Int J Contemp Math Sci, Vol 3, 2008, no 23, 1103-1112 On Extreme Bernoulli and Dependent Families of Bivariate Distributions Broderick O Oluyede Department of Mathematical Sciences Georgia Southern University, Statesboro, GA 30460, USA Boluyede@GeorgiaSouthernedu Mavis Pararai Department of Mathematics Indiana University of Pennsylvania Indiana, PA 15705, USA Abstract The obective and purpose of this note is to generate bivariate distributions via extreme Bernoulli distributions and obtain results on positively and negatively dependent families of bivariate binomial distributions generated by extreme Bernoulli distributions Some distributional properties and results are presented The factorial moment generating functions, correlation functions, conditional distributions and the regression functions are given Mathematics Subect Classifications: 60E05; 60F99 Keywords: Extreme Bernoulli distributions; Conditional distributions; Bivariate binomial distributions 1 Introduction Bernoulli distribution is an important distribution that arises in many applications in probability and statistics There are several distributions and models that take into account the concept of dependence in the parameters for certain classes of distributions Lancaster 1958), Hamdan and Jensen 1976), Takeuchi and Takemura 1987) obtained expressions for association correlation) functions in the form of series bilinear in suitable orthogonal polynomials
1104 B O Oluyede and M Pararai Oluyede 1994) studied a family of bivariate binomial distributions generated by extreme Bernoulli distributions, see also references therein, including results by Marshall and Olkin 1985) on bivariate distributions generated by the Bernoulli distribution Hamdan 1972) provided applications of canonical expansion of the bivariate binomial distribution In this paper, we present new probability distributions generated by extreme Bernoulli distributions and discuss the properties of these distributions including the factorial moment generating functions, correlation functions, conditional distributions and the regression functions In section 2, some basic results leading to the generated distributions are presented In section 3, distributional results including their properties and some comparisons are presented Section 4 offers useful results on families of positive and negative dependent distributions including some comparisons In Section 5, a brief discussion and concluding remarks are given 2 Extreme Bernoulli Distributions and Some Basic Results Let P p i ) and Q q i )ber c probability matrices with r r p i q i i1 i1 c c for 1, 2,,c, p i q i for i 1, 2,, r 1 1 Then P maorizes Q, denoted by P > m Q if k h1 p h) k h1 q h), k 1, 2,,rc 1, where p h) and q h), h 1, 2,,k, are elements of P and Q respectively, arranged in a nondecreasing order Now let p 11, p 10, p 01 and p 00 be the probabilities of classification of individuals into one of four mutually exclusive classes AB, AB, AB, AB A and B denotes negation) An example will be to classify patients as diseased or healthy using standard diagnostic as opposed to surgical procedures The number of occurrences of A and B, denoted by X and Y are random variables having the binomial distributions with parameters n, p 1+ ) and n, ) respectively, where p 1+ p 11 + p 10 and p 11 + p 01 The oint probability function of X and Y is given by Aitken and Gonin 1935) The distribution is given by P X x, Y y) n! s s!x s)!y s)!n x y + s! ps 11 px s 10 py s 01 p n x y+s 00, 1) for x, y 0, 1, 2,, n When sampling without replacement from a finite population, a bivariate hypergeometric distribution is obtained leading to a bivariate binomial distribution in the limit
