Sequences of elliptical distributions and mixtures of normal distributions

Transcription

1 Sequences of ellipical disribuions and mixures of normal disribuions E. Gómez-Sánchez-Manzano a,, M. A. Gómez-Villegas a,,, J. M. Marín b, a Deparameno de Esadísica e Invesigación Operaiva I, Fac. de CC. Maemáicas, Universidad Compluense, 84 Madrid, Spain b Deparameno de Esadísica, Universidad Carlos III, 893 Geafe (Madrid), Spain Absrac Two condiions are shown under which ellipical disribuions are scale mixures of normal disribuions wih respec o probabiliy disribuions. The issue of finding he mixing disribuion funcion is also considered. As a unified heoreical framework, i is also shown ha any scale mixure of normal disribuions is always a erm of a sequence of ellipical disribuions, increasing in dimension, and ha all he erms of his sequence are also scale mixures of normal disribuions sharing he same mixing disribuion funcion. Some examples are shown as applicaions of hese conceps, showing he way of finding he mixing disribuion funcion. Key words: Bayesian mehods, Ellipically conoured disribuion, Ellipical disribuion, Laplace ransform, Scale mixures of normal disribuions. AMS subjec classificaion: Primary 6-, secondary, 6E, 6F5. Inroducion In his paper wo condiions for an absoluely coninuous ellipical disribuion o be a scale mixure of normal disribuions, wih respec o a probabiliy disribuion, are shown, and he issue of finding he mixing disribuion funcion Corresponding auhor. addresses: eusebio_gomez@ma.ucm.es (E. Gómez-Sánchez-Manzano), ma.gv@ma.ucm.es (M. A. Gómez-Villegas), jmmarin@es-econ.uc3m.es (J.M.Marín). Suppored by Gran BFM3-479; Miniserio de Ciencia y Tecnología, Spain. Preprin submied o Elsevier Science 7 January 5

2 is approached; also, i is shown ha any scale mixure of normal disribuions can be viewed as a erm of a sequence of ellipical disribuions. The proof of he main resuls relies on he concep of an expansive sequence of ellipical disribuions, ha we sae previously, and on he exisence of some disribuions of higher dimensions shown by Gupa and Varga [8] and Eaon [4]. Scale mixures of normal disribuions are an imporan class of ellipical disribuions. They share good properies wih normal disribuions, are easy o work wih, and are useful o robusify saisical procedures usually based on normal disribuions. In a Bayesian framework, hey can be used in simulaion mehods. So, a characerizaion and a way o reduce ellipical disribuions o scale mixures of normal disribuions is very valuable. There are many publicaions on ellipical disribuions. For a comprehensive sudy of ellipical disribuions, Kelker [], Chu [3], Cambanis e al. [], Fang e al. [5], Fang and Zhang [6], Gupa and Varga [8] can be seen. A survey abou absoluely coninuous ellipical disribuions can be found in Gómez e al. [7]. Some resuls on scale mixures of normal disribuions and heir relaionship wih ellipical disribuions can be found in Chu [3], Eaon [4], Fang e al.[5],andgupaandvarga[8,9].some of hese approaches do no keep o probabiliy mixing disribuions (as we do in his paper) bu allow weighing funcions ha can urn ou o ake negaive values. Andrews and Mallows [] sudy condiions for a unidimensional symmerical disribuion o be a scale mixure of normal disribuions. The firs condiion ha we show for an absoluely coninuous ellipical disribuion o be a scale mixure of normal disribuion is based upon he successive derivaives of is funcional parameer g, and includes he ellipical disribuion in a sequence. In his way, hese resuls give an inerpreaion of derivaives of g in a probabilisic framework, exend he heorem of Andrews and Mallows [], reinerpre, in he framework of mixures of normals, proposiion of Eaon [4] and supplemen, in some sense, heorem 4..3 of Gupa and Varga [8]. The second condiion refers o Laplace ransforms, and also shows he mixing disribuion funcion. These resuls paricularize some aspecs of a heorem of Chu [3] (see also Gupa and Varga [8]) and exend he lemma in Andrews and Mallows []. In secion we inroduce he conceps of expansive and semi-expansive sequences of ellipical disribuions and esablish condiions under which an ellipical disribuion can be a erm of an expansive sequence. These conceps provide a general heoreical framework for he subsequen sudy of mixures of normal disribuions.

