Characteristic Functions of L 1 -Spherical and L 1 -Norm Symmetric Distributions and Their Applications

Transkript

1 Joural of Multivariate Aalysis 76, (2001) doi jmva , available olie at httpwww.idealibrary.com o Characteristic Fuctios of L 1 -Spherical ad L 1 -Norm Symmetric Distributios ad Their Applicatios Kai Wag Ng The Uiversity of Hog Kog, Hog Kog hrtwhucc.hu.h ad Guo-Liag Tia 1 Peig Uiversity ad Chiese Academy of Scieces, Beijig, People's Republic of Chia Received December 10, 1997 I this article we obtai the characteristic fuctios (c.f.'s) for L 1 -spherical distributios ad simplify that of the L 1 -orm symmetric distributios to a expressio of a fiite sum. These forms of c.f.'s ca be used to derive the probability desity fuctios (p.d.f.'s) of liear combiatios of variables. We shall show that this gives a uified approach to the treatmet of the liear fuctio of i.i.d. radom variables ad their order statistics associated with double-expoetial (i.e., Laplace), expoetial, ad uiform distributios. Some applicatios i reliability predictio, radom weightig, ad serial correlatio are also show Academic Press AMS 1991 subject classificatios 60E10, 62E15, 62G30, 62H99, 62N05. Key words ad phrases characteristic fuctio, L 1 -orm symmetric distributios, L 1 -spherical distributios, order statistics, radom weightig method, serial correlatio. 1. INTRODUCTION Osiewalsi ad Steel (1993) itroduced the class of multivariate L p -spherical distributios, the symmetry is imposed through the desity fuctio. A importat special class of L p -spherical distributios is geerated by idepedet samplig from expoetial power distributio (Box ad Tiao, 1973, Chap. 3). For p=1 the sample comes from doubleexpoetial distributio, for p=2 it correspods to samplig from a ormal, ad for p=+ it is from a uiform distributio. A -variate radom vector x is said to have ad L p -spherical distributio, deoted by 1 Preset address Departmet of Biostatistics ad Epidemiology, St. Jude Childre's Research Hospital, 332 North Lauderdale St., Memphis, TN guoliag.tia stjude.org X Copyright 2001 by Academic Press All rights of reproductio i ay form reserved. 192

2 FUNCTIONS OF L 1 DISTRIBUTIONS 193 xts(, p; G), if x = d R*}w, w has the uiform distributio o the surface of the L p -sphere i R, F, p = {x=(x 1,..., x ) T &<x j <+, x j p =1 =, (1.1) ad R*, beig idepedet of w, is uivariate oegative radom variable with c.d.f. G. Based o the symmetry of a stochastic represetatio, Gupta ad Sog (1997a, b) recetly studied the properties of the L-spherical distributio. The c.f. of the L p -spherical distributio has ot bee available. Yue ad Ma (1995) developed a family of the multivariate versios of the Weibull distributios, called the multivariate L p -orm symmetric distributios, which are extesios of the family of multivariate L 1 -orm symmetric distributios studied by Fag ad Fag (1988). A -dimesioal radom vector z is said to have a L p -orm symmetric distributio, deoted by ztl(, p; G), if z = d R*}u, u is uiformly distributed o the L p -orm closed simplex i R, + T, p = {x=(x 1,..., x ) T x j 0, x p j =1 =, (1.2) ad R*, beig idepedet of u, is uivariate oegative radom variable with c.d.f. G. Whe p=1, we deote F,1 ad T,1 by F ad T respectively. The c.f. of the uiform distributio o T, i.e., utu(t ), is give by Fag et al. (1990, p. 116) as follows E(e ittu )=1() e it j=0 i j 1(+ j) r 1 +}}}+r &1=j &1 ` (t &t ) r. (1.3) Note that the right side of (1.3) is a summatio with ifiite terms. I this paper we shall employ the partial-fractio expasio, the CKS (Cambais, Keeer, ad Simos) formula ad the HG (Hermite Geocchi) formula to obtai for the first time the c.f. for L 1 -spherical distributios i Theorem 1 of Sectio 3. Our secod cotributio is to simplify (1.3) as fiite summatios for three differet situatios (see Theorem 2). Aalogous results are developed for the c.f.'s of ztl(, 1; G) ad ytu(v ), V V (1) is a special case of the ope simplex i R +, V (c)= {x=(x 1,..., x ) T x j 0, x j c =, (1.4) c is a positive costat. I Sectio 4, we use the c.f. to obtai the p.d.f. of the liear combiatios of variables for three ids of cases. This leads to a uified approach to the treatmet of the liear fuctio of i.i.d.

