CHARACTERIZATIONS OF SOME POLYNOMIAL VARIANCE

J. Appl. Math. & Computing Vol. 19(2005), No. 1 - 2, pp. 427 - 438

CHARACTERIZATIONS OF SOME POLYNOMIAL VARIANCE FUNCTIONS BY d-PSEUDO-ORTHOGONALITY ´ CELESTIN C. KOKONENDJI

Abstract. From a notion of d-pseudo-orthogonality for a sequence of polynomials (d ∈ {2, 3, · · · }), this paper introduces three different characterizations of natural exponential families (NEF’s) with polynomial variance functions of exact degree 2d − 1. These results provide extended versions of the Meixner (1934), Shanbhag (1972, 1979) and Feinsilver (1986) characterization results of quadratic NEF’s based on classical orthogonal polynomials. Some news sets of polynomials with (2d−1)-term recurrence relation are then pointed out and we completely illustrate the cases associated to the families of positive stable distributions. AMS Mathematics Subject Classification : 62E10, 42C05. Key words and phrases : Bhattacharyya matrix, d-orthogonality, natural exponential family, Sheffer polynomial, stable distribution.

1. Introduction It is well-known that (natural) exponential families of probability measures play a central role both in statistics and applied probability [12]. Let us first recall briefly the definition of a natural exponential family; for more details see [8, Chapter 2], the contribution of Casalis [10, Chapter 54], or [13]. If µ is a σ-finite positive measure on R (not necessarily a probability), we define its cumulant function Kµ by Z Kµ (θ) = ln exp{θx}µ(dx) R

on its (canonical parameter) domain Θ = {θ ∈ R : Kµ (θ) < ∞}. Let us assume that both µ and Θ are not degenerate (i.e., µ is not concentrated at one point and the interior of Θ is not empty), and hence Kµ is known to be strictly convex on the interior intΘ of Θ. We denote by ψµ the inverse of the first derivative Received May 14, 2004. Revised October 14, 2004. c 2005 Korean Society for Computational & Applied Mathematics and Korean SIGCAM.

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function Kµ0 . The natural exponential family (NEF) F = F (µ) generated by µ is the set F = {P (m, F ); m ∈ MF = Kµ0 (intΘ)}, where each P (m, F ) is a probability distribution with mean m such that its density with respect to µ can be written as fµ (x; m) = exp{xψµ (m) − Kµ (ψµ (m))}.

(1)

We then have the classical result of polynomial expansions of the density function (1); see [13, Chapter 3]. Proposition 1. Let F be a NEF on R and let µ = P (m0 , F ) with fixed m0 in MF . Then we have successively: (i) for all n ∈ N, ∂n fµ (x; m) = Pn (x; m)fµ (x; m), ∂mn where Pn is a polynomial in x of exact degree n; (ii) for all m ∈ MF , Pn+1 (x; m) = ψµ0 (m)(x − m)Pn (x; m) + Rn (x; m), where Rn is a polynomial in x of degree ≤ n; (iii) there exists r > 0 such that for all m ∈ (m0 − r; m0 + r), fµ (x; m) =

X (m − m0 )n Pn (x; m0 ). n!

n∈N

These polynomials expansions (Pn )n∈N associated with a NEF belong to the class of Sheffer’s [22] polynomials and they are such that, for all m ∈ MF , P0 (x; m) = 1 and P1 (x; m) = ψµ0 (m)(x − m). They have provided the first classification of NEF’s by the orthogonality property [16], which is limited as property to characterize other classes of NEF’s. Another way to classify NEF’s is to use a particular form of the variance function VF of P (m, F ) defined as Z m 7→ VF (m) = Kµ00 (θ) = 1/ψµ0 (m) = (x − m)2 fµ (x; m)µ(dx). R

[Note that Kµ , ψµ and VF are analytic functions on their respective domains]. Together with the mean domain MF , the variance function VF characterizes the family F within the class of all NEF’s [17]; it does not depend on a particular generating measure, and it presents an expression simpler than the density of P (m, F ). Among the different forms of VF in literature, the most basic is of

