CHARACTERIZATIONS OF A FAMILY OF BIVARIATE PARETO DISTRIBUTIONS

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1 STATISTICA, anno LXXV, n. 3, 205 CHARACTERIZATIONS OF A FAMILY OF BIVARIATE PARETO DISTRIBUTIONS P. G. Sankaran Department of Statistics Cochin University of Science and Technology, Cochin N. Unnikrishnan Nair Department of Statistics Cochin University of Science and Technology, Cochin Preethi John Department of Statistics Cochin University of Science and Technology, Cochin. Introduction In the literature, Pareto distributions have been etensively employed for modeling and analysis of statistical data under different contets. Orginally, the distribution was first proposed as a model to eplain the allocation of wealth among individuals. Later, various forms of the Pareto distribution have been formulated for modeling and analysis of data from engineering, environment, geology, hydrology etc. These diverse applications of the Pareto distributions lead researchers to develop different kinds of bivariatemultivariatepareto distributions. Accordingly, Mardia 962 introduced two types of bivariatemultivariate Pareto models which are referred as bivariate Pareto distributions of first kind and second kind respectively. Since then there has been a lot of works in the form, alternative derivation of bivariate Pareto models, their etensions, inference, characterizations and applications to a variety of fields. Various types of bivariate multivariate Pareto distributions discussed and studied in literature include those of Lindley and Singpurwalla 986, Arnold 985, Arnold 990, Sankaran and Nair 993, Hutchinson and Lai 990, Langseth 2002, Balakrishnan and Lai 2009 and Sankaran and Kundu 204. The models discussed above are individual in nature and are appropriate for a particular data set that meet the specified requirements. However, when there is little information about the data generating process, it is desirable to start with a family of distributions and then choose a member of the family that agrees with the patterns in the data. Motivated by this fact, we introduced a family of bivariate Pareto distributions arising from a generalization of the univariate dullness property Talwalker 980 that characterized the univariate Pareto law.see Sankaran et al. 204 Corresponding Author. sankaran.pg@gmail.com

2 276 P. G. Sankaran, N. Unnikrishnan Nair and Preethi John The major aim of the present paper is to develop characterizations of the above family of Pareto distributions. One of the problems that is usually addressed while eamining the characteristic properties of the bivariate distribution is to investigate how far the characterizations of the corresponding univariate version can be etended to the bivariate forms. Among the characterizations of the univariate Pareto I law, one with important practical applications is the dullness property. The bivariate version of the dullness property is employed to characterize the family of Pareto distributions. A second concept that has applications in economics is income gap ratio which is used for developing indices of affluence Sen 988. The bivariate generalization of the concept is proposed and characterizations using this concept are derived. Another function of interest that has applications in reliability also is the bivariate generalized failure rate. Characterizations of the family of distributions using the generalized failure rate are also discussed. As the bivariate versions of these functions are not unique, different versions of these concepts lead to various types of bivariate Pareto distributions which are members of the family. The rest of the paper is organized as follows. In Section 2, we present a family of bivariate Pareto distributions characterized through a generalized version of the univariate dullness property. In Section 3, we introduce bivariate versions of dullness property and present characterizations using these versions. The bivariate version of income gap ratio and related concepts in economics are discussed in Section 4. The forms of these functions for various bivariate Pareto distributions are given. Section 5 discusses bivariate generalized failure rate. Characterizations using the generalized failure rate are also developed. Finally, Section 6 provides brief conclusions of the study. 2. Bivariate Pareto family Let X, Y be a non-negative random vector having absolutely continuous survival function F, y = P X >, Y > y. Assume that Z is a non-negative random variable with continuous and strictly decreasing survival function Ḡz and cumulative hazard function Hz defined by Hz = log Ḡz. Then the family of distributions specified by the survival function F, y = [g, y],, y > is a bivariate Pareto family if and only if there eist a function g, y satisfying Hlog g, y = Ha log + Hb log y 2 Notice that g, y is a function of, y in R 2 + following properties a g, y = y b, g, = a, b g, y =, g, =, = {, y, y > 0} satisfying the c since H. is increasing and continuous, g, y is also increasing and continuous in and y and

3 Characterizations of a family of bivariate Pareto distributions 277 d it is assumed that g, y satisfies the inequality 2 g,y g g y 2 g y 0. Further, 2 holds if and only if P Z > log g, y Z > a log = P Z > b log y. 3 The proof of this is given in Sankaran et al Notice that the marginal distributions of X and Y are Pareto I distributions with survival functions F X = a, > and F Y y = y b, y >. The property 3 is referred as etended dullness property. The family includes various known and unknown bivariate distributions. The members of the family can be derived by choosing appropriate univariate distribution of Z. Table provides some members of family according to different choices of the distribution of Z. It may be noted that the properties of F, y can be inferred from the corresponding properties of Ḡz. The members of the family listed in Table, include bivariate Pareto with independent marginals, Mardia s962 Type model and bivariate Burr distribution. For more properties one could refer to Sankaran et al Bivariate dullness property We first discuss the univariate dullness property. Let Z be a non-negative random variable representing the income in a population. Then the distribution of Z is said to have dullness property if P Z > y Z > = P Z > y 4 for all, y. This means that the conditional probability that true income Z is at least y times the reported value is the same as the unconditional probability that Z has at least income y. In otherwords, the distribution of error in an income is independent of the reported value. It is proved that the property 4 holds if and only if the distribution of Z is Pareto I Talwalker 980. There have been many works on models that accounts for underreporting, characterizations and related concepts, like those of Wong 982, Xekalaki 983, Korwar 985, Artikis et al. 994, Pham-Gia and Turkkan 997 and Nadarajah Sometimes it is more convenient to work with an equivalent form of dullness property, as the property 4 is not helpful in practice to verify whether a given data set follows Pareto distribution or not. In this contet, we have the following result. Theorem. The random variable Z satisfies dullness property 4 if and only if m = EZ Z > = µ 5 for all >, where µ = EZ <. Proof. Assume that 4 holds. Then we can write S, y = SSy for all, y > 6

