)i(standard)h (Cauc)m(h)m(y)g(densities,)324 2684 y(that)i(is,)893 2848 y Fs(f)935 2807 y Fr(\()p Fy(1)p Fr(\))1029 2848 y Fp(\()p Fs(x)p Fp(\))27 b(=)1371 2780 y Fs(1)p 1299 2825 194 4 v 1299 2843 a Fo(p)p 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b(is)g(w)m(ell)f(kno)m(wn)g(\(see,)j(e.g.,)f([4]\))f(that)324 3755 y Fo(k)p Fs(f)416 3719 y Fr(\()p Fy(1)p Fr(\))531 3755 y Fp(-)23 b Fs(f)671 3719 y Fr(\()p Fy(1)p Fr(\))786 3755 y Fc(?)f Fs(g)914 3770 y Fy(c)953 3755 y Fo(k)27 b Fk(!)g Fs(0)33 b Fw(as)h Fs(c)27 b Fk(!)g Fs(0)p Fw(,)33 b(and)f(therefore,)i(for)g(a)e(su\036cien)m(tly)h(small)d Fs(c)p Fw(,)j(the)324 3888 y Fs(L)380 3903 y Fy(1)450 3888 y Fw(distance)d(b)s(et)m(w)m(een)h Fs(f)g Fw(and)f Fs(f)1526 3852 y Fr(\()p Fy(1)p Fr(\))1651 3888 y Fw(can)g(b)s(e)h (made)f(arbitrarily)d(small,)h(in)i(particular,)324 4021 y Fo(k)p Fs(f)416 3985 y Fr(\()p Fy(1)p Fr(\))537 4021 y Fp(-)e Fs(f)p Fo(k)41 b Fs(<)h Fo(k)p Fs(f)982 3985 y Fr(\()p Fy(1)p Fr(\))1103 4021 y Fp(-)28 b Fs(f)1248 3985 y Fr(\()p Fy(2)p Fr(\))1342 4021 y Fo(k)p Fs(=2)f Fp(-)h Fs(\017)p Fw(.)69 b(Nev)m(ertheless,)44 b(it)d(is)g(easy)h(to)f (sho)m(w)h(that)e(for)324 4154 y(an)m(y)f(small)e Fs(c)p Fw(,)j(the)g(probabilit)m(y)c(of)k(error)g(of)g(the)f(maxim)m(um)c(lik) m(eliho)s(o)s(d)h(detector)324 4287 y(con)m(v)m(erges)e(to)f(one.)44 b(Indeed,)33 b(on)g(the)g(one)g(hand,)797 4531 y Fq(E)875 4361 y Fk(\016)979 4464 y Fs(1)p 970 4508 67 4 v 970 4599 a(n)1111 4410 y Fy(n)1063 4436 y Fk(X)1077 4646 y Fy(`)p Fr(=)p Fy(1)1224 4531 y Fw(log)17 b Fs(f)1412 4490 y Fr(\()p Fy(1)p Fr(\))1505 4531 y Fp(\()p Fs(X)1610 4546 y Fy(`)1642 4531 y Fp(\))1681 4361 y Fk(\017)1844 4531 y Fp(=)2002 4395 y Fk(Z)2074 4531 y Fs(f)p Fp(\()p Fs(x)p Fp(\))g Fw(log)g Fs(f)2451 4490 y Fr(\()p Fy(1)p Fr(\))2544 4531 y Fp(\()p Fs(x)p Fp(\))p Fs(dx)1844 4825 y Fp(=)83 b(-)17 b Fw(log)2240 4735 y Fo(p)p 2323 4735 111 4 v 90 x Fs(2\031)22 b Fp(-)2563 4758 y Fs(1)p 2563 4802 50 4 v 2563 4893 a(2)2640 4689 y Fk(Z)2712 4825 y Fs(f)p Fp(\()p Fs(x)p Fp(\))p Fs(x)2936 4784 y Fy(2)2976 4825 y Fs(dx)1844 5002 y Fp(=)83 