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2 Characterizations of the Pareto distribtion in the presence of otliers U. J. Dixit 1 Department of Statistics, University of Mmbai, Mmbai-India M. Jabbari Nooghabi 2 Department of Statistics, Ferdowsi University of Mashhad, Mashhad-Iran Abstract Here we have given some characterizations for the Pareto distribtion in the presence of otliers. It is proved that a necessary and sfficient condition for f(x) to be a Pareto density fnction in the presence of otliers is that the statistics X (r) and X (s) X (1 r < s n) are independent. Frther, we have derived some another (r) characterizations of the Pareto distribtion in the presence of otliers. Key Words: Pareto distribtion, Power distribtion, Characterization, Order statistics, Otliers. 1 Introdction Ahsanllah and Kabir [2] proved that necessary and sfficient condition for f(x) to be a Pareto distribtion is that the statistics X (r) and X (s) X (r) (1 r < s n) are independent. Dallas [3] proved that for a cmlative distribtion fnction (CDF) G(y) (y β), if E(Y r Y > c) = E ( ) r Y c β holds then Y has a Pareto distribtion. In this paper, we assme that the random variables (X 1, X 2,..., X n ) are sch that k of them are distribted with probability density fnction (pdf) f 2 (x; α, β, ) = α()α, 0 < x, α > 0, β > 1, > 0, (1) xα+1 and remaining (n k) random variables are distribted as f 1 (x; α, ) = αα, 0 < x, α > 0. (2) xα+1 One shold note that two sets of the observation (i.e. k and n k) are independent. Bt X 1, X 2,..., X n are not independent becase of the model of otliers (for more details see [5, 7, 8]). Also, we may note that or assmptions are based of Dixit s model for 1 lhasdixit@yahoo.co.in 2 jabbarinm@yahoo.com, jabbarinm@m.ac.ir 1

3 the otliers problem and it is totaly different than the mixtre models which considers X 1, X 2,..., X n are independent. Here, we have extended the approaches of Ahsanllah and Kabir [2] and Dallas [3] for the homogenos case of the Pareto distribtion and derived some characterizations of the Pareto distribtion in the presence of otliers. 2 Prereqisite Reslts Assme that X (1) < X (2) <... < X (n) be the order statistics of a random sample of size n sch that k ot of n are coming from pdf f 2 (or CDF F 2 ) and the remaining (n k) follow the pdf f 1 (or CDF F 1 ). The CDF and pdf of r th (1 r n) order statistic are H X(r) (x) = n i=r m 6 j=m 5 C(k, j)[f 2 (x)] j [1 F 2 (x)] k j C(n k, i j)[f 1 (x)] i j [1 F 1 (x)] n k i+j, (3) where m 5 = max(0, i n + k) and m 6 = min(k, i) and m 2 h X(r) (x) = kf 2 (x) {C(k 1, j)[f 2 (x)] j [1 F 2 (x)] k j 1 C(n k, r 1 j) j=m 1 [F 1 (x)] r j 1 [1 F 1 (x)] n k r+j+1 } + (n k)f 1 (x) {C(k, j)[f 2 (x)] j j=m 3 [1 F 2 (x)] k j C(n k 1, r 1 j)[f 1 (x)] r j 1 [1 F 1 (x)] n k r+j }, (4) where m 1 = max(0, k + r n 1), m 2 = min(k 1, r 1), m 3 = max(0, k + r n), m 4 = min(k, r 1), respectively (for more details see [4, 5, 6, 8]). Frther, the joint CDF and pdf of (X (r), X (s) ) (1 r < s n) are H X(r),X (s) (x, y) = n j w 10 j=s i=r m=w 9 t 10 m 4 l=t 9 {C(k, m)c(k m, l)[f 2 (x)] m [F 2 (y) F 2 (x)] l [1 F 2 (y)] k m l C(n k, i m)c(n k i + m, j i l)[f 1 (x)] i m [F 1 (y) F 1 (x)] j i l [1 F 1 (y)] n k j+m+l }, (5) where w 9 = max(0, i n + k), w 10 = min(k, i), t 9 = max(0, j n + k m) and t 10 = min(k m, j i) and h X(r),X (s) (x, y) = k(k 1)f 2 (x)f 2 (y) w 2 j=w 1 t 2 i=t 1 {C(k 2, j)c(n k, r 1 j)[f 2 (x)] j [F 1 (x)] r 1 j C(k 2 j, i)c(n k r + j + 1, n s i)[1 F 2 (y)] i [1 F 1 (y)] n s i [F 2 (y) F 2 (x)] k j i 2 [F 1 (y) F 1 (x)] s r k+i+j+1 } + (n k)(n k 1)f 1 (x)f 1 (y) 2 w 4 j=w 3 t 4 i=t 3 {C(n k 2, j)c(k, r 1 j)

4 [F 1 (x)] j [F 2 (x)] r 1 j C(n k 2 j, i)c(k r + j + 1, n s i)[1 F 1 (y)] i [1 F 2 (y)] n s i [F 1 (y) F 1 (x)] n k 2 i j [F 2 (y) F 2 (x)] s r+k n+i+j+1 } + k(n k)f 1 (x)f 2 (y) w 6 j=w 5 t 6 i=t 5 {C(n k 1, j)c(k 1, r 1 j) [F 1 (x)] j [F 2 (x)] r 1 j C(n k j 1, i)c(k r + j, n s i)[1 F 1 (y)] i [1 F 2 (y)] n s i [F 1 (y) F 1 (x)] n k i j 1 [F 2 (y) F 2 (x)] s r n+k+i+j } + k(n k)f 2 (x)f 1 (y) w 8 j=w 7 t 8 i=t 7 {C(k 1, j)c(n k 1, r 1 j) [F 2 (x)] j [F 1 (x)] r 1 j C(k j 1, i)c(n k r + j, n s i)[1 F 2 (y)] i [1 F 1 (y)] n s i [F 2 (y) F 2 (x)] k i j 1 [F 1 (y) F 1 (x)] s r k+i+j }, (6) where w 1 = max(0, r n + k 1), w 2 = min(k 2, r 1), t 1 = max(0, k s + r j 1) and t 2 = min(k j 2, n s), w 3 = max(0, r k 1), w 4 = min(n k 2, r 1), t 3 = max(0, n s k + r j 1) and t 4 = min(n k j 2, n s), w 5 = max(0, r k), w 6 = min(n k 1, r 1), t 5 = max(0, n s k +r j 1), t 6 = min(n k j 1, n s), w 7 = max(0, r n + k), w 8 = min(k 1, r 1), t 7 = max(0, k s + r j 1), t 8 = min(k j 1, n s), respectively. One shold note that if k = 1 the joint pdf of (X (r), X (s) ) is given in Sinha [10]. Also, if we pt f 1 = f 2 and F 1 = F 2 then all pdfs and CDFs are redced to homogeneos cases. The following eqations are named as Pexider s eqations. and f(xy) = g(x) + h(y), (7) f(xy) = g(x)h(y). (8) For solving these eqations, the following Theorem has taken from Aczel [1] (Theorem 4. in p. 144) and Kczma [9] (Theorem in p. 358). Theorem 2.1. The general soltions, with f continos in a point of (7) and (8), respectively, both spposed for positive x and y, are and f(t) = c ln(αβt), g(t) = c ln(αt), h(t) = c ln(βt), (α > 0, β > 0, t > 0), (9) f(t) = abt c, g(t) = at c, h(t) = bt c, (t > 0), (10) respectively, spplemented with the following trivial soltions in case of (8). f(t) = 0, g(t) = 0, h(t) arbitrary, and f(t) = 0, g(t) arbitrary, h(t) = 0. (11) 3

