Characterizations involving U|(U+V=m) for certain discrete distributions

Journal of Statistical Planning and Inference 109 (2003) 31 – 41

www.elsevier.com/locate/jspi

Characterizations involving U |(U + V = m) for certain discrete distributions Adrienne W. Kemp University of St Andrews, The Mathematical Institute, North Haugh, KY 16 9SS Fife, Scotland, UK

Abstract The paper gives a number of generalizations of the Moran characterization of the binomial distribution as the distribution of U |(U + V = m) where U and V have independent Poisson distributions. These involve q-series distributions including two new q-analogues of the binomial distribution. c 2002 Elsevier Science B.V. All rights reserved.  Keywords: Heine distribution; Euler distribution; q-Poisson distribution; q-Binomial distribution; q-Negative binomial distribution; Absorption distribution; Basic hypergeometric series; q-Series; Generalized Rogers–Szeg8o polynomials; Generalized Stieltjes–Wigert polynomials; q-Laguerre polynomials; Wall polynomials; Cauchy functional equation

1. Introduction Rao and Shanbhag (1994) in their book Choquet-Deny Type Functional Equations prove the following theorem. Theorem A (Rao and Shanbhag, 1994, Theorem 7.4.5; Shanbhag and Kapoor, 1993). Let B1 ; B2 ; : : : be a sequence of nondegenerate independent (0 –1 valued) Bernoulli random variables. Further let (X; Y ) be a random vector of nonnegative integer-valued components such that P{X ¿ 1} ¿ 0 and for almost all n ¿ 0
n   P{Y = r|X = n} = P Bi = r ; r = 0; 1; : : : ; n: i=1

E-mail address: [email protected] (A.W. Kemp). c 2002 Elsevier Science B.V. All rights reserved. 0378-3758/02/$ - see front matter
PII: S 0 3 7 8 - 3 7 5 8 ( 0 2 ) 0 0 2 9 6 - 3

32

A.W. Kemp / Journal of Statistical Planning and Inference 109 (2003) 31 – 41

Then the following are equivalent: (i) P{Y = r|X − Y = 0} = P{Y = r|X − Y = 1}, P{X − Y = r|Y = 0} = P{X − Y = r|Y = 1}; r = 0; 1; : : : : (ii) Conditionally upon X − Y ∈ {0; 1; 2}, Y and X − Y are independent. (iii) Conditionally upon Y ∈ {0; 1; 2}, Y and X − Y are independent. (iv) The random variable X is such that  n   1 + m ; n = 1; 2; : : : P{X = n} = Kn 1 +  + · · · + m−1 m=1

= K;

n=0

and Bi ’s satisfy P{Bi = 1} =

i ; 1 + i

i = 1; 2; : : : ;

where K; ;  and  are appropriate positive constants (with  not necessarily equal to 1). (If  = 1, we have X as Poisson and Bi ’s as identically distributed.) (v) X − Y and Y are independent. The authors remarked, “According to the deFnitions in Benkherouf and Bather (1988), the distributions of Y and X − Y in (iv) of Theorem 7.4.5 are, respectively, Heine and Euler distributions if  ¡ 1, and vice versa if  ¿ 1”. Further work on the Heine and Euler distributions appears in Kemp (1992a,b) and Benkherouf and Alzaid (1993). n Let B = i=1 Bi . Then the distribution of B has the probability generating function (pgf) −n n+1 z) 1 0 ( ; −; ; − GB (z) = when  ¡ 1; (1) −n n+1 ) 1 0 ( ; −; ; − −n 1 0 (q ; −; q; −z) (2) when  = q−1 ¿ 1: = −n 1 0 (q ; −; q; −) These are both pgf’s of the q-binomial distribution of Kemp and Newton (1990) and Kemp and Kemp (1991). Thus we obtain the following corollary to Rao and Shanbhag’s Theorem: (i) iI Y |(X = n) has a q-binomial distribution with pgf (1) and Y and X − Y are independent random variables (rv’s), then Y has a Heine distribution with parameter  and X − Y has an (independent) Euler distribution with parameter , where = = , and (ii) iI Y |(X = n) has a q-binomial distribution with pgf (2) and Y and X − Y are independent rv’s, then Y has an Euler distribution with parameter and X − Y has a Heine distribution with parameter , where = = =q. This corollary is a q-analogue of the Moran (1952) characterization of the binomial distribution as the conditional distribution of Y |(X = n) where Y and X − Y are independent; Y and X − Y then both have Poisson distributions. In the above q-analogue, however, the two component distributions are diIerent (although both can be regarded as q-analogues of the Poisson distribution (Kemp, 1992a)). The same feature appeared

