Journal of Multivariate Analysis 7, 2226 (999) Article ID jmva.999.826, available online at httpwww.idealibrary.com on Spherically Symmetric Logistic Distribution Nikolai A. Volodin The Australian Council for Educational Research, Melbourne, Victoria, Australia Received October 23, 997 In this paper exact formulae of the probability density function for the spherically symmetric distribution with marginal logistic are given. They are entirely different for odd and even dimensions. For an odd number of dimensions it is possible to express them by elementary functions but for an even number of dimensions, it is possible only by an infinite series of functions. These series, however, are very convenient for computations and could be useful in practice. 999 Academic Press AMS 99 subject classifications 62H, 62H5, 6E5, 62E7. Key words and phrases logistic distribution; spherically symmetric distribution; approximation.. INTRODUCTION The purpose of this paper is to provide a complete description of the family of multivariate spherically symmetric distributions with marginal logistic. This problem has been discussed in a number of papers and the most recent results and references can be found in Arnold [2, 3]. Define a spherically symmetric distribution in the n-dimensional space R n to be a distribution whose probability density function (pdf) can be represented in the form p(x, x 2,..., x n )= f n (x T n 7 & x n ), where 7 is a positive definite matrix and x T n =(x, x 2,..., x n ). It is obvious that all marginal distributions should be the same when 7=I. Letf(x) be the marginal density distribution in this case. The notation s n =x T n 7 & x n will also be used. It is worth noting that 7 is not the covariance matrix but it is proportional to it. Using results from Bochner [4] and from McGraw and Wagner [7] the next theorem concerning the form of spherical distributions was given in Volodin [8] (original paper in Russian in 988). 22 47-259X99 3. Copyright 999 by Academic Press All rights of reproduction in any form reserved.
SPHERICAL LOGISTIC DISTRIBUTION 23 Theorem. For n=2m+, m=,2,...the following formula holds, f n (s n )= (&)m? m 7 2 d m f(- s n ); () and for n=2m, m=, 2,... the following formula is true, f n (s n )= 2}(&)m? m 7 2 d m f(- z 2 +s n ) dz. (2) Now apply formulae () and (2) to logistic distribution with probability density function f(x)= 2(+cosh(x)). 2. CASE OF ODD NUMBER OF DIMENSIONS n=2m+, m=, 2,... This case follows directly from (). The general formula is f n (s n )= (&)m? m 7 2 d m The exact formulae for n=3, 5, 7 are sinh(- s 3 ) f 3 (s 3 )= 4? 7 2 - s 3 (+cosh(- s 3 )), 2 2(+cosh(- s n )). f 5 (s 5 )= sinh(- s 5 )&- s 5 (2&cosh(- s 5 )) 8? 2 7 2 s 32(+cosh(- s, 5 5 )) 2 3(+cosh(- s 7 ))(sinh(- s 7 )&- s 7 (2&cosh(- s 7 ))) &s 7 sinh(- s 7 )(5&cosh(- s 7 )) f 7 (s 7 )= 6? 3 7 2 s 52 (+cosh(- s. 7 7 )) 3 It is easy to see that the formula for f 3 (s 3 ), which was given above is an alternative form for f 3 (s 3 )= tanh (- s 32) sech 2 (- s 3 2) 8? 7 2 - s 3, which was derived in Arnold and Robertson [] (see formula (3.4) in Arnold [3]).