3 194 NG AND TIAN radom variables ad their order statistics associated with expoetial ad uiform distributios. The formula (5.5.4) of David (1981, p. 103) is a direct cosequece of our (4.13). The applicatios i reliability predictio, radom weightig ad serial correlatio are show i Sectio PRELIMINARIES I order to derive the characteristic fuctios i the ext sectio, we shall collect some useful formulae about the partial-fractio expasio ad other multi-fold itegrals o the ope simplex V (c) Partial-Fractio Expasio. A lot of useful formulae ca be obtaied by combiig the surface itegral formula with the partialfractio idetity (Hazewiel, 1990, p. 311) N(x) (x&b 1 )}}}(x&b ) = N(b ) 2(b 1,..., b ) x&b, (2.1) 2 (b 1,..., b )=> j{, (b &b j ) &1, N(x) is a polyomial of degree r, 0r&1, ad b j {b for j,,...,, j{. Especially, taig N(x)=&x i (2.1) ad settig x=0, we have 0= 2 (b 1,..., b ). (2.2) 2.2. CKS Formula. Let h( } ) ad g( } ) be measurable fuctios o R 1 + ad R 1 respectively, ad further let g( } ) have the (&1)th absolutely cotiuous derivatives. It ca be show by iductio ad through the use of (2.2) that (Cambais et al. 1983, p. 225) R + h \ = x j+ g(&1) \ s j x j+ dx 2 (s 1,..., s ) h(u) g(us ) du. (2.3) From the proof of Theorem 3.1 i Cambais et al. (1983, pp ), we ow that (2.3) ca derive the itegral h \ R + x j+ ` cos(s j x j ) dx= 0 2 (s 2,..., 1 s2)}b (s 2 ), (2.4)

4 FUNCTIONS OF L 1 DISTRIBUTIONS 195 B (t)={(&1) (&1)2 t (&1)2 cos(u - t) h(u) du, 0 (&1) (&2)2 t (&1)2 si(u - t) h(u) du, 0 odd, eve. We shall call (2.4) the CKS formula i the followig. A alterative versio of (2.3) is give by V (c) h \ Aother importat formula is x j+ g(&1) \ s j x j+ dx = 2 (s 1,..., s ) c h(u) g(us ) du. (2.5) 0 h \ x V (c) j+ dx= 1 (&1)! c h(u) u &1 du, (2.6) which ca be obtaied by usig (5.10) of Fag et al. (1990, p. 115) HG Formula. The classical HermiteGeocchi (HG) formula (Karli et al., 1986, p. 71) ca be stated as T g (&1) 0 \ s j x j+ dx=- g(s ) 2 (s 1,..., s ), (2.7) dx deotes the volume elemet of T ad g( } ) has the same meaig as i (2.3). The followig three lemmas will be used i the sequel ad their proofs are omitted. Lemma 1. Assume that t # R 1 +, b # R1, ad b{0. Let! =! (t; b) t 0 ebx } x &1 dx, the! (t; b)=!(&b) & &! t e bt (&bt) & (&)!. (2.8) Lemma 2. Assume that t # R 1 +, a, b # R1, ad b{0. Let ' =' (t; a, b) t 0 cos(a+xb)}x &1 dx ad `=`(t; a, b) t 0 si(a+xb)}x &1 dx. Deote the largest iteger ot exceedig x by [x]. The

5 196 NG AND TIAN [(+1)2]&2 ' =si(a+tb) =0 [(+1)2]&2 +cos(a+tb) =0 (&1) (&1)! b 2+1 t &2&1 (&2&1)! (&1) (&1)! b 2+2 t &2&2 (&2&2)! +(&1) [(+1)2]&1 (&1)! b } C (t; a, b), (2.9) [(+1)2]&2 `=&cos(a+tb) =0 [(+1)2]&2 +si(a+tb) =0 (&1) (&1)! b 2+1 t &2&1 (&2&1)! (&1) (&1)! b 2+2 t &2&2 (&2&2)! +(&1) [(+1)2]&1 (&1)! b } D (t; a, b), (2.10) C (t; a, b)= {si(a+tb)&si(a), (tb) si(a+tb)+cos(a+tb)&cos(a), D (t; a, b)= {cos(a)&cos(a+tb), &(tb) cos(a+tb)+si(a+tb)&si(a), odd, eve. odd, eve. Lemma 3. Assume that t # R 1 +, b # R1, ad defie The we have &1 $ (t) cos V&1 (t) \t& $ (t)= (t2)&1 (&1)! &1& _ j=0 \ &1 =0 &1 u j+ ` 2 \&1 t + & the fuctio ' + j is give by (2.9). cos(u j ) du j. &1& j + (&t)&j ' + j (t; t, &0.5), (2.11) 3. CHARACTERISTIC FUNCTIONS Cosider x=(x 1,..., x ) T which has a L 1 -spherical distributio, i.e., xts(, 1;G). The the c.f. of x is give by E(e itt x )=E(e itt R*}w )=, w (r*t 1,..., r*t ) dg(r*), (3.1) 0