A.W. Kemp / Journal of Statistical Planning and Inference 109 (2003) 31 – 41

33

in Kemp’s (2001) characterization of the absorption distribution (Kemp, 1998) as the conditional distribution of U |(U + V = m) where U and V are independent; here U had a q-binomial distribution and V had a Heine distribution. The present paper gives further characterization theorems for discrete q-series distributions, including two other q-analogues of the binomial distribution and a generalization of the absorption distribution. The focus is on conditional distributions that involve two kinds of q-Laguerre polynomials (the Rogers–Szeg8o polynomials and the Stieltjes–Wigert polynomials) and their generalizations.

2. Notation and denitions Throughout the paper we use the Gasper and Rahman (1990) deFnition of a q-series (basic hypergeometric series): A B (a1 ; : : : ; aA ; b1 ; : : : ; bB ; q; z) =

∞  j=0

(a1 ; q)j : : : (aA ; q)j z j (b1 ; q)j : : : (bB ; q)j (q; q)j

 × (−1)j q

 B−A+1 j 2



;

where (a; q)0 = 1, (a; q)j = (1 − a) : : : (1 − aqj−1 ) and |q| ¡ 1 (this is not the earlier Bailey (1935) and Slater (1966) deFnition). The Heine, Euler, q-binomial, q-negative binomial, and absorption distributions have the following pgf’s. Heine (Benkherouf and Bather, 1988; Kemp, 1992a,b): GH (z) =

0 0 (−; −; q; −z) 0 0 (−; −; q; )

;

Euler (Benkherouf and Bather, 1988; Kemp, 1992a,b): GE (z) =

1 0 (0; −; q; z) 1 0 (0; −; q; )

;

q-binomial (Kemp and Kemp, 1991): GB (z) =

; −; q; −q n z) ; −n n 1 0 (q ; −; q; −q )

1 0 (q

−n

q-negative binomial (Dunkl, 1981): GNB (z) =

1 0 (; −; q; z) 1 0 (; −; q; )

;

Absorption (Kemp, 1998): GA (z) = qmn 2 1 (q−n ; q−m ; 0; q; qz):

34

A.W. Kemp / Journal of Statistical Planning and Inference 109 (2003) 31 – 41

The Rogers–Szeg8o polynomials, (Andrews, 1976; Fine, 1988), are   m  m Hm (t) = KR t j = KR 2 0 (q−m ; 0; −; q; tqm ); j q j=0 by the generalized Rogers–Szeg8o polynomials we mean   m  m Hm∗ (t) = KR∗ t j (a; q)j = KR∗ 2 0 (q−m ; a; −; q; tqm ): j j=0 q The Stieltjes–Wigert polynomials, (Szeg8o, 1975), are   m  m sm (t) = Ks (−t)j qj( j+1=2) ; j j=0 q i:e:

sm (−tq

−3=2

) = KS

  m  m j=0

j

t j qj( j−1) = KS 1 1 (q−m ; 0; q; tqm ) q

and the generalized Stieltjes–Wigert (q-Laguerre) polynomials, (Gasper and Rahman, 1990) are   m  m ∗ ∗ sm (t) = Ks (−t)j qj( j+1=2) =(b; q)j ; j q j=0   m  m ∗

∗ sm (−tq−3=2 ) = KS

i:e:

j=0

j

t j qj( j−1) =(b; q)j = KS∗ 1 1 (q−m ; b; q; tqm ): q

The Wall polynomials are a special case of the little q-Jacobi polynomials, (Gasper and Rahman, 1990):   m  m qj( j−1)=2 −m pm (t; a; b; q) = KW 2 1 (q ; 0; aq; q; qt) = KW (−q1−m t)j : (aq; q)j j j=0

