RELIABILITY FOR SOME BIVARIATE BETA DISTRIBUTIONS

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1 RELIABILITY FOR SOME BIVARIATE BETA DISTRIBUTIONS SARALEES NADARAJAH Received 24 June 24 In the area of stress-strength models there has been a large amount of work as regards estimation of the reliability PrX <Y). The algebraic form for PrX <Y)has been worked out for the vast majority of the well-known distributions when X and Y are independent random variables belonging to the same univariate family. In this paper, we consider forms of R when X,Y) follows a bivariate distribution with dependence between X and Y. In particular, we derive eplicit epressions for R when the joint distribution is bivariate beta. The calculations involve the use of special functions. 1. Introduction Bivariate beta distributions are tractable lifetime models in many areas, including life testing and telecommunications. In the contet of reliability, the stress-strength model describes the life of a component which has a random strength Y and is subjected to random stress X. The component fails at the instant that the stress applied to it eceeds the strength and the component will function satisfactorily whenever Y>X.Thus, PrX <Y) is a measure of component reliability. It has many applications especially in engineering concepts, such as structures, deterioration of rocket motors, static fatigue of ceramic components, fatigue failure of aircraft structures, and the aging of concrete pressure vessels. Two eamples are i) the receptor of a communication system operates only if it is stimulated by a source where magnitude Y is greater than a random lower threshold X for the system. In this case, R is the probability that the receptor operates, ii) if X and Y are future observations on the stability of an engineering design, then R would be predictive probability that X is less than Y. Similarly, if X and Y represent lifetimes of two electronic devices, then R is the probability that one fails before the other. Because of these applications, the calculation and the estimation of PrX <Y)isimportant for the class of bivariate beta distributions. The calculation of R has been etensively investigated in the literature when X and Y are independent random variables belonging to the same univariate family of distributions. The algebraic form for R has been Copyright 25 Hindawi Publishing Corporation Mathematical Problems in Engineering 25:1 25) DOI: /MPE.25.11

2 12 Reliability for some bivariate beta distributions worked out for the majority of the well-known distributions, including normal, uniform, eponential, gamma, beta, etreme value, Weibull, Laplace, logistic, and the Pareto distributions Nadarajah [5, 6, 7, 8, 9]; Nadarajah and Kotz [1]). In this paper, we calculate R when X and Y are dependent random variables from five fleible families of bivariate beta distributions. We will assume throughout this paper that X,Y) has a bivariate beta distribution with joint probability density function pdf) f and joint survivor function F. Onecanwrite f, y)dyd. 1.1) Our calculations of R make use of a number of special functions. They are the incomplete beta function ratio defined by I a,b) = 1 t a 1 1 t) b 1 dt; 1.2) Ba,b) the Gauss hypergeometric function 2 F 1 )definedby 2F 1 a,b;c;) = k= a) k b) k c) k and, the generalized hypergeometric function 3 F 2 )definedby 2F 2 a,b,c;d,e;) = k= a) k b) k c) k d) k e) k k k! ; 1.3) k k!, 1.4) where f ) k = f f +1) f + k 1) denotes the ascending factorial. The properties of these special functions being used can be found in [2, 11, 12, 13]. 2. Dirichlet distribution Dirichlet distribution has the joint pdf specified by f, y) = K a 1 y b 1 1 y) c 1 2.1) for, y, + y 1, a>, b>, and c>, where Γa + b + c) K = Γa)Γb)Γc). 2.2) The reliability in 1.1) can be epressed as 1 K a 1 = K = KBc,b) a 1 1 ) b+c 1 1 y b 1 1 y) c 1 dyd /1 ) z b 1 1 z) c 1 dzd a 1 1 ) b+c 1 I 1 2)/1 ) c,b)d, 2.3)

