- 1 - Non-Central Multivariate Chi-Square and Gamma Distributions
Thomas Royen
TH Bingen, University of Applied Sciences e-mail: [email protected]
A ()p -1 -variate integral representation is given for the cumulative distribution function of the general p - variate non-central gamma distribution with a non-centrality matrix of any admissible rank. The real part of products of well known analytical functions is integrated over arguments from ( , ). To facilitate the computation, these formulas are given more detailed for p 3. These ()p -1 -variate integrals are also derived for the diagonal of a non-central complex Wishart matrix. Furthermore, some alternative formulas are given for the cases with an associated “one-factorial” ()pp - correlation matrix R, i.e. R differs from a suitable diagonal matrix only by a matrix of rank 1, which holds in particular for all (3 3) - R with no vanishing correlation.
1. Introduction and Notation
The following notations are used: stands for with nn,..., . The spectral norm of any ()n n1... np n 10p square matrix B is denoted by ||B || and ||B is its determinant. For a symmetrical matrix, A 0 means positive definiteness of A, and I or I p is always an identity matrix. The symbol also denotes a non-singular cova- 1 ik 1 ik riance matrix () ik with () , and Rr ()ik is a non-singular correlation matrix with Rr ( ). The abbreviation Lt stands for “Laplace transform” or “Laplace transformation”. Formulas from the NIST Handbook of Mathematical Functions are cited by HMF and their numbers.
The p - variate non-central chi-square distribution with ν degrees of freedom, an “associated” covariance matrix 2 and a “non-centrality” matrix (the p (,,)- distribution) is defined as the joint distribution of the diago- nal elements of a non-central Wp (,,) - Wishart matrix. To avoid confusion with different multivariate distribu- tions having univariate (non-central) chi-square marginal distributions, this distribution can also be called a (non- central) “Wishart chi-square distribution”. Without loss of generality (w.l.o.g.) we can standardize to a correla-
T tion matrix R. The ()p - matrix Z of the corresponding Wishart matrix ZZ has independent NRpj(,) - 1 T normally distributed columns and 2 MM with the ()p - matrix M (1 ,..., ). Therefore, rank() min(p , ). (In the literature, frequently 1 is called “non-centrality” matrix.) This distribution can be applied to power calculations for many statistical tests with several correlated chi-square test statistics, (see e.g. section 6 in [13]), but also in wireless communication at least the trivariate case occurs. For p 3 the corresponding cu-
2 mulative distribution function (cdf) is not easy to compute. The complexity of the p (,,)R - distribution (or of ______Key words and phrases: non-central multivariate chi-square distribution, non-central multivariate gamma distri- bution 2010 Mathematics Subject Classifications: 62H10, 62E15 - 2 -
the slightly more general p (,,) R - distribution defined in (2.2) below) depends - apart from the dimension p - essentially on two ranks, the rank k of and the minimal possible rank m of RV , where V is a suitable non-singular - not necessarily positive definite (pos. def.) - diagonal matrix. For many applications, only distribu- tions with rank( ) 1 are required, e.g. with identical expectation vectors j in the above matrix M. These 2 cases can be reduced to central p1(,) - distributions by theorem 4.1 in [13] or by a similar theorem in [8]. For p 2 see formula (3.24) and (3.25) in section 3. However, a bivariate correlation matrix is always “one-fac- 2 torial”, i.e. m 1. The p (,,)R -cdf with m 1 and V 0 was already derived in [13] from a suitable re- presentation of the Lt of the corresponding probability density (pdf). For the corresponding p (,,) R - cdf see [14]. These distributions are included in section 3 and extended to cases with an indefinite V. In particular, all
R33 with no vanishing correlation are “one-factorial” in this extended sense or limit cases with |V | 0. In [2] a 2 13 special 3 (,,)R - pdf is found in formula (49) with rank( ) 1 and r 0. This is a limit case of the distri- butions with a “one-factorial” R33 with v 22 0. A simpler form of the p (,,) R - cdf, p 3, of such limit cases is also derived in section 3.
p The most general p (,,) R - cdf is represented by a p - variate integral over the cube p (,] in section 4 and by a (p 1) - variate one over p1 in section 5. These latter integrals are written in a more detailed form in formula (5.7) and (5.8) for p 2 and p 3 to facilitate the direct computation. At least for the above mentioned cases with m 1 and an indefinite V33 these bivariate integrals are supposed to be more favourable for a numerical evaluation than the corresponding formulas in section 3. A further advantage is the visibility of the integrand if p 3.The extension with a non-central complex Wishart matrix is found in section 6.