Extreme Bernoulli distributions 1105 The extreme Bernoulli distributions can be displayed as follows [ ] p E 1 1+ 0 if p p 1+ 1 p 1+ ; +1 [ ] 0 p1+ E 2 if p 1 p 1+ p 1+ 1 +1 2 ; [ ] p+1 p E 3 1+ if p 0 1 p 1+ ; 1+ and [ ] p1+ + p E 4 +1 1 1 if 1 1 p 1+ 0 2 p 1+ 2) The extreme Bernoulli distributions E k, k 1, 2, 3, 4 can be obtained as follows Let p i ), i 0, 1 0, 1, be a probability matrix with p 1+ p 11 +p 10, p 11 + p 01 If p 1+, then p 1+, 0) > m p 11,p 10 ) and p 1+, 1 ) > m p 01,p 00 ) Similarly, if p 1+, then,p 1+ ) > m p 11,p 10 ) and 1 p 1+, 0) > m p 01,p 00 ) The results now follows from the fact that max0,p 1+ + p 1+ 1) p 11 minp 1+, ) The bivariate distributions generated to E k, k 1, 2, 3, 4 are as follows Suppose n independent drawings are made from these populations and let X i, i, 0, 1 be the number of occurrences of event whose cell probabilities are given by E k, k 1, 2, 3, 4 For E 1, P E1 X 11 n 1,X 10 n 2,X 01 n 3 ) n!pn 1 1+ p 1+ ) n 3 1 ) n n 1+n 3 ), n 1!n 3!n n 1 + n 3 ))! 3) where n i 0, i 1,2,3,andn 1 + n 2 + n 3 n With X X 11 + X 10 and Y X 11 + X 01, the oint distribution of X and Y is given by n!p x 1+ p 1+ ) y x 1 ) n y, n y x 0 P E1 X x, Y y) x!y x)!n y)! 4) 0, otherwise For E 2, n!p x 1+ py +1 1 p 1+ ) n x y 0 x, y n, P E2 X x, Y y) x!y!n x y)! x + y n 0, otherwise For E 3, n!p y +1 p 1+ ) x y 1 p 1+ ) n x, n x y 0 P E3 X x, Y y) x!x y)!n x)! 0 otherwise 5) 6)
1106 B O Oluyede and M Pararai For E 4, n!p 1+ + 1) x 1 p 1+ ) n y 1 ) n x 0 x, y n, P E4 X x, Y y) x!n y)!n x)! x + y n 0, otherwise 7) We now examine obtain the regression functions EY X x) and EX Y y) for the distributions given in 1) and those generated by E 1 and E 2 respectively The regression functions for the distributions generated by E 3 and E 4 can be obtained similarly obtained For the distribution given by 1), after some simplification, the conditional distribution of X given Y y is P X Y y) minx,y) max0,x+y n) n y x ) y ) ) ) y ) x ) n x y+ p11 p01 p10 p00 8) This is the distribution of the sum of two random variables One distributed as binomial with parameters y and p 11, and the other as binomial with parameters n y and p 10 The regression function is given by EX Y y) n ) p10 + y p11 p ) 10 9) For the bivariate distribution generated by E 1, the regression function is EX Y y) y p 1+ ) and for E2, the corresponding regression function is given by EY X x) n x) p+1 1 p 1+ 3 Some Distributions and Properties ) 10) In this section, we examine some distributions and their properties including those generated by αi +1 α)e i,i1,2,3,4,0 α 1, where I is the fourfold independence table with entries p hk p h+ p +k,h,k0, 1 First, consider the convex combination of P E1 x, y) and P E2 x, y) The corresponding distribution P E x, y) has the covariance function CovX, Y )np 1+ α ) The regression function is given by EY X x) n α ) + xαp 1+ ) 11) 1 p 1+ 1 p 1+ Now consider the family generated by a linear combination of the independence model I and E i, i 1, 2, 3, 4 Here, we consider the family generated