Here

KR ; KR∗ ; Ks ; KS ; Ks∗ , 

m j



= q

KS∗ ,

q

and KW are normalizing constants and

(q−m ; q)j (q; q)m = (−1)j qmj−j( j−1)=2 : (q; q)j (q; q)m−j (q; q)j

(3)

Theorem B (Patil and Seshadri, 1964; see also Kagan et al., 1973, Theorem 13.4.4). Let X and Y be independent discrete r.v.’s and c(x; x + y) = P[X = x|X + Y = x + y]. If c(x + y; x + y)c(0; y) h(x + y) = : c(x; x + y)c(y; y) h(x)h(y) where h is a nonnegative function, then f(x) = f(0)h(x)eax ;

g(y) = g(0)k(y)eay

A.W. Kemp / Journal of Statistical Planning and Inference 109 (2003) 31 – 41

35

where 0 ¡ f(x) = P[X = x];

0 ¡ g(y) = P[Y = y];

k(y) =

h(y)c(0; y) : c(y; y)

Patil and Seshadri’s proof depends on the Cauchy functional equation ‘(x + y) = ‘(x)‘(y) (for which the solution is ‘(x) = eax where a is a constant). 3. The characterizations The rv’s X and Y have diIerent interpretations in Theorems A and B. Let Y =U and X − Y = V in Theorem A and let X = U and Y = V in Theorem B. Also let C denote the appropriate normalizing constant wherever it appears. Note that if X |(X + Y = m) has the distribution with pgf G(z) then the distribution of Y |(X + Y = m) has pgf z m G(z −1 ), i.e. the reversed distribution. We begin with a direct proof of the corollary to Rao and Shanbhag’s Theorem. Theorem 1. The distribution with pgf −m ; −; q; −qm z=0) 1 0 (q G1 (z) = −m ; −; q; −qm =0) 1 0 (q

(4)

(a q-binomial distribution) is the distribution of U |(U + V = m), where U and V are independent, if and only if U and V have a Heine distribution and an Euler distribution, with pgf’s 0 0 (−; −; q; −z) 1 0 (0; −; q; 0z) and ; (−; −; q; −)  0 0 1 0 (0; −; q; 0) respectively. Proof. If U and V are independent and have the postulated Heine and Euler distributions, then 0 m−u Cu qu(u−1)=2 × P[U = u|U + V = m] = (q; q)u (q; q)m−u u  C0 m (q−m ; q)u −qm = (q; q)m (q; q)u 0 from (3), and the corresponding pgf is G1 (z). Conversely, if U |(U + V = m) has the pgf G1 (z), then from Theorem B c(u + v; u + v)c(0; v) Cu+v q(u+v)(u+v−1)=2 C0 v × = c(u; u + v)c(v; v) (q; q)u+v (q; q)v −1  u u(u−1)=2 v 0 C q Cv qv(v−1)=2 × × (q; q)u (q; q)v (q; q)v =

h(u + v) ; h(u)h(v)

36

A.W. Kemp / Journal of Statistical Planning and Inference 109 (2003) 31 – 41

where h(u) = qu(u−1)=2 =(q; q)u . Thus k(v) = (0=)v =(q; q)v and P[U = u] = C1 qu(u−1)=2 eau =(q; q)u ; P[V = v] = C2 (0=)v eav =(q; q)v yielding a Heine distribution with parameter  = ea for U and an Euler distribution with parameter 0 for V . Remark 1a. The reverse of the q-binomial distribution with pgf (4) where 0 ¡ q ¡ 1 (corresponding to (1)) is another q-binomial distribution; the new pgf, also with 0 ¡ q ¡ 1, is −m ; −; q; −0qz=) 1 0 (q G1R (z) = (5) −m ; −; q; −0q=)  (q 1 0 (corresponding to (2)). The distributions of U and V , when V |(U + V = m) has the pgf (5), are therefore the same as those when U |(U + V = m) has the pgf (4). Remark 1b. The pfg (4) can be restated as   m  m G1 (z) = C (z=0)j qj( j−1)=2 j j=0 q and as G1 (z) =

m 

(1 + qi−1 z=0)=(1 + qi−1 =0):