More general integrals over p for convolutions of non-central gamma distributions were already given in [16]. In particular, for p 1 an integral over (0, ) is found there for the cdf of a pos. def. quadratic form of (non-cen- tral) Gaussian random variables. For p 2 e.g. a bivariate integral is obtained for the cdf of Jensen’s bivariate
2 chi-square distribution (see also [5] and [17]). A survey on formulas for the central p (,)R - distribution and some similar distributions is found in the appendix of [3].
2. Some Preliminaries and Special Functions
Definition. Any non-singular covariance matrix pp is called “positive m-factorial” or only “m-factorial” if m is the lowest possible rank of A in a representation V AAT with any pos. def. diagonal matrix V , where A may contain a mixture of real or pure imaginary columns. If m is the lowest possible rank in such a represen- tation with a non-singular not necessarily pos. def. diagonal matrix V , then is called “real m-factorial”.
2 T j If pp is m-factorial with VW then W W Ip BB , where BWA has p rows b and m columns
T 2 b and - w.l.o.g. - bb0, . Each pp 0 is at most (p -1)- factorial with WI p , where 0 is the lowest eigenvalue of . - 3 -
2 The p (,,)R - pdf has the Lt
ˆ /2 1 fttR(1 ,...,p ; , , ) | I p 2 RT | etr( 2 TI ( p 2 RT ) ), (2.1)
1 T T diag( t1 ,...,tp ) 0, , 2 MM , rank( ) min( p , ).
The p - variate non-central standardized gamma - pdf (in the extended sense of Krishnamoorthy and Parthasa- rathy [6]) - the p (,,) R - pdf - can be defined by the Lt
1 gtˆ(,...,;,,)|1 tp R IRT p |etr( TIRT ( p ) ). (2.2)
2 For positive integers 2 this distribution differs from the p (,,)R - distribution only by a scale factor 2, but this distribution is also defined for some non-integer values 2. The function g( x1 ,..., xp ; , R , ) with the Lt from (2.2) is not for all 0 a pdf, but additionally to 2 1,...,pp 2 , 3, rank( ) 2 , at least all real values 2 pp 1, 2, are admissible with any 0, see e.g. [7] or [9] (and also [11] if rank( ) 1).
Recently, in [20] the central p (,) R - distribution was also shown to exist for all values 2 [(p 1) / 2]
T without any restrictions for R. For an m-factorial R with WRW Ip BB the p (,) R - cdf is given by
p E G( w2 x ,1 bjj S b T ) , 2 or 2 mp 1 [( 1) / 2], (2.3) j1 jj2 2
with the univariate non-central gamma cdf G (,) x y from (2.8) and the expectation referring to the WImm(2 , ) -
Wishart matrix S2 , see [13] or [15]. If B is real, then in (2.3) all non-integer values 2 m 1 0 are admis- sible too. Hence, the p (,,) R - distribution also exists for all real values 2 m with such m-factorial R and rank( ) 1. Formula (2.3) was also used in [19] to extend the proof of the Gaussian correlation conjecture
1 in [18]. If ||I RT is infinitely divisible, then ||I RT is the Lt of a p (,) R - pdf for all 0 and then formula (2.2) is the Lt of a p (,,) R - pdf for all values 21 if rank( ) 1. Sufficient and necessary conditions for R entailing infinite divisibility are found in [1] and [4]. Special infinitely divisible p (,) R - distri- butions arise by a one-factorial RW 2 aaT with a real column a.
2 If R is real one-factorial with a real or pure imaginary column a and possibly with one wk 0, we have the relations
T 22 T WRW Ip bb with wj1 a j , b j w j a j , : | WRW | 1 b b ,
11T ().WRW Ip bb (2.4) We need the following functions: The univariate gamma density
1 1 x g ( x ) (() )xe , , x 0, (2.5) with shape parameter and the corresponding cdf - 4 -
x G()()(,)(,)/() x g d P x x (2.6) 0 with the notations from HMF 8.2.4 for the incomplete gamma function ( ,x ), which can be extended to , cut along the negative real axis. Later on, we shall use the fact that xx (,) is a single valued function on .