24 NIKOLAI A. VOLODIN 3. CASE OF EVEN NUMBER OF DIMENSIONS n=2m, m=, 2,... The main problem in this case is to find an expression of the integral J(s n )# dz 2(+cosh(- z 2 +s n )) in a more convenient form which allows us to work with Lemma. For x> f n (s n )= 2}(&)m d m dz? m 7 2 2(+cosh(- z 2 +s n )). Proof. As x> then J(x)= &?x 2 [? 2 (2k&) 2 +x] &32. J(x)= = = = dz 2(+cosh(- z 2 +x)) = exp(&- z 2 +x) (&) k k& exp(&k - z 2 +x) dz (&) k& k - x exp(&- z 2 +x) dz (+exp(&- z 2 +x)) 2 (&) k& } k } exp(&(k&) - z 2 +x) dz exp(&k - x cosh(u)) cosh(u) du, where substitution z=- x sinh(u) has been made. For the Macdonald function K & (x) (a cylindrical function of an imaginary argument) the following integral representation (see 8.432. in [5]) is true, K & (x)= exp(&x cosh(t)) cosh(&t) t, which gives for J(x) J(x)=- x From 8.526.2 in [5] it is not difficult to extract that (&) k K (kx)= 2\ C+ln \ x 4?++ +? (&) k& kk (k - x). (3) l={ - x 2 +? 2 (2l&) 2l?= 2&
SPHERICAL LOGISTIC DISTRIBUTION 25 and taking into consideration that (ddx) K (x)=&k (x) it is easy to show that (&) k& kk (kx)= dx\ d = 2x &?x The last formula and (3) prove lemma. (&) k K (kx) + l= [? 2 (2l&) 2 +x 2 ] &32. Now when the following representation for f 2m (s 2m ) has been derived f 2m (s 2m )= 2(&)m&? m& 7 2 d m ds m 2m { it is easy to show by induction that or &32= s 2m [? 2 (2k&) 2 +s 2m ] f 2m (s 2m )= (2m&)!! 2{ 2 m&? m& 7 2m [? 2 (2k&) 2 +s 2m ] &(2m+)2 &s 2m (2m+) f 2m (s 2m )= (2m&)!! 2 m&? m& 7 2 [? 2 (2k&) 2 +s 2m ] &(2m+3)2= 2m? 2 (2k&) 2 &s 2m [? 2 (2k&) 2 +s 2m ] (2m+3)2. It is difficult to say how useful the two last formulae are for theoretical purposes but for calculations they are quite useful because it is easy to estimate the remainder for a given number of terms which were chosen for calculations. It is also possible to obtain a formula for f 2 (s 2 ) from f 3 (s 3 ) by integrating by one of the variables. ACKNOWLEDGMENTS The author acknowledges Dr. R. J. Adams' comments and suggestions which improved the presentation of this paper. REFERENCES. B. C. Arnold and C. A. Robertson, ``Elliptically Contoured Distributions with Logistic Marginals,'' Technical Report No. 8, Department of Statistics, University of California, Riverside, CA, 989.
26 NIKOLAI A. VOLODIN 2. B. C. Arnold, Multivariate logistic distributions, in ``Handbook of the Logistic Distributions'' (N. Balakrishnan, Ed.), pp. 23726, Decker, New York, 992. 3. B. C. Arnold, Distributions with logistic marginals andor conditionals, in ``Distributions with Fixed Marginals and Related Topics'' (L. Ru schendorf, B. Schweitzer, and M. D. Taylor, Eds.), IMS Lecture Notes Monograph Series, Vol. 28, pp. 532, Inst. Math. Sci., Hayward, CA, 996. 4. S. Bochner, ``Lectures on Fourier Integrals,'' Princeton Univ. Press, Princeton, NJ, 959. 5. I. S. Gradshteyn and I. M. Ryshik, ``Tables of Integrals, Series, and Products'' (Allan Jeffrey, Ed.), Academic Press, New York, 995. 6. N. L. Johnson, S. Kotz, and N. Balakrishnan, ``Continuous Univariate Distributions,'' Vol. 2, 2nd ed., Wiley, New York, 995. 7. D. K. McGraw and J. F. Wagner, Elliptically symmetric distributions, IEEE Trans. Inform. Theory 4 (968), 2. 8. N. A. Volodin, On some classes of spherically symmetric distributions, ``Stability Problems for Stochastic Models,'' J. Soviet Math. 57 (99), 389392.