3 In secion 3 he wo condiions are shown. Finally, in secion 4, some examples and applicaions o he sudy of ellipical disribuions are shown. Sequences of ellipical disribuions We show he definiion and some properies of ellipical disribuions, inroduce he conceps of expansive and semi-expansive sequences of ellipical disribuions, and sudy some of heir properies.. Some properies of ellipical disribuions We deal wih ellipical disribuions ha are absoluely coninuous. Definiion (Ellipical disribuion). If µ is an n-dimensional vecor, Σ is an n n posiive definie symmeric marix and g is a non-negaive Lebesgue measurable funcion on [, ) such ha < hen he n-dimensional densiy f given by f(x; µ, Σ,g)= π n Γ n n g() d <, (.) R n g() d Σ g (x µ) Σ (x µ) (.) is said o be ellipical wih parameers µ, Σ and g. If vecor X has densiy (.), we say ha X has he ellipical disribuion (e.d.) E n (µ, Σ,g) and wrie X E n (µ, Σ,g). We will refer o funcion g by he name of funcional parameer. The paramerizaion of an ellipical disribuion is no sricly unique. Suppose ha X E n (µ, Σ,g); hen X E n (µ, Σ,g ) iff here exis wo posiive numbers a and b, such ha µ = µ, Σ = aσ and g () =bg(a) for almos all (Gómez e al. [7]; see also Fang and Zhang [6]). We will say ha wo real funcions g and g are equivalen if g = bg a.e. for some b>; we will denoe g g. Thus, he funcional parameer g of an e.d. E n (µ, Σ,g) can be replaced wih anoher g (keeping he same parameers µ and Σ) iff g g. For each vecor x =(x,..., x n ), we will denoe x (p) =(x,..., x p ), for p n; also, Σ (p) will denoe he upper-lef p p submarix of he n n marix Σ. 3

4 If X E n (µ, Σ,g) and X (p) =(X,..., X p ), wih p<n,hen (see Gómez e al. [7] and Fang and Zhang [6]) where g (p) is he funcion given by g (p) () = X (p) E p µ(p), Σ (p),g (p), (.3) = In paricular, if p = n, hen Z g (n ) () = w n p g( + w) dw (.4) (w ) n p g(w) dw. (.5) g(w) dw (.6) and, hence, g(n ) () = g() for each coninuiy poin of g. The following lemmadevelopshissubjec.wedenoeby n m he null n m marix and by I n he ideniy n n marix. Lemma (Derivaive of marginal parameer). Le X E n ( n,i n,g) and Y E n+ (n+),i n+,h for some n. Equaliy in disribuion X = d Y (n) holds iff g h. PROOF. Y (n) E n n,i n,h (n), where, from (.6), h(n) () = R herefore, h (n) = h a.e. If X d = Y (n), hen g h (n) and g h (n) h. h (w) dw; Reciprocally, if g h, hen here is a posiive number b such ha bg = h = h (n) a.e. and, consequenly, bg = h (n) + c a.e., where c is a consan. Bu c =because bg and h (n) are funcional parameers of n-dimensional e.d. s, and hen, from (.), R n bg () d < and R herefore, n cd= n bg () d n h (n) () d < ; n h (n) () d <, andiispossibleonlyifc =. Thence bg = h (n) a.e. and X E n ( n,i n, h (n) ).. Expansive and semi-expansive sequences The conceps of expansive and semi-expansive sequences of ellipical disribuions are inroduced. A sequence is expansive (alernaively: semi-expansive) 4

5 if each one of is erms is equivalen o a marginal disribuion of he nex erm (al.: of he erm following he nex erm). These conceps allow us o esablish relaions beween he derivaives of he funcional parameer of a disribuion and he funcional parameers of some disribuions poserior o i in he sequence. And his permis us o derive condiions under which e.d. s are scale mixures of normal disribuions. Theorem 4 shows a condiion for a sequence of e.d. s o be semi-expansive and heorem 6 shows a condiion for erms of expansive sequences. When a erm of a sequence is a vecor or a marix we use superscrips o denoe is posiion in he sequence and subscrips o denoe is componens. Definiion 3 (Expansive and semi-expansive sequences). For all m {,,...}, le E m (µ m, Σ m,g m ) be an m-dimensional ellipical disribuion and le X m =(X m,...,x m m) E m (µ m, Σ m,g m ). (i) The sequence of disribuions {E m (µ m, Σ m,g m )} is said o be expansive if, for all m {,,...}, X m = d X m+ (m) = X m+,..., Xm m+, (.7) namely,ifhedisribuionofvecorx m is equal o he (marginal) disribuion of a subvecor of vecor X m+. (ii) The sequence {E m (µ m, Σ m,g m )} is said o be semi-expansive if, for all m {,,...}, X m = d X m+ (m) = X m+,..., Xm m+. (.8) Clearly, all expansive sequences are semi-expansive and a semi-expansive sequence is expansive if, in addiion, X m d = X m+ (m) for all odd m. The nex heorem shows a condiion, based on he funcional parameer g, for a sequence o be semi-expansive. (On noaion: we use someimes signs such as ġ or g simply o denoe funcions, wih no reference o derivaives.) Theorem 4 (Condiion for semi-expansiviy). The sequence of e.d. s {E m ( m,i m,g m )} is semi-expansive iff here exiss a sequence {g m} of funcions such ha for all m {,,...} and >. gm g m, (.9) gm+ = (gm), (.) 5