b(-)p Fk(1)34 b Fs(;)1918 5251 y Fw(6)p eop %%Page: 7 7 7 6 bop 324 548 a Fw(and)33 b(on)f(the)i(other)f(hand,)710 850 y Fq(E)788 679 y Fk(\016)892 782 y Fs(1)p 884 826 67 4 v 884 918 a(n)1024 728 y Fy(n)976 755 y Fk(X)990 964 y Fy(`)p Fr(=)p Fy(1)1137 850 y Fw(log)17 b Fs(f)1325 808 y Fr(\()p Fy(2)p Fr(\))1418 850 y Fp(\()p Fs(X)1523 865 y Fy(`)1555 850 y Fp(\))1594 679 y Fk(\017)1757 850 y Fp(=)1916 714 y Fk(Z)1988 850 y Fs(f)p Fp(\()p Fs(x)p Fp(\))g Fw(log)f Fs(f)2364 808 y Fr(\()p Fy(2)p Fr(\))2457 850 y Fp(\()p Fs(x)p Fp(\))p Fs(dx)1757 1143 y Fp(=)84 b(-)17 b Fw(log)f Fs(\031)22 b Fp(-)2334 1007 y Fk(Z)2406 1143 y Fs(f)p Fp(\()p Fs(x)p Fp(\))17 b Fw(log)o Fp(\()p Fs(1)22 b Fp(+)g Fs(x)2983 1102 y Fy(2)3024 1143 y Fp(\))p Fs(dx)1757 1320 y(>)84 b Fp(-)p Fk(1)33 b Fs(:)324 1552 y Fw(Therefore,)28 b(the)d(strong)g(la)m(w)g(of)h(large)e(n)m(um)m(b)s (ers)h(implies)e(that)h(for)i(su\036cien)m(tly)g(large)324 1685 y Fs(n)p Fw(,)33 b(the)g(maxim)m(um)28 b(lik)m(eliho)s(o)s(d)i (detector)j(errs)h(with)e(probabilit)m(y)e(one.)324 1868 y Fx(tests)41 b(based)g(on)i(density)e(estima)-9 b(tes.)56 b Fw(An)37 b(alternativ)m(e)e(w)m(a)m(y)i(of)h(p)s(erforming)324 2001 y(robust)j(detection)g(is)g(based)h(on)g(estimating)c(the)k (densit)m(y)-8 b(.)69 b(Indeed,)44 b(suc)m(h)e(meth-)324 2134 y(o)s(ds)34 b(ha)m(v)m(e)i(b)s(een)f(prop)s(osed)g(in)f(the)g (literature.)48 b(F)-8 b(or)35 b(example,)e(Zabin)g(and)i(W)-8 b(righ)m(t)324 2266 y([11)o(])35 b(in)m(v)m(estigate)f(maxim)m(um)d (lik)m(eliho)s(o)s(d)h(detection)i(based)i(on)f(k)m(ernel)h(densit)m(y) f(es-)324 2399 y(timates.)42 b(Once)34 b(again,)e(it)g(is)h(easy)h(to)f (sho)m(w)g(that)g(these)h(metho)s(ds)d(do)i(not)g(ac)m(hiev)m(e)324 2532 y(the)40 b(robustness)h(of)g(the)g(prop)s(osed)f(metho)s(d)f(in)h (the)g(sense)i(of)f(the)f(theorem,)h(and)324 2665 y(they)g(cannot)f (comp)s(ete)f(with)g(the)h(simplicit)m(y)d(of)k(the)g(prop)s(osed)f (metho)s(d.)64 b(Ho)m(w-)324 2798 y(ev)m(er,)34 b(detection)f(based)g (on)g(densit)m(y)g(estimates)f(ma)m(y)g(b)s(e)i(necessary)g(if)f(ev)m (en)i(larger)324 2931 y(h)m(yp)s(othesis)24 