3 Saralees Nadarajah 13 where the transformation z = y/1 ) has been applied. Using the relationship 2.3)canberewrittenas I z α,β) = z α αbα,β) 2 F 1 α,1 β;α +1;z), 2.4) K 1 c 2 a 1 1 ) b 1 1 2) c 2F 1 c,1 b;c +1; 1 2 ) d 1 = K 1 z a 1 1 z) a 1 c 2 2 z) a+b+1 2 F 1 c,1 b;c +1;z)dz, 2.5) where the transformation z = 1 2)/1 ) has been applied. Application of Lemma A.3 showsthattheintegralin 2.5) can be calculatedtogive KΓa + b)γc +1) ac 2 Γa + b + c) 3 F 2 a,a + b +1,a + b;a +1,a + b + c; 1). 2.6) Elementary epressions for R can be obtained if one assumes that b or c is an integer. If b is an integer, then, by using the property it follows from 2.3)that I z α,b) = b Γα + i 1) Γα)Γi) zα 1 z) i 1, 2.7) KBc,b) = KBc,b) b = KBc,b)2 1 a b a 1 1 ) b+c 1 b Γc + i 1) Γc)Γi) Γc + i 1) 1/2 a 1 1 ) b i 1 2) c d Γc)Γi) = KBc,b)2 1 a b = KBc,b)2 1 a b = KBc,b)2 1 a b ) 1 2 c ) i 1 d 1 1 Γc + i 1) 1 2 i y i+a 2 1 y) c 1 y ) b i dy Γc)Γi) 2 Γc + i 1) 1 b i ) 2 i y i+a 2 1 y) c ) b i y k 1) k dy Γc)Γi) k 2 k= b i ) b i 1) k Γc + i 1) 1 k 2 k= i+k y i+k+a 2 1 y) c dy Γc)Γi) b i ) b i 1) k Γi + c 1)Γi + k + a 1) k 2 i+k, Γi + k + a + c 1)Γi) k= 2.8) where the transformation y = 2 has been applied. Thus, for integer values of b, one can epress R as a finite sum of elementary functions. On the other hand, if c is an integer,

4 14 Reliability for some bivariate beta distributions then, by using the property I z c,β) = 1 it follows from 2.3)that one can write c KBc,b) a 1 1 ) 1 b+c 1 where = KBc,b) I c Γb + i 1) Γb)Γi) Ji), Γβ + i 1) Γβ)Γi) i 1 1 ) β, 2.9) c Γb + i 1) Γb)Γi) ) 1 2 i 1 ) b d ) I = a 1 1 ) b+c 1 d, Ji) = a+b 1 1 ) c i 1 2) i 1 d. 2.11) By the definition of the incomplete beta function ratio, and after transforming y = 2 Ji) can be calculated as I = Ba,b + c)i 1/2 a,b + c) 2.12) 1 Ji) = 2 a+b) y a+b 1 1 y) i 1 1 y ) c i dy 2 1 c i ) ) c i y k = 2 a+b) y a+b 1 1 y) i 1 1) k dy k 2 k= c i = 2 a+b) ) c i 1 2) k y k a+b+k 1 1 y) i 1 dy k= c i ) = 2 a+b) c i 2) k Ba + b + k,i). k k= 2.13) Substituting 2.12)and2.13)into2.1), one obtains the elementary epression c c i KBc,b) Ba,b+c)I 1/2 a,b+c) 2 a+b) for integer values of c. k= c i k ) 1) k Γb+i 1)Γa+b+k) 2 k Γb)Γa+b+i+k) 2.14)