6 PROOF. (The if par) For all m, le X m =(X m,..., X m m) E m ( m,i m,g m ). From (.9) and (.) we have g m g m+ ; hence, by virue of lemma, X m d = X m+ (m). (The only if par) Forallm {,,...}, le X m E m ( m,i m,g m ). Firs, we are going o obain differeniable funcions g m such ha g m g m. For all m, by virue of (.8) and (.3) (see also (.6)), X m E m ( m,i m, ġ m ), wih ġ m () = R g m+ (w) dw; hence ġ m g m ; and funcion ġ m is coninuous. Again by (.8) and (.3), we obain ha ġ m g m, where g m () = ġ m+ (w) dw. Thus g m g m, and funcion g m is differeniable. R For all m, by obaining again anoher equivalen funcion by means of (.8) and (.3), we find ha here exiss a consan b m+ > such ha g m () = R b m+ g m+ (w) dw and, consequenly, g m = b m+ g m+. We add b = b =, and for all k {,,,...} we define funcions g+k and g+k as follows: g +k = g +k = ky b +j j= ky b +j j= g +k, g +k. All he elemens of sequence {g m} saisfy (.9), because g m g m g m. We check ha hey also saisfy (.). For all k {,,,...} we have ha g +k = ky b +j j= k+ Y b +j j= = g +k = ky b +j j= g +(k+) = g +k+. Similarly, i is proved ha g +k = g +k+. ( b +k+ g +k+ )= By applying ieraively (.) for m =, 3, 5,... andhenform =, 4, 6,... we see ha a sequence {E m ( m,i m,g m )} is semi-expansive iff here exis funcions g m g m such ha for k {,,...}. g+k =( ) k (g) (k), (.) g+k =( ) k (g) (k), (.) 6

7 We noice, by (.) and (.), ha a semi-expansive sequence {E m ( m,i m,g m )} is deermined by g and g. I is also deermined by any pair of erms g n and g n, wih n even and n odd, of he sequence {g m }, because from hem equivalen funcions o g and g can be obained by applying any of he expressions (.4) or (.5). An expansive sequence {E m ( m,i m,g m )} is deermined by any erm, g n, of he sequence {g m }, because from his one equivalen funcions o g,g 3 and g can be obained successively by applying (.4) or (.5), (.), and (.4) or (.5), respecively. To prove heorem 6 we need o esablish firs he following lemma. Lemma 5 (Disribuions of higher dimension). Le E n (µ, Σ,g) be an ellipical disribuion, wih n. If ( ) k g (k) () (.3) for k {,,...} and >, hen,forallk {,,,...}, < n+k ( ) k g (k) () d < (.4) and, hence, he disribuion E n+k µ n+k, Σ n+k,g n+k, wih g n+k =( ) k g (k), does exis for each vecor µ n+k R n+k and each posiive definie symmeric (n +k) (n +k) marix Σ n+k. PROOF. We prove (.4) by inducion on k. From (.), he saemen is rue for k =. Le, now, k>; we suppose ha he saemen is rue for k and we prove i for k. By inegraing by pars we have n+k ( ) k g (k) () d = (.5) = lim n+k ( ) k g (k ) () (.6) lim ( ) k g (k ) () (.7) n+k à n +k +! n+(k ) ( ) k g (k ) () d. (.8) The inegrals in (.5) and (.8) exis and are non-negaive. Hence, here exis he limis (.6) and (.7). Clearly, hey are no negaive. We prove ha boh limis are zero by reducion o absurd. 7