5 3 Characterization of the Pareto distribtion in the presence of otliers Theorem 3.1. Let X be a random variable having an absoltely continos CDF F (x). A necessary and sfficient condition that X follows the Pareto distribtion in the presence of otliers as given by (1) and (2) is that for some r and s (1 r < s n) the statistics X (r) and X (s) X (r) are independent. Proof. Necessity: From (6) we can get the joint pdf of X (r) and X (s). Sbstitting U = X (r) and V = X (s) X (r) in (6), we can obtain the joint pdf of U and V as h U,V (, v) = h X(r),X (s) (, v). Then after some simplification (replace h U,V (, v) with h(, v)) h(, v) = α 2 α(n r+1) β αk v α(s n 1) 1 [1 v α ] s r 1 [ ( ) w2 α(r n 1) 1 k(k 1) t 2 α ] j [ ( ) α ] r 1 j A 1 β αj 1 1 j=w 1 i=t 1 [ ( ) w 4 + (n k)(n k 1)β α(r 1) t 4 α ] j [ ( ) α ] r 1 j A 2 β αj 1 1 j=w 3 i=t 3 [ ( ) w 6 + k(n k)β α(r 1) t 6 α ] j [ ( ) α ] r 1 j A 3 β αj 1 1 j=w 5 i=t 5 [ ( ) w 8 t 8 α ] j [ ( ) α ] r 1 j + k(n k) A 4 β αj 1 1 i=t 7, (12) j=w 7 where A 1 = C(k 2, j)c(n k, r 1 j)c(k 2 j, i)c(n k r + j + 1, n s i), A 2 = C(n k 2, j)c(k, r 1 j)c(n k 2 j, i)c(k r + j + 1, n s i), A 3 = C(n k 1, j)c(k 1, r 1 j)c(n k j 1, i)c(k r + j, n s i), (13) A 4 = C(k 1, j)c(n k 1, r 1 j)c(k j 1, i)c(n k r + j, n s i). Therefore, it establishes the independence of U and V. Sfficiency: Here we assme that U and V are independent. The joint pdf of U and V is h(, v) = k(k 1)f 2 ()f 2 (v) w 2 j=w 1 t 2 i=t 1 {A 1 [F 2 ()] j [F 1 ()] r 1 j [1 F 2 (v)] i 4

6 [1 F 1 (v)] n s i [F 2 (v) F 2 ()] k j i 2 [F 1 (v) F 1 ()] s r k+i+j+1 } + (n k)(n k 1)f 1 ()f 1 (v) w 4 j=w 3 t 4 i=t 3 {A 2 [F 1 ()] j [F 2 ()] r 1 j [1 F 1 (v)] i [1 F 2 (v)] n s i [F 1 (v) F 1 ()] n k 2 i j [F 2 (v) F 2 ()] s r+k n+i+j+1 } + k(n k)f 1 ()f 2 (v) w 6 j=w 5 t 6 i=t 5 {A 3 [F 1 ()] j [F 2 ()] r 1 j [1 F 1 (v)] i [1 F 2 (v)] n s i [F 1 (v) F 1 ()] n k i j 1 [F 2 (v) F 2 ()] s r n+k+i+j } + k(n k)f 2 ()f 1 (v) w 8 j=w 7 t 8 i=t 7 {A 4 [F 2 ()] j [F 1 ()] r 1 j [1 F 2 (v)] i [1 F 1 (v)] n s i [F 2 (v) F 2 ()] k i j 1 [F 1 (v) F 1 ()] s r k+i+j }, (14) where A 1, A 2, A 3 and A 4 are given in (13). By sing some elementary algebra we have h(, v) = k(k 1)f 2 ()f 2 (v)[f 1 ()] r 1 [1 F 1 (v)] n s [F 1 (v) F 1 ()] s r k+1 w 2 [F 2 (v) F 2 ()] k 2 t 2 j j [ F2 () 1 F1 () 1 F 2 (v) {A 1 i=t 1 F 1 () 1 F 2 () F 2 (v) F 2 () j=w 1 i j [ F1 (v) F 1 () F1 (v) F 1 () 1 F 1 (v) 1 F 1 () 1 F 2 () F 2 (v) F 2 () + (n k)(n k 1)f 1 ()f 1 (v)[f 2 ()] r 1 [1 F 2 (v)] n s [F 1 (v) F 1 ()] n k 2 w 4 [F 2 (v) F 2 ()] s r+k n+1 t 4 j j [ F1 () 1 F2 () 1 F 1 (v) {A 2 i=t 3 F 2 () 1 F 1 () F 1 (v) F 1 () i [ F2 (v) F 2 () 1 F 2 (v) j=w 3 1 F 1 () F 1 (v) F 1 () ] j } ] j j F2 (v) F 2 () } 1 F 2 () + k(n k)f 1 ()f 2 (v)[f 2 ()] r 1 [1 F 2 (v)] n s [F 1 (v) F 1 ()] n k 1 w 6 [F 2 (v) F 2 ()] s r+k n t 6 j j [ F1 () 1 F2 () 1 F 1 (v) {A 3 j=w 5 i=t 5 F 2 () 1 F 1 () F 1 (v) F 1 () i j [ ] j F2 (v) F 2 () F2 (v) F 2 () 1 F 1 () } 1 F 2 (v) 1 F 2 () F 1 (v) F 1 () + k(n k)f 2 ()f 1 (v)[f 1 ()] r 1 [1 F 1 (v)] n s [F 2 (v) F 2 ()] k 1 w 8 [F 1 (v) F 1 ()] s r k t 8 j j [ ] i F2 () 1 F1 () 1 F 2 (v) {A 4 j=w 7 i=t 7 F 1 () 1 F 2 () F 2 (v) F 2 () i j [ ] j F1 (v) F 1 () F1 (v) F 1 () 1 F 2 () }. (15) 1 F 1 (v) 1 F 1 () F 2 (v) F 2 () Also from (4) and after some simplification, the marginal pdf of U = X (r) is as h 1 (). h 1 () = [1 F 2 ()] k 1 [F 1 ()] r 1 [1 F 1 ()] n k r+1 D, (16) 5 ] i ] i ] i