b(classes)h(need)h(to)e(b)s (e)h(considered.)41 b(A)24 b(stronger)h(notion)e(of)i(robust)g(detec-) 324 3063 y(tion)37 b(is)h(obtained)e(if)i(one)h(requires)f(go)s(o)s(d)f (detection)h(whenev)m(er)g(the)h(true)f(densit)m(y)324 3196 y(is)46 b(closer)g(to)h(the)f(nominal)d(densit)m(y)j(than)g(to)g (an)m(y)h(other)f(densit)m(y)h(in)e(the)i(\034nite)324 3329 y(collection.)42 b(F)-8 b(ormally)g(,)29 b(this)j(leads)h(to)g (the)g(h)m(yp)s(otheses)p 834 3483 79 4 v 834 3562 a Fs(H)912 3577 y Fy(j)968 3562 y Fp(=)28 b Fn(f)p Fs(f)f Fp(:)44 b Fo(k)p Fs(f)22 b Fp(-)g Fs(f)1491 3520 y Fr(\()p Fy(j)p Fr(\))1573 3562 y Fo(k)27 b Fs(<)h Fw(min)1786 3622 y Fy(i)p Fl(6)p Fr(=)p Fy(j)1938 3562 y Fo(k)p Fs(f)22 b Fp(-)g Fs(f)2191 3520 y Fr(\()p Fy(i)p Fr(\))2274 3562 y Fo(k)p Fn(g)p Fs(;)116 b(j)28 b Fp(=)g Fs(1;)17 b(:)g(:)g(:)d(;)j(k) 32 b(:)324 3820 y Fw(This)c(problem)e(ma)m(y)i(b)s(e)h(solv)m(ed)f(b)m (y)h(using)f(a)h(nonparametric)c(estimate)i Fs(f)3141 3835 y Fy(n)3222 3820 y Fw(of)i Fs(f)f Fw(and)324 3953 y(accepting)p 773 3874 V 38 w Fs(H)851 3968 y Fy(j)918 3953 y Fw(if)39 b Fo(k)p Fs(f)1108 3968 y Fy(n)1186 3953 y Fp(-)26 b Fs(f)1329 3917 y Fr(\()p Fy(j)p Fr(\))1410 3953 y Fo(k)39 b Fw(is)g(minimal)34 b(among)j(the)i Fo(k)p Fs(f)2599 3968 y Fy(n)2676 3953 y Fp(-)27 b Fs(f)2820 3917 y Fr(\()p Fy(i)p Fr(\))2903 3953 y Fo(k)p Fw(,)40 b Fs(i)e Fp(=)f Fs(1;)17 b(:)g(:)g(:)d(;)j(k)p Fw(.)324 4086 y(\(Break)30 b(ties)g(b)m(y)g(selecting)f(the)h(smallest)e (index.\))42 b(A)30 b(suitable)e(c)m(hoice)j(is)e(the)i(k)m(ernel)324 4218 y(estimate)g(de\034ned)i(b)m(y)1390 4406 y Fs(f)1432 4421 y Fy(n)1483 4406 y Fp(\()p Fs(x)p Fp(\))28 b(=)1762 4339 y Fs(1)p 1754 4383 67 4 v 1754 4475 a(n)1895 4285 y Fy(n)1846 4312 y Fk(X)1862 4521 y Fy(i)p Fr(=)p Fy(1)2007 4406 y Fs(K)2074 4421 y Fy(h)2122 4406 y Fp(\()p Fs(x)23 b Fp(-)f Fs(X)2399 4421 y Fy(i)2429 4406 y Fp(\))p Fs(;)324 4674 y Fw(where)48 b Fs(K)53 b Fp(:)g Fq(R)889 4637 y Fy(d)995 4674 y Fk(!)g Fq(R)1214 4637 y Fr(+)1325 4674 y Fw(is)48 b(a)g(\034xed)h(k)m(ernel)f(fuction)g(with)2702 4593 y Fk(R)2774 4674 y