ON CHARACTERIZATION OF POWER SERIES DISTRIBUTIONS BY A MARGINAL DISTRIBUTION AND A REGRESSION FUNCTION

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1 Ann. Inst. Statist. Math. Vol. 41, No. 4, (1989) ON CHARACTERIZATION OF POWER SERIES DISTRIBUTIONS BY A MARGINAL DISTRIBUTION AND A REGRESSION FUNCTION A. KYRIAKOUSSIS 1 AND H. PAPAGEORGIOU 2 1Department of Mathematics, University of Athens, Panepistemiopolis, Athens , Greece 2Department of Mathematics, University of Kansas, Lawrence, KS 66045, U.S.A. and Department of Mathematics, University of Athens, Panepistemiopolis, Athens , Greece (Received May 23, 1988; revised January 18, 1989) Abstract. The conditional distribution of Y given X = x, where X and Y are non-negative integer-valued random variables, is characterized in terms of the regression function of X on Y and the marginal distribution of X which is assumed to be of a power series form. Characterizations are given for a binomial conditional distribution when X follows a Poisson, binomial or negative binomial, for a hypergeometric conditional distribution when X is binomial and for a negative hypergeometric conditional distribution when X follows a negative binomial. Key words and phrases: Characterizations, regression function, Poisson, binomial, negative binomial, hypergeometric. 1. Introduction Korwar (1975) considered two types of characterization problems for distributions of non-negative integer-valued random variables (r.v.) X and Y, when they follow a Poisson, binomial or negative binomial distribution. He derived characterizations of the marginal distribution of X by the conditional distribution of Y given X, and the linear regression of X on Y and characterizations of the conditional distribution of Y given X by the marginal distribution of X and the linear regression of X on Y. Extensions and generalizations of Korwar's characterizations by conditional distributions and regression functions were considered by, among others, Dahiya and Korwar (1977), Khatri (1978a, 1978b), Cacoullos and Papageorgiou (1983, 1984), Papageorgiou (1983, 1984 and 1985) and Kyriakoussis (1988). In this paper we are concerned with characterizing the conditional distribution of Y given X = x by the marginal distribution of X and the regression function E(XIY) of X on Y, when X has a power series 671

2 672 A. KYRIAKOUSSIS AND H. PAPAGEORGIOU distribution with parameter 0 and E(XIY) is of a power series form with variable 0. Korwar's characterizations of a conditional binomial when X is a Poisson, binomial or negative binomial, as well as additional characterizations of a conditional hypergeometric or negative hypergeometric, are derived as illustrative examples. 2. The main result Denote the marginal probability functions of the r.v.'s X and Y by p(x) = P(X = x) and f(y)= P(Y= y), respectively. Also denote the conditional probability function of Y given X= x byp(ylx) and that of X given Y= y by p(xly). THEOREM 2.1. Distribution Let X be distributed according to the Power Series a(x) O x p(x) - 2x a(x)o x, x= 0, 1,2,... Then if the regression function of X on Y is of the form M E(X I Y = y) = m(y; O) - j=~o bj(y)oj for all O, with bo(y) = y, y = 0, 1,2,... and r a positive integer or infinity, the conditional distribution of Y [X can be determined uniquely. PROOF. We have m(y; O) = ~x Xp(xty) = 2 xp(y[x)p(x) x f(y) Hence m(y; O) ~x p(y]x)p(x) = ~x xp(y[x)p(x), or from the theorem's assumptions (2.1) m(y; O) ~p(ylx)a(x)o x = ~ xp(ylx)a(x)o x Writing

3 CHARACTERIZATION OF POWER SERIES DISTRIBUTIONS 673 (2.2) ~ P(Yl x) a(x)o x = g(y; 0), equation (2.1) becomes g'(y; 0) m (y; 0) g(y; O) 0 ' where g'(y; O) = (0/O0)g(y; 0). Consequently, g(y; O) = c(y) exp { f m(y; O) do} 0 where c(y) is only a function ofy. Hence (2.3) g(y; O) = c(y)oy exp { j~l bj(y)oj /j } However, which may be defined by their exponential generating function as from the exponential Bell polynomials, Bn = B,(bl, b2,...,b~), where objo1/j ' Bo-- l (2.4) ~b(0) --j, equation (2.3) becomes where n=0 B,O"/n! = exp [~b(0) - ~b(0)] g(y; O) = c(y)o y Y_, B,(y)O"/n! n=0 B,(y) = B,,(b~(y), l!b2(y),...,(n - 1)!b,(y)), n = 1,2,..., Bo(y) = 1. Explicit expressions for B, = B,(bl, b2,..., b,), as functions of bl, b2,..., b,, are given in Kendall and Stuart ((1969), p. 69) and David and Barton ((1962), p. 42) where the role of the cumulants k; is played here by the bi as defined in (2.4) and the Bell polynomials themselves are the moments it;. From equation (2.2), we have