7 where and D = kf 2 () m 2 j=m 1 B 1 + (n k)f 1 () j [ F2 () 1 F1 () F 1 () 1 F 2 () 1 F2 () m4 1 F 1 () j=m 3 B 2 ] j [ F2 () F 1 () B 1 = C(k 1, j)c(n k, r 1 j), B 2 = C(k, j)c(n k 1, r 1 j). ] j j 1 F1 (), 1 F 2 () (17) Therefore from independency of U and V, we can write h 2 (v) = h(, v) h 1 (), (18) where h 2 (v) is pdf of V. Letting p = p(, v) = 1 F 1(v) 1 F 1 () and q = q(, v) = 1 F 2(v) 1 F 2, we obtain () w 2 h 2 (v) = {k(k 1)p n s [1 p] s k r+1 [1 q] k 2 t 2 A 1 j=w 1 i=t 1 A 4 j=w 7 i=t 7 [ F2 () F 1 () ] j j 1 F1 () 1 F 2 () p i [1 p] i+j q i [1 q] i j f 2 () q v r 1 r 2 F2 () 1 F1 () + (n k)(n k 1) [1 p] n k 2 q n s [1 q] k+s r n+1 F 1 () 1 F 2 () w 4 t 4 j j F1 () 1 F2 () A 2 q i [1 q] i+j p i [1 p] i j f 1 () p j=w 3 i=t 3 F 2 () 1 F 1 () v r 1 r 2 F2 () 1 F1 () + k(n k) q n s [1 q] k+s r n [1 p] n k 1 F 1 () 1 F 2 () w 6 t 6 j j F1 () 1 F2 () A 3 p i [1 p] i j q i [1 q] i+j f 1 () q j=w 5 i=t 5 F 2 () 1 F 1 () v w 8 + k(n k)p n s [1 p] s k r [1 q] k 1 t 8 j j F2 () 1 F1 () F 1 () 1 F 2 () p i [1 p] i+j q i [1 q] i j f 2 () p v }D 1. (19) From the assmption, we know that U and V are independent. So h 2 (v) is independent of and by sing the lemma in Ahsanllah and Kabir [2] p = p(, v) = g 1 (v) and 6

8 q = q(, v) = g 2 (v) (we say fnctions of v only) and the remaining parts shold be constant. Therefore { 1 F1 (v) = [1 F 1 ()]g 1 (v),, 1 < v, > 0, (20) 1 F 2 (v) = [1 F 2 ()]g 2 (v),, 1 < v, > 0, β > 1. It is clear that these are version of Pexider s eqation. So from Theorem 2.1 we can solve them. Since F 1 (x) and F 2 (x) are CDFs continos for all x [, ) and x [, ), respectively. We may conclde that { 1 F1 (x) = c 1 x α, x, > 0, 1 F 2 (x) = c 2 x α, x, > 0, β > 1, (21) where c 1, c 2 and α are constant. After replacing these soltions in (19) and sing some simplification we get where h 2 (v) = αv α(n s+1) 1 [1 v α ] s r 1 H[c 2 D] 1, (22) w 2 H = k(k 1)c 2 t 2 A 1 j=w 1 i=t 1 w 4 + (n k)(n k 1)c 1 w 6 + k(n k)c 1 + k(n k)c 2 t 8 A 3 j=w 5 i=t 5 i=t 7 A 4 t 6 [ 1 c2 α 1 c 1 α t 4 A 2 j=w 3 i=t 3 [ 1 c2 α 1 c 1 α [ 1 c2 α 1 c 1 α ] j (c1 ) j c 2 [ 1 c2 α 1 c 1 α ] r 1 j (c1 ) r 2 j ] j (c1 ) j. c 2 c 2 ] r 1 j (c1 ) r 2 j We know that C(n, j) = 0 if j > n, then by sing some elementary algebra H[c 2 D] 1 = (n r)c(n r 1, n s) and the right side of (22) is only depend on v and it is pdf of V. Finally, from the property of CDF, α > 0, c 1 = α and c 2 = () α. Ths sfficiency is established and the proof is complete. Theorem 3.2. Let X be a random variable with CDF F (x) = bf 2 (x) + bf 1 (x) sch that F 1 (x) (x ) and F 2 (x) (x ) are CDFs, where b = k n, b = 1 b, > 0 and β > 1. If E(X α X > c) = be 2 ( Xc ) α + be ( ) Xc α 1, (23) holds for some α > 0 then F (x) is the Pareto distribtion in the presence of otliers. We assme that E(X α ) <. Proof. Proof is similar as given in Dallas [3]. In the process to prove the theorem, 7 c 2