The non-central gamma density g (,) x y with non-centrality parameter y is given by
yn gxy(,)eyy gx ()eexy()()x ( 1)/2 I 2 xy gxFxy ()(;) (2.7) n0 n n! y 1 0 1
with the modified Bessel function I 1 and the hypergeometric function 01F . The corresponding non-central cdf is yn G(,)() x y ey G x . (2.8) n0 n n! Besides,
Gxy* (,):(,)()(;)ey Gxy g xF kxy (2.9) k1 k 01 will be used.
For n we have the relations
n G( x , y )1 (erf()()x y erf x y)ey g ( x ) F (1 k ; xy ), (2.10) 1/2nk2 k1 1/2 0 1 2
n GxyGxy(,) (,)ey gxF ()(;),1 k xy (2.11) 1nk 1k1 1 0 1
Gnn( x , y ) 1 Q ()2yx , 2 with the Marcum Q - function Qn of order n. Furthermore, with
0,z 0 1 Gz0 (): 2 ,0z (2.12) 1,z 0 we have the integral representation, (see section 2 in [13]),
ycos( n ) xy cos(( n 1) ) G( x , y )exy( x)n/2 1 e2xy cos( )dG ( x y ), n , y 0. (2.13) n y 0 y2 xy cos( ) x 00
The functions G (,) x y can be extended to xy, too with the principal value of x if .
Furthermore, with xy,, we need the functions
1 1 xy ( 1) 1y (G()() x y e G xy ),y 1, 0 (xy , ) Gx ( ) yn 1 ( GxyeGxy ( ) 1 xy ( 1) ( ) ), 1, , (2.14) n0 n 1y 11 xg( x ) (1 x ) G ( x ), y 1. verified by Lt. - 5 - 3. Distributions with “One-Factorial“ Correlation Matrices and the Trivariate Distributions
Some earlier results are compiled here and extended by theorem 1. The p (,,) R - cdf is given for all non- singular “real one-factorial” correlation matrices R from (2.4) and any admissible rank of 0 and additionally 22 for the limit cases with wakk10 for one index k. In particular, all non-singular Rr33 ()ik with r12 r 13 r 23 0 are real one-factorial or limit cases of such matrices. We have the following subclasses: pure imagi- 2 2 nary or real column a with maxa j 1, real column a with one value ak 1 and the limit cases with one value ak 1 (replacing a by a avoids ak 1). An essential simplification is obtained for rank( ) 1. The 2 cases with maxa j 1 were already given in [13] and [14]. The totally different unified (p 1) - variate integral representation in section 5 is suitable for feasible computations for p 3 and might be preferred for a numerical 2 evaluation at least for cases with ak 1 or for the here not included case with one vanishing correlation.
The key for the formulas in theorem 1 for the p (,,) R - cdf are suitable representations of the Lt from (2.2) with R as in (2.4), which were already given by the formulas (2.16) and (2.17) in [13] for the corresponding 2 2 p (,,)R - distribution with maxa j 1. With
2TT 2 2 2 R W aaWRW I pp bb, W diag ( w1 ,..., w ), bj w j a j , (3.1) 2 (one wk 0 admissible),
11 21 2 DRR, zj(1 w j t j ) , uj w j t j z j 1 z j , Z diag( z1 ,..., zpp ), U I Z , (3.2)
T diag( t1 ,..., tp ) 0, we have the following identities
1 |Ipp RT | etr() T ( I RT ) )
T T b UW WUb |Z | (1 b Ub ) etr( W WU )expT (3.3) 1b Ub
T TT 11 a ZDZa exp(aDaZ ) | | (1 bUb ) etr( WDWU )expT . (3.4) 1b Ub
TT With rank( ) 1, ( i j ), D dd ( d i d j ), these formulas become
T 2 p ()b UW |Z | (1 bT Ub ) exp 22 w u exp (3.5) j1 j j j T 1b Ub and
T 2 p ()a Zd exp() (aTT d )2 | Z | (1 b Ub ) exp d 2 w 2 u exp , (3.6) j1 j j j T 1b Ub respectively. With |WRW | 1 bT b we can also write
1bTT Ub(1 1 b Zb ). (3.7) - 6 -
22 If there is a negative value wakk1, we obtain a negative and the above formulas will be used later with
21 1 T takk( 1) entailing zk 0 and 1 b Zb 0.