5 3. Connor and Mosimann s generalized Dirichlet distribution Saralees Nadarajah 15 Connor and Mosimann generalized Dirichlet distribution [1] has the joint pdf given by f, y) = K a 1 y b 1 1 ) c b d 1 y) d 1 3.1) for, y, + y 1, a>, b>, d>andc>b+ d,where K = For this distribution, R can be epressed as Γa + c)γb + d) Γa)Γb)Γc)Γd). 3.2) 1 K a 1 1 ) c b d = K = KBd,b) a 1 1 ) c 1 1 /1 ) y b 1 1 y) d 1 dyd w b 1 1 w) d 1 dwd a 1 1 ) c 1 I 1 2)/1 ) d,b)d, 3.3) where the transformation w = y/1 ) has been applied. Using the relationship 2.4), 3.3)canberewrittenas K a 1 1 ) c d 1 1 2) d 2F 1 = 2 a+c) K 1 d,1 b;1+d; ) d z d 1 z) a 1 1 z/2) c+a) 2F 1 d,1 b;1+d;z)dz, 3.4) which follows by transforming z = 1 2)/1 ). Now, an application of Lemma A.3 shows that 3.4)can be reduced to KΓd +1)Γa + b) 3F 2 a,a + c,a + b;a +1,a + b + d; 1). 3.5) aγa + b + d) As in Section 2, simpler epressions for R can be obtained for certain special cases. In particular, if b is an integer, then, by using 2.7), one can show that KΓb) b 2 a Γb + d) Γd + i 1)Ba + i,d +1) 2 i 2F 1 d c + i,a + i;a + d + i +1; 1 ). 3.6) Γi) 2 Also, if d is an integer, then, by using 2.9), one can show that Γa + b) d iγb + i 1) KBd,b) Ba,c)I 1/2 a,c) 2 a+b Γb) Γa + b + i +1) 2 F 1 b c + i +1,a + b;a + b + i +1; 1 ) )

6 16 Reliability for some bivariate beta distributions 4. Inverted Dirichlet distribution The inverted Dirichlet distribution has the joint pdf specified by for, y, a>, b>, and c>, where f, y) = K a 1 y b y) a+b+c 4.1) K = For this distribution, R can be epressed as Γa + b + c) Γa)Γb)Γc). 4.2) K a 1 = K = K a + c a ) a+b+c y b y) a+b+c dyd y b 1 1+y/1 + ) ) a+b+c dyd c+1) 2F 1 a + b + c,a + c;a + c +1; 1+ ) d, 4.3) where the last step follows by an application of Lemma A.4. Ontransformingy = 1 + )/,4.3)canberewrittenas K y 1) c 1 2F 1 a + b + c,a + c;a + c +1; y)dy. 4.4) a + c 1 Now, the application of Lemma A.1 shows that 4.4)can be reduced to K Bc, c) 3 F 2 a + b + c,a + c,1;a + c +1,c +1; 1) a + c Γc)Γa + b) + aa + c)γa + b + c) 3 F 2 1 c,a + b,a;1 c,a +1; 1). 4.5) An elementary epression for R can be obtained if b is an integer. In this case, using the property one can rewrite 4.4)as 2F 1 γ α,γ β;γ;z) = 1 z) α+β γ 2F 1 α,β;γ;z), 4.6) K y 1) c 1 y +1) 1 a b c 2F 1 1 b,1;a + c +1; y)dy, 4.7) a + c 1

7 Saralees Nadarajah 17 which can be calculated, using the definition of the Gauss hypergeometric function, as K b 1 y 1) c 1 y +1) 1 a b c 1 b) k y) k dy a + c 1 a + c +1) k = K a + c b 1 k= b 1 1) k 1 b) k a + c +1) k k= l= 1 k= y k y 1) c 1 y +1) 1 a b c dy = K 1) k 1 b) k 1 + z) k z c 1 z +2) 1 a b c dz a + c a + c +1) k= k = K b 1 1) k 1 b) k k k )z a + c a + c +1) k= k l l z c 1 z +2) 1 a b c dz l= = K b 1 ) k k 1) k 1 b) k z a + c l l+c z) 1 a b c dz a + c +1) k= l= k b 1 ) K k k 1) k 1 b) k 2 l 1 = 2 a+b 1 w a + c) l c+l 1 1 w) a+b l 2 dw a + c +1) k= l= k b 1 ) K k k 1) k 1 b) k 2 l = 2 a+b 1 Bc + l,a + b l 1), a + c) l a + c +1) k 4.8) where the transformations z = y 1andw = 1/2 + z) have been applied. A simpler epression than 4.5) can be obtained also if a + c is an integer. In this case, by using the property that for m>n, 2F 1 n,β;m;z) = zn m n 1 m 1)!Γβ m + n) 1 z) n m+β n 1)!Γβ) one can rewrite 4.7)as k= + z n m n 1 m 1)!Γm n β) m n 1)!Γm β) 1 n) k m n) k m β n +1) k k! k= n m +1) k n) k β m + n +1) k k! ) z 1 k z ) z 1 k, z 4.9) where Ka + c 1)!Γ1 a b c) 1) a+c I Γ1 b) Jk) = I = 1 1 a+c 1 K a + b + c k= y a+c) y 1) c 1 dy, 1 a c) k 2 a b c) k Jk), 4.1) y 1) c ) y k+1 y +1) a+b+c k 1 dy. On transforming w = 1/y, I can be calculated as 1 I = w c 1 1 w) a 1 dw = Bc,a) 4.12)