Fs(K)53 b Fp(=)g Fs(1)p Fw(,)f Fs(h)h(>)h(0)48 b Fw(is)324 4806 y(a)42 b(smo)s(othing)e(factor,)k(and) f Fs(K)1506 4821 y Fy(h)1554 4806 y Fp(\()p Fo(\001)p Fp(\))g(=)g(\()p Fs(1=h)2022 4770 y Fy(d)2069 4806 y Fp(\))p Fs(K)p Fp(\()p Fo(\001)p Fs(=h)p Fp(\))p Fw(.)71 b(If)43 b Fs(h)g Fw(is)f(c)m(hosen)h(suc)m(h)h(that)324 4939 y Fs(h)33 b Fk(!)h Fs(0)i Fw(and)h Fs(nh)965 4903 y Fy(d)1046 4939 y Fk(!)c(1)k Fw(as)g Fs(n)c Fk(!)h(1)p Fw(,)j(then)f(it)g(is)h(w)m(ell)e(kno)m(wn)h(\(see)i(Devro)m(y)m(e)f (and)1918 5251 y(7)p eop %%Page: 8 8 8 7 bop 324 548 a Fw(Gy)m(\366r\034)25 b([4]\))g(that)g(the)h(estimate) e(is)h(univ)m(ersally)g(consisten)m(t,)h(that)f(is,)i Fq(E)13 b Fo(k)p Fs(f)3117 563 y Fy(n)3182 548 y Fp(-)8 b Fs(f)p Fo(k)28 b Fk(!)f Fs(0)324 681 y Fw(for)33 b(an)m(y)g(densit)m (y)-8 b(.)44 b(Also,)32 b(Devro)m(y)m(e)i([3)o(])f(sho)m(ws)g(that)f (for)i(an)m(y)f Fs(\017)28 b(>)g(0)p Fw(,)1107 913 y Fq(P)p Fn(f)p Fo(k)p Fs(f)1289 928 y Fy(n)1366 913 y Fp(-)22 b Fs(f)p Fo(k)g Fp(-)g Fq(E)14 b Fo(k)p Fs(f)1827 928 y Fy(n)1906 913 y Fp(-)23 b Fs(f)p Fo(k)k(\025)h Fs(\017)p Fn(g)g Fo(\024)g Fs(e)2498 872 y Fr(-)p Fy(n\017)2637 849 y Fi(2)2675 872 y Fy(=2)2751 913 y Fs(:)324 1146 y Fw(Using)f(these)h(prop)s(erties,)g(it)f(is)h(easy)g(to)g(see)h(that) e(the)h(detection)f(metho)s(d)f(based)i(on)324 1279 y(the)f(k)m(ernel)h (densit)m(y)g(estimate)d(is)i(consisten)m(t)h(in)f(the)g(sense)i(that)e (the)h(probabilit)m(y)c(of)324 1411 y(error)36 b(con)m(v)m(erges)h(to)e (zero)h(exp)s(onen)m(tially)d(for)j(all)e Fs(f)d Fo(2)2446 1345 y Fh(S)2515 1371 y Fy(k)2515 1436 y(j)p Fr(=)p Fy(1)p 2647 1333 79 4 v 2647 1411 a Fs(H)2725 1426 y Fy(j)2753 1411 y Fw(.)52 b(In)35 b(order)h(to)f(sho)m(w)324 1544 y(this)d(supp)s(ose)i(that)e Fs(f)27 b Fo(2)p 1274 1465 V 28 w Fs(H)1352 1559 y Fy(1)1392 1544 y Fw(,)33 b(and)g(put)1294 1777 y Fs(\017)27 b Fp(=)h Fw(min)1506 1837 y Fy(j>1)1663 1777 y Fo(k)p Fs(f)22 b Fp(-)h Fs(f)1917 1736 y Fr(\()p Fy(j)p Fr(\))1998 1777 y Fo(k)f Fp(-)h Fo(k)p Fs(f)e Fp(-)i Fs(f)2421 1736 y Fr(\()p Fy(1)p Fr(\))2514 1777 y Fo(k)p Fs(:)324 2035 y