The identity (3.3) follows by straightforward calculation from
TTT1 1 1 1 1 T()()() Ip RT T Z aa T TZ I p aa TZ TZ I p T aa TZ 1 a TZa
1 T 11 TZ Ip T W bb W TZ . 1b Ub For the equality of the representations in (3.3) and (3.4) only the equation
aTT ZDZa b UW WUb aT D a tr( W11 DW U ) tr() W WU (3.8) 11bTTU b b Ub has to be verified. The left hand side can be written as
bT ()() I U W11 DW I U b bTW 1 DW 1 b tr(). W 1 DW 1 U 1 bT Ub After having inserted
W1 DW 1()()()() WRW 1 W W WRW 1 I 1 bbTT W W I 1 bb the equality (3.8) can be verified by elementary calculations.
2 For the limit case : 1 ak 0 the limit
()n ty lim (1 t ) exp exp( ty ) 0 1t
1 1 1 of the Lt of gn (,) x y implies
11 0,xy limG n ( x , y ) . (3.9) 0 1,xy
d , ji T **ii For a Da dij a i a j, d ij ji 1i j p 2,dij we need
T n ()a Da p nnj1 * ij d( n11 ,..., np ) a j , d ( n ,..., n p ) d ij , (3.10) n!!(2n ) j1 nij 1i j p where the summation is taken over the n with n n n, j 1,..., p . ij ji j ij i j ji
2 T Theorem 1. The p (,,) R - cdf with the non-singular correlation matrix R W aa
T (WRW I ppbb), W diag ( w1 ,..., w ) with possibly one imaginary value wk , is given by
G( x1 ,..., xp ; , R , )
p exp(aDaT ) dnn ( ,..., ) aGnj (w2 xdwbygydy , 2 2 ) ( ) , (3.11) 0 nn0 (2 ) 1 p j1 jnj j j jj j j n
2 1 1 maxaj 1, D ( d ij ) R R , - 7 - or by
exp(aT Da)( etr DW 2 )
p n dn(,...,) n aGnj * (w2 xdw , 2 1 byg 2 ) (), ydy (3.12) 0 nn0 (2 ) 1 p j1 jnj j j jj j j n
22 T * y maxaajj 1 or max 1, |WRW | 1 b b, Gnn(,)(,) x y e G x y from (2.9).
TT With rank( ) 1, ( i j ), D dd ( d i d j ) these formulas are simplified to
G( x1 ,..., xp ; , R , )
nj p ()ad exp(())adT 2(2n )! jj G(w 2 xdwbygydy , 2 2 2 )() (3.13) nn0 (2 ) j1 n ! n j j j j j j n 0 n! j
p Gwx()2, 2 wbybwy 2 2 2 1/2 cos()()() f gyddy , (3.14) 00 j1 j j j j j j j j
111 2( 1) 2 fB ( ) (,),22 () sin maxa j 1, or to p exp( (aT d )2 )exp d 2 w 2 j1 jj
nj p ()ad n (2n )! jj G* (w2 x,)() d 2 w 2 1 b 2 y g y dy (3.15) nn0 (2 ) j1 n ! n j j j j j j n 0 n! j p exp( (aT d )2 )exp d 2 w 2 j1 jj
p Gwxdw*2(), 22 12 by 2 1/2 bdwy 11/2 cos()()(), f gyddy (3.16) 00 j1 j j j j j j j j
22 maxaajj 1 or max 1.
2 If exactly one value ak 1, then
G( x1 ,..., xp ; , R , )
x exp(aDaT )k dnn ( ,..., ) aGnj (w2 xdwbygydy , 2 2 ) ( ) , (3.17) 0 nn0 (2 ) 1 p jnj j j jj j j n jk and, if rank( ) 1, this is simplified to
p n x ()ad j exp(())adT 2k (2n )! jj G(w 2 xdwbygydy , 2 2 2 )() (3.18) 0 nn0n! (2 ) nj! nj j j j j j n j1 j k Gwx()2, 2 wby 2 2 2 bwy 1/2 cos()()() f gydyd , (3.19) j j j j j j j j jk
2 1/2 (,)|y [,],00 y , k y 2 k y cos() x k .
