A SURVEY OF MEIXNER'S HYPERGEOMETRIC DISTRIBUTION

C. D. Lai

(received 12 August, 1976; revised 9 November, 1976)

Abstract. Meixner's hypergeometric distribution is defined and its properties are reviewed. Some bivariate density functions of this class are also obtained.

Introduction

The Meixner classes of distributions were first characterized by Meixner ([16], [17]) and then studied by Eagleson [2] who also generated a class of bivariate distributions with marginals belong­ing to the Meixner classes. Lancaster [12] simplified the proofs of Meixner's characterization theorems and applied the theory to obtain joint distributions in the normal, binomial, gamma and Poisson classes.

In fact, the Meixner classes have turned up in several other contexts, especially in characterization theorems on regression. In this paper, we shall attempt to give a connected account of Meixner's hypergeometric distribution.

1. Meixner classes

Let X be a random variable possessing distribution and moment generating functions, F and <f>. Let {P b e a system of orthogonal

polynomials associated with F, P (x) = xn + terms of lower degree.

Math. Chronicle 6(1977) 6-20.

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X is said to belong to a Meixner class defined by u{t), ifoo

(1) K (x,t) = I tn/nl = exp [aru(t)] / <f>(u(t)),n=0

where u(t) = (t + possibly terms of higher degree). From the theorem of Lagrange on the inversion of power series, there is an analytic function v(.u) = (u + possibly terms of higher degree), such that i>(w(t)) = t; v is said to be inverse to u, It follows from Lancaster [12] (a simplified proof of the original Meixner’s result) that

(2) f i = 1 - Xt - rt*.

Let (1 - Xt - kt2) = (1 - at) (1 - Bt). By considering different possible values of a and B > Meixner ([16], [17]) characterized the classes of distributions in the following manner:

(i) Normal distribution: (1 - Xt - <t2) = 1,

(ii) Poisson distribution; (1 - \t - <t2) = 1 + t,

(iii) Gamma distribution: (1 - Xt - <t2) = (1 + £)2,

(iv) ■ Positive binomial: (1 - Xt - kt2) = (1 - pt)(1 + qt)where p and q are real and non-negative: p + q = 1,

Cv) Negative binomial: (1 - Xt - ki2) = (1 + Pt)(1 + Qt) where P, Q are real: Q - P = 1, and

(vi) Meixner’s hypergeometric distribution:

Cl - Xt ~ <t2) = (1 - at) (1 - Bt)where a f B , k f 0, a and 3 are complex conjugates.

Lemma 1 . Let F be a distribution function in the Meixner classes and $(w) = log <f>(w). Then

(3) (i) $f(u) = j = biv{u) , bi = var^, i>(w(t)) = t,

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(4) (ii)

(5) (iii)

(6) (iv)

(7) (v)

(vi)

to a Meixner class with the same w(t), then so does Z = ,X + Y, and

?n (z) " E $ F,l-k(i/)* Wha'e ?n * ^korthogonal polynomials of the Meixner class3 and

(vii) every system of orthogonal polynomials with a distribution in the Meixner classes is complete.

Proof: See [16], [17] , [2], [3], and [12].From (2) and (3), we get

(8) t"(a) - -- I = ij(l - U - <t2).

As k < 0 (except for the positive binomial distribution), it can be deduced from [l2j that all Meixner variables are infinitely divisible. However, a simple proof is available. More precisely, we have

Lemma 2. If < 5 0, then the Meixner distribution is infinitely divisible.

k < 0, or if < > 0, b\/k is a positive integer,

d * d t ^ = t h l 1 ( 1 ’ X t “ < t 2 ) >

pn+ 1 0 * 0 ■ ( a ? + n \ ) P n (x ) + « ( - & ! + ( « - 1 ) k )

P_l(x) = 0 for n = 0,1,2,.....

«(*) - log / (a-B).

♦ [«(*)] =1 /6 -1

l/ot(1-at)

if X and Y are independent random variables and belong

Proof: Favard [5] shows that there is always a distribution

determined by C6) provided k < 0, It follows immediately that (6) still determines a distribution if b\ is replaced by b\/ N, where N is any positive integer. So every Meixner distribution (except the positive binomial) is infinitely divisible.

As we pointed out in the Introduction, Meixner classes have appeared in many other contexts. Laha and Lukacs [9] characterized a family of distributions in the following manner: Let X\, X2» .

be a random sample of size n from a population with distrib­ution function F(x). Let A = X\ + X2 + ►.• + X^ = nX. Consider

n n na quadratic statistic Q = ][ £ a.. X.X. + \ b .X. . Laha and

0 0 0 i=1 0-1 0-1

Lukas studied all the populations (in which the moment of order two of X\ exists) which have the property that Q has a quadratic regression on A , i.e.

(9) E[g|A] = 80 + BjA + 82A2.

As in [16], they characterized the Meixner classes via differential equations (of similar form to (8) ) for the characteristic functions. (See (4.4), (4.7), (5.2), (5.4) and (5.6) of their paper) .

Another possible way to characterize a family, consisting of the Cauchy distribution and Meixner distributions, was discussed by Bolger and Harkness [l]. They showed that for any two independent random variables in the family, it is necessary that E[^ll^ = yl - Mt//X and V(*i|l=z/) = (XiX2/\2)u{y) where

J = + X2, Xj and \2 are positive constants with Xj + = X and u(y) is a polynomial in y of degree at most two. In fact, this w(.) plays the same role as the function in (2).

Earlier, Tweedie [23] also characterized the same class by

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considering the regression of the sample variance on the sample mean. Similarly, Khatri [8] characterized the bivariate distrib­utions in the Meixner classes by imposing different conditions onthe coefficients y.., and $. , where

Y33 ' :

F, [s ... I X ] = a ... + 6.T., + & . ,X. + ny .., X .X., , for33 ~ 33' 3 3 ' 3' 3 33 3 3

3 > 3' = 1,2,...,p. Here, 1 = and S = (s^.,) : p x p

denote the sample mean vector and sample covariance matrix

respectively based on n independent observations of the vector X.

2. Heixner's hyperneometric distributions

In this section, our main objective is to show that the Meixner*s hypergeometric distributions can be identified as the two distributions of Harkness and Harkness [7]. We shall also show that two particular Meixner*s hypergeometric distributions are derivable from normal, exponential and Pollaczek distributions.

2.1. Definition and Meixner's constant

The Meixner's hypergeometric density function is defined by

(10) fix) = m(-3/a)*/a~e r[(6x-b1)/$(a-3)]r[(ca:-&1)/a(a-6)],

where a and £ are complex conjugates; -« < x < « , |arg“" | -

where m is a constant of proportionality. (See [16]). In other words, f(x) is proportional to the product of two gamma functions of complex argument. If we let a = bi/aB, then (10) can be more conveniently written as

(11) f(x) = m(-B/a)‘ (x-atL x-a&I a-eJI a-SjIt is of interest that without a knowledge of actual form of

distribution, Meixner's recurrence relation (6) enables the polynomials to be computed. However, it seems desirable to obtain

IQ

an explicit form of m.

Theorem 1. The Meixner 's hyper geometric constant is

(12) m = 1 / {2irir(a) 1 - |-aS/Co-B)

(6-a)}.

Proof: Recall the identity (Mellin-Barnes integral)ry+i«

(13) r(u) (1+t)~U 2-rri =y-i°°

(See [4, p.256]). Let -s

r(-s)r(w+s)t ds , 0 > y > Re(l-w)

x-aQ x-a8 •a-3 ’ U+S “ a-3 '

It follows immediately that u = a. If we further let t =(13) becomes

Y+

r x-a3 I a-3ioo v J

x-a 8a-3 t ds - 2iri T(a) 1-*

-a

from which we obtain

f rr"aS]r x-a8 f-B)J.» I I a~̂ \ I »J

x/8-adx - 2iri r(a) 1 - -ai

a8/a-8(3-a).

Hence the result follows.It follows from [16] that the moment generating function of

the .hypergeometric distribution isft — (y<f>(£) = {(a-8)/(ae - 8e )} . That is, its characteristic

function is

(14) f(t) = e^^^ (coshci + ~ sinhet}~a.

This function is studied quite extensively by Laha and Lukacs [9],

2.2 Meixner's hypergeometric distribution and generalized

hyperbolic secant distribution.

The function (coshet)(a > 0 and c real), is known as a

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a |tx>

characteristic function of a certain random variable in Il3jf Harkness and Harkness [7] showed that the density function corresponding to (coshc?£)~a is given by

(15) f{.x,o,d) = [c-aT{a)Tl2a~l r

In particular,

a ix rfa ia;]

J + 2c.1 2 " 2c\

(16) f{x,c,a) =coshra:

, if e = 1/2, a - 1,

(17) f[x,o,d) = 2x , if o - 1/2, a = 2, sinhirc

More generally, for positive integral values n of a , we have

(18) f(x,c,2n+l) =

for n = 1,2,... and

(19) f[x,c, 2n) =

O 2̂Z *" 1 TT'T*2____ sech—(2n)!c 2c

r,«- H n-1 ( 24 x _ , irx --------- csch~—(2n-l)!2e2 C

IT { -2— + r2 }r=1 4c2 ^

n = 1,2,---

Harkness and Harkness called the distribution determined by (15)

the generalized hyperbolic secant distribution. As a matter of fact, (16) - (19) had already been studied by Meixner [17].(See (40) and (41) of his paper). By putting a = ci (hence B = -ei) in (11), it is not difficult to see that (15) is indeed a special case of (11).

2.3 Hyperbolic secant and hyperbolic cosecant distributions Although, in general, the Meixner's hypergeometric distribution defined by (11) is not very well known, its two special cases the hyperbolic secant and the hyperbolic cosecant distributions (corresponding to (16) and (17) respectively) have occurred quite frequently in the statistical literature. It may be of interest to

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note that these two particular distributions are derivable from other distributions.

(i) Normal distribution and hyperbolic secant distribution

Let W - y where X and Y are independently distributed as

tf(0,a2). It is well known that \W\ is a Cauchy distribution. It then follows that 1og|f/1 is a hyperbolic secant distribution.(See [14], [7] and [22]).

(ii) Pollaczek distribution and hyperbolic secant distrib­ution

Pollaczek [20, p.395] defined a class of polynomials pj^GEja) by the generating function

(20) I ?(y (*j o) n=0

(1 - tela/2)"X+lx(l - te‘1“/2)"X"lx,

Here 0 < a < tt and these polynomials are orthogonal in the inter­val < x < «° with respect to the density function

(21) f(x;X,a) - e'Clt'2“)3: |r(X + i*) | .

. (See [23]).For X = 1/2 and a = tt/2, (20) and (21) become

(22) I pW(x;cO[2jn=0

f§r ■ 1 + it/2 h1 - it/2 1t2i 1 - it/2 1 + it/2and

(23) can also be obtained from Meixner's result; as from (1) and (7), we deduce that

I ?n(x)tn/n\ = 71=0

i-gt y t a~3)a: r (i-gt)l/3 ~Uat ' (1-at) 1/a

-bj/a-3

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which reduces to (23) when a = j and b± = j .

(iii) Perks distribution and hyperbolic secant distribution

The British actuary Perks [19] proposed a general function for graduating life-table data. His density function has the form:

m . j n _

f{x) = T a.e~J x / y b .e 3 x , where a., b. and 9 areJ J L J / «7 «7 J

J=0real parameters. Take 1, n = 2, 6 = 1, ao = 0* £>o = ^2 anc*

£ ‘ f- then = CSeeAs for the hyperbolic cosecant distribution, it can be derived

from negative exponential distribution, [9], Moreover, it is also a particular case of Perks' and Pollaczek’s distributions,

2.4 Some properties of hyperbolic secant distribution The generalized hyperbolic secant distribution has many properties which are similar to, but rather weaker than, those of the normal distribution. We shall now examine some of them.

(I) When a = bi/< = bj/aB = (2m+l) (m=0,l,... ) and a = ci,(18) shows that f(x,c,2ml) is a symmetric density function.Since X = 0, it follows from the Meixner recursion relation (6) that(24) Pn+1C*) = x Pn(x) + n((2m+n)K) (*)

from which we can deduce by an inductive argument that

(25) P2nM = P2n(-x) and P2 m l (-x) - -P2 w l W fBrn-0.1,2.... .

The above conclusion follows also from the result in [15]. Obviously, the normal distribution (with zero mean) also satisfies (25).

(II) A curious property of the hyperbolic secant distribution

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noted by Feller ([6], p,477), is that it exhibits a "self­reciprocal pair": the density function and its characteristic function differ only by obvious parameters. (The normal distrib­ution is the prime example of this phenomenon).

(III) Let \|r(t) = where vj/j (t) = (cosh£) and

tyl(t) = (coshpt)-1 , |p| < 1. It can be shown that \ji{t) is another infinitely divisible characteristic function with probability density function given by

f261 fCx) = c o s (nP /2l P°.s,h.Crcc/2) » < x < « Ini < 1f COS (7Tp) + COsh(Trx) * * I I

(See [4, p.3l] for the Fourier inversion formula). The variance of this distribution is 1-p2. The normal distribution also exhibits this property, but more strongly. More precisely, if oi2 > C22 i and we define

^l(^)tO) 55 ~̂2~(t')~ = exp(-^ai2t2} exp-{ho22t2} = exp{ -^(ai2-a22) t2 },

then \)t(t) is another normal characteristic function.

(IV) Let be a symmetric bivariate hyperbolic secant probability density function which has a diagonal expansion of the form:

oo(2 7 ) I p 0 far) a Cif5

n= 0where 6 (a?) &re the orthonormal polynomials obtained from (1). Let U = X + 7, V - X - Y. Clearly cov(U,V) = 0. It follows from[15] that U and V are semi-independent in the sense that E[i/|y] = 0 and E [Fj i/] = 0. However, in the case of the normal distribution, U and V are independent.

(V) It is known that the mean and the variance in a sample

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from a normal population are stochastically independent. Let Xi,X2,.••»̂ n be a random sample from the generalized hyperbolic

n _ nsecant population. Let L * £ X. - nX, S - £ J.2.

i=l i=l

Suppose Q is a quadratic statistic: Q = aL2 + bS. Lukacs [13] showed that there exists no Q which is independent of the statistic L.

3. Bivariate Meixner's hypergeometric distributions

We shall now consider a few classes of bivariate hypergeometric distributions which are of considerable interest.(a) Consider the function of the form:

(28) \K£,s) = cosht coshpt cosh(Xt + s)

-a, a > 0, |p J < 1 and |x| < 1.

cos

Clearly, \|f(t,s) is the characteristic function of a bivariate Meixner’s hypergeometric distribution (asymmetric). The density corresponding to (28) (<2=1) is given by.

■ |*~2~ cosh[j (x - \y) ] cosech^p-

2{cosh(irp) + cosh[iT(x-Xz/) ]}

It can be verified that E[x|Y=z/] = \y and var(x|y=z/) = 1 - p2.(b) We know from Section 2 that a hyperbolic secant distribution is derivable from the logarithm of the absolute value of the ratio of two independent and identical variates.

Let U =*1 *1x2 and V = ?2 where cov(Ji,J2) = 0> cov(Yi.Ya) = °*

Also, (Xi,Yi) and ( 2̂ ,̂ 2) are two identical bivariate normals with1 p"positive definite covariance matrix M = P 1 p < 1. Miller

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[18, p.54] showed that the joint density function of U and V is given by

■ p2)* 1 1 t y l M l o»)V.l . ( l « 2 ) ( l « 2 ) i i J m 0

i+j (i+j) I p21 0 \?ti+h)T{o+h) (l+u2)(l+yz)

Let X = logU and Y = logV. Then the joint density of X and Y is (30)

4ci-p2) 2ex+y ^ r* .2 n Wir(l+e2x) (l+e2#) . . _t,J=0

IW (£+<7) ! r p2 i[ o JT ti+h)T (j+h) (l+e2̂ ) (1+e2#)(ex+^)2j.

Let Jl(x,i/) = • It: can be shown that

n 2 ( x , y ) f ( x ) f ( y ) d x d y = y y y y C W ) ( m*n) 1 r r ( £ l S + ? f l p .2 (z'*f *r'*_K' jjiyj y n 44f^ (l+j+m+n)i r (t+%) r (j+*s) r (m+h) r (n+ )̂^Jmn

which is finite for |p2| < 1. It follows from [ll, p.95] that (30) can be exhibited in the biorthogonal form,(c) Eagleson [2] obtained a class of bivariate Meixner's hyper­geometric distributions which can be expanded in a diagonal form (27), where {p^} is given by the formula

f3n p = .n r(a)r(2a+n)

In particular, for a - 1, we have p = 1 in which case, (26)n (w+1)reduces to a bivariate hyperbolic secant distribution; while for

3 12 = 2, pn * (n+3) (n+2) * reduces to a bivariate hyperbolic

:osecant density function.(d) Let f(x,y) be a bivariate hyperbolic secant density function /hich is of the form (27) , Let C denote the set of all sequences >f correlation coefficients {p^} in (27) such that

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p0 = 1 > pi2 > p22 and I p 2 < « , Clearly, C is a convexn= 0

set. It is of interest to note that the sequence

{pn = pn, n-0,1,..., |p| < 1} does not belong to C , (See [23] for a discussion.) In this regard, the Meixner's hypergeometric distribution behaves rather differently from the rest of the Meixner classes. (For a discussion, see [10].) The question of identifying the extremal points of C of this distribution remains an open problem.

Acknowledgement: I am very grateful to the referee whose helpful comments on the original form of this paper have substantially improved its clarity. I also wish to express my thanks to ProfessorsH.O. Lancaster and D, Vere-Jones for their help in this work.

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