8 If he limi (.6) would be equal o c>, here should be a poin > such ha for every (, ) i would be n+k ( ) k g (k ) () > c ; hence i would be n+(k ) ( ) k g (k ) () d c Z d =, which is false, by he recurrence hypohesis. Similarly, if he limi (.7) would be equal o c>, i would be a such ha n+(k ) ( ) k g (k ) () d c d =. Therefore, he inegral (.5) is equal o he addend (.8), which, by he recurrence hypohesis, is posiive and finie. Thenexheoremshowshaane.d.E n (µ, Σ,g) can be a erm of an expansive sequence iff he successive derivaives of is funcional parameer g are alernaively posiive and negaive. Theorem 6 (Condiion for erms of expansive sequences). Le E n (µ, Σ,g) be an ellipical disribuion, wih n. There is an expansive sequence of e.d. s whose n-h erm is he disribuion E n (µ, Σ,g) iff here is a funcion g = g a.e. such ha for k {,,,...} and >. ( ) k g (k) () (.9) In his case, one such sequence is he sequence {E m (µ m, Σ m,g m )} given by he following specificaions: µ m = µ (m) if m n µ (.) µ (m n) if m>n For m<n Σ m = g m () = Σ (m) Σ n m m n I m n if m n if m>n (.) (w ) n m g(w) dw; (.) 8

9 and for k {,,,...} g n+k =( ) k g (k), (.3) g n+k+ () =( ) k+ (w ) g (k+) (w) dw. (.4) PROOF. Firs we prove ha if condiion (.9) is saisfied hen (.) (.4) really describe a disribuion E m (µ m, Σ m,g m ) for every m, and ha sequence {E m (µ m, Σ m,g m )} is expansive and is n-h erm is he disribuion E n (µ, Σ,g). Le X n = (X n,..., Xn) n E n (µ, Σ,g). Obviously, X n E n (µ, Σ, g). For any m < n here exiss he disribuion E m (µ m, Σ m,g m ) because his is jus he disribuion of subvecor X(m) n. There exiss also he disribuion E n (µ n, Σ n,g n ); i is he same as E n (µ, Σ,g). For µ k {,,...}, by virue of lemma 5, here exiss he disribuion E n+k n+k, Σ n+k,g n+k, specified by (.), (.) and (.3). We also see ha for k {,,,...} here exiss he disribuion E n+k+ µ n+k+, Σ n+k+, g n+k+ ), specified by (.), (.) and (.4): if X n+k+ E n+k+ µ n+k+, Σ n+k+,g n+k+, hen, from (.3), he disribuion of subvecor X n+k+ (n+k+) = X n+k+,..., Xn+k+ n+k+ is jus En+k+ µ n+k+, Σ n+k+,g n+k+. Hence, here exiss he sequence {E m (µ m, Σ m,g m )}, and is n-h erm is he disribuion E n (µ, Σ,g). Le us see ha sequence {E m (µ m, Σ m,g m )} is expansive. For all m, le X m E m (µ m, Σ m,g m ). For m<nwe have X m = d X(m) n d = X m+ (m) ; and for k {,,,...} we have ha X n+k+ = d X n+k+ (n+k+).toseehaxn+k = d X n+k+ (n+k) we may suppose, wihou loss of generaliy, ha µ = n and Σ = I n ; hen, by virue of (.3) and lemma, we have ha X n+k = d X n+k+ d (n+k), and Xn+k+ (n+k) = X n+k+ (n+k). Now we prove he reciprocal. Le {E m (µ m, Σ m,g m )} be an expansive sequence whose n-h erm is he disribuion E n (µ, Σ,g). Then he sequence {E m ( m,i m,g m )} is expansive, oo, and, by virue of heorem 4, here exiss a sequence of funcions {g m} saisfying (.9) and (.) for every m. Since (.9) is verified for m = n, here exiss a consan b> such ha g = g n = bg n a.e. Le g = bg n and, for all m, le g m = bg m.iisobviousha g = g a.e. and, besides, for all k, ( ) k g (k) =( ) k (bg n) (k) = b ( ) k (g n) (k) = bg n+k, where he las equaliy is obained by inducion on k, saringak =and by applying (.). 9