2 2 1/2 2 1 1/2 Remarks. With yx () kcos( ) ( k k sin ) and 1 arcsin() |kk | x we obtain
2 ( ,y ) | 0 y y ( ), 0 , 0 kk x 2 ( ,y ) | y ( ) y y( ), 1 , k x k , k 0 (3.20) 2 ( ,y ) | y ( ) y y( ), 0 1 , k x k , k 0 - 8 -
For p 3 the coefficients d( n1 ,..., np ) are * nij ()dij 1 d(,),,() n1 n 2 ni n 12 even n ii n i n 12 (3.21) nij!2 0n12 min( n 1 , n 2 ) ij
n [][]nn12/2 /2 []n3 /2 * ij ()dij 1 dnnn(,,)1 2 3 , nij ( nnn i j k )(n ii nnijkij jj kk ),,,. (3.22) nij!2 n110 n 2200 n 33 ij
Proof of Theorem 1. All the representations for the cdf in theorem 1 can be verified by the Lt of the correspond-
22 ding densities. The Lt gˆ n (,) t y of w gn (,) w x y is
(1w2 t ) (n ) exp() w 2 t ( 1 w 2 t ) 1 y ze n uy , u 1 z , which also holds for w2 0 if tw2 0 and consequently z 0. Then, from the formulas (3.11), (3.12) we obtain the Lt in (3.3) and (3.4), from the formulas (3.13), (3.14) the Lt in (3.5), and from (3.15), (3.16) the Lt in (3.6). For the verification of the formulas (3.14), (3.16) the integral representation of the modified Bessel functions
I 1 from HMF10.32.1 is used for the integration over within the corresponding integral for the Lt. For the 22 2 verification of the cases with one wakk1 0, the variable tk of the Lt is supposed to be larger than wk , which is equivalent to zk 0. If for the moment the integrals for the corresponding asserted representations of the pdf are denoted by h( x1 ,..., xp ), we obtain with a sufficiently large 0 a Lt of exp( xxip hx( 1 ,..., ) coinciding with the Lt from (2.2) with TIT p , 0, instead of T. Therefore, h( x1 ,..., xp ) has the Lt. in (2.2), 2 which proves the corresponding representation. Finally, the formulas for the limit cases with ak 1 follow from (3.9).
2 11TT The relationship between a p (,,)R - pdf with 22MM xpp1, Mp ( y 1 ,..., y ), x 1 y2 and a central 2 (,) - pdf with j1 j p1 R T T 1 is described in theorem 4.1 in [13] (for a similar theorem see [8]). For the corresponding densities, this theorem gives the relation
11 1 f(,..., x1 xp ;,,) R ()22 g /2 ( x p 1 ) f (,..., x 1 x p 1 ;,). (3.23)
Here, a slightly more general version is shown for the relation between a p1(,) - pdf and a p (,,) 0 - T pdf with 0 and x T . T p1 1
1 T Theorem 2. The conditional density ()g ( xpp1 ) g (,..., x 1 x 1 ;,) is equal to g( x1 ,..., xpp ; , 0 , x 1 ).
T Proof. With Tpp diag( t1 ,..., t ) the Lt of g ( xp1 )(,...,;, g x 1 x p 0 , x p 1 ) is given by - 9 -
|IT | exp xTIT (T ( )1 tgxdx ( ) p0 p p 1 p p 0 p p 1 p 1 p 1 0 T 1 |Ip 0 T p |1 t p 1 T p ( I p 0 T p ) ,
which coincides with the Lt ||ITp1 of g( x11 ,..., xp ; , ) since
TTT11 |ITtITp1 |()|1 pppppp 1 0TTT tt 1 () 1 1 ppppp |=()| 1 tITt 1 0 () 1 1 p |
11T 11T (11tp1 ) | I p 0 T p || I p ( t p 1 ) ( I p 0 T p ) T p | and |Ip (1 t p10 ) ( I p T p ) T p |
11T 11()(). tp10T p I p T p
With the correlation r in a (2 2) - correlation matrix we also use the notation 2 ( ,r , ). With the central case from formula (3.11) and theorem 2 with 1 r 0 r 1 and a one-factorial correlation matrix
21/2 21/2 21/2 21/2 2 TT R diag()()(1 1 ) ,(1 2 ) ,1 diag (1 1 ) ,(1 2 ) ,1 W aa, WRW I 3 bb ,
T we obtain the 23(,,rx ) - cdf 2 Gxx(,;,, rx T )() ()gx 1 G (1) 2 1 wxbywgwxby 2 , 2 2 (,)g(). 2 2 ydy (3.24) 1 2 3 3 ()j j j j 3 3 3 3 0 j1 2 In a similar way we can apply formula (3.12) for the central case if maxa j 1.