4 674 A. KYRIAKOUSSIS AND H. PAPAGEORGIOU c~ x~=op(ylx)a(x)ox= c(y) ~ Bx-y(y)OX/(x- y)!. r=),, Equating coefficients of 0 x, we obtain (2.5) 1 Bx-y(y) p(ylx)=c(y) a(x) (x-y)!' y=o,l,2,...,x, x=o,l,2,..., p(ylx)=o for x<y. To determine the coefficients c(y), note first that, from (2.5), for a given x, y ranges from zero to x, so that in turn x i~_op(ilx) = I, x = O, 1,2,... ; equivalently, From the relation (2.5), we have y-1 P(YlY) = 1-2 p(ijy). i=0 (2.6) y-1 By-i(i) c(y)= ct(y) - i~:oc(i) (f-- -~v. ' y= 1,2,..., and (2.7) c(0) = a(0), because p(olo) = 1. From (2.6) and (2.7) the remaining coefficients c(1),c(2),.., can be determined uniquely. Conversely, if the conditional distribution of Y IX = x and the marginal distribution of X are known, then the joint distribution of (X, Y) is determined; therefore the marginal distribution of Y, the conditional distribution of X [ Y and the regression E(X I Y) can also be determined. Characterizations of some well-known distributions are given as illustrative examples when the r.v. X follows a Poisson, binomial or negative binomial distribution. For clarity these results are summarized in Table 1.

5 0<a<1, v=0,1,2,... for all0=a/(1-a), n=0,i,2,..., v-y, y=0,1,2,...,x, p=1-q O y = 0, 1,2.... and v < v some 0<q< 1-a O<p< 2, N=0,1,.... for allo=p/(1-p), n=0,1,2,..., N-v y=0,1,2,..., min {x, v} Y = 0, 1,.. and some positive integer -~ v<n O z CA d Table 1. Bell polynomials and conditional distributions. Marginal Distribution Exponential Regression Bell Polynomials P(X=x) E(XIY = Y)=m(Y ;0) B"(Y) P(YIX = x) C1 e 0"/ x! y + Oq 4" x ) p'q' > Y ~ 0>0 for all 0 > 0, y = 0, 1, 2,.., n=0,1,2,... y = 0, 1, 2,..., x, p=1-q n and some O < q < 1 H ~x)do-a) z ( Y+vgO)l(I+40) n!(vny~q" ~ y ~P'rg =-r Conditional Distribution X v ) (- 1)'a ( I - a)x (v v40)l (1-90) n! v (_ 1)'q' p r q=-r / Y rn O<a<I, v>0 for all0=1-a, n=0,1,2,... y=0,1,2,...,x, p=i-q "7 y = 0, I, 2,... and ~' some 0<q<l M (N)Pz(I p) N-x Y+(N-v)0/(0+1) n!( Nn v) (y)(n-y)~(n) ( x )(-1)P,(I-P)s Y+(N-v)0/(1-0) n!( n+v ) (- I)" ( y v )( x-y v )'( x ) 0<p<1, N>0 for all0=1-p, n=0,1,2,... y=0,1,2,...,x y = 0, 1, 2,... and for some positive integer v<n a U

6 676 A. KYRIAKOUSSIS AND H. PAPAGEORGIOU Acknowledgement The authors are grateful to the referee for his suggestions. REFERENCES Cacoullos, T. and Papageorgiou, H. (1983). Characterizations of discrete distributions by a conditional distribution and a regression function, Ann. Inst. Statist. Math., 35, Cacoullos, T. and Papageorgiou, H. (1984). Characterizations of mixtures of continuous distributions by their posterior means, Scand. Actuar. J., Dahiya, R. C. and Korwar, R. M. (1977). On characterizing some bivariate discrete distributions by linear regression, Sankhyd Set. A, 39, David, F. N. and Barton, D. E. (1962). Combinatorial Chance, Griffin, London. Kendall, M. G. and Stuart, A. (1969). The Advanced Theory of Statistics, Vol. 1, Distribution Theory, Griffin, London. Khatri, C. G. (1978a). Characterization of some discrete distributions by linear regression, J. Indian Statist. Assoc., 16, Khatri, C. G. (1978b). Characterization of some multivariate distributions by conditional distributions and linear regression, J. Indian Statist. Assoc., 16, Korwar, R. M. (1975). On characterizing some discrete distributions by linear regression, Comm. Statist., 4, Kyriakoussis, A. (1988). Characterization of bivariate discrete distributions, Sankhyd Ser. A, 50, Papageorgiou, H. (1983). On characterizing some bivariate discrete distributions, Austral J. Statist., 25, Papageorgiou, H. (1984). Characterizations of multinomial and negative multinomial mixtures by regression, Austral. J. Statist., 26, Papageorgiou, H. (1985). On characterizing some discrete distributions by a conditional distribution and a regression function, Biometrical J., 27,