6 FUNCTIONS OF L 1 DISTRIBUTIONS 197 R* has c.d.f. G ad, w is the c.f. of wtu(f ). Therefore, it suffices to ivestigate, w. A special feature of this characteristic fuctio is that it is a real fuctio. We have the followig mai result. Theorem 1. Let, w (t 1,..., t ) be the c.f. of wtu(f ), the, w (t 1,..., t ) ={1() 1() (&1) (&1)2 t &1 cos(t )}2 (t 2 1,..., t2 ), (&1) (2)&1 t &1 t 2 j {t2, for j,,...,, j{. si(t )}2 (t 2 1,..., t 2 ), odd, eve, (3.2) Proof. The techique i provig this case is similar to that of Lemma 7.1 i Fag et al. (1990, p. 185). Let x 1,..., x be a i.i.d. sample from double-expoetial with p.d.f. (2*) &1 exp[&* &1 x ], *>0, &<x<, that is, x 1,..., x t iid DE(*), x=(x 1,..., x ) T. It follows from Theorem 1.1 i Sog ad Gupta (1997a, b) that w=(w 1,..., w ) T = d \ x 1< x j,..., x < x j + T tu(f ). (3.3) Without loss of geerality, we tae *=1 ad obtai the c.f. of w as, w (t 1,..., t ) =E exp[it T w]=e exp {itt x < = R = R + x j = exp t 1x 1 +}}}+t x {i x 1 + }}} + x = 2& exp[&( x 1 + }}} + x )]dx exp[&(x 1 +}}}+x )] ` cos \ t j x j x 1 +}}}+x + dx,

7 198 NG AND TIAN i the last step we have used the symmetry of the itegrad. The trasformatio y j =x j x i=1 i, 1j&1, y = x i=1 i, has the Jacobia J(x 1,..., x y 1,..., y )=y &1. Therefore, we have, w (t 1,..., t ) =1() V&1 &1 cos \ _t 1& &1 y j+& ` cos(t j y j ) dy j. (3.4) Puttig h(u)=cos(t (1&u)) } I [0, 1] (u) i the CKS formula (2.4), I D ( } ) represets the idicator fuctio of domai D, we have &1, w (t 1,..., t )=1() 2 (t 2 1,..., t2 &1 )}B &1(t 2 ), B &1 (t)={(&1) (&2)2 t (&2)2 (- t si(- t)&t si(t ))(t&t 2 ), &1 odd, (&1) (&3)2 t (&2)2 } - t(cos(t )&cos(- t))(t&t 2 ), &1 eve. Notig the idetity (&1) (&3)2 t &1 2 (t 2,..., 1 t2 )=0, is odd, (3.5) which ca be obtaied by taig N(x)=(&1) (&1)2 x (+1)2 i (2.1) ad settig x=0, we have, whe is odd,, w (t 1,..., t ) &1 =1() 2 (t 2 1,..., t 2 )}(&1)(&3)2 t &1 (cos(t )&cos(t )). This implies the first expressio of (3.2) by virtue of (3.5). Liewise the secod expressio of (3.2) ca be reached. K Remar 1. (i) If t 2 j =t2,for,...,, the we have, w (t,..., t)=1() t &(&1) $ (t), (3.6) the fuctio $ (t) is give by (2.11). I fact, if t=0, of course,, w (0,..., 0)=1. Sice, w (&t,..., &t)=, w (t,..., t), we may assume without loss of geerality that t>0. Formula (3.6) follows immediately by usig (3.4) ad (2.11).

8 FUNCTIONS OF L 1 DISTRIBUTIONS 199 (ii) I priciple, the c.f. of wtu(f ) for other cases ca be obtaied by taig appropriate limits of (3.2) because of uiform cotiuity i real space R. For example, let be odd. The from (3.2) we have, whe t 2 1 =}}}=t2 r =t2,2r<, adt 2,..., r+1 t2, t2 are distict,, w (t 1,..., t ) 1()(&1) (&1)2=0 r } + ` j=r+1 =r+1 (t 2 &t 2 j )&1 t &1 cos(t ) (t 2 &t2 ) r 2 (t 2 r+1,..., t2 ), 0 r lim t 1 t,..., t r t r t &1 cos(t ) 2 (t 2 1,..., t2 r ). If we could fid a geeral expressio for 0 r, other cases follow from (3.2). I fact, this idea will be employed i showig Theorems 2 ad 3. However, there is a techical difficulty for the preset theorem ad its corollary. For ay specific r, oe ca wor out 0 r. For example, 0 2 = 1 2[(&1) t &3 cos(t)&t &2 si(t)], 0 3 = 1 8 [(&t&3 +(&1)(&3) t &5 )cos(t) &((&1) t &4 +(&2) t &5 ) si(t)]. But it is ot easy to fid the recursive patter of 0 r, hece iductio caot be used i this situatio. A direct attac o the geeral patter of 0 r without iductio would be eve harder. Next, let us cosider the uiform distributio iside the L 1 -sphere i R, E = {x=(x 1,..., x ) T x # R, x j 1 =. If vtu(e ), the the joit p.d.f. of v is 2 &!}I E (v) (see Gupta ad Sog, 1997a, Example 2.7). I this case, we ca represet it as v = d R*}w with R*tBe(, 1) beig idepedet of wtu(f ); that is, vts(, 1;G) with G(x)=x } I [0, 1] (x). We use Be(a, b) to deote the beta distributio with parameters a ad b. Applyig (3.1) ad Theorem 1, we immediately obtai