(6)

(7)

j=1

As q → 1, these tend to the pgf of a binomial distribution. Suppose now that U and V both have Heine distributions, with parameters 1 and 2 , respectively. Theorem 2. The distribution with pgf −m ; 0; q; 1 qz=2 ) 1 1 (q G2 (z) = −m ; 0; q; 1 q=2 ) 1 1 (q

(8)

is the distribution of U |(U + V = m), where U and V are independent, i; U and V have Heine distributions with pgf’s 0 0 (−; −; q; −i z) ; i = 1; 2; 0 0 (−; −; q; −i ) respectively. Proof. It is straightforward to show that if U and V have these Heine distributions and are independent, then  u C2m qm(m−1)=2 (q−m ; q)u −1 P[U = u|U + V = m] = qu(u+1)=2 (q; q)m (q; q)u 2 and the pgf of U |(U + V = m) is G2 (z). The converse follows from Theorem B with h(u) = qu(u−1)=2 =(q; q)u and k(v) = qv(v−1)=2 (2 =1 )v =(q; q)v .

A.W. Kemp / Journal of Statistical Planning and Inference 109 (2003) 31 – 41

Remark 2a. Pgf (8) for Theorem 2 can be restated as   m  m (1 q1−m z=2 )j qj( j−1) ; G2 (z) = C j j=0 q

37

(9)

this is a Stieltjes–Wigert polynomial. As j → 1 (9) tends to a binomial polynomial, so G2 (z) is therefore the pgf of a second q-analogue of the binomial distribution. It is possible to interpret it as the outcome of a stationary birth–death process. There is no summation formula, such as (7), for (9). Remark 2b. The pgf of the reversed distribution is the same as (8), but with 1 and 2 interchanged. Consider now the situation where U |(U + V = m) has a pgf that can be expressed as a generalized Stieltjes–Wigert polynomial. Theorem 3. The distribution with pgf G3 (z) =

1 1 (q 1 1

−m

; q n+1−m ; q; qz=4)

(10)

(q−m ; q n+1−m ; q; q=4)

is the distribution of U |(U + V = m), where U and V are independent, i; U has a Heine distribution and V has a q-binomial distribution with pgf’s 0 0 (−; −; q; −z)

and

0 0 (−; −; q; −)

1 0 (q 1 0

−n

; −; q; −q n 4z)

(q−n ; −; q; −q n 4)

;

respectively. Proof. If U and V have these distributions and are independent, then u  C(−4q n )m (q−n ; q)m (q−m ; q)u qu(u−1)=2 −q P[U = u|U + V = m] = ; (q; q)m (q; q)u (q n+1−m ; q)u 4 so the pgf of U |(U + V = m) is G3 (z). The converse follows from Theorem B with h(u) = qu(u−1)=2 =(q; q)u and k(v) = (q−n ; q)v (−4q n =)v =(q; q)v . Remark 3a. The pgf (10) restated as a generalized Stieltjes–Wigert (q-Laguerre) polynomial is   m  m qj( j−1) (q1−m z=4)j : (11) G3 (z) = C n+1−m (q ; q)j j j=0

q

As n → ∞, this tends to a Stieltjes–Wigert polynomial and the distribution of V tends to a Heine distribution. Remark 3b. In Theorems 1 and 2 the reversed form of the distribution of U |(U + V = m) had the same form as the unreversed distribution (like the binomial distribution).

38

A.W. Kemp / Journal of Statistical Planning and Inference 109 (2003) 31 – 41

In Theorem 3; however, the reversed form of (10) is G3R (z) = z m G3 (z −1 ) =

; q−n ; 0; q; 4q n+1 z=) ; −m ; q−n ; 0; q; 4q n+1 =) 2 1 (q

2 1 (q

−m

this is a generalization of the pgf of the absorption distribution (Kemp, 1998), for which 4q n = . It cannot be expressed in terms of a Stieltjes–Wigert type polynomial. From Theorem 3 the distributions of U and V , given that they are independent and that V |(U + V = m) has the distribution that is the reverse of (10), are q-binomial and Heine, with pgf’s 1 0 (q 1 0

−n

; −; q; −q n 4z)

