SOME RECURRENCE RELATIONS BETWEEN PRODUCT MOMENTS OF ORDER STATISTICS OF A DOUBLY TRUNCATED PARETO DISTRIBUTION

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1 Sankhyā : The Indian Journal of Statistics 1995, Volume 57, Series B, Pt. 1, pp. 1-9 SOME RECURRENCE RELATIONS BETWEEN PRODUCT MOMENTS OF ORDER STATISTICS OF A DOUBLY TRUNCATED PARETO DISTRIBUTION By A. P. KHURANA and V. D. JHA Devi Ahilya Vishwavidyalaya SUMMARY. Product moments of order statistics from a doubly truncated Pareto distribution and some recurrence relations between them are obtained. 1. Introduction The Pareto distribution provides reasonably good fit to the distribution of income property values (Malik (1970)). It has been observed that Pareto curve gives good fit at the extremities of the income range while the fit is rather poor over the entire range (Johnson and Kotz (1970)). This calls for the study of a truncated Pareto distribution. Various recurrence relations between raw and central moments of different distributions are available in the literature. The main utility and advantage of such recurrence relations between moments is that having obtained one of the moments, the moments of higher/lower order can be easily obtained without indulging into copious computational work. Some recurrence relations between moments of order statistics have been reported by Balakrishnan and Joshi (1982) for doubly truncated and by Jha (1983) for untruncated Pareto and power function distributions. Joshi (1978) obtained some recurrence relations between moments of order statistics for a truncated exponential distribution. Khurana and Jha (1990) have obtained exhaustive recurrence relations, linear in argument, between the moments of order statisfied from a doubly truncated Pareto distribution. The recurrence relations between the product moments of order statistics from untruncated and right or left truncated Pareto distributions can be easily obtained by setting both or one of the proportions of truncation Q and 1 P equal to zero. Paper received. June 1990; revised June AMS classification 62G30. Key words. Pareto distribution, recurrence relations, product moments, order statistics, Appe; s function, Kampe De Feriet s function, contiguous functions relations, doubly truncated distribution, right truncated distribution, left truncated distribution, partial differential equations.

2 2 a.p. khurana and v.d. jha The Appel s functions of two variables have been generalized by Kampe De Feriet (1926). The Kampe De Feriet s function of two variables, in notaions of Burchnall and Chaundy (1941), is defined by = m,n=0 F λ:µ ν:σ λ Π i=1 (a) : (b) : (b ); (c) : (d) : (d ); x, y µ [(a i ) m+n Π (b j ) m (b j ) n] j=1 ν σ Π (c k ) m+n Π [(d 1 ) m.(d 1 ) n] k=1 l=1 x m y n m! n!,...(1.1) where (a) stands for λ parameters a 1,...,a λ, (b) for µ parameters b 1,...,b µ,(b ) for µ parameters b 1,...,b µ,(c) for ν parameters c 1,...,c ν,(d) for σ parameters d 1,...,d σ and (d ) for σ parameters d 1,...,d σ, and (α) s = α(α +1)...(α + s 1), s 1, (α) 0 =1,α 0;...(1.2) none of c, d, d is zero or a negative integer and provided λ v<1+σ µ or λ v =1+σ µ = p with x 1/p + y 1/p...(1.3) < 1 for p>0 and x < 1, y < 1 for p 0. Following Gauss s definition of contiguous hypergeometric function as defined in Rainville (1971), a function is said to be contiguous to F if it has equal number of parameters in the numerator and denominator as in F and all corresponding parameters are equal except one pair and this pair of parameters differs only by unity. Thus (a +1) :(b) : (b ); F 1:2 1:1 (c) : (d) : (d ); x, y...(1.4) is contiguous to F and we shall denote this by F (a+). Similarly F (a ), F (b+), F (b ) etc. are defined. It is observed that the product moments of order statistics of a doubly truncated Pareto distribution are expressible in terms of Kampe De Feriet s function of two variables, F1:1 1:2 defined by (1.1). Khurana and Kale (1991) have obtained some contiguous functions relations for F1:1 1:2. In this paper we have obtained the product moments of order statistics from a doubly truncated Pareto distribution and developed some recurrence relations between them, using contiguous functions relations for F1:1 1:2.