9 200 NG AND TIAN Corollary 1. Let v (t 1,..., t ) be the c.f. of vtu(e ), the v (t 1,..., t ) ={!! (&1) (&1)2 t &2 si(t )}2 (t 2 1,..., t2 ), (&1) 2 t &2 cos(t )}2 (t 2 1,..., t2 ), odd, eve, t 2 j {t2, for j,,...,, j{. Now let us cosider the c.f. of a L 1 -orm symmetric distributio. Let z=(z 1,..., z ) T have such a distributio, i.e., ztl(, 1;G), the the c.f. of z is give by E(e ittz )=E(e )= ittr*}u 8 u (r*t 1,..., r*t ) dg(r*), 0 R* has c.d.f. G ad 8 u is the c.f. of utu(t ). If the momet geeratig fuctio (m.g.f.) E(e st u ) of radom vector u is available, we ca obtai the c.f. of u by replacig s with it. For this reaso, we derive the m.g.f. of U(T ) to sigify that all momets exist. Theorem 2. Let 8 u (s 1,..., s ) be the m.g.f. of utu(t ). (i) If s j {s, for j,,...,, j{, the 8 u (s 1,..., s )=(&1)! e s 2 (s 1,..., s ). (3.7) (ii) (iii) If s j =s, for,...,, the 8 u (s,..., s)=exp(s). If s 1 =}}}=s r =s, 2r<, ad s, s r+1,..., s are distict, the 8 u (s 1,..., s )=(&1)! { e s (r&1)! + =r+1 ` j=r+1 (s&s j ) &1 e s (s &s) &r 2 (s r+1,..., s ) =. (3.8) Proof. (i) Sice the joit p.d.f. of utu(t ) is (&1)!- } I T (u), the the m.g.f. of u is give by 8 u (s 1,..., s )=E(e st u )= (&1)! - T exp { s j u j= du.

10 FUNCTIONS OF L 1 DISTRIBUTIONS 201 Applyig the HG formula (2.7) to the fuctio g(t)=exp(t), we obtai (3.7) immediately. (ii) (iii) It is trivial. Cosiderig the limit of (3.7), we have 8 u (s 1,..., s ) (&1)! =0 r *} ` + j=r+1 =r+1 (s&s j ) &1 e s (s &s) &r 2 (s r+1,..., s ), (3.9) 0 r * lim s 1 s,..., s r s r e s 2 (s 1,..., s r ). I particular, we ca obtai 0 2 *=e s, 0 3 *= es 2!, 0 4*= es 3!,.... Therefore 0 r *=e s (r&1)!. By substitutig this ito (3.9), we obtai (3.8). K Now we tur to the c.f. of U(V ). Agai, we ca do more by derivig the m.g.f. of U(V ). Theorem 3. Let 9 y (s 1,..., s ) be the m.g.f. of ytu(v ). (i) If s j {s, for j,,...,, j{, the 9 y (s 1,..., s )=! s &1 (es &1) 2 (s 1,..., s ). (3.10) (ii) If s j =s, for,...,, the 9 y (s,..., s)=!(&s) & &! e s (&s) & (&)!. (3.11)

11 202 NG AND TIAN (iii) If s 1 =}}}=s r =s, 2r<, ad s, s r+1,..., s are distict, the 9 y (s 1,..., s ) r =! &e s {\(&s)&r + =r+1 (&s) & (r&)! + ` j=r+1 (s&s j ) &1 s &1 (es &1)(s &s) &r 2 (s r+1,..., s ) =. (3.12) Proof. (i) As the joit p.d.f. of ytu(v )is!}i V (y), we have E(e sty )=! V exp { s j y j= dy. (3.13) If all s 1,..., s are distict, we see from (3.13) ad (2.5) that E(e sty )=! 2 (s 1,..., s ) 1 exp( ys ) dy 0 =! 2 (s 1,..., s ) s &1 (es &1), which implies (3.10). (ii) If all s j =s, for,...,, we have from (3.13) ad (2.6) E(e sty )=! V exp {s y j= dy= 1 So we obtai (3.11) by Lemma 1. (iii) Cosiderig the limit of (3.10), we have 0 e sx } x &1 dx=! (1; s). 9 y (s 1,..., s )! =0 r ** } ` + j=r+1 =r+1 (s&s j ) &1 s &1 (es &1)(s &s) &r 2 (s r+1,..., s ), (3.14) 0 r ** lim s 1 s,..., s r s r s &1 (es &1) 2 (s 1,..., s r ).