9 we shold note that the soltion of the differential eqation cp (c) = γp (c) (P (c) = 1 F (c)) is P (c) = Ac α, where A is a constant, γ = αδ/(δ 1) > 0 and δ = b ( ) α X df 2 (x) + b ( X ) α df 1 (x). (24) Comparing the soltion with the assmption imply that A = b() α + b α and the proof is complete. 4 An actal example Here, we have given an example of motor insrance company from Dixit and Jabbari Nooghabi [7]. From the example, we know that the data follow the Preto distribtion in the presence of otliers. So by sing Theorem 3.1, we can check the sfficiency. Assming r = 3 and s = 12, we have x (r) =63000, and x (s) x (r) = So, sing the copla method and independent test by package copla in R, the reslt is as follows: Global Cramer-von Mises statistic: with p-vale Combined p-vales from the Mobis decomposition: from Fisher s rle, from Tippett s rle. Therefore, X (r) and X (s) X (r) are independent, becase of the p-vale is grater than 0.05, as significant level of the test. So, we can conclde that the data follow the Pareto distribtion in the presence of otliers. 5 Acknowledgements The athors are thankfl to the two referees for their valable comments and sggestions. This research was spported by a grant from Ferdowsi University of Mashhad, No. MS92299MEH. References [1] Aczel, J. (1966). Lectres on Fnctional Eqations and their Applications. Academic Press, New York. [2] Ahsanllah, M. and Kabir Ltfl A. B. M. (1973). A Characterization of the Pareto Distribtion. The Canadian Jornal of Statistics, 1(1), [3] Dallas, A. C. (1976). Characterizing the Pareto and power distribtions. Ann. Inst. Statist. Math., 28,

10 [4] Dixit, U. J. (1987). Characterization of the gamma distribtion in the presence of k otliers. Blletin Bombay Mathematical Colloqim, 4, [5] Dixit, U. J. (1989). Estimation of parameters of the Gamma Distribtion in the presence of Otliers. Commn. Statist. Theory and Meth., 18, [6] Dixit, U. J. (1994). Bayesian approach to prediction in the presence of otliers for Weibll distribtion, Metrika, 41, [7] Dixit, U. J. and Jabbari Nooghabi, M. (2011). Efficient estimation in the Pareto distribtion with the presence of otliers, Statistical Methodology, 8(4), [8] Dixit, U. J. and Jabbari Nooghabi, M. (2011). Efficient Estimation of the parameters of the Pareto Distribtion in the Presence of Otliers, Commnications of the Korean Statistical Society, 18(6), [9] Kczma M. (2008). An Introdction to the Theory of Fnctional Eqations and Ineqalities. Second edition, edited by Attila Gilányi, Birkhäser Verlag AG, Berlin. [10] Sinha, S. K. (1973). Distribtions of order statistics and estimation of mean life when an otlier may be present. The Canadian Jornal of Statistics, 1(1),