10 Noe ha if funcion g menionedinheorem(6)isconinuouson(, ), hen we need no use funcion g, and we can pu g insead of g in expressions (.9) and (.) (.4). Noice how expressions (.3) and (.4) allow us o inerpre he funcional parameers g m, for m n, in he sequence {E m (µ m, Σ m,g m )}, in erms of he successive derivaives of parameer g. 3 Ellipical disribuions as scale mixures of normal disribuions We approach he issue of characerizing he e.d. s ha are scale mixures of normal disribuions and he one of finding he corresponding mixing disribuion funcion. Firsly, some basic definiions and resuls concerning scale mixures of normal disribuions are shown. Then we show a characerizaion of ellipical disribuions ha are scale mixures of normal disribuions and pu hem in relaion wih expansive sequences of e.d. s. Nex we show a second characerizaion and deal wih he subjec of finding he mixing disribuion funcion. 3. Basics on scale mixures of normal disribuions and relaed ellipical disribuions We denoe N n ( ; µ, Σ) he n-dimensional normal densiy wih parameers µ and Σ; we wrie X N n (µ, Σ) if vecor X has densiy N n ( ; µ, Σ). Definiion 7 (Scale mixure of normal disribuions). If µ is an n-dimensional vecor, Σ is an n n posiive definie symmeric marix and H is a (unidimensional) probabiliy disribuion funcion such ha H() =, hen he n-dimensional densiy f given by f(x; µ, Σ,H)= = N n x; µ, v Σ dh(v) (3.) Z =(π) n Σ ½ v n exp ¾ v (x µ) Σ (x µ) dh(v) (3.) is said o be a scale mixure of normal densiies {N n (x; µ, v Σ) v (, )} wih mixing disribuion funcion H. If vecor X has densiy (3.), we say ha he disribuion of X is he scale mixure of normal disribuions (s.m.n.d.) SMN n (µ, Σ,H) and wrie X SMN n (µ, Σ,H).

11 Noe ha in he above definiion we use only probabiliy disribuion funcions as mixing disribuions. Some wider approaches o he idea of mixure (see for insance Chu [3]) allow he use of anoher kind of weighing funcion. For each vecor x 6= µ, we have f(x; µ, Σ,H) <, since funcion N n (x; µ, v Σ), as a funcion of v, is bounded, because i is coninuous and boh is limis, a and a, are. On he conrary, he value f(µ; µ, Σ,H)=(π) n Σ v n dh(v), (3.3) which is proporional o he momen R v n dh(v), may be. A vecor X =(X,..., X n ) has he disribuion SMN n (µ, Σ,H) iff i admis he following equaliy in disribuion X d = µ + VZ, (3.4) where Z N n (, Σ) and V, he mixure variable, is a unidimensional random variable independen of Z and having disribuion funcion H. Then, he disribuion of X condiional o V = v is (X V = v) N n (µ, v Σ). If X (p) =(X,..., X p ), wih p n, hen he disribuion of X (p) condiional o V is X (p) V = v N p µ(p),v Σ (p). Hence, X (p) SMN p µ(p), Σ (p),h, (3.5) where he mixing disribuion funcion is H, he same as for X. From (3.) we see ha any scale mixure of normal disribuion is an ellipical disribuion (his subjec will be developed in corollary 9). The reciprocal is no rue; some counerexamples are shown in secion 4. A firs general characerizaion of e.d. s ha are s.m.n.d. s can be saed in erms of he usual quadraic form as follows. Le X E n (µ, Σ,g). IisknownhaX has a sochasic represenaion of he form X = d µ + A Q U (n), (3.6) (see deails in Gómez e al. [7] and Fang and Zhang [6]) wih Q =(X µ) Σ (X µ). Ifwecompare(3.6)wih(3.4)wededucehaX SMN n (µ, Σ,H) iff Q = d V J, (3.7) where V is he mixure variable, whose disribuion funcion is H, and J is a variable independen of V, wih J χ n; his is equivalen o saying ha he densiy of Q is a mixure of gamma densiies n G o v, n v (, )

12 wih mixing disribuion funcion H. (See also Gupa and Varga [8], Corollary ) In his case, from (3.7), he momens of he modular variable R = d Q and hose of vecor X (see Gómez e al. [7]) can be expressed as funcions of he momens of he mixure variable V : E [R s ]= s Γ n+s Γ n E [V s ], Var [X] =E h V i Σ, γ [X] =n(n +) E [V 4 ] (E [V ]), where Var[X] and γ [X] denoe he covariance marix and he kurosis coefficien (see Mardia e al. []) of vecor X, respecively. Two more pracical characerizaions of e.d. s ha are s.m.n.d. s are shown in he following secions. 3. A necessary and sufficien condiion Ellipical disribuions ha are scale mixures of normal disribuions are characerized and pu in relaion wih expansive sequences of ellipical disribuions. The nex heorem shows ha a necessary and sufficien condiion for an e.d. E n (µ, Σ,g) o be a s.m.n.d. is he alernaion of sign of he successive derivaives of is funcional parameer g; his is jus he same condiion esablished in heorem 6 for being a erm of an expansive sequence. The paricularizaion of heorem 8 o n =coincides wih he heorem esablished in Andrews and Mallows [] for univariae symmerical disribuions. Theorem 8 (Condiion for being a mixure). Le E n (µ, Σ,g) be an ellipical disribuion, wih n, and le X =(X,..., X n ) E n (µ, Σ,g). There is a mixing disribuion funcion H such ha X SMN n (µ, Σ,H) iff here exiss a funcion g = g a.e. such ha for k {,,,...} and >. ( ) k g (k) () (3.8)