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course the polynomial variance function [2] with degree δ ∈ N that, to make short, we denote by NEF’s with δ-PVF: VF (m) =

δ X

αk mk , αk ∈ R.

k=0

In his paper, Morris [17] characterized all NEF’s on R with quadratic variance functions (i.e. δ = 0, 1, 2) in six “types” (normal, Poisson, gamma, binomial, negative binomial and cosine hyperbolic), which are associated to six orthogonal polynomials (Hermite, Charlier, Laguerre, Krawchouk, Meixner-type I and Pollaczek, respectively). For instance, these associated orthogonal polynomials have been obtained in three different ways: Meixner [16] using “exponential generating function”, Feinsilver [5] building polynomials from successive derivatives of probability density, and Shanbhag [20] [21] using some characterizations based on the diagonality of the infinite Bhattacharyya matrix. The extensions to multidimensional quadratic NEF’s are done by [18] and [19]. Let us precise that two NEF’s F (µ) and F (ν) are said to be of the same type if there exist an affinity ϕ and a real λ > 0 such that ν = ϕ(µ∗λ ), where ∗ denotes the convolution product. It is also interesting to note that Morris’ classification was discovered a few years earlier in approximation theory [18]; see [7] for generalizations, and for the exact relation between NEF’s and approximation theory see [3]. A few years later, different classes of polynomial NEF’s were considered and similarly classified: the cubic NEF’s (3-PVF) are described in six types [14], the power NEF’s with VF (m) = mp [8] and NEF’s with VF (m) = m + mp [9]. It would be interesting to carry on the characterization of NEF’s by using one of several generalizations of the notion of orthogonal polynomials. Thus we introduce the following extension of orthogonality, which is not the common notion of d-orthogonality in the literature. Definition 1. Let d ∈ {2, 3, · · · }. A sequence of real polynomials (Pn )n∈N is said to be d-pseudo-orthogonal with respect to a probability measure µ (denoted µ − d-pseudo-orthogonal) if, for all n and q in N∗ , Z Pn (x)Pq (x)µ(dx) = 0 for n ≥ dq, q ∈ N, and (2) R

Z h i ∃q ≥ 0, n ∈ (q + 1)/d; dq − 1 ∩ N such that Pn (x)Pq (x)µ(dx) 6= 0. R

(3)

This definition extended to d = 1 corresponds to the quasi-othogonality of order 1. Recall that the classical µ-orthogonality is defined by Z Pn (x)Pq (x)µ(dx) = 0 R

when n 6= q, and it is the quasi-orthogonality of order 0. The aim of this paper is to provide some charaterizations of the real NEF’s by this d-pseudo-orthogonality

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definition, for all d ∈ {2, 3, · · · }. This is also motivated by the recent paper [6] of Hassairi and Zarai, which concerns the real cubic NEF’s in full detail with d = 2. Remember that for δ-PVF with δ ≥ 4, all the types of NEF’s are not described yet. Most of these NEF’s do not possess densities in closed analytical forms. Hence, given d ∈ {2, 3, · · · }, we are going to determine a new set of d-pseudo-orthogonal polynomials associated with a NEF from a particular form of its variance function. This new setting provides a way to treat and to use these classes of NEF’s, for example to connect d-pseudo-orthogonal polynomials with stochastic processes [5]. In the three following sections we give, respectively, the analogous characterizations of Feinsilver [5], Meixner [16] and Shanbhag [20] [21] using the d-pseudoorthogonality, as defined in Definition 1 instead of classical orthogonality. Our results concern the classes of NEF’s with (2d − 1)-PVF, where d ∈ {2, 3, · · · } is the same d of the d-pseudo-orthogonality. Their proofs will be willingly short because they are technically similar to those of the literature, except when we need the d-pseudo-orthogonality property. The last section is devoted to some remarks and the most important examples concerning the power NEF’s. 2. Characterization in the Feinsilver sense In this section we show our key result. It also provides the (2d − 1)-term recurrence relation of the associated d-pseudo-orthogonal polynomials. Theorem 1. Let F be a NEF on R and µ an element of F with mean m0 . Consider the polynomials (Pn )n∈N associated to F and defined by Pn (x) = Pn (x; m0 ) = ∂ n fµ (x; m)/∂mn |m=m0 . Then, for all d ∈ {2, 3, · · · }, the three following statements are equivalent: (i) the polynomials (Pn )n∈N are µ − d-pseudo-orthogonal; (ii) F is (2d − 1)-PVF; (iii) there exist real numbers (ak )k=0,1,··· ,2d−1 such that, for all n ≥ 2(d − 1), h i h i a0 Pn+1 (x) = x − (na1 + m0 ) Pn (x) − n (n − 1)a2 + 1 Pn−1 (x) −