8 18 Reliability for some bivariate beta distributions and, on transforming z = y 1, Jk) can be calculated as z Jk) c 1 = z) k+1 dz 2 + z) a+b+c k 1 = Ba + b,c) 2 F 1 k +1 a b c,a + b;a + b + c; 1), 4.13) where the last step follows by application of Lemma A.5. Substituting 4.12) and4.13) into 4.1), one obtains the simpler epression KΓa)Γc)Γ1 a b c) 1) a+c Γ1 b) 2 F 1 k +1 a b c,a + b;a + b + c; 1) a+c 1 KBa + b,c) 1 a c) k a + b + c 2 a b c) k k= 4.14) for integer values of a + c. 5. Lee s generalized inverted Dirichlet distribution Lee generalized inverted Dirichlet distribution [3] has the joint pdf given by bk) f, y) = K a 1 bky) a 1 ) 1 k 1 + bk) a+c 1 + bky) a+c 2 F 1 a + c,a + c;c; 1 + bk)1 + bky) 5.1) for, y, a>, b>, c>, and k>, where K = b2 k c+2 Γ 2 a + c) Γ 2 a)γ ) c) Using the definition of the Gauss hypergeometric function, the corresponding form of R can be epressed as where Kbk) 2a 2 Im) = m= a + c) m a + c) m 1 k) m Im), 5.3) c) m m! a bk) a+c+m An application of Lemma A.4 reduces this integral to y a 1 dyd. 5.4) 1 + bky) a+c+m Im) = bk) a+c+m) a c m 1 c + m 1 + bk) a+c+m 2 F 1 a + c + m,c + m;c + m +1; 1 ) d bk = bk) 2a y 2a+2m 1 c + m 1 + y) a+c+m 2 F 1 a + c + m,c + m;c + m +1; y)dy, 5.5)

9 Saralees Nadarajah 19 where the last step follows after transforming y = 1/bk). The integral in 5.5) can be calculatedbya direct applicationof Lemma A.2;so,it followsfrom5.3)that K bk) 2 a + c) m a + c) m 1 k) m c + m)c) m m! m= B2c +2m,a c m) 3 F 2 a + c + m,c + m,2c +2m;c + m +1,c + m a +1;1) c + m)γ2a)γc + m a) + 3F 2 2a,a,a + c + m;a +1,a c m +1;1). aγa + c + m) 5.6) As in Section 4, simpler epressions for R can be obtained in certain special cases. In particular, if a is an integer, then, using 4.6), one can show that K bk) 2 a 1 m= n= 1) n a + c) m a + c) m 1 k) m 1 a) n B2c +2m + n,2a n 1). c + m)c) m c + m +1) n m! 5.7) Also, if c is an integer, then, using 4.9), one can show that K bk) 2 a + c) m a + c) m 1 k) m c + m)c) m m! Γa)Γ 2 c + m)γ1 a c m) 1) c+m Γa + c + m)γ1 a) m= c+m c m) n B2c +2m n 1,2a) a + c + m 1 2 a c m) n n=. 5.8) 6. Beta Stacy distribution The beta Stacy distribution Mihram and Hultquist [4]) has the joint pdf specified by f, y) = ) c y c a bc Γb)Bp,q) p 1 y bc p q y ) q 1 ep a 6.1) for y>, a>, b>, <c<, p>, and q>. By definition, it is clear that 1.