3 some recurrence relations 3 2. Product moments of order statistics of a doubly truncated pareto distribution Let X 1:N X 2:N...X N:N be the order statistics obtained by rearranging the sample of size N from a doubly truncated Pareto distribution defined by [ (1/(P Q)) av f(x) = a /x a+1, L<x<U; a, v > 0; P >Q,L>v; 0 elsewhere...(2.1) where L = v/(1 Q) 1/a,U= v/(1 P ) 1/a and Q and 1 P are proportions of truncation on the left and on the right of the Pareto distribution with p.d.f φ(x) = [ av a /x a+1, 0 <v<x,a>0; 0 elsewhere...(2.2) The distribution function F (x) for the truncated Pareto distribution is given by F (x) =(1/(P Q)) (1 Q v a /x a )....(2.3) We shall denote by X r:n and X s:n the r-th and s-th (1 r<s N) smallest order statistics and E(Xr:N i.xj s:n )byµi,j,(i, j 0). For simplicity we shall write µ for µ 1,1. The joint density of X r:n and X s:n (r<s) for the doubly truncated Pareto distribution is given by g (x, y) =C{F (x)} r 1 {F (y) F (x)} s r 1 {1 F (y)} N s f(x)f(y), L<x<y<U...(2.4) where C =Γ(N +1)/Γ(r)Γ(s r)γ(n s +1)....(2.5) The product moment L i+j m=0 n=0 works out to be Γ(N + 1)Γ(n + r)γ(m + s r)(j/a) m (m + i/a + j/a) n Γ(N + 1 +m + n)γ(r)γ(s)m!n! { } m+n P Q. 1 Q...(2.6) Using Γ(α + m)/γ(α) =(α) m and (α + m) n =(α) m+n /(α) m...(2.7) and introducing (1) n in the numerator and denominator, we can write it in the form (i/a + j/a):{j/a, r} : {s r, 1} : L i+j = F1:1 1:2 P Q 1 Q, P Q 1 Q...(2.8) (N +1):{i/a + j/a, 1) : = F (say)

4 4 a.p. khurana and v.d. jha We thus find that the product moments of order statistics of a doubly truncated Pareto distribution can be expressed in the form of a Kampe De Fariet s function of two variables with equal arguments (P Q)/(1 Q). It can easily be verified that the conditions mentioned in (1.3) hold for λ = ν = σ = 1 and µ = 2 and that (P Q)/(1 Q) Product moments of order statistics of an untruncated pareto distribution In (2.6), setting Q and 1 P equal to zero, we obtain = vi+j m=0 n=0 (i/a + j/a) m+n (j/a) m (r) n (s r) m (1) n....(3.1) (N +1) m+n (i/a/j/a) m (1) n m!n! The summation on the right hand side may be easily evaluated by using twice the known result (Rainville (1971)) for the hypergeometric function F (a, b; c;1)= m=0 (a) m (b) m (c) m m! =Γ(c)Γ(c a b)/γ(c a)γ(c b),...(3.2) for Re (c a b) > 0 and for c neither zero nor a negative integer. Thus (3.1) may be written as = vi+j m=0 (j/a) m (s r) m (N +1) m m! n=0 (i/a + j/a + m) n (r) n (N + m +1) n n! i+j Γ(N + 1)Γ(N i/a j/a +1 r)γ(n +1 j/a s) = v Γ(N +1 i/a j/a)γ(n +1 j/a r)γ(n +1 s)....(3.3) It we let i = 1 and j = 1 in (3.3), we obtain the well known result (Johnson and Kotz (1970), pp. 241) 2 Γ(N + 1)Γ(N 2/a +1 r)γ(n +1 j/a s) µ = v Γ(N +1 2/a)Γ(N +1 1/a r)γ(n +1 s)....(3.4) 4. Product moments of order statistics of a singly truncated pareto distribution (i) Right truncated distribution. In (2.6), setting Q = 0, using (2.7) and introducing (1) n in the numerator and denominator, we obtain the product moments for the order statistics from a right truncated Pareto distribution. = vi+j m=0 n=0 (i/a + j/a) m+n (j/a) m (r) n (s r) m (1) n p m+n...(4.1) (N +1) m+n (i/a + j/a) m (1) n m!n!