2d−2 X

ak+1 Ak+1 n Pn−k (x),

k=2

with Akn = n(n − 1) · · · (n − k + 1). Furthermore, in this case we have 2d−1 X VF (m) = ak (m − m0 )k . k=0

Proof. (i) ⇒ (ii): If, for (m, m) e ∈ (m0 − r; m0 + r)2 , we consider n    o g(m, m) e = exp kµ ψµ (m) + ψµ (m) e − kµ ψµ (m) − kµ (ψµ (m) e ,

(4)

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431

then the µ−d-pseudo-orthogonality of the polynomials (Pn )n∈N and Proposition 1 (iii) imply that Z X (m − m0 )n (m e − m0 ) q Pn (x)Pq (x)µ(dx) g(m, m) e = n!q! R n,q≥0 X = 1+ an,q × (m − m0 )n (m e − m0 ) q , q≥1,n∈[(q+1)/d;dq−1]∩N

where an,q = (n!q!)−1

Z

Pn (x)Pq (x)µ(dx). R

Since ψµ (m0 ) = 0, taking the first derivative of (4) with respect to m and putting m = m0 , then we get X ψµ0 (m0 )(m e − m0 ) = an,1 × (m e − m0 ) n , n∈[2/d;d−1]∩N

for all m e ∈ (m0 − r; m0 + r); and, hence, one has: a1,1 = ψµ0 (m0 ) 6= 0 and an,1 = 0 for n ∈ [2; d − 1] ∩ N (which may be considered as a property of this dpseudo-orthogonality related to a NEF). A second derivative of (4) with respect to m, for m = m0 leads us to the following expression of the variance function: VF (m) e

−2 00 = a−2 e − m0 ) − ( m e − m0 ) 2 1,1 VF (m0 ) − a1,1 ψµ (m0 )(m X +2a−2 an,2 (m e − m0 ) n , 1,1 n∈[3/d;2d−1]∩N

for all m e ∈ (m0 − r; m0 + r). Therefore, by extension of VF on its domain MF and the µ − d-pseudo-orthogonality of (Pn )n∈N which means that a2d−1,2 6= 0, we obtain that F is (2d − 1)-PVF. (ii) ⇒ (iii): We assume (ii) that there exist real numbers (ak )k=0,1,··· ,2d−1 2d−1 P such that VF (m) = ak (m − m0 )k for fixed m0 in MF . Substituting θ for k=0

ψµ (m) in (1), Proposition 1 (iii) may be written as exp(θx) =

X Pn (x) (kµ0 (θ) − m0 )n n!

!

exp{kµ (θ)},

n∈N

where its first derivative with respect to θ is equivalent to X (m − m0 )n xPn (x) n! n∈N i X Pn (x) h = n(m − m0 )n−1 kµ00 (θ) + (m − m0 )n m n! n∈N

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

=

2d−1 X

X Pn (x)  n ak (m − m0 )  k=0 n! n∈N +m0 (m − m0 )n

n+k−1

+ n(m − m0 )

n+1



 .

Identifying the coefficients of (m − m0 )n , it follows that: h i xPn (x) = a0 Pn+1 (x) + (na1 + m0 )Pn (x) + n (n − 1)a2 + 1 Pn−1 (x) +

2d−1 X

ak Akn Pn−k+1 (x)

k=3

and, hence, the expression of (iii) is easily deduced. (iii) ⇒ (i): Proceeding in three steps, we may show by induction on the order n or q that: Z Step 1: for all n ∈ N∗ ,

Pn (x)µ(dx) = 0.

R

Step 2: (the most delicate to establish): for all d ∈ {2, 3, · · · }, there exist real s numbers βn,q,d such that, for all n, q ∈ N∗ verifying n ≥ dq, X 0 s xq Pn (x) = βn,q,d Pn−2q(d−1) (x) + βn,q,d Ps (x), n−2q(d−1)+1≤s≤n+q 0 where βn,q,d = 0 if n = 2q(d − 1) (≥ dq because d ∈ {2, 3, · · · }). Step 3: there exist real numbers (αk )0≤k≤n such that αn = [1/VF (m0 )]n and X Pn (x) = αn xn + α k xk . 0≤k≤n−1

Therefore, given these three conditions, we obtainZ by integration that, for d ∈ xq Pn (x)µ(dx) = 0. Hence, {2, 3, · · · } and for all n, q ∈ N∗ such that n ≥ dq, R

the sequence (Pn )n∈N is µ − d-pseudo-orthogonal. This concludes the proof.  3. Characterization in the Meixner sense This section is devoted to the characterization of the µ − d-pseudo-orthogonal polynomials on R with the “exponential generating function”. Theorem 2. With the assumptions of Theorem 1, let d ∈ {2, 3, · · · } and let (Qn )n∈N be a sequence of µ − d-pseudo-orthogonal polynomials such that Qn = Qn (.; m0 ) has degree n. Then the three following statements are equivalent: (i) the generating function of the (Qn )n∈N is exponential; i.e., there exist r > 0 and two real analytic functions a and b defined on (−r; r) such that for all z ∈ (−r; r), X zn Qn (x) = exp{xa(z) + b(z)}; n! n≥0

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(ii) F is (2d − 1)-PVF; (iii) there exist t > 0 such that, for all n ∈ N , Qn (x) = tn Pn (x), where (Pn )n∈N is defined in Theorem 1. In this case, we have a(z) = ψµ (tz + m0 ) and b(z) = −Kµ (a(z)). Proof. Without loss of generality, we can assume Q0 = 1. (i) ⇒ (iii): From Proposition 1 (iii), there exists r > 0 such that, for all z ∈ (−r; r), we have (from the µ − d-pseudo-orthogonality of Qn (x))     Z Z X zn X zn   Qn (x) µ(dx) = Qn (x) Q0 (x)µ(dx) n! n! R R n≥0 n≥0 Z = Q20 (x)µ(dx) = 1. R

Furthermore, from the generating function (i) associated to (Qn )n≥0 we obtain   Z Z X zn  Qn (x) µ(dx) = exp{xa(z) + b(z)}µ(dx) n! R R n≥0

= exp{Kµ (a(z)) + b(z)}. Hence, b(z) = −Kµ (a(z)).

(5)

Proceeding in a similar manner to the above, we get the following equality:   Z  Z X zn 2   Qn (x) Q1 (x)µ(dx) = Q1 (x)µ(dx) z. (6) n! R R n≥0

The polynomial Q1 (x) is of degree 1 in x; therefore, there exist u ∈ R∗ and v ∈ R such that

Since

Z

Z

R

Q1 (x) = ux + v.

(7)

Q1 (x)Q0 (x)µ(dx) = Q1 (x)µ(dx) = 0, we then have v = −um0 and R Z Z Q21 (x)µ(dx) = u2 (x − m0 )2 µ(dx) = u2 VF (m0 ). (8) R

R

Moreover, it follows from (5), (6) and (7) that: Z  Z 2 Q1 (x)µ(dx) z = Q1 (x) exp{xa(z) + b(z)}µ(dx) R R Z n o = u (x − m0 ) exp xa(z) − Kµ (a(z)) µ(dx) hR i = u Kµ0 (a(z)) − m0 ,

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and we deduce that (8) can be written as i h u2 VF (m0 )z = u Kµ0 (a(z)) − m0 . Therefore, setting t = uVF (m0 ) ∈ R∗ , we have Kµ0 (a(z)) = tz + m0 , that is a(z) = ψµ (tz + m0 ) and, finally we obtain n o X zn Qn (x; m0 ) = exp xψµ (tz + m0 ) − Kµ (ψµ (tz + m0 )) n! n≥0

X (tz)n Pn (x; m0 ). n!

=

n≥0

(i) ⇐ (iii) and (ii) ⇔ (iii) follow obviously from Theorem 1.



4. Characterization in the Shanbhag sense Let F = F (ν) be a NEF generated by a positive measure ν. Given an open set I ⊆ R and a C ∞ -diffeomorphism a : I → Θ(ν), we may consider another parametrization of F = F (ν) by F = {P (a(z); ν); z ∈ I}, where the density of P (a(z); ν) with respect to ν is hν (x; z) = exp{xa(z) − Kν (a(z))} = fν (x; Kν0 (a(z))). Then for all n ∈ N and z ∈ I, ∂n hν (x; z) ∂z n is a polynomial in x of degree n, independent of the choice of the generator ν of F = F (ν), and satisfies the recurrence relation Sn (x; z) = [hν (x; z)]−1

Sn (x; z) = [xa0 (z)]n + Tn−1 (x; z), with S0 (x; z) = 1, S1 (x; z) = [x − Kν0 (a(z))]a0 (z) and where Tn−1 (x; z) is a polynomial in x of degree ≤ n − 1. We can check it directly or from Proposition 1 (i, ii) for which ν is not necessarily a probability measure. For all z ∈ I, we then call Bhattacharyya matrix the infinite matrix B(z) = (Bn,m (z))n,m∈N , where Z Sn (x; z)Sm (x; z)P (a(z); ν)(dx). (9) Bn,m (z) = R

Definition 2. Let d ∈ {2, 3, · · · }. An infinite matrix B = (Bn,m )n,m∈N is said to be d-semi-orthogonal if, for all (n, m) ∈ N2 such that n ≥ dm, Bn,m = 0 and Bn,0 = 0.

Characterizations of some polynomial variance functions

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Obviously, given d ∈ {2, 3, · · · }, the Bhattacharyya matrix B(z) is d-semiorthogonal if and only if (Sn (.; z))n≥0 are P (a(z); ν)−d-pseudo-orthogonal polynomial. Now, we can state our last result. We omit the proof, which is technically similar to [19], [20] and [21]. Theorem 3. Let d ∈ {2, 3, · · · } and let ν be a generating measure of NEF F . If a : I → Θ(ν) is a C ∞ parametrization of F and B(z) is the Bhattacharyya matrix defined in (9), then the four following statements are equivalent: (i) (ii) (iii) (iv)

for all z ∈ I, B(z) is d-semi-orthogonal; there exists z ∈ I such that B(z) is d-semi-orthogonal; for all z ∈ I, B1,2 (z) = 0 and B2,2d (z) = 0; F is (2d − 1)-PVF and there exists (u, v) ∈ R∗ × R such that a(z) = ψν (uz + v); i.e., the mean of P (a(z); ν) is uz + v. 5. Final remark and examples

There exist many NEF’s with polynomial variance functions of degree 2d − 1, for d ∈ {2, 3, · · · }. For example, to build such variance functions, we can consider the Bar-Lev criterion [1] stated slightly informal as follows: VF (m) = m∆(m2 ) on MF = (0; r), where ∆ is a polynomial with non-negative coefficients and r ∈ (0; ∞]. However it seems hard, indeed even inaccessible, to obtain “explicitly” their densities or cumulant functions when the degree of VF is greater than or equals 3 (see [9] and [14]). Note that the cumulant function Kµ is necessary to the computation of our sequence polynomials, and the density or generating measure µ is important to precise the d-pseudo-orthogonality with respect to this µ. The calculation of the sequence polynomials Pn (x; m0 ) can be done by using Umbral Calculus [3] or by mean of the Fa` a di Bruno formula [11] as follows: Pn (x; m0 ) X = k1 +···+nkn =n

(10) kj n  j Y n! ∂ [xψµ (m) − Kµ (ψµ (m))]|m=m0 . k1 ! · · · kn ! j=1 ∂mj

Let us conclude this paper by pointing out the most interesting cases from positive stable families, which are generating by the probability measures  αk ∞ dx X (α − 1)k Γ(1 + αk) −1 sin(−kπα), x > 0, µα (dx) = πx k!αk (α − 1)x k=1

where 0 < α < 1; see [4] for their probabilistic interpretations and other properties. Instead of the index parameter α, it is also convenient to introduce the “power” parameter p, defined by (p − 1)(1 − α) = 1;

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C´ elestin C. Kokonendji

that means p > 2 for 0 < α < 1. The well-known special case of positive stable families is the inverse Gaussian type, which corresponds to p = 3 or α = 1/2, with dx µ1/2 (dx) = √ x−3/2 exp{−1/(2x)}, x > 0. 2π For fixed p > 2 or 0 < α < 1, the NEF Fp = F (µα ) generated by µα is such that Θ(µα ) = (−∞; 0], Kµα (θ) = (α − 1)[θ/(α − 1)]α /α, ψµα (m) = (α − 1)m1/(α−1) = m1−p /(1 − p), and VFp (m) = mp

on MFp = (0; ∞);

see [8, Chapter 4] for the complete classification with p ∈ R. Hence, for all p = 2d − 1 with d ∈ {2, 3, · · · }, we associate to the NEF Fp = F (µα ) the sequence of µα − d-pseudo-orthogonal polynomials (Pn (x; m0 ))n≥0 . The general expression of which is obtained from (11) as Pn (x; m0 ) =

X k1 +2k2 +···+nkn =n

n Y m3−2d−j m2−2d−j Aj2−2d Aj3−2d n! 0 0 − x k1 ! · · · kn ! j=1 2 − 2d 3 − 2d

!kj

,

Ajk = k(k − 1) · · · (k − j + 1) with A0k = 1. Taking m0 = 1 and then, for all d ∈ {2, 3, · · · }, VF2d−1 (m) = m2d−1 =

2d−1 X k=0

Ak2d−1 (m − 1)k , k!

the (2d − 1)-order recurrence relation (Theorem 1 (iii)) is given by   Pk+2d−1 (x; 1) = x − A1k+2d−2 A12d−1 − 1 Pk+2d−2 (x; 1)   − k + 2d − 2 + A2k+2d−2 A22d−1 /2 Pk+2d−3 (x; 1) −

2d−4 X τ =0

+3 +3 Aτk+2d−2 Aτ2d−1 Pk+2d−4−τ (x; 1), k ≥ 0, (τ + 3)!

with the initial conditions: P0 (x; 1) = 1, P1 (x; 1) = x − 1, P2 (x; 1) = x2 − 5x + 3, and     Pj (x; 1) = x − A1j−1 A12d−1 − 1 Pj−1 (x; 1) − j − 1 + A2j−1 A22d−1 /2 Pj−2 (x; 1) −

j−3 +3 +3 X Aτj−2−τ Aτ2d−1 Pj−3−τ (x; 1), j = 3, · · · , 2d − 2. (τ + 3)! τ =0

When d = 2, we obtain precise results of the inverse Gaussian NEF [6, Table 1].

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Acknowledgments The author wishes to thank the Editors and an anonymous referee for their perceptive comments. My special thanks go both to A. Hassairi and to M. Zarai for showing me their preprints.

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C´ elestin C. Kokonendji

21. D. N. Shanbhag, Diagonality of the Bhattacharyya matrix as a characterization, Theor. Probab. and Appl. 24 (1979), 430-433. 22. I. M. Sheffer, Concerning Appell sets and associated linear functional equations, Duke Math. J. 3 (1937), 593-609. C´ elestin C. Kokonendji received his Ph. D at Paul Sabatier University of Toulouse (France) under the direction of G´erard Letac. Since 1995 he has always been at the University of Pau. His research interests center on the exponential families in Statistics and Applied Probability. Also, he sometimes does statistical consulting. Department of Statistics, IUT STID - LMA CNRS, University of Pau, Avenue de l’universit´e, 64000 Pau, France e-mail: [email protected]