10 11 Reliability for some bivariate beta distributions Appendi Some technical lemmas required for the calculations above are noted below. Lemma A.1Prudnikov et al.[13, equation )]). For α> and β>, y where α 1 y) β 1 2F 1 a,b;c; w)d = Bβ,1 α β)y α+β 1 3F 2 a,b,α;c,α + β; wy) + Kw 1 α β 3F 2 1 β,a α β +1,b α β +1;2 α β,c α β +1; wy), A.1) K = Γc)Γa α β +1)Γb α β +1)Γα + β 1). A.2) Γa)Γb)Γc α β +1) Lemma A.2Prudnikov et al.[13, equation )]). For α>, α 1 + z) ρ 2F 1 a,b;c; w)d = z α ρ Bα,ρ α) 3 F 2 a,b,α;c,α ρ +1;wz) + Kw ρ α 3F 2 a α + ρ,b α + ρ,ρ;c α + ρ,ρ α +1;wz), Γc)Γa α + ρ)γb α + ρ) K =. Γa)Γb)Γc α + ρ) Lemma A.3Prudnikov et al.[13, equation )]). For c> and β>, y c 1 y ) β 1 1 z) ρ 2F 1 a,b;c; ) d y = Kyc+β 1 1 yz) ρ 3 F 2 β,ρ,c a b + β;c a + β,c b + β; ) yz, yz 1 A.3) A.4) where K = Γc)Γβ)Γc a b + β) Γc a + β)γc b + β). A.5) Lemma A.4 Gradshteyn and Ryzhik [2, equation )]). For ν >µ>, u µ 1 uµ ν d = 1 + β) ν β ν ν µ) 2 F 1 ν,ν µ;ν µ +1; 1 ). A.6) βu Lemma A.5 Gradshteyn and Ryzhik [2, equation )]). For µ>λ>, λ ) µ+ν + β) ν d = Bµ λ,λ) 2 F 1 ν,µ λ;µ;1 β). A.7)

11 Acknowledgments Saralees Nadarajah 111 The author is grateful to Professor Samuel Kotz George Washington University, USA) and to Professor Marianna Pensky University of Central Florida, USA) for introducing him to this area of research. References [1] R. J. Connor and J. E. Mosimann, Concepts of independence for proportions with a generalization of the Dirichlet distribution,j. Amer.Statist. Assoc ), [2] I. S. Gradshteynand I. M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, California, 2. [3] P. A. Lee, The correlated bivariate inverted beta distribution, Biometrical J ), no. 7, [4] G. A. Mihram and R. A. Hultquist, A bivariate warning-time/failure-time distribution,j.amer. Statist.Assoc ), [5] S. Nadarajah, Reliability for beta models, Serdica Math. J ), no. 3, [6], Reliability for etreme value distributions, Math. Comput. Modelling 37 23), no. 9-1, [7], Reliability for lifetime distributions, Math. Comput. Modelling 37 23), no. 7-8, [8], Reliability for Laplace distributions,math. Probl. Eng.1 24), [9], Reliability for logistic distributions, Engineering Simulation 26 24), [1] S. Nadarajah and S. Kotz, Reliability for Pareto models, Metron61 23), no. 2, [11] A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integrals and Series. Vol. 1, Gordon& Breach Science Publishers, New York, [12], Integrals and Series. Vol. 2, Gordon & Breach Science Publishers, New York, [13], Integrals and Series. Vol. 3, Gordon & Breach Science Publishers, New York, 199. Saralees Nadarajah: Department of Statistics, University of Nebraska, Lincoln, NE 68583, USA address: snadaraj@math.iupui.edu