ON THE NONCENTRAL DISTRIBUTION OF THE RATIO OF THE EXTREME ROOTS OF THE WISHART MATRIX

The distribution of the ratio of the extreme latent roots of the Wishart matrix is useful in testing the sphericity hypothesis for a multivariate normal population. Let X be a p x n matrix whose columns are distributed independently as multivariate normal with zero mean vector and covariance matrix Further, let S XX’ and let 11 > > i > 0 be the characteristic roots of S. P Thus S has a noncentral Wishart distribution. In this paper, the exact distribution of fp 1 ip/lI is derived. The density of fp is given in terms of zonal polynomials. These results have applications in nuclear physics also.

matrix is useful in testing the sphericity hypothesis for a multivariate normal population.In the central (null) case, Sugiyama (1970) derived the density of the ratio of the smallest to the largest root of the Wishart matrix when the associated covariance matrix is the identity matrix.Waikar and Schuurmann   (1973)  derived an alternate expression which is much superior to that given by Sugiyama (1970) from the point of view of computing and in fact we computed some tables of the percentage points which are also included in the above paper.In this paper, the author has derived an exact expression for the ratio of the smallest to the largest root of the noncentral Wishart matrix.This research has applications in nuclear physics see Wigner (1967) ].
Anderson 2, p. 409] relates the Wishart and non-central Wishart distributions to each other and to neighboring areas of multivariate analysis.James 3, p.475]   gives a brief exposition of this area of multivariate analysis.

PRELIMINARIES
If A is a square, nonsingular matrix its inverse and determinant are denoted respectively by A -I and The transpose, trace and exponential of the trace of a matrix B are denoted respectively by B', tr B and etr B. Also I and 0 de- P P note respectively a p x p identity matrix and a p x p null matrix.In addition, we define as in James (1964).pFq(al' ap; b l,...,bq;S) and F bl,...,bq; S,T) P q(aI, ,ap where S and T are p x p symmetric matrices and K (k l,...,kp) is a partition of the integer k satisfying (i and finally C (S) is the zonal polynomial as defined in James (1964) and satis- fies (tr S) k C (S).A special case of the above is K IF0(a;S) II SI -a P Note that if one of the a. 's above is a negative integer say a, -n then l F for k _> pn + 1 all the coefficients vanish so that the function p q reduces to a (finite) polynomial of degree pn (see Constantine (1963) p. 1276).Further, throughout the paper, whenever a partition say K (kl,..., kp) of a nonnegative integer k is defined, it will be implied that (i) k I > > k > 0 and (ii The following three lemmas are needed in the sequel. LEMMA 2.1.(bl,...,bp) is a partition of the integer k + b.LEMMA 2.2.Let G be as defined in Lemma 2.1 and further let G I diag(l,G).
Let K (kl,...,kp+l) be a partition of a nonnegative integer k.Then where (tl,...,t) is a partition of t.P The above two lemmas are stated in Khatri and Pillai (1968).The g-coeffi- cients in (2.1) and the b-coefficients in (2.2) were tabulated by Khatri and   Pillai (1968) for various values of the arguments and can be obtained from them.
Throughout this paper, the following notations will be used: The following lemma can be proved by making trivial modification in the proof of the Lemma given in Sugiyama (1967).
LEMMA 2.3.Let R diag(r I, rp_I) where 0 < r I <...< rp_ I < i and let R I diag (rl,...,rp_l,l).Further let K (k I, kp) be a partition of the positive integer k.Then p(p-l)/4 P r(a + k i (i 1)/2).P i=l LEMMA 2.4.Let A be any p x p matrix and let (kl,...,k) be a partition P of a nonnegatlve integer k.Then k ,v p where (gl gp) is a partition of g.
The above lemma is stated in Constantine (1963) and some tabulations of a- coefficients are also given in the same paper.
3. DENSITY OF THE RATIO 'OF THE SMALLEST TO THE LARGEST ROOT OF THE WISHART MATRIX, Let X be a p x n matrix whose columns are distributed independently as multi- variate normal with zero mean vector and covariance matrix ). and let p _< n.Fur- ther, let S XX' and let > 2 >'''> > 0 be the characteristic roots of S. diag (f2'''" f and K (kl,...,kp) is a partition of the integer k.
p On using Lemmas 2.2 and 2.4 to expand C [ diag(l I F)] and further, writing etr(1F) as 0F0(1 F) and then expanding it we can rewrite the above density as h(l,f2,...,f ): P V.B. WAIKAR k(p,n) II -n/2 [(n-2)/2 e-P1 IFI II FI (n-p-l)/2 (33) C (I) P t=O T g=O y y where = (a I ap_I) is a partition of the integer a, T (tl,...,tp_I) is a partition of the integer t, y (gl,...,g p I is a partition of g and the b and a are given by Lemmas 2.2 and 2.4 respectively.Now note that C=(F) Cy(-F) L (-I) g g,yn C n(F) where (nl,...,np_I) is a partition of a + g and the coefficients g,y are given by Lemma 2.1.Further ii Fl(n-p-1)/2 IF 0(_(n p i)/2, F) Also then Cn(F) C(F) L gn C (F)   where B (b I, bp_I) is a partition of (a + g) + d and the coefficients g, are given by Lemma 2.1.Thus the density in (3.n (fi-f Cs(F) 0 < i < ' 0 < f2 <'''< f < i.

P
Now, on making the transformation i i' ri fl/fp and then integrating out r2,...,rp_1 over the surface 0 < r 2 <...< rp_ 1 (using Lemma 2.3) we get the joint density of i and f as P ((p + 2)/2)is given by Lemma 2.3.Now, on integrating out i (0 < i < )' we get the marginal density of f 1 /1 as P P CK (I ...1 -1) An important observation is that in the special case when (np 1)/2 is an integer, the summation over d becomes finite (see Remark i) in (3.6) which means the noncentral density of f i /I involves only two infinite sums.

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space.Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon.Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles.His original model was then modified and considered under different approaches and using many versions.Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).We intend to publish in this special issue papers reporting research on time-dependent billiards.The topic includes both conservative and dissipative dynamics.Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned.Mathematical papers regarding the topics above are also welcome.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable: '''''p) and k(p,n) HP /2/(2Pn/2pp(n/2) Pp(p/2)).Now, on making the transformation AI AI' fi i Ai/li 2 p in (3.1), we obtain the joint density of Al, a partition of the integer d.