(q−n ; −; q; −q n 4)

and

0 0 (−; −; q; −z) 0 0 (−; −; q; −)

;

respectively. This generalizes Kemp’s (2001) characterization of the absorption distribution as the distribution of U |(U + V = m) where U and V are q-binomial and Heine with 4q n = . When U and V are independent and have a Heine and a q-negative binomial distribution, respectively, then the pgf for U |(U + V = m) again involves a generalized Stieltjes–Wigert polynomial. Theorem 4. The distribution with pgf ; q1−m =; q; −qz=5) ; −m ; q1−m =; q; −q=5) 1 1 (q   m  m qj( j−1) (−q1−m z=5)j =C (q1−m =; q)j j

G4 (z) =

1 1 (q

−m

j=0

(12) (13)

q

is the distribution of U |(U + V = m), where U and V are independent, i; U has a Heine distribution and V has a q-negative binomial distribution with pgf’s 0 0 (−; −; q; −z) 1 0 (; −; q; 5z) and ;  (−; −; q; −) 0 0 1 0 (; −; q; 5) respectively. Proof. As for Theorem 3. Remark 4. The reversed form of (12) is 2 1 (q 2 1

−m

; ; 0; q; −q5z=)

(q−m ; ; 0; q; −q5=)

:

Now suppose that U and V both have Euler distributions, with parameters 01 and 02 , respectively. Theorem 5. The distribution with pgf G5 (z) =

; 0; −; q; 01 qm z=02 ) −m ; 0; −; q; 01 qm =02 ) 2 0 (q

2 0 (q

−m

(14)

A.W. Kemp / Journal of Statistical Planning and Inference 109 (2003) 31 – 41

39

is the distribution of U |(U + V = m), where U and V are independent, i; U and V have Euler distributions with pgf’s 1 0 (0; −; q; 0i z) ; i = 1; 2; 1 0 (0; −; q; 0i ) respectively. Proof. If U and V have these Euler distributions and are independent, then  u C02m (q−m ; q)u −01 qm q−u(u−1)=2 P[U = u|U + V = m] = (q; q)m (q; q)u 02 and the pgf of U |(U + V = m) is G5 (z). The converse follows from Theorem B with h(u) = eau =(q; q)u and k(v) = eav (02 =01 )v =(q; q)v . Remark 5a. Pgf (14) can be restated as   m  m G5 (z) = C (01 z=02 )j ; j j=0 q

(15)

this is a Rogers–Szeg8o polynomial. As j → 1, (15) also tends to a binomial polynomial, so therefore G5 (z) is the pgf of a third q-analogue of a binomial distribution. There is no summation formula corresponding to (7) for (15). Remark 5b. The reverse of this distribution has the pgf (14) with 01 and 02 interchanged. The distribution with pgf (14) is closely related to one that has appeared in the physics literature under the name “q-deformed binomial distribution” (Jing, 1994). The pgf of Jing’s distribution is −m ; 0; q1−m =; q; 0qz=5) 2 1 (q G6 (z) = (16) −m ; 0; q1−m =; q; 0q=5) 2 1 (q with 0 = 5. Theorem 6. The distribution with pgf (16) is the distribution of U |(U +V =m), where U and V are independent, i; U and V have an Euler and a q-negative binomial distribution with pgf’s 1 0 (0; −; q; 0z) 1 0 (; −; q; 5z) and ; 1 0 (0; −; q; 0) 1 0 (; −; q; 5) respectively. Proof. Similar to the previous proofs. Remark 6a. The pgf G6 (z) can be restated as a Wall polynomial:   m  m qj( j−1)=2 (−q1−m 0z=5)j G6 (z) = C : (q1−m =; q)j j q j=0

40

A.W. Kemp / Journal of Statistical Planning and Inference 109 (2003) 31 – 41

Remark 6b. The reverse form G6R (z) of the Jing pgf (16) is expressible as a generalized Rogers–Szeg8o polynomial: −m ; ; −; q; 5qm z=0) 2 0 (q G6R (z) = (17) −m ; ; −; q; 5qm =0) 2 0 (q   m  m (; q)j (5z=0)j : (18) =C j j=0 q There is no formula corresponding to (7) for (18). The generalized Rogers–Szeg8o polynomials also arise when U has a q-binomial distribution and V has an Euler distribution. Theorem 7. The distribution with pgf −m −n ; q ; −; q; −4qm+n z=0) 2 0 (q G7 (z) = −m ; q−n ; −; q; −4qm+n =0) 2 0 (q   m  m =C (q−n ; q)j (−4q n z=0)j ; j j=0 q