Alternatively, we can use the absolutely convergent series for a trivariate density in formula (3.5b) in [12]. With 1 ij ij ii jj 1/2 Rr () and Q ( qij ), q ij r ( r r ) , this leads to 2 T 1 33 33 2 1 jj Gxxrx(,;,,1 2 3 )||g() Q x 3 ( r g( rxG 3 ) () (1 jj ) rx j1 2 (1) NqmmmQr (,,,,)g 33 () rxG 33() ()1, 2 1 rx jj (3.25) 1 2 3M mm3 3 M j j j ) NM2 m1 m 2 m 3 N j1 min(m ) 2m 1 j 2N ( M m ) 3 M[/2], N N 2,(,,,,) M q m m m Q q2mk N , N1 2 3 3 ij () m0 (2m )! ( m m )! k1 Nj j1 i j,,. k i j
A further alternative is the integral over (,)0 in (5.7), which also holds for rank( ) 2. If theorem 2 is applied with p 3, again the formulas (3.14) and (3.16) are obtained.
4. A p-Variate Integral Representation for the General Non-Central Distribution Function If a p - variate pdf or cdf with a known characteristic function is not explicitly available, it can be represented by a p - variate integral over p by means of the Fourier inversion formula under rather general assumptions. In this - 10 -
section an alternative p - variate integral representation for the general p (,,) - cdf (and the pdf) is given, which requires only integration over (,). p A further advantage is the always possible reduction of this in- tegral to a (p 1) - variate one as shown in the following section. This facilitates the computation at least for p 3 with a visible integrand. The simple idea is the representation of the required probability by an integral over a p - variate Fourier series (or a - variate Laurent series). The Lt
1 1 1 1 |ITTITITITp | etr()() ( p ) etr( ) | p | etr ( p ) (4.1)
with any non-singular covariance matrix and T diag( t1 ,.., tp ) 0 is not for all 0 and all non-centrality matrices 0 the Lt of a pdf. The following integral representation also holds for real measures whose density has the above Lt. For admissible values and rank() of a p (,,) - pdf see the explanations behind (2.2) and (2.3). For a suitable form of the Lt we use the matrices
1 1 1 1 BID(v ) p , v (4.2) with any scale factor v, leading to ||B || < 1, e.g.
1 11 v)2 (min max (4.3)
1 with the extremal eigenvalues of , which leads to ||B || (max min )( max min ) . With
11 Z diag( z1 ,..., zp ), z j (1 v t j ) , (4.4) we obtain
-1 -1 -1 -1 -1 |ITIp | =| p vv| T |vΣ||(v) IIT p p v| |v|| BZ | |v|| ZIBZ || p | and 1 1 1 1 1 1 1 1 1 tr(vv)()()()Ipp T tr( B Z )v tr( Z I BZ ). D Hence, the Lt from (4.1) can be written as p 11 |v | etr( )| ZIBZ | |p | etr() ZIBZD ( p ) zbz j (1 ,..., z p ) (4.5) j1 with a p - variate analytical function
n1 np b( y1 ,..., yp ) ( n 1 ,..., n p ) y 1 ... y p , (4.6)
1 which is abs. convergent if max|yBj | || || .
Theorem 3. With the functions from (2.14) the p (,,) - cdf is given by
etr()Y()1I BY 1 1 D p G( x ,..., x ; , , ) |v | etr( 1 ) (2 ) p p (v x , y ) d , 1 p 1 j j j ||I BY j1 p p
p i j p ( , ] , v from (4.3), the matrices B and D from (4.2), Y diag( y1 ,..., ypj ), y re , r || B ||.
- 11 -
Proof. The asserted integral representation of the theorem is equivalent to a corresponding representation for the p (,,) - pdf with the derivatives (vxyjj , ). Comparing this representation with the Lt from (4.5) x j we obtain
ppnn (2) p (,...,)n n yjj v g (v) x y d ... d 11pjj11 j nj j j p p p (,...,n n ) v g (v) x , 1 p j1 nj j
n p p which is abs. convergent because of lim maxvgn (v xjj ) | n n 0 for every 0. The Lt n j1 j j1 pp of this series is z ( n ,..., n ) znj , coinciding with the Lt in (4.5). Integration over xx,..., jj11j 1 p j 1 p concludes the proof.