5 some recurrence relations 5 (ii) Left truncated distribution. Similarly in (2.6), setting 1 P = 0, we obtain the product moments for the order statistics from a left truncated Pareto distribution, = v i+j (1 Q) 1/a+j/a m=0 n=0 (i/a + j/a) m+n (j/a) m (r) n (s r) m (1) n. (N +1) m+n (i/a + j/a) m (1) n m!n!...(4.2) This shows that the product moments of the r-th and s-th order statistics for a left truncated Pareto distribution are also same as the corresponding product moments for an untruncated Pareto distribution with the lower limit v of the untruncated Pareto distribution replace by the new lower limit L = v/(1 Q) 1/a of the left truncated distribution. 5. Recurrence relations between product moments of order statistics of a doubly truncated pareto distribution In this section we have used some of the contiguous functions relations obtained by Khurana and Kale (1991) for developing some recurrence relations between the product moments of a doubly truncated Pareto distribution. Relation 1. setting (j/a s + r) =(j/a)µi a,j+a (s r) r,s+1:n....(5.1) Proof. In the contiguous functions relation for F = F 1:2 1:1 namely (b 1 + b 2 )F = b 1 F (b 1 +) b 2 F (b 2 +),...(5.1a) b 1 = j/a, b 1 = r; b 2 = s r, b 2 =1;a = i/a + j/a, d = i/a + j/a, d =1,c= N + 1 and x = y =(P Q)/(1 Q)...(5.1b) and using (2.6) and (2.7), we readily obtain the relation (5.1). It can be noticed that the recurrence relation (5.1) is independent of the proportions of truncation Q and 1 P, therefore it is satisfied by the product moments of order statistics from singly truncated and from untruncated Pareto distribution as well. Relation 2. L 2a (i/a + j/a N) = j a. s r N +1.P Q 1 Q µi+a,j+a r,s+1:n+1 + La (i/a + j/a)µ i+a,j...(5.2) can, similarly, be obtained by using the contiguous functions relation (a c +1)F = af (a+) (c 1)F (c ),...(5.2a)

6 6 a.p. khurana and v.d. jha Relation 3. il a = jla µ i a,j+a j s r P Q N +1 1 Q µi,j+a r,s+1:n+1....(5.3) can be obtained by using the conditiguous functions relation Relation 4. 1 P 1 Q (j/a s + r)µi,j (b 1 d +1)F = b 1 F (b 1 +) (d 1)F (d ),...(5.3a) = (i/a + j/a s + r)µi,j r,s 1:N (i/a)µi+a,j a + (j/a s+r)(i/a+j/a N 1) N+1 can be obtained by using the contiguous functions relation [ ] P Q 1 Q (5.4) (b 1 b 2 )(1 x)f = (b 1 d)f (b 1 ) (b 2 d)f (b 2 ) +(b 1 b 2 )(a c)c 1 xf (c+)...(5.4a) The other contiguous functions relations do not yield any useful recurrence relations between the product moments because of the dummy parameters b 2,d =1 in the Kampe De Feriet s function F1:1 1:2. 6. Recurrence relations for product moments of order statistics from untruncated pareto distribution Four recurrence relations for product moments of order statistics for untruncated Pareto distribution are readily obtained from (5.1) to (5.4) on setting Q and 1 P equal to zero. They are (j/a s + r) =(j/a)µi a,j+a (s r)µi,j r,s+1:n....(6.1) v 2a (i/a + j/a N) = j a. s r N +1.µi+a,j+a r,s+1,n+1 + va (i/a + j/a)µ i+a,j....(6.2) iv a = jva µ i a,j+a j s r N +1 µi,j+a r,s+1,n+1....(6.3) (i/a+j/a s+r) r,s 1:N =(i/a)µi+a,j a s + r)(i/a + j/a N 1) +(j/a N (6.4) Some other recurrence relations for product moments of order statistics from untruncated Pareto distribution can easily be obtained by simple rearrangement and manipulation of the expression (3.3). They are

7 some recurrence relations 7 v a N i/a j/a r = N i/a j/a µi+a,j....(6.5) v a = µi+a,j (r/n +1)µi+a,j r+1,s+1:n+1....(6.6) v 2a = va µ i+a,j r(n +1 s) (N + 1)(N i/a j/a) µi+a,j+a +1...(6.7) µ i+a,j = N s j/a N r j/a µi,j+a...(6.8) v a (N s j/a)(n i/a j/a r) = +a (N r j/a)(n i/a j/a)....(6.9) = N s j/a r,s+1:n...(6.10) N s N r i/a j/a = r+1,s:n N r j/a....(6.11) µi+a,j r,s+α:n = µi+a,j µi,j r,s+α:n....(6.12) 7. Recurrence relations for product moments of order statistics from right truncated pareto distribution Four recurrence relations for product moments of order statistics for right truncated Pareto distribution are readily obtained from (5.1) to (5.4) on setting Q equal to zero. They are (j/a s + r) =(j/a)µi a,j+a (s r)....(7.1) v 2a (i/a + j/a N) = j a. s r.p µi+a,j+a r,s+1:n+1 N +1 (i/a + j/a)µ i+a,j...(7.2) iv a = jva µ i a,j+a j s r.p µi,j+a r,s+1:n+1 N +1...(7.3) (1 P )(j/a s + r) =(i/a + j/a s + r) r,s 1:N (i/a)µi+a,j a (j/a s + r)(i/a + j/a N 1) + P +1...(7.4) N +1