12 FUNCTIONS OF L 1 DISTRIBUTIONS 203 I particular, we have 0 2 ** =s &2 (se s &e s +1), 0* 3 *=s &3 ( 1 2 s2 e s &se s +e s &1), 0 4 **=s &4 ( 1 6 s3 e s & 1 2 s2 e s +se s &e s +1),.... The geeral patter of 0 r ** is ot quite obvious. Notig that the limit of (3.10) is (3.11), i.e., lim s 1 s,..., s s =(&s) & &e s s &1 (e s &1) 2 (s 1,..., s ) (&s) & (&)!. (3.15) We have 0 r **=(&s) &r &e s r (&s)& (r&)! by replacig with r i (3.14). By substitutig of 0 r ** ito (3.14), we obtai (3.12). K 4. DENSITY FUNCTIONS FOR LINEAR FORMS We first cosider the liear fuctio associated with U(F ) ad the double-expoetial distributio. Let x 1,..., x t iid DE(1) ad x=(x 1,..., x ) T. From (3.3), we ow that w=x x tu(f ). We are ofte iterested i the exact p.d.f. of liear fuctio such as w=a T w= a w ad x=a T x= a x. (4.1) Sice &x = d x ad &w = d w, we ca assume without loss of geerality that all a 1,..., a are positive. The cofiguratio classificatios for [a 1,..., a ] are as follows. Case 1. All a are differet, say, 0<a 1 <}}}<a. Case 2. All a are equal, say, a 1 =}}}=a =1. Case 3. At least two of a are equal. For example, (a 1,..., a 5 ) T = (0.5,0.5,1,4,4) T ; For Cases 1 ad 2, we have the followig results.

13 204 NG AND TIAN Theorem 4. Case 1. The p.d.f.'s ofw=a T w ad x=a T x are respectively give by f 1 (w)= g 1 (x)= { } &1 &2 2a \ 1& w I a + [&a, a ] (w), (4.2) 1 { } exp 2a x \& a +, &<x<+, (4.3) { =a 2(&1) 2 (a 2,..., 1 a2 ),,...,. (4.4) Case 2. give by The p.d.f.'s ofw= w ad x= x are respectively f 2 (w)= g 2 (x)= \ } w &1 (1& w ) &&1, w 1, (4.5) 2B(, &) \ } x &1 exp(& x ), &<x<+, (4.6) 21() \ = \2&&1 &1 +< 22&&1,,...,. (4.7) Proof. Case 1. It suffices to cosider the situatio whe is odd. From (3.2), the c.f. of w=a T w is. w (t)=e(e itw )=E(e itat w )=, w (ta 1,..., ta ) =1() (&1) (&1)2 a &1 2 (a 2 1,..., a2 )}t&(&1) cos(ta ). I terms of the iversio theorem, the p.d.f. of w is give by f 1 (w)= 1 2? + & = 1 2? } 1() e &itw. w (t) dt (&1) (&1)2 a &1 2 (a 2 1,..., a2 )}I 1, (4.8)

14 FUNCTIONS OF L 1 DISTRIBUTIONS 205 I 1 + & e &itw } t &(&1) cos(ta ) dt= + e &itw (e ita +e &ita ) dt. & 2t &1 By meas of residue theorem i complex aalysis, Stuart ad Ord (1987, p. 362) derived the complex itegral + e ibz & z dz= { &2?i b &1 (&1)!, 0, if if b0, b>0, (4.9) which ca be employed to give I 1 = {?(&1)(&1)2 (a & w ) &2 (&2)!, 0, w a, otherwise. (4.10) By substitutig (4.10) ito (4.8), we obtai (4.2). Defie ` x, the `tga(, 1) ad idepedet w (Gupta ad Sog, 1997a, Theorem 1.1). So x=w } ` ad ` is idepedet of w. The p.d.f. of x=a T x is give by g 1 (x)= 0 = which implies (4.3). f 1\ x t+ 1 t } t&1 e &t dt 1() { } &1 2a x x a \ 1& ta + &2 t &2 e &t dt, 1() Case 2. c.f. to p.d.f. It ca be show similarly by usig the iversio theorem from K Some isights ito (4.2)(4.6) are give i Remar 2 below. We recall some distributios related to the beta distributio. Deote by xt Be( p, q; a), the beta distributio with scale a, ifx has p.d.f. 1 a } B( p, q)\ x p&1 q&1 a+ \ 1&x, 0xa, a>0. a+ The p.d.f.'s of the symmetric beta distributio ad the symmetric beta distributio with scale a are respectively defied as