2 2 2 2 Remarks. More generally, the scale factor v w can be replaced by a scale matrix W diag( w1 ,..., wp ) 0. Then, with
2 2 1 1 TWTW, W WW , W WWz , j (1 wt j j ) , diagzzB (1 ,..., p ), W I p , and 1 1 1 1 1 1 DWW WWW (4.7) the Lt in (4.1) becomes
pp |WW | etr( 11 )| ZIBZ | | | etr() ZIBZD ( ) z ( nn ,..., ) znj , (4.8) p pjj11 j 1 p j
1 -1 -1 -1 1 where the series is abs. convergent if maxzjp || B || || W W I || , i.e. for all sufficiently large values t j if ||B || 1. This can occur in particular with a “natural” scaling if
1 1 1 Q() qij W W (4.9)
1 ij 2 jj is a correlation matrix with () and wj .
Then, with any scale matrix W and the matrices BD, from (4.7), the p (,,) - cdf is given by
G( x1 ,..., xp ; , , )
etr()Y()1I BY 1 1 D p |W W | etr( 12 ) (2 ) p p ( w x , y ) d , (4.10) 1 j j j j ||I BY j1 p p
i j yBj re, r || ||.
The proof is the same as for theorem 3, but by the series expansion and the termwise Lt the correct Lt is
1 obtained first only for sufficiently large values t j with max z j ||B || , if ||B ||
- 12 -
5. The (p-1)-Variate Integral Representation
In the integral representation from (4.10) we can perform the integration over one argument j , w.l.o.g. jp . With
1Bpp b p 1 1 1 1 D pp d p BWWIDWW(),, p TT (5.1) bp b pp d p d pp
i j Ypp diag( y11 ,..., y p ), y j re , j ( , ], (5.2)
T 1 y0 y 0( y 1 ,..., yp 1 ) b p ( Y pp B pp ) b p b pp , (5.3)
TTT11 qqyy(,...,11p )(( bYB p pp pp ),1)(( DbYB p pp pp ),1), (5.4)
L ( y11 ,..., yp )
1 1 1 1 1 etr((YIBYDIBYYBDYBYpppp11 pp pp ) pp ) | pp pp |etr( ( pp pp ) pp ) | pp pp |||, pp (5.5)
* and the functions G from (2.9) and from (2.14) we obtain
Theorem 4. With the above notations the p (,,) - cdf is given by
G( x1 ,..., xp ; , , )
etr(1 ) 1 L( y ,..., y ) p1 1 p1 G* ()(11 ywx )2 ,( yq ) 1 ( wxydrB 2 , ) , || ||. p1 00p p j j j j |WW | (2 )p1 (1y0 ) j1
* 2q 1 2 2 Remark. lim (1yG0 ) (1 ywx 0 )p p , () ( 1) wxF p p ) 0 1 wqx p p ). y 1 1y 0 0
For the proof we use two lemmas:
Lemma 1. With the notations (5.1) to (5.4) the equation
11 q etr( (YBDYBYBDYB ) ) | | etr ( (pp pp ) pp ) | pp pp | exp() (y p y0 ) (5.6) yyp 0 holds for r| yj | > || B ||.
Proof. For
Abpp p AYB: T bp y p b pp we obtain with the Schur complement
T 1 |AAybbAb | = |pp | ( p pp p pp p ) | YByy pp pp |( p 0 ), - 13 -
A1()() yyAbbA 1 1T 1 yyAb 1 1 A1 pp p00 pp p p pp p pp p 1T 1 1 ()()yp y00 b p A pp y p y and
1 1 1 1TTT 1 1 1 1 tr()trAD() ADpp pp ()( yy p 00 AbbAD pp p p pp pp Abd pp p p )()( yy p d pp bAd p pp p )
11 tr(App D pp ) (y p y0 ) q , which implies equation (5.6).