13 PROOF. (i) (The if par) By virue of heorem 6, here exiss an expansive sequence of e.d. s {E m (µ m, Σ m,g m )}, whose n-h erm is he disribuion E n (µ, Σ,g). Therefore, here exiss, by virue boh of heorem 4..3 of Gupa and Varga [8] and heorem of Eaon [4], a mixure funcion H, such ha X SMN n (µ, Σ,H). (Theonlyifpar) If disribuion E n (µ, Σ,g) is a s.m.n.d., hen he cied heorem of Gupa and Varga [8] implies ha here exiss an expansive sequence of e.d. s {E m (µ m, Σ m,g m )} whose n-h erm is he disribuion E n (µ, Σ,g). The resul follows now from heorem 6. Noe ha, once again, if funcion g is coninuous we can do wihou he funcion g and direcly pu g in (3.8). The nex corollary shows ha any s.m.n.d. SMN n (µ, Σ,H) is also an e.d. E n (µ, Σ,g), included in a sequence of e.d. s having funcional parameers ha are funcions of he successive derivaives of g, and which are also s.m.n.d. s, all wih he same mixing disribuion funcion H. Corollary 9 (Mixures inside sequences). Le SMN n (µ, Σ,H) be a scale mixure of normal disribuions, wih n. (i) A vecor X saisfies X SMN n (µ, Σ,H) iff X E n (µ, Σ,g), wih ½ g() = v n exp ¾ v dh(v). (3.9) (ii) The sequence {E m (µ m, Σ m,g m )} of ellipical disribuions described by (.) (.4), replacing g wih g, is expansive, is n-h erm is he disribuion E n (µ, Σ,g) and, in addiion, i verifies ha if X m E m (µ m, Σ m,g m ) hen X m SMN m (µ m, Σ m,h), where H is he same mixure funcion as in SMN n (µ, Σ,H). PROOF. (i) I follows immediaely from (3.). (ii) Funcion g saisfies (3.8), herefore, by virue of heorem 6, he sequence {E m (µ m, Σ m,g m )} is expansive and is n-h erm is he disribuion E n (µ, Σ,g). Besides, for all m, if X m E m (µ m, Σ m,g m ) hen here exiss, by virue of heorem 4..3 of Gupa and Varga [8], a mixing funcion H m such ha X m SMN m (µ m, Σ m,h m ). Now, he mixing funcion H m is he same for all m, because if m<rhen X m is disribued as a sub-vecor of X r and (see (3.5)) H m = H r. Hence, for all m, H m = H n = H and X m SMN m (µ m, Σ m,h). 3

14 Remark. Le E n (µ, Σ,g) be an e.d. equivalen o SMN n (µ, Σ,H). The funcional parameers g m of he disribuions of he sequence alluded in corollary 9(ii) can be obained from g and g in his way: g +k =( ) k g (k), g +k =( ) k g (k), for k {,,...}. As for he saring poins, g and g, if n> hen g () = R (w ) n g(w) dw and g () = R (w ) n g(w) dw; if n =,and we suppose ha g = g is coninuous (in any case, we can replace g wih anyone of is equivalens: he funcion g of heorem 8 or he funcion defined by (3.9)), we make g 3 = g and calculae g as g () = R (w ) g3 (w) dw = R (w ) g (w) dw; and if n =, we make g () = R and, if g is no coninuous, we can calculae g as g () = R dw. (w ) g(w) dw (w ) g (w) 3.3 A second condiion; calculaion of he mixing disribuion funcion If a given e.d. E n (µ, Σ,g) is known o be a s.m.n.d. SMN n (µ, Σ,H) he issue arises of finding he corresponding mixing disribuion funcion H. According o definiion 7, his should be a probabiliy disribuion funcion. We obain H from he inverse Laplace ransform of he funcional parameer g. In his way, he corresponding s.m.n.d. is fully deermined. Theorem, based upon heorem 8, shows firs ha a condiion for an e.d. E n (µ, Σ,g) o be a s.m.n.d., in he sense of definiion 7, is ha is funcional parameer is he Laplace ransform of a disribuion funcion M of a measure wih suppor in (, ). The heorem also shows he relaion beween he mixing disribuion funcion H and he disribuion funcion M. Thus he heorem paricularizes some aspecs of a heorem of Chu [3] (see also Gupa and Varga [8]) and exends he lemma in Andrews and Mallows []. Theorem (Condiion; mixing disribuion funcion). Le E n (µ, Σ, g) be an ellipical disribuion, wih n, and le X E n (µ, Σ,g). There is a mixing disribuion funcion H such ha X SMN n (µ, Σ,H) iff here exiss a measure disribuion funcion M, wih M(x) =for x, such ha g() = e y dm (y) (3.) for almos all, namely, such ha g is equivalen o he Laplace ransform of M a.e. In his case, he mixing disribuion funcion H is relaed o he 4

