Asymptotic Normality in Monte Carlo Integration

Transcription

1 MATHEMATICS OF COMPUTATION, VOLUME 30, NUMBER 136 OCTOBER 1976, PAGES Asymptotic Nomality in Monte Calo Integation By Masashi Okamoto Abstact. To estimate a multiple integal of a function ove the unit cube, Habe poposed two Monte Calo estimatos /'j and J'2 based on 2N and 4N obsevations, espec- 2 2» tively, of the function. He also consideed estimatos Dy and D2 of the vaiances of/j and J'2, espectively. This pape shows that all these estimatos ae asymptotically nomally distibuted as N tends to infinity. 1. Intoduction. Monte Calo integation is a method to estimate the value of a definite integal of a given eal-valued function ove a finite egion (say, a cube) by obseving the value of the function only at a finite numbe of points in the egion which ae chosen suitably and stochastically. Kitagawa [4] poposed seveal estimating methods but he was concened mainly with the case when the function has a cetain pio distibution. Habe [1], [2] poposed a mesh estimato of the integal and then impoved it by means of the idea of "antithetic vaiâtes" due to Hammesley and Moton [3]. SpecificaUy, let /be a eal-valued function defined ove the unit cube Gs in the s- space and set / = jg f. Let A ( = 1,..., A) be a family of conguent subcubes aising by patitioning Gs so that the inteval [0, 1] on the x'-axis is divided into n equal subintevals fo each i = 1,..., s, N = nx ns. Let x be a andom point in A chosen independently fo each and let x' = 2c,. - x, c stands fo the cente of A. Then <1J> and '!-* /(*,) 1 f(x) + f(x') (1.2) 2 = A? Z-2- = 1 ae unbiased estimatos of/ based on N and 2A obsevations of/, espectively. To estimate the vaiances of/, and J2 we need eplications of obsevations. Let z be anothe andom point in A, the z being chosen independently othe and also of the x's. Define similaly z' 2c - z. Then (1.3),, 1 ± /(*,) + /(z) 1 *kx 2 and (1 4) /-' I V f(x) + f(x') + f(z) + f{z') 2 N x A of each Received August 26, 1975; evised Januay 27, AMS (MOS) subject classifications (1970). Pimay 65D30; Seconday 62E20, 60F05. Key wods and phases. Monte Calo integation, Habe's estimatos, asymptotic nomality. Lyapunov condition. 831 Copyight Ameican Mathematical Society

2 832 MASASHI OKAMOTO ae again unbiased estimatos of/ based on 2N and AN obsevations, espectively. using these andom points, the vaiances of J\ and f2 can be estimated unbiasedly by (L5) ^= 1 íf(x)-f(z)}2 and 4M '6) P' ' ^,?, j 2-1 j espectively. =\ Fo k = 1 and 2 let Ck denote the set of all eal-valued functions defined ove Gs and having continuous fcth ode patial deivatives. In the sequel we say just 'W o " to indicate that n fo evey i = I,..., s. Put n = min(jj,..., ns). Habe poved that if/e C1, then (1.7) va(/j) = T2N(Jt) + ottnn2)-1) as N -* -, By and also that if/ C2, then (1.8) va(/2) = t2n(j2) + o^a«4)"1) asjv^«2, L v ± f Z-^V TiV^l^ 12A ff-, î JGs\bxi) ' (1.9) T2.,_L_ j2 f i- f /-^-V "l 2' 1440A)2-4 Jc,^(xi)V + 5 ± _l_ ; pv_,,./=! (n.«.)2-^vaxw Futhemoe, since J'k (k = 1,2) is the aveage of two independent ealizations of Jk, its vaiance is half as lage as that of Jk. In estimating / by J'k, Habe used Dk as a measue of the eo, assuming that J'k is appoximately nomally distibuted. The pupose of this pape is to show, fist, that Jk (k = 1,2) is asymptotically nomally distibuted with mean / and vaiance T2N(Jk) as N >. This implies the asymptotic nomality of J'k. Next, it is shown that Dk(k= nomally distibuted with mean vax(j'k) and a pope vaiance. 1,2) is asymptotically 2. Asymptotic Nomality of /, and J2. Befoe stating the theoems we intoduce some notations. Let I=E{f(x)}=Nf /, K = K» fy - X, - C f,=f(c), t'-^ic) dx' and ff-jül-icu dxdx'

3 MONTE CARLO INTEGRATION 833 x = (x1,..., Xs), fo /,/ = 1,..., s, and = 1,..., N. The L2-nom will be denoted by. Theoem 1. Assume /EC1. As N,JX is asymptotically nomally distibuted with mean I and vaiance T2N(Jy) defined in (1.9). Poof. Since / = 2* j I/N, it holds that (2.1) A-' = Z {/(*, )-/,} A Taylo seies expansion of f(x) aound the point c gives N =i N (2.2) f(x)=f+ 5'//+/?l, í=i the emainde RX has the following popety in view of the unifom continuity of df/dx' in Gs: fo any e > 0, \Rl\ < e 6 fo evey, povided N is sufficiently lage. Again, by a Taylo expansion we have \R2\ < e/n fo evey, povided N is sufficiently lage. Substitution of (2.2) and (2.3) into (2.1) yields (2-4) J!-I = SN+RN, (2.5) ^=Â7ZtôX ^=Í (*l-*2)- / = 1 i= 1 = 1 Since X/,) = /, to pove the theoem it emains only to veify the following thee popositions: (OvaiS^-T^/^asA' «>, the symbol ~ means that the atio of the two sides tends to one, (ii) the sequence SN satisfies the Lyapunov condition of the cental limit theoem (see, e.g., Loève [5, p. 275]), and (iii) vai(rn) = o((nn2)'1) as N -* o. In fact, in the ight-hand side of the identity /, -/ SN - E(SN) RN - E(RN) TN TN TN the fist tem conveges to the standadized nomal distibution because of (i) and (ii), as the second tem conveges in pobability to zeo because of (iii) and the Chebyshev inequality. Now, pat (i) is essentially equivalent to Theoem 3 of Habe [1] o (1.7); but a poof is given hee fo the completeness of the poof. By definition, o' has a unifom distibution ove the inteval [- 1/(2«,), 1/(2«,.)] independently of each othe; and hence, va(s') = l/(12n2), which implies that

4 834 MASASH1 OKAMOTO vaís»,) = E i -T(^')2~^i) as TV-> oc. 12A2 ~\ ~\ nf Re (ii). Define XN = 2J=, «',//, then (2.6) - E(X2N)=N2va:(SN). Fom the inequality \ô'\ < l/(2n() it follows that (2.7) pu*,i» < (p/>/y~ f J& ax'" 0(Nn~3). By (i),(2.6) and (2.7) [Z^l^l3] /[Z^Ä)]372 = OÍTV-1/2) = 0(1), which is the Lyapunov condition fo 5^. Re (iii). Since R2 ae constant, va(/^) = va(/?l)//v2 < I>(/?2,.)//V2 < e2/(/v«2). This implies (iii), since e can be made abitaily small. Remak. Q.E.D. The asymptotic nomality may be poved by applying the cental limit theoem diectly to 7, in the fom (1.1), not indiectly to SN. This appoach, howeve, equies asymptotic expansions of E {f2(x)} and I up to the tems of ode n~2 so that a stonge assumption / G C2 is needed instead of /G C1. This emak is valid also with Theoems 2, 3 and 4 fo which a much stonge assumption / C* (k = 4, 4 and 6, espectively) is equied. Theoem 2. Assume /G C2.,4s A, /2 is asymptotically nomally distibuted with mean I and vaiance t2n(j2) defined in (1.9). Poof. Similaly as (2.1) (2.8) ^2-í = Z {f(x)+f(x')-2i}. =l Expanding f(x) and f(x') in Taylo seies aound the point c, we find (2.9) f(x) =f+í, Kf + \ i KKf + RU> i= 1 L ij= 1 f{x')=f-± Kfl + \ f S'M+K, i=l i,j=l \Rl\ and \R\\ ae bounded fom above by e 5 2 fo evey, povided N is sufficiently lage. Taking one moe tem in the expansion (2.3), we have (2.10) I=f+t 2fï+R2, í=i 24h? \R2\ < e/n2 fo evey. Substitution of (2.9) and (2.10) into (2.8) yields

5 MONTE CARLO INTEGRATION 835 ^2 ~~ I ~ $N + ^JV ' (2.11) ^=^E ^v =i \i,/=i /=i 12n? J 1 " =l (Äi + /i;-2ä2). Just as fo Theoem 1 we have only to pove the following thee popositions: (i) va^) ~ 2N(J2) as N -* -, (ii) SN satisfies the Lyapunov condition, and (iii) va(rn) = o((an4)-1) as N. Pat (i) is equivalent to the main theoem in Habe [2] o (1.8), while (ii) and (iii) can be veified by using easoning simila to that in the poof of Theoem 1. Q.E.D. Coollay 1. Fo k = 1 o 2, assume fgck. As N, Jk follows asymptotically a nomal distibution with mean I and vaiance (2.12) T%(j'k) = i 2N(Jk). 3. Asymptotic Nomality of Dx and D2. Though the asymptotic nomality of Dx o D2 may not be so impotant in pactice as that of any estimato of/ itself, it can be poved along a simila line of aguments fo the latte. Fist let us conside Theoem 3. Assume / G C1. As N, D\ is asymptotically nomally distibuted with mean va(/j) and vaiance 72( )2)= 1 Ulf /-V-Y + 10 T l f ibf bf\ N ' 2880A3) éik}jo'\9x*) tk^(nñ)2)g\^bxi) 2 W Poof. Similaly to (2.2) it is the case that i f(z)=f+'lsífí+r2> i=l = ttî,- > ) = *, -<V. \R2\ < «HM fo evey. Substitution of (2.2) and (3.1) into (1.5) yields (3-2) D2 = TN +RN, (3.3) 1 N / s tn = 2 ZÍZU). AN1 =\ \ =l and t = «', - V RN = -^ Z \(Rl-R2)2+^Rl-R2)t<f\ ANZ =\ ( 1=1 ]

6 836 MASASHI OKAMOTO Since E(D\) = va(/',), a poof of Theoem 3 can be educed to the following: (i) va(tn) ~ t2n(d2) as N-+, (ii) TN satisfies the Lyapunov condition, and (iii) va^) = o((n3n4yi) as N >. Re (i). Fist we have Since fo evey i, j and -, (3.4) (TN) = ^Z (ZvfJ- EW'^i- 6n( 15/14 36Wj.f and since the expectations of any othe monomial of the tj's of ode 2 o 4 vanish, we find i I I \ I <M-m ^m i fi \ 2 This implies Zvif-)2=^\iz(^ +ioz/ Substitution of the last fomula into (3.4) poves (i). Re (ii). Define Then a staightfowad and hence, calculation gives Y,E\XN\3 = 0(Nn~6); [pl^l3] /[Z^ÁV,)]3/2 = OÍA"1/2) = 0(1). Re (iii). Fom (3.3) it follows that l*jvi < ~h Z (H*ii + iifid2 + 2(iiô,ii + iifii) i Z l/íll; 4A2 * n i ) and hence, vaí/í^) < fi/?2,) = o((a V)"1). Q.E.D. Remak. Though t^/',) defined in (2.12) is indeed the leading tem in the asymptotic expansion of va(/j) in n~l, the phase "mean vai(j[)" in the statement

7 MONTE CARLO INTEGRATION 837 of Theoem 3 cannot be eplaced by "mean 2N(fxy\ in geneal. The eason is that the diffeence va(/',) - t2n(j\) is o((nn2y~l) and hence is not negligible in geneal as compaed with N(D2) which is of ode (A3'2«2)-1. Similaly, we can pove the following: Theoem 4. Assume f EC2. As N, D22 is asymptotically nomally distibuted with mean va(/2) and vaiance t2n(d\) which is a linea combination of seven tems of the fom y i f / a2/ a2/ \2 i,/,lb=i N3(ntnfttkn,)2 JGs\dx'dx> bxkdx' J some of i,j, k and I ae constained to be equal. The oot-squae tansfomation of the andom vaiable D\ (k = 1,2) induces the following: Coollay 2. Fo k = 1 o 2, assume /G Ck. As N, Dk is asymptotically nomally distibuted with mean o(fk) and vaiance lát2n(d2.)/yax(j'k), o(j'k) stands fo the standad deviation of J'k. Depatment of Applied Mathematics Faculty of Engineeing Science Osaka Univesity Toyonaka, Osaka 560, Japan 1. S. HABER, "A modified Monte-Calo quadatue," Math. Comp., v. 20, 1966, pp MR 35 # S. HABER, "A modified Monte-Calo quadatue. II," Math. Comp., v. 21, 1967, pp MR 38 # J. M. HAMMERSLEY & K. W. MORTON, "A new Monte Calo technique: Antithetic vaiâtes," Poc. Cambidge Philos. Soc, v. 52, 1956, pp MR 18, T. KITAGAWA, "Random integations," Bull. Math. Statist., v. 4, 1950, pp MR 14, M. LOÈVE, Pobability Theoy, 3d ed., Van Nostand, Pinceton, N.J., MR 34 # 3596.