(19)

is the distribution of U |(U + V = m), where U and V are independent, i; U and V have a q-binomial distribution and an Euler distribution with pgf’s −n n 1 0 (q ; −; q; −4q z) 1 0 (0; −; q; 0z) and ; −n n)  (q ; −; q; −4q 1 0 1 0 (0; −; q; 0) respectively. Proof. As previously. Remark 7. The reverse pgf, G7R (z), is ; 0; q n+1−m ; q; −q0z=4) ; −m ; 0; q n+1−m ; q; −q0=4) 2 1 (q   m  m qj( j−1)=2 (q1−m 0z=4)j ; =C (q n+1−m =; q)j j

G7R (z) =

2 1 (q

−m

j=0

(20) (21)

q

which is similar to (16). References Andrews, G.E., 1976. The Theory of Partitions. Addison-Wesley, Reading, MA. Bailey, W.N., 1935. Generalized Hypergeometric Series. Cambridge University Press, London. Benkherouf, L., Alzaid, A.A., 1993. On the generalized Euler distribution. Statist. Probab. Lett. 18, 323–326. Benkherouf, L., Bather, J.A., 1988. Oil exploration: sequential decisions in the face of uncertainty. J. Appl. Probab. 25, 529–543.

A.W. Kemp / Journal of Statistical Planning and Inference 109 (2003) 31 – 41

41

Dunkl, C.F., 1981. The absorption distribution and the q-binomial theorem. Commun. Theor.-Meth. A 10, 1915–1920. Fine, N.J., 1988. Basic Hypergeometric Series and Applications. American Mathematical Society, Providence, RI. Gasper, G., Rahman, M., 1990. Basic Hypergeometric Series. Cambridge University Press, Cambridge. Jing, S.C., 1994. The q-deformed binomial distribution and its asymptotic behaviour. J. Phys. A: Math. Gen. 27, 493–499. Kagan, A.M., Linnik, Y.V., Rao, C.R., 1973. Characterization Problems in Mathematical Statistics. Wiley, New York. Kemp, A.W., 1992a. Heine-Euler extensions of the Poisson distribution. Commun. Theor.-Meth. 21, 571–588. Kemp, A.W., 1992b. Steady state Markov chain models for the Heine and Euler distributions. J. Appl. Probab. 29, 869–876. Kemp, A.W., 1998. Absorption sampling and the absorption distribution. J. Appl. Probab. 35, 1–6. Kemp, A.W., 2001. A characterization of a distribution arising from absorption sampling, In: Probability and Statistical Models with Applications, Ch.A. Charalambides, M.V. Koutras, N. Balakrishnan (Eds.), Chapman and Hall/CRC, Boca Raton, pp. 239–246. Kemp, A.W., Kemp, C.D., 1991. Weldon’s dice data revisited. Amer. Statistician 45, 216–222. Kemp, A.W., Newton, J., 1990. Certain state-dependent processes for dichotomised parasite populations. J. Appl. Probab. 27, 251–258. Moran, P.A.P., 1952. A characteristic property of the Poisson distribution. Proc. Camb. Phil. Soc. 48, 206–207. Patil, G.P., Seshadri, V., 1964. Characterization theorems for some univariate probability distributions. J. Roy. Statist. Soc. B 26, 286–292. Rao, C.R., Shanbhag, D.N., 1994. Choquet-Deny Type Functional Theorems with Applications to Stochastic Models. Wiley, New York. Shanbhag, D.N., Kapoor, S., 1993. Some questions in characterization theory. Math. Scientist 18, 127–133. Slater, L.J., 1966. Generalized Hypergeometric Functions. Cambridge University Press, Cambridge. Szeg8o, G., 1975. Orthogonal Polynomials. American Mathematical Society, Providence, RI.