4 278 P. G. Sankaran, N. Unnikrishnan Nair and Preethi John TABLE Some members of bivariate Pareto family Type Survival function of the distribution of Z Survival function of the bivariate Pareto distribution Ḡ z = ep λ z, z > 0 F, y = a y b ; > y > ; a, b > 0 2 Ḡ 2 z = ep[ θe αz ]; z 0; α, θ > 0 F2, y = aα + y bα α ;, y >, α, a > 0 3 Ḡ 4 z = + βz α F4, y = a c log y y b,, y >, a, b > 0; 0 c 4 Ḡ 5 z = 2 + e z σ, z > 0, σ > 0 F5, y = [ 2 α + y β + α y β ] σ ; α = a σ > 0, σ > 0, β = b σ > 0 5 Ḡ 6 z = + z c k, z > 0; c, k > 0 F6, y = ep[ a log c b log y c ab log log y c ] c 6 Ḡ 7 z = 2e z σ, z > 0; σ > 0 F7, y = + 2 a y b a y b 7 Ḡ 8 z = e λzα α, λ > 0, z > 0 F8, y = ep[ λ {λa log α + λb log y α } α ] 8 Ḡ 9 z = p, z > 0; λ > 0, 0 < p <, q = p F9, y = q + p aλ qy bλ q e λz λ q 9 Ḡ 0 z = + eλz α, α, λ > 0 F0, y = + α α + aλ α + y bλ α λ

5 Characterizations of a family of bivariate Pareto distributions 279 where S = P Z >. Integrating 6 from to, we obtain or Sy dy = S Since left side of 7 is EZ Z >, we get Sy dy St dt = Sµ 7 m = µ which leads to 5. Conversely, when 5 hold, we obtain 7 and hence S = St dt µ 8 Integrating 8, we obtain St dt = c µ 9 where c is the integrating constant. Differentiating 9 with respect to, we get Since S =, we obtain c = µ and thus S = c µ µ µ 0 S = µ µ which means that the distribution of Z is Pareto I. From Talwalker 980 it follows that the Pareto distribution satisfies 4, which completes the proof. Now we propose bivariate versions of 5 and eamine whether they characterize members of the bivariate Pareto family. A natural etension of 5 with respect to the random vector X, Y is the vector m, y, m 2, y where and m, y = EX X >, Y > y m 2, y = EY X >, Y > y. 2 We first observe that the joint survival function F, y of X, Y can be determined from and 2.

6 280 P. G. Sankaran, N. Unnikrishnan Nair and Preethi John The joint survival function F, y is obtained as m t, y m 2,t t F, y = ep m t, t dt t m 2, t t dt 3 = ep y m 2,t t m 2, t t dt m t,y t m 2 t, y t dt 4 The proof follows from Nair and Nair 989 by using the relationship between m, 2, m 2, 2 and bivariate mean residual life functions. Definition 2. The distribution of the random vector X, Y is said to have bivariate dullness property one BDP- if and only if for, y > P X > t X >, Y > y = P X > t Y > y, t > and P Y > ys X >, Y > y = P Y > s X >, s >. Definition 3. The distribution of X, Y satisfies bivariate dullness property two BDP-2 if and only if for, y > P X > t, Y > ys X >, Y > y = P X > t, Y > s. 5 Now we have the following theorems characterizing bivariate Pareto distributions belonging to the family by two versions of the bivariate dullness property. Theorem 4. Let X, Y be a bivariate random vector as described in Section 2, with µ X = EX < and µ Y = EY <. Denote µ X y = EX Y > y and µ Y = EY X >. Then the following statements are equivalent. a F 4, y = a+c log y y b ;, y >, a, b >, 0 < c a b b X, Y satisfies BDP-. c m, y = µ X y and m 2, y = yµ Y Proof. To prove a b, we have P [X > t X >, Y > y] = t a+c log y y b a+c log y y b = t a+c log y = P X > t Y > y The second part is proved similarly. To establish b c, we note that the first statement in b is same as F t, y F, y = F t, y P Y > y = F X t Y > y 6 where F X t Y > y is the conditional survival function of X given Y > y. Integrating 6 with respect to t in,, we obtain EX X >, Y > y = EX Y > y

7 Characterizations of a family of bivariate Pareto distributions 28 or m, y = µ X y. The epression for m 2, y is proved similarly. Now we establish c a. From c, we have and m t, = t t tµ Xy = µ X, m 2, t = µ Y t m t, y t = tµ X y t m 2, t t = tµ Y t. Substituting the above epressions in 3 and 4 and equating the resulting epressions, we get, after some simplifications, µ X µ X log + µ Y µ Y log y = µ Y µ Y log y + µ Xy µ X y log which leads to the functional equation µx µ X µ Xy µ X y To solve 7, we rewrite it as log = µy µ Y µ Y log y 7 µ Y log µ Y µ Y µ Y µ Y = log y µ X µ X µ X y µ X y 8 The rightleft side of 8 is a function of y alone and therefore the equality of the two sides hold good for all, y > if and only if each side must be a constant say c. Hence µ Y µ Y = µ Y µ Y c log and Using 9 in 3 or 4, we get Taking µ X = a a and µ Y = µ X y µ X y = µ X c log y 9 µ X F