15 206 NG AND TIAN 1 2}B( p, q) x p&1 (1& x ) q&1, p&1\ 1& x a + 1 2a } B( p, q) } x a} x 1, q&1, x a, a>0, ad they are symbolized by Sbe( p, q) ad Sbe( p, q; a). Remar 2. (i) We ow that each compoet w of w has the same symmetric beta distributio with parameters 1 ad &1, i.e., w t Sbe(1, &1), thereby, a w tsbe(1, &1; a ). Equatio (4.2) idicates that the distributio for the sum of depedet variables with Sbe(1, &1; a ) is the mixture of Sbe(1, &1; a ). Liewise, (4.3) implies that the distributio for the sum of idepedet variables with DE(a ) (a x tde(a )) is the mixture of DE(a ). (ii) Formula (4.5) ad (4.6) deote the mixtures of the symmetric beta distributio Sbe(, &) ad the symmetric gamma distributio SGa(, 1) respectively. The latter coicides with the result listed o p. 24 of Johso ad Kotz (1970). Now let us cosider the exact distributio of liear fuctio y=a T y= a y, y=(y 1,..., y ) T tu(v ). The relatioship betwee the c.f. of y=a T y ad the c.f. of y is. y (t)=9 y (ta 1,..., ta ). By meas of (3.10), we have. y (t)=! _ } exp(ita )&1 (ita ), (4.11) _ =a &1 2 (a 1,..., a ),,...,. (4.12) I aalogy with (4.8), the p.d.f. of y=a T y is give by h 1 ( y)=! 2? I 2 = + & By (4.9), we have for some fixed a, _ 1 (ia ) } I 2, e it(&y+a ) t dt& + e &ity & t dt. I 2 = { &2?i ( y&a ) &1 }(&1) (&1)!, &2?i (&y+a ) &1 (&1)!, if a >0, 0ya, if a <0, a y0,

16 FUNCTIONS OF L 1 DISTRIBUTIONS 207 that is, I 2 = 2?i (a & y) &1 } sg(a )}I [mi(0, a ), max(0, a (&1)! )]( y). These results are summarized ito the followig theorem. Theorem 5. Let y=(y 1,..., y ) T tu(v ). If all a ({0) are differet, the the p.d.f. of y=a T y= a y is h 1 ( y)= _ } a \ 1& y &1 a + } sg(a )}I [mi(0, a ), max(0, a )]( y), (4.13) weights [_,,..., ] are give by (4.12) ad sg( } ) deotes the sig fuctio. Remar 3. (i) Whe a >0, the formula (5.5.4) i David (1981, p. 103) coicides with (4.13), which idicates that h 1 ( y) is the mixture of Be(1, ; a ), the beta distributio with scale a. (ii) If a 1 =}}}=a =1, it is easy to see y= y tbe(, 1) by viewig (3.11). Correspodig to Theorem 5, we have Theorem 6. Let u=(u 1,..., u ) T tu(t ). If all a ({0) are differet, the the p.d.f. of u=a T u= a u is h 2 (u)= _ } &1 a \ 1& u &1 a + } sg(a )}I [mi(0, a ), max(0, a )](u), (4.14) weights [_,,..., ] are give by (4.12). Fially, we preset a uified approach to liear fuctios of variables. We shall adopt the followig otatios for samples ad their order statistics y=(y 1,..., y ) T tu(v ), y (1) }}}y (), u=(u 1,..., u ) T tu(t ), u (1) }}}u (), z=(z 1,..., z ) T tl(, 1;G), z (1) }}}z (),!=(/ 1,...,! +1 ) T t iid E(1),! (1) }}}! (+1), '=(' 1,..., ' ) T t iid U[0, 1], ' (1) }}}' (),

17 208 NG AND TIAN E(*) deotes the expoetial with p.d.f. * &1 exp(&* &1!), *>0,!>0. We shall illustrate that all distributios of the liear fuctios b y (), b u (), b z (), c z, ad c! b! (), b ' (), ca be reduced to the distributio of a y as give by (4.13) or to that of a u as give by (4.14). (i) b y () a y. From Example 5.1 of Fag et al. (1990, p. 121), we ow that ytu(v ) belogs to the class of the L 1 -orm symmetric distributio. Therefore, the coclusio stated i Theorem 5.12 of Fag et al. (1990, p. 126) is also available to y. Defie y * (&+1) ( y () & y (&1) ), y (0) =0,,...,, called ormalized spacigs of y, the ( y 1 *,..., y *) T = d y, which implies b y () = d a y, a = j= b j (&+1). (4.15) (ii) b u () a u. Sice utu(t ) also belogs to the family of the L 1 -orm symmetric distributio, similar to (4.15), we obtai b u () = d a u, a = j= b j (&+1). (iii) b z () ad c z a y. Now z=(z 1,..., z ) T tl(, 1;G) which implies z = d R*}u, R*tG( } ) idepedet of utu(t ), hece b z () = d R*} b u (), c z = d R*} c u. (iv) b! () ad c! a y. Because ( y 1,..., y ) T = d +1 \ 1<! +1!,...,! < T! +,

18 FUNCTIONS OF L 1 DISTRIBUTIONS 209 the b! () = d c! = d c!, c = j= b j (&+1), +1 \! + } a y, a =c, +1! tga(+1, 1) idepedet of a y. (v) b ' () a y.nowy = d (' () & ' (&1) ), ' (0) =0,,...,, the b ' () = d a y, a = j= b j. 5. APPLICATIONS I this sectio, we shall demostrate the usefuless of the precedig results with three importat examples predictig the reliability of compoets i a system, the radom weightig method, ad serial correlatio. Example 1 (Predictio Problem i Reliability). Cosider the oparametric problem of predictig a T x (2) based o the first m observatios x (1), a is a (&m)_1 scalar vector ad x=(x T, (1) xt (2) )T t S(, 1;G). It is easy to see that t(x)=a T x (2) m x is scale-ivariat (see, Gupta ad Sog, 1997a, Theorem 6.2). Hece we ca tae x= (x 1,..., x ) T, x 1,..., x t iid DE(1). If all compoets of a are 1, the the p.d.f. of xa T x (2) = x =m+1 from (4.6) is give by &m g 2 (x)= \ &m, } x &1 exp(& x ), &<x<. 21() Sice m x tga(m, 1) ad is idepedet of a T x (2), the p.d.f. of t(x) is h(u)= 1 g 2 (uv)} 1(m) vm&1 e &v vdv 0 &m = u &1 1 \ &m, } 2B(, m) } (1+ u ) +m. (5.1)

19 210 NG AND TIAN I this case we ca obtai the predictio iterval [L 1, U 1 ] of a T x (2) = =m+1 x for a give cofidece coefficiet 1&, 1&=P {L 1 x U =m+1 1= Hece =P { L 1 m x t(x) U 1 m x =. [L 1, U 1 ]= _\ m x + } L 2, \ m L 2 ad U 2 are determied by virtue of (5.1) by 1&=P[L 2 t(x)u 2 ]= U 2 x + } U 2&, (5.2) L 2 h(u) du. (5.3) I the same fashio we may also get the predictio iterval of a T x (2) whe a i {a j, i{ j, by meas of (4.3). Example 2 (Radom Weightig Method). Sice Efro's (1979) wellow paper appeared there has bee cosiderable wor o resamplig methods. Amog all of these techiques, the bootstrap is the simplest ad most attractive oe, ad the radom weightig method is a alterative which is aimed at estimatig the error distributio of estimators. Let x =++e,, 2,..., (5.4) be a measure model, [e,, 2,...] are radom errors of measuremets. It is assumed that [e 1, e 2,...] are i.i.d. with a commo distributio fuctio F(x) satisfyig xdf(x)=+ ad (x&+) 2 df(x)= _ 2 >0, ad that + ad _ 2 are uow. The commo estimator for + is the sample mea x, with sample size. To costruct a cofidece iterval for +, we eed to ow the distributio of the error x &+. The mai idea of the radom weightig method is to costruct a distributio based o samples x 1,..., x, to mimic the distributio of x &+. Letu=(u 1,..., u ) T t U(T ) be idepedet of x 1,..., x, ad defie D *=- (x &x ) u, (5.5) which is the weighted mea of - (x &x ) with radom weight u. Zheg (1987, 1992) shows that D * (x 1, x 2,...), the coditioal distributio of D *

20 FUNCTIONS OF L 1 DISTRIBUTIONS 211 give (x 1, x 2,...), is close to the distributio of the error - (x &+) whe is large, i.e., with probability oe, D * (x 1, x 2,...) w L N(0, _ 2 ), (5.6) the otatio w L stads for covergece i law, provided that i model (5.4) the errors [e,,2,...] are i.i.d. with E(e )=0 ad Var(e )=_ 2 <. Our iterest here is to fid the exact (coditioal) p.d.f. of D * (x 1, x 2,...). I fact, the (coditioal) p.d.f. of D * (x 1, x 2,...) is a mixture of beta distributios with scale by virtue of (4.14) with a =- (x &x ). Example 3 (Serial Correlatio Problem). Cosider the followig model of time series (Johso ad Kotz, 1970, p. 233) X t =\X t&1 +Z t, \ <1, t=1, 2,..., (5.7) the Z$ t s are mutually idepedet uit ormal variables, ad further, Z t is idepedet of all X for <t. Defie a modified ocircular serial correlatio coefficiet as R 1, 1 = &1 2&1 (X &X 1)(X +1 &X 1)+ X 1= &1 =+1 2 (X &X 1) 2 + =+1 2 X ; X 2= &1 (X &X 2)(X +1 &X 2) (X &X 2) 2, (5.8) =+1 The exact distributio of R 1, 1 has bee obtaied by Pa (1968) for the case whe the correlatio betwee X i ad X j is \ for i& j =1, ad is 0 otherwise. I this case it ca be show that R 1, 1 is distributed as ( &1 *! )( &1! ), the!'s are mutually idepedet variables, each distributed as stadard expoetial, i.e., (! 1,...,! ) T t iid E(1), ad * 2&1 =cos(2?(+1)),, 2,..., [2], while * 2, * 4,..., * 2[(&1)2] are roots of the equatio X. (1&*) &2 [(& 1 2 ) & 1 2 D &1(*)&(+1&*) D (*)]

21 212 NG AND TIAN with D (*)= \ &1 2 * + [2+1] \ +1 2&1+ (&1)&1 (* 2 &1) &1. It's easy to see that R 1, 1 has the same distributio as &1 * u whose p.d.f. is give by (4.14) with (u 1,..., u &1 ) T tu(t &1 ). ACKNOWLEDGMENTS The authors express their gratitude to Drs. G. Wei, H. B. Fag, M. Y. Xie, ad R. X. Yue for their commets. Special thas go to Dr. R. Z. Li for valuable suggestios. The authors are grateful to the reviewers for costructive commets which have greatly improved Theorems 2 ad 3. This wor was partially supported by a FRG grat from the Hog Kog Baptist Uiversity, the Statistics Research ad Cosultacy Cetre of HKBU, ad The Chiese Academy of Scieces. REFERENCES 1. G. E. P. Box ad G. C. Tiao, ``Bayesia Iferece i Statistical Aalysis,'' Addiso Wesley, Readig, S. Cambais, R. Keeer, ad G. Simos, O -symmetric multivariate distributios, J. Multivariate Aal. 13 (1983), H. A. David, ``Order Statistics,'' 2d ed., Wiley, New Yor, B. Efro, Bootstrap methods Aother loo at the jacife, A. Statist. 7 (1979), K. T. Fag ad B. Q. Fag, Some families of multivariate symmetric distributios related to expoetial distributio, J. Multivariate Aal. 24 (1988), K. T. Fag, S. Kotz, ad K. W. Ng, ``Symmetric Multivariate ad Related Distributios,'' Chapma ad Hall, Lodo, A. K. Gupta ad D. Sog, L p -orm spherical distributio, J. Statist. Plaig Iferece 60 (1997a), A. K. Gupta ad D. Sog, Characterizatio of p-geeralized ormality, J. Multivariate Aal. 60 (1977b), M. Hazewiel, ``Ecyclopedia of Mathematics,'' Vol. 9, Kluwer, Dordrecht, N. L. Johso ad S. Kotz, ``Distributio i Statistics Cotiuous Uivariate Distributio2,'' Vol. 3, Wiley, New Yor, S. Karli, Micchelli, ad Y. Riot, Multivariate splies A probabilistic perspective, J. Multivariate Aal. 20 (1986), J. Osiewalsi ad M. F. J. Steel, Robust Bayesia iferece i L q -spherical models, Biometria 80 (1993), J. J. Pa, Distributio of the serial correlatio coefficiets with o-circular statistics, i ``Selected Traslatios i Mathematical Statistics ad Probability,'' Vol. 7, pp , America Mathematics Society, Providece, RI, D. Sog ad A. K. Gupta, L p -orm uiform distributio, Proc. Amer. Math. Soc. 125 (1997), A. Stuart ad J. K. Ord, ``Kedall's Advaced Theory of Statistics Distributio Theory,'' 5th ed., Vol. 1, Oxford Uiv. Press, Oxford, 1987.

22 FUNCTIONS OF L 1 DISTRIBUTIONS X. N. Yue ad C. S. Ma, Multivariate L p -orm symmetric distributios, Statist. Probab. Lett. 24 (1995), Z. G. Zheg, Radom weightig methods, Acta Math. Appl. Siica 10 (1987), Z. G. Zheg, Radom weightig method Aother approach to approximate the error distributio of estimators, i ``The Developmet of Statistics Recet Cotributios from Chia'' (X. R. Che, K. T. Fag, ad C. C. Yag, Eds.), pp , Logma Scietific, Harlow, 1992.