Extreme Bernoulli distributions 1107 by A i 1 α)i + α)e i, 0 α 1, i 1, 2, 3, 4 Suppose n independent drawings are made from the population A 1 1 α)i + αe 1, and let X hk, h, k 0, 1, be the number of occurrences of the event whose cell probabilities are given by A 1 Then, the oint distribution of X and Y is given by P A1 X x, Y Y ) minx,y) max0,x+y n) n! [1 α)p 1+ + αp 1+ ] [αp 1+ ] x!x )! [1 α)p 0+ + α p 1+ )] y [1 α) p 0+ + α1 )] n x y+ y )!n x y + )! x, y 0, 1, 2,, n, 0 α 1, p 1+ 12) The conditional distribution of X given Y y is given by P A1 X x Y y) minx,y) max0,x+y n) y ) n y x ) 1 α)p1+ + αp 1+ ) ) y 1 α)p0+ + α p 1+ ) αp1+ ) n x y+ 1 α)p0+ + α1 p 1+ ) ) x This is the distribution of the sum of two random variables One distributed as binomial ) y, 1 α)p 1+ +αp 1+ and the other as binomial n y, αp1+ ) The regression function is EX Y y) nαp 1+ + y 1 α)p 1+ + αp 1+ αp 1+ 13) Also, the variance of X given Y y is given by VarX Y y) n y) αp 1+ + y 1 α)p 1+ + αp 1+ αp 1+ 14) For the extreme Bernoulli distribution generated by A 2 1 α)i + αe 2, suppose n independent drawings are made from this population and let X hk, h, k 0, 1, be the number of occurrences of the event whose cell probabilities are given by A 2 Then the oint distribution of X and Y is given by
1108 B O Oluyede and M Pararai P A2 X x, Y Y ) minx,y) max0,x+y n) n! [1 α)p 1+ ] [1 α)p 1+ + αp 1+ ] x!x )!y )!n x y + )! [1 α)p 0+ + α ] y [1 α) p 0+ + α1 )] n x y+ x, y 0, 1, 2,, n, 0 α 1 15) The corresponding conditional distribution of X given Y y is given by P A2 X x Y y) minx,y) max0,x+y n) y ) n y x ) 1 α)p1+ ) ) y 1 α)p0+ + α αp1+ +1 α)p 1+ ) n x y+ 1 α)p0+ + α1 p 1+ ) The regression function reduces to EX Y y) n [ 1 α)p 1+ + αp 1+ The factorial moment generating function is given by ] ) x y αp 1+ 16) Mt 1,t 2 ) [1 α)p 0+ + α1 p 1+ ) + 1 α)p 1+ + αp 1+ )t 1 from which the correlation +1 α)p 0+ + α )t 2 +1 α)p 1+ t 1 t 2 ] n, CorrX, Y ) α ) 1/2 p1+ p 0+ Proposition For the bivariate binomial distribution generated by A 2 1 α)i + E 2, α CorrX, Y ) 0, if p 1+ + 1 17)
Extreme Bernoulli distributions 1109 4 Positive and Negative Dependent Families of Distributions In this section, we present positive and negative dependent families of bivariate binomial distributions generated by the extreme Bernoulli distributions discussed in section 2 For the distributions generated by {Γ :p 11 p 1+ }, we have, P 2 X x, Y Y ) minx,y) max0,x+y n) n! [1 α)p 1+ + αp 1+ ] [αp 1+ ] x!x )!y )!n x y + )! [1 α) p 1+ )+α ] y [α p 0+ +1 α)1 )] n x y+ x, y 0, 1, 2,, n, p 1+, 0 α 1 18) The corresponding conditional distribution of X given Y y is given by P 2 X x Y y) minx,y) max0,x+y n) y ) n y x ) 1 α)p1+ + αp 1+ ) y 1 α)p+1 )+αp 0+ αp1+ The regression function is given by ) n x y+ 1 α)p0+ + α1 ) ) x EX Y y) nαp 1+ + y αp 1+ +1 α)p 1+ αp 1+ 19) The factorial moment generating function is Mt 1,t 2 )[αp 0+ +1 α)1 )+αp 1+ t 1 +αp 0+ +1 α) p 1+ ))t 2 +αp 1+ +1 α) )t 1 t 2 ] n The correlation CorrX, Y ) is given by ) 1/2 p1+ CorrX, Y )1 α) 0 p 0+ Proposition For the bivariate binomial distribution generated by A 2 1 α)i + E 2, 0 CorrX, Y ) 1 α), if p 1+ + 1 20) )