15 disribuion funcion M as follows: M(y) = R n g() d Γ n Z (,y] u n dh (u). (3.) PROOF. If (3.) holds hen, for k {,,,...} and >, ( ) k g (k) () = y k e y dm(y), and X SMN n (µ, Σ,H) for some mixing disribuion funcion H. If X SMN n (µ, Σ,H), hen, by equaing (.) o (3.) we obain ha g () = R s n g(s) ds n Γ n ½ v n exp ¾ v dh(v) (3.) for almos all, and now, by making he change u = v, we obain g () = = R s n g(s) ds Γ n e y dm (y) exp { u} u n dh (u) (3.3) for almos all, where M is as in (3.). If we know he inverse Laplace ransform M of g, we may use (3.) o find H, as we will show in secion 4. Furhermore, he nex corollary shows an explici expression for he densiy h of H, whenever i exiss. I is obained easily by differeniaing in (3.). Corollary (Mixing densiy funcion). Suppose ha (3.) (and (3.)) holds. Then disribuion H is absoluely coninuous, wih densiy h, iff M is absoluely coninuous. In his case g() = e y m (y) dy wih m(y) =M (y) a.e., for almos all, and m (y) = R s n g(s) ds 3 Γ n y n 3 h (y), (3.4) 5

16 a.e. and, herefore, h (v) = R n Γ n µ n g() d vn 3 m v. (3.5) Remark. Someresulsabouhemomen R v n dh (v) can be obained. If (3.) (and (3.)) holds hen (see (3.) and also (3.3)) n v n Γ n dh (v) = R s n g(s) ds g() =(π) n Σ f(µ; µ, Σ,g). I will be R v n dh (v) < iff g () < or, equivalenly, if f(µ; µ, Σ,g) <. Remark. FuncionM, given by (3.), is a disribuion funcion of a measure. Is limi a infiniy is lim y M(y) =g() (pu =in (3.3) and compare wih (3.)). Therefore, M is a disribuion funcion of a finie measure iff g() < (or R v n dh (v) < or f(µ; µ, Σ,g) < ). In his case, if we have chosen he funcional parameer g so ha g() = (he reparamerizaion g () =g()/g() can do i), funcion M is a probabiliy disribuion funcion. 4 Some examples and applicaions Firs, in ligh of he obained resuls, we are going o realize he uiliy of he previous conceps for a comprehensive reamen of he relaionship beween e.d. s and s.m.n.d. s. Nex, we consider hree families of ellipical disribuions and sudy heir possible represenaion as scale mixures of normal disribuions. We pu q(x) =(x µ) Σ (x µ). 4. Examples of mixures Adegeneraemixure. If we make H(v) =I [v, )(v) (he disribuion funcion of a probabiliy degenerae in v ), for some v (, ), in (3.), hen f ( ; µ, Σ,H) is he normal densiy N n ( ; µ, vσ). This is an ellipical densiy wih funcional parameer g() =exp n v o.wemaycheckhag, in fac, saisfies (3.8): ( ) k g (k) () = k v k e v. Also, he resuls 6