Lemma 2. Let be q any number, Sr y | | y | r and y0 \{1} any number with |y0 | r | y |, then
1 11 * 1 exp((yy 0)q ) (x , y )( y y0 ) y dy (1 y 0 ) G () (1 y 0 ) x ,(1 y 0 ) q . 2i Sr
Proof. With the binomial series we get
y1 () nk y k 11 (,)x y dyg() x ym 0 yn 1 dy n mk00m ()! nk y 22iSS x()yy 0 i rr 1 gx()( nky )kn/()k!g ()exp()(1 xxy y ) (1 yg ) (1 yx ). () n n k0 n 0 0 0 n 0 k0 Multiplication by qnn / ! and summation over n yields
*1 (1y0 )x G ( (1 y 0 ),(1 x y 0 )) q ).
Integration over x provides the assertion.
Proof of Theorem 4. If y p in Y is replaced by a variable y with any value |y |, then the equation
T 1 |Y B | | Ypp B pp |()y b pp b p ( Y pp B pp ) b p 0
T 1 always has a unique solution y y0 bp(), Y pp B pp b p b pp where ||yr0 since ||Bpp || || B || r | y j |,
1jp 1. Hence, with lemma 1 and lemma 2, theorem 4 is obtained by integration over p in the integral from (4.10).
For p 2 we obtain with
2 0 1 1 2 ,w2 2(1), 2B ,() D d W 1 1 1 W 1 , i ||, 2 j j ij y re, r 1 2 2 0
21 the cdf G(,;,,)(1 x12 x )etr( ) `
1 2 1 2 1 1 exp(d11 y )* (1 y ) x21 d 11 y 2 d 12 d 22 y x G,,. yd (5.7) (12yy 1 ) 2 (1 2 ) 2 2 (1 2 ) 0 21 - 14 -
12ij jj 1 1 1 For p 3 we find with ( ), wj , the correlation matrix Q(),, qij W W B Q I3 D
22 1 1 1 1 q23 y 1 q 13 y 22 q 12 q 13 q 23 i q(,) y12 y (dij ) W W , y0 ( y 1 , y 2 ) 2 , yj re, j 1, 2, r || B ||, y1 y 2 q 12 1 y0 ( y 1 , y 2 )
1 (,,)(,,)qy qqqy qqq 22 yyDqy qqqy qqq yy T 13 2 12 23 23 1 12 13 12 1 2 13 2 12 23 23 1 12 13 12 1 2 L(,) y y 2 2 2 2 , and 12 yyq1 2 12 yyqyqyqqqq 1 2 23 1 13 2 2 12 13 23 12
d y d y2 d q q2 22 1 11 2 12 12 12 exp2 1 the cdf y1 y 2 q 12 y 1 y 2
1 G(,,;,,)||etr( x1 x 2 x 3 Q )
2 1 L (,)yy12 * 33 q( y12,) y jj 2 G(1 y0 ( y 1 , y 2 )) x 3 , xjj , y d 1 d 2 . (5.8) 2 ((,)1yyy ) 1 y ( y , y ) 0 0 1 2 0 1 2 j1
21 21 The values (1 y ) and ()1 q12 ( y 1 y 2 ) are defined by the binomial series and the functions (,)xy * 1 and (1y0 ) G ( (1y 0 ) x ,(1 y 0 ) q ), x 0 , are single-valued functions. If, e.g., one element qik 0, 2 ik , then ||QI -3 || 1. If ||QI -3 || 1, then also theorem 4 can be applied with WI vp , v from (4.3), B
11 v I3 with ||B || < min(r , 1 ) , and y0 ,, q L from (5.3), (5.4), (5.5).
6. The Extension to the Distribution of the Diagonal of a Non-Central Complex Wishart Matrix
Let X be a ()p - matrix with independent CN pj(,) - distributed circular complex Gaussian column
** T vectors xj . The diagonal of XX, ( X X ), has a pdf with the Lt from (4.8) with the notations from (4.7) and instead of , but now with a pos. def. Hermitian covariance matrix and a pos. semi-def. Hermitian “non- * centrality” matrix MM M (1 ,..., ). For the corresponding cdf F( x1 ,..., xp ; , , ) we obtain again the integral representation from (4.10), but with replaced by . The (p 1) - variate integral representation of theorem 4 can also be retained for F( x1 ,..., xp ; , , ) with the same proof if we replace within the notations
T T * * (5.1), (5.3), (5.4) the row vectors bp and d p by bp and d p respectively, and q by
* 1 1 * 1 * 1 qy(11 ,..., yp ) bADAb p pp pp pp p dAb p pp p bAd p pp p d pp
with the non-Hermitian (pp 1) ( 1) - matrix AppYB pp pp.
- 15 -
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