8 8 a.p. khurana and v.d. jha 8. Recurrence relations for product moments of order statistics from left truncated pareto distribution Four recurrence relations for product moments of order statistics for left truncated Pareto distribution are also readily obtained from (5.1) to (5.4) on putting 1 P equal to 0. They are (j/a s + r) =(j/a)µ i a,j+a (s r) r,s+1:n.l2a (i/a + j/a N)...(8.1) = j a. s r N+1.µi+a,j+a r,s+1:n+1 + La (i/a + j/a)+a....(8.2) il a = jla µ i a,j+a j s r N+1 µi,j+a r,s+1:n+1.(i/a + j/a s + r)µi,j r,s 1:N...(8.3) =(i/a)µ i+a,j a + (j/a s+r)(i/a+j/a N 1) N+1 +1,...(8.4) which shows that the recurrence relations for the product moments of the r-th and s-th order statistics for a left truncated Pareto distribution are also same as those for an untruncated Pareto distribution with the lower limit v of the untruncated distribution replaced by the new lower limit L = v/(1 Q) 1/a of the left truncated distribution. 9. Partial differential equations satisfied by Writing µ for and differentiating equation (2.6) partially with respect to P and Q, we obtain, after some manipulations, the differential equation (1 P ) µ µ +(1 Q) p Q = i + j µ,... (9.1) a which is in the standard form of Lagrange s linear equation Xx + Yy= R. Differentiating (9.1) partially with respect to P and Q and eliminating P and Q from the resulting equations and (9.1) itself, we obtain (i/a + j/a +1){( µ P )2 2 µ Q 2 +( µ Q )2 2 µ P 2 2 µ µ 2 µ P Q P Q } (i/a + j/a){ 2 µ P µ Q 2 ( 2 µ P Q )2 }...(9.2) which is again of a standard form, Rr + Ss + Tt+ U(rt s 2 )=V

9 some recurrence relations 9 where R, S, T, U and V are functions of P, Q, µ, µ/ P and µ/ Q and can be solved by Monge s method (Forsyth (1954)). Acknowledgement. The authors are thankful to the referee for valuable suggestions and comments. References Balakrishnan, N. and Joshi, P. C. (1982). Moments of order statistics from doubly truncated Pareto distribution. J. Indian Statist. Assoc, 20, Burchnall, J. L. and Chaundy, T. W (1941). Expansion of Appel s double hypergeometric function II. Quart. J. Math. Oxford Ser., 12, Forsyth, A. R. (1954). A Treatise on Differential Equations, MacMillan and Co. Ltd., London. Jha, V. D. (1983). Recurrence relations between the moments of Pareto and Power function distributions. Research Journal Science University of Indore, 8, Johnson, N. L. and Kotz, S. (1970). Distributions in Statistics, Continuous Univariate Distributions, Vol. 1 and 2, Houghton Mifflin, Boston. Joshi, P. C. (1978). Recurence relations between moments of order statistics from exponential and truncated exponential distributions. Sankhyā, Ser. B, 39, Kampe de Feriet, J. Apple and Paul (1926). Fonctions Hypergeometriques et Hyperspheriques, Polynomes d hermites, Paris, Gauthier-Villars Et C i.e., Editeurs. Khurana, A. P. and Kale, P. P. (1991). Some contiguous functions relations for the Kampe de Feriet s function F1:1 1:2. Jr. Indian. Acad. Math., 13, No. 1, Khurana, A. P. and Jha, V. D. (1990). Recurrence relations between moments of order statistics from a doubly truncated Pareto distribution. Sankhyā, Ser. B. (to appear). Malik, H. J. (1970). Estimation of the parameters of the Pareto distributions. Metrika, 16, Rainville, E. D. (1971). Special Functions, Chelsea Publishing Company, New York. School of Computer Science and Electronics Devi Ahilya Vishwavidayalaya Khanwa Road Indore India