1110 B O Oluyede and M Pararai The bivariate binomial distributions generated by {Δ :p 11 p 1+ }, is P 3 X x, Y Y ) minx,y) max0,x+y n) n![αp 1+ ] [αp 1+ +1 α)p 1+ ] x!x )!y )!n x y + )! [1 α) + αp 0+ ] y [α p 0+ +1 α)1 p 1+ )] n x y+ x, y 0, 1, 2,, n, p 1+, 0 α 1 21) The corresponding conditional distribution of X given Y y is given by P 3 X x Y y) minx,y) max0,x+y n) y ) n y x ) ) αp1+ ) y 1 α)p+1 + αp 0+ αp1+ +1 α)p 1+ αp0+ +1 α)1 p 1+ ) The regression function is given by ) n x y+ ) x EX Y y) n αp 0+ +1 α)p 1+ + y αp 1+ +1 α)p 1+ + αp 0+ 22) Proposition For the bivariate binomial distribution generated by A 2 1 α)i + E 2, CorrX, Y ) 1 α) ) 1/2 p1+ 0 23) p 0+ Let P C X x, Y y) be the bivariate binomial distribution generated by αe 1 +1 α)e 2, 0 α 1 If y<xand x + y n, then the oint probability function is given by P αe1 +1 α)e 2 X x, Y y) P C X x, Y y) n x ) p x1+1 p 1+ ) n x minx,y) max0,x+y n) n x y ) x ) ) αp1+ 1 αp ) x ) y 1+ p+1 αp 1+ 1 p ) n x y+ +1 αp 1+, p 1+ p 0+ p 0+ x, y 0, 1, 2,, n, 0 α 1 24) p 1+
Extreme Bernoulli distributions 1111 The following result provides a lower bound for the oint cumulative distribution function P C X x, Y y) in terms of the bivariate binomial distributions generated by E 1 and E 2 respectively Theorem For y x and p 1+, P C X x, Y y) P E1 X x, Y y)p E2 X x, Y y) provided α Proof: For y x, and p 1+, P C X x, Y y) P E1 X x, Y y)p E2 X x, Y y)k x, y) where { n n k0 h0 } τ n k, h) φ k x, p 1+ )φ h y, ), k!h! 0 if k h αp 1+ p 1+ ) τ n k, h) ) k, if k h, n k [ K y x)!x! n x y)!y! 1 p 1+ ) n x ][ ] n x y 1 x, y), y! n x)! p 1+ ) y x p x 1+ 1 p 1+ and φ k x, p 1+ ) is the kth Krawchouk polynomial Note that { [ ] n y x)!x! n x y)!y! 1 p 1+ lim n K x, y) n lim y! n x)! 1 p 1+ [ ] x [ ] y } p+1 p 1+ )1 p 1+ ) 1 p1+ 1, p 1+ 1 p 1+ ) p 1+ so that P C X x, Y y) P E1 x, y)p E2 x, y) { n } n τ n k, h) φ k x, p 1+ )φ h y, ) k!h! k0 h0 Consequently, for α, P C X x, Y y) P E1 X x, Y y)p E2 X x, Y y) Corollary Fory x, p 1+, and α, P C X x, Y y) P E1 X x, Y y)p E2 X x, Y y)
1112 B O Oluyede and M Pararai 5 Concluding Remarks Several bivariate binomial distributions generated by extreme Bernoulli distributions are derived Important properties of these distributions including factorial moment generating functions and the regression functions are presented Families of positive and negative dependent bivariate binomial distributions generated by the extreme Bernoulli distributions as well as their properties are presented The representations of convex combinations of these extreme distributions are given in terms of Krawchouk polynomials Some comparisons and dependence of the random variables X and Y are explored Also, bounds on certain oint probabilities are given ACKNOWLEDGEMENT: The authors expresses their gratitude to the editor and referee for many useful comments and suggestions References [1] A C Aitken, and H T Gonin, On fourfold sampling with and without replacement, Proceedings of Royal Society of Edinburg, 55, 1935), 114-125 [2] M A Hamdan, Canonical expansion of the bivariate binomial distribution with unequal marginal index, International Statistical Review, 40, 1972), 277-280 [3] M A Hamdan, and D R Jensen, A bivariate binomial distribution with some applications, Australian Journal of Statistics, 18, 1976), 163-169 [4] H O Lancaster, The structure of bivariate distributions, Annals of Mathematical Statistics, 29, 1958), 719-736 [5] A W Marshall, and I Olkin, A family of bivariate distributions generated by Bernoulli distribution, Journal of the American Statistical Association, 80, 1985), 332-338 [6] B O Oluyede, A family of bivariate binomial distributions generated by extreme Bernoulli distributions, Communications in Statistics, 235), 1994), 1531-1547 [7] K Takeuchi, and A Takemura, On sum of 0-1 random variables multivariate) Annals of the Institute of Statistical Mathematics, 39, 1987), 307-324 Received: December 10, 2007