17 of secion 3.3 permi us o ge back H from g. In his case, g is he Laplace ransform of disribuion funcion M(y) = I [ v, ) (y). From (3.) we have ha Z I h Z (y) = n n v n, u dh (u) = v n h v n dh (v), (,y] (y), v and hence R [z, ) v n dh (v) =v n I (,v ] (z). This implies ha P H ((,v )) = P H ((v, )) = and P H ({v })=, where P H is he probabiliy whose disribuion funcion is H. Therefore, H(v) =I [v, )(v). A uniform mixure. If n = 3 and H has densiy h(v) = I (,) (v), hen f(x; µ, Σ,H)=(π) 3 Σ (q(x)) exp n q(x)o for x 6= µ and f(µ; µ, Σ, H) =. This is an ellipical densiy wih funcional parameer g() = exp n o. Again we may check: ( ) k g (k) () =k! (k+) e P kj= j! j. And we also can ge back H from g: by making use of he inverse Laplace ransform we obain ha m(y) = I (, ) (y), and from (3.5) we ge h (v) =I (,) (v). NoehainhiscaseM (y) =yi [, ) (y), andhen lim y M(y) =f(µ; µ, Σ,H)=. APareomixure. If n =3and H has densiy h(v) =v I [, ) (v), hen he mixure densiy (3.) is jus he E 3 (µ, Σ,g) densiy wih g() = ( + ) exp n o for >. In his case we have f(µ; µ, Σ,H)= 7 π 3 Σ < k+ µ P k+ j= j! j. Again we check: ( ) k g (k) () = (k+)! e (k+)! µ k+ e P j= j! j =. Now we obain H from g: bymakinguseofhe inverse Laplace ransform we obain ha m(y) =yi (, (y), and now (3.5) ) yields h (v) =v I [, ) (v). In his case lim y M(y) < and f(µ; µ, Σ,H) <, oo. 4. Applicaion o he sudy of ellipical disribuions Pearson ype VII disribuion. Γ m+n f(x) = (mπ) n Γ m Σ For any posiive ineger n, le +m q (x) m+n (4.) for some m (, ). This is a E n (µ, Σ,g) densiy wih g() = + m We have µ ( ) k g (k) () = + m+n k Γ m+n + k m m m+n Γ( m+n ) ; m+n. 7

18 herefore, (4.) is a SMN n (µ, Σ,H) densiy for some H. We obain H from g: he use of Laplace ransform yields m(y) =m m+n Γ m+n e my y m+n, and from (3.5) we ge h (v) =m m n Γ m ( m ) exp mv o v m. If V is he mixure variable, hen W = mv has he gamma G, m disribuion. If m is an ineger, hen (4.) is a Suden s densiy, and variable W has he χ m disribuion. If m =, (4.) is a Cauchy densiy. Pearson ype II disribuion. For any n, le f(x) = Γ n + m + π n Γ (m +) Σ ( q (x)) m I [,) (q (x)), for some m (, ). This an insance of an e.d. ha is no a s.m.n.d., because is funcional parameer, g() = ( ) m I [,) (), does no saisfy condiion (3.8), for he lack of coninuiy a of some of is derivaives g (k). Acually, g (k) ( + )=for all m (, ); bu, for m<, we have g( )= ; for any non-negaive ineger m, we have g (m) ( )=( ) m m!; andfor posiive non-ineger m, g (bmc) ( ) is if he ineger par, bmc, of m is even, and is if bmc is odd. Logisic disribuion. Here a densiy is shown ha, as opposed o he previous disribuion, is posiive everywhere, bu is no a scale mixure of normal densiies eiher. For any n we consider he densiy (see Fang e al. [5]) f(x) = π n R n Γ n e (+e ) d Σ exp { q (x)} ( + exp { q (x)}). Here g() = e ( + e ) and condiion (3.8) is no saisfied, because ( ) g () () =e ( + e 4e )(+e ) 4 < for, ln 3. Acknowledgemens The auhors hank he valuable commens and help from Professor José Mendoza. References [] D. R. Andrews and C. L. Mallows, Scale mixure of normal disribuions, J.R.Sa.Soc.Ser.BSa.Mehodol.36 (974) 99. 8

19 [] S. Cambanis, S. Huang, and G. Simons, On he heory of ellipically conoured disribuions, J. Mulivariae Anal. (98) [3] K. C. Chu, Esimaion and decision for linear sysems wih ellipical random processes, IEEE Trans. Auoma. Conrol 8 (973) [4] M. L. Eaon, On he projecion of isoropic disribuions, Ann. Sais. 9 (98) [5] K. T. Fang, S. Koz, and K. W. Ng, Symmeric Mulivariae and Relaed Disribuions, Chapman and Hall (London, 99). [6] K. T. Fang, andy. T. Zhang, Generalized Mulivariae Analysis, Springer- Verlag (Beijing, 99). [7] E. Gómez, M. A. Gómez-Villegas and J. M. Marín, A survey on coninuous ellipical vecor disribuions, Rev. Ma. Complu. 6 (3) [8] A. K. Gupa and T. Varga, Ellipically Conoured Models in Saisics, Kluwer Academic Publishers (Dordrech, 993). [9] A. K. Gupa and T. Varga, Normal mixure represenaions of marix variae ellipically conoured disribuions, Sankhyā, Ser. A 57 (995) [] D. Kelker, Disribuion heory of spherical disribuions and a locaion-scale parameer generalizaion, Sankhyā, Ser. A 3 (97) []K.V.Mardia,J.T.KenandJ.M.Bibby,Mulivariae Analysis, Academic Press (London, 979). 9