Fw(Then)399 2267 y Fq(P)p Fn(f)p Fw(error)s Fn(g)83 b Fo(\024)g Fq(P)p Fn(f)p Fo(9)p Fs(j)31 b(>)d(1)g Fp(:)f Fo(k)p Fs(f)1509 2282 y Fy(n)1582 2267 y Fp(-)c Fs(f)1722 2226 y Fr(\()p Fy(1)p Fr(\))1815 2267 y Fo(k)28 b(\025)g(k)p Fs(f)2090 2282 y Fy(n)2164 2267 y Fp(-)22 b Fs(f)2303 2226 y Fr(\()p Fy(j)p Fr(\))2385 2267 y Fo(k)p Fn(g)818 2425 y Fo(\024)83 b Fp(\()p Fs(k)23 b Fp(-)f Fs(1)p Fp(\))17 b Fw(max)1333 2485 y Fy(j>1)1499 2425 y Fq(P)p Fn(f)p Fo(k)p Fs(f)1681 2440 y Fy(n)1758 2425 y Fp(-)22 b Fs(f)1897 2384 y Fr(\()p Fy(1)p Fr(\))1990 2425 y Fo(k)28 b(\025)g(k)p Fs(f)2265 2440 y Fy(n)2339 2425 y Fp(-)22 b Fs(f)2478 2384 y Fr(\()p Fy(j)p Fr(\))2560 2425 y Fo(k)p Fn(g)818 2630 y Fo(\024)83 b Fp(\()p Fs(k)23 b Fp(-)f Fs(1)p Fp(\))17 b Fw(max)1333 2690 y Fy(j>1)1499 2630 y Fq(P)p Fn(f)p Fo(k)p Fs(f)1681 2645 y Fy(n)1758 2630 y Fp(-)22 b Fs(f)p Fo(k)g Fp(+)g Fo(k)p Fs(f)g Fp(-)h Fs(f)2320 2589 y Fr(\()p Fy(1)p Fr(\))2413 2630 y Fo(k)28 b(\025)g(k)p Fs(f)21 b Fp(-)i Fs(f)2849 2589 y Fr(\()p Fy(j)p Fr(\))2930 2630 y Fo(k)f Fp(-)h Fo(k)p Fs(f)3192 2645 y Fy(n)3265 2630 y Fp(-)g Fs(f)p Fo(k)p Fn(g)818 2819 y Fo(\024)83 b Fp(\()p Fs(k)23 b Fp(-)f Fs(1)p Fp(\))p Fq(P)p Fn(f)p Fs(2)p Fo(k)p Fs(f)1513 2834 y Fy(n)1589 2819 y Fp(-)g Fs(f)p Fo(k)28 b(\025)g Fs(\017)p Fn(g)819 2976 y Fp(=)84 b(\()p Fs(k)23 b Fp(-)f Fs(1)p Fp(\))p Fq(P)p Fn(f)p Fo(k)p Fs(f)1463 2991 y Fy(n)1539 2976 y Fp(-)g Fs(f)p Fo(k)g Fp(-)h Fq(E)13 b Fo(k)p Fs(f)2000 2991 y Fy(n)2080 2976 y Fp(-)22 b Fs(f)p Fo(k)28 b(\025)g Fs(\017=2)21 b Fp(-)i Fq(E)13 b Fo(k)p Fs(f)2828 2991 y Fy(n)2908 2976 y Fp(-)22 b Fs(f)p Fo(k)p Fn(g)818 3134 y Fo(\024)83 b Fp(\()p Fs(k)23 b Fp(-)f Fs(1)p Fp(\))p Fs(e)1331 3093 y Fr(-)p Fy(n=2)p Fr(\([)p Fy(\017=2)p Fr(-)p Fj(E)s Fl(k)p Fy(f)1815 3101 y Fi(n)1868 3093 y Fr(-)p Fy(f)p Fl(k)p Fr(])2004 3069 y Fb(+)2057 3093 y Fr(\))2084 3069 y Fi(2)2125 3134 y Fs(;)324 3366 y Fw(where)32 b(the)f(last)g(inequalit)m(y)f(follo)m(ws)g(from)h(the)g (ab)s(o)m(v)m(e-men)m(tion)f(inequalit)m(y)g(of)i(De-)324 3499 y(vro)m(y)m(e)48 b([3)o(].)85 b(The)48 b(consistency)f(of)g Fs(f)1752 3514 y Fy(n)1851 3499 y Fw(assures)h(that)e(for)h(a)g (su\036cien)m(tly)g(large)f Fs(n)p Fw(,)324 3632 y Fq(E)13 b Fo(k)p Fs(f)476 3647 y Fy(n)552 3632 y Fp(-)20 b Fs(f)p Fo(k)27 b Fs(<)h(\017=4)j Fw(and)g(for)h(suc)m(h)g Fs(n)p Fw(,)g Fq(P)p Fn(f)p Fw(error)s Fn(g)27 b Fo(\024)h Fp(\()p Fs(k)19 b Fp(-)g Fs(1)p Fp(\))p Fs(e)2562 3596 y Fr(-)p Fy(n\017)2701 3573 y Fi(2)2739 3596 y Fy(=32)2850 3632 y Fw(.)44 b(Ho)m(w)m(ev)m(er,)32 b(since)324 3765 y Fq(E)13 b Fo(k)p Fs(f)476 3780 y Fy(n)546 3765 y Fp(-)f Fs(f)p Fo(k)28 b Fw(ma)m(y)e(tend)i(to)g(zero)g(at)g(an)g(arbitrarily)d(slo)m (w)i(rate)h(\(see)g(Devro)m(y)m(e)h([2]\),)f(the)324 3898 y(error)39 b(exp)s(onen)m(t)h(is)f(not)g(uniform:)55 b(it)38 b(dep)s(ends)i(on)f Fs(f)p Fw(.)63 b(It)39 b(is)g(kno)m(wn)g (\(see)h(Barron)324 4031 y([1)o(],)f(LeCam)d([9)o(]\))i(that)f(for)h (the)g(h)m(yp)s(otheses)p 2095 3952 V 39 w Fs(H)2173 4046 y Fy(j)2239 4031 y Fw(it)e(is)i(imp)s(ossible)c(to)k(construct)g (a)324 4163 y(test)33 b(with)f(a)h(uniform)e(error)i(exp)s(onen)m(t.) 324 4506 y Ft(References)374 4744 y Fw([1])48 b(Barron)32 b(A.R.)f(\(1989\).)h(Uniformly)e(p)s(o)m(w)m(erful)i(go)s(o)s(dness)g (of)h(\034t)g(tests.)g Fe(A)m(nnals)528 4877 y(of)k(Statistics)p Fw(,)c(17:107\025124.)1918 5251 y(8)p eop %%Page: 9 9 9 8 bop 374 548 a Fw([2])48 b(Devro)m(y)m(e,)i(L.)c(\(1983b\).)e(On)j (arbitrary)e(slo)m(w)h(rates)g(of)h(global)d(con)m(v)m(ergence)528 681 y(in)26 b(densit)m(y)i(estimation.)c Fe(Zeitschrift)33 b(f\374r)f(W)-8 b(ahscheinlichkeitsthe)i(orie)33 b(und)528 814 y(verwandte)39 b(Gebiete)p Fw(.)34 b(62:475-483.)374 1029 y([3])48 b(Devro)m(y)m(e,)k(L.)47 b(\(1991\).)f(Exp)s(onen)m(tial) h(inequalities)e(in)i(nonparametric)e(esti-)528 1162 y(mation.)39 b(in:)62 b Fa(Nonparametric)40 b(F)-8 b(unctional)41 b(Estimation)p Fw(,)g(ed.)85 b(G.)43 b(Roussas,)528 1295 y(pp.)38 b(31\02544,)g(NA)-8 b(TO)38 b(ASI)h(Series,)h(Dordrec)m(h)m (t:)55 b(Klu)m(w)m(er)38 b(A)m(cademic)f(Publish-)528 1428 y(ers.)374 1644 y([4])48 b(Devro)m(y)m(e,)36 b(L.)g(and)f(Gy)m (\366r\034,)h(L.)f(\(1985\).)f Fe(Nonp)-6 b(ar)g(ametric)42 b(Density)f(Estima-)528 1777 y(tion:)49 b(the)38 b Fs(L)1026 1792 y Fy(1)1104 1777 y Fe(View)p Fw(.)33 b(Wiley)-8 b(,)32 b(New)g(Y)-8 b(ork.)374 1993 y([5])48 b(Devro)m(y)m(e,)27 b(L.)d(and)g(Lugosi,)i(G.)e(\(1996\).)g(A)g(univ)m(ersally)f (acceptable)i(smo)s(othing)528 2125 y(factor)d(for)h(k)m(ernel)f (densit)m(y)h(estimates.)d Fe(A)m(nnals)28 b(of)f(Statistics)p Fw(,)d(24:2499\0252512.)374 2341 y([6])48 b(Devro)m(y)m(e,)35 b(L.)f(and)g(Lugosi,)f(G.)i(\(1997\).)e(Nonasymptotic)e(univ)m(ersal)j (smo)s(oth-)528 2474 y(ing)27 b(factors,)i(k)m(ernel)g(complexit)m(y)-8 b(,)26 b(and)i(Y)-8 b(atracos)28 b(classes.)g Fe(A)m(nnals)34 b(of)f(Statis-)528 2607 y(tics)p Fw(,)f(25:2626-2635.)374 2823 y([7])48 b(Devro)m(y)m(e,)33 b(L.)f(and)g(Lugosi,)f(G.)i (\(2000\).)e Fe(Combinatorial)37 b(Metho)-6 b(ds)39 b(in)e(Den-)528 2956 y(sity)h(Estimation)p Fw(.)32 b(Springer,)h(New)f(Y)-8 b(ork,)33 b(2000.)374 3171 y([8])48 b(Ho)s(e\033ding,)g(W.)e(\(1963\).) g(Probabilit)m(y)e(inequalities)g(for)j(sums)e(of)i(b)s(ounded)528 3304 y(random)29 b(v)-6 b(ariables.)29 b Fe(Journal)36 b(of)f(the)i(A)m(meric)-6 b(an)37 b(Statistic)-6 b(al)37 b(Asso)-6 b(ciation)p Fw(,)528 3437 y(58:13\02530.)374 3653 y([9])48 b(Le)f(Cam,)i(L.)e(\(1973\).)f(Con)m(v)m(ergence)i(of)g (estimates)e(under)h(dimensionalit)m(y)528 3786 y(restrictions.)31 b Fe(A)m(nnals)39 b(of)e(Statistics)p Fw(,)d(1:38\02553.)324 4002 y([10])48 b(Y)-8 b(atracos,)26 b(Y.G.)g(\(1985\).)f(Rates)i(of)g (con)m(v)m(ergence)h(of)f(minim)m(um)21 b(distance)26 b(esti-)528 4134 y(mators)j(and)j(K)m(olmogoro)m(v's)d(en)m(trop)m(y)-8 b(.)31 b Fe(A)m(nnals)37 b(of)f(Statistics)h Fw(,)32 b(13:768\025774.)324 4350 y([11])48 b(Zabin,)33 b(S.M.)h(and)g(W)-8 b(righ)m(t,)33 b(G.A.)g(\(1994\).)g(Nonparametric)e(densit)m(y)k (estima-)528 4483 y(tion)h(and)i(detection)g(in)f(impulsiv)m(e)e(in)m (terference)k(c)m(hannels\025P)m(art)f(I)s(I:)f(Detec-)528 4616 y(tors.)32 b Fe(IEEE)39 b(T)-8 b(r)i(ansactions)38 b(on)h(Communic)-6 b(ation)p Fw(,)32 b(42:1698\0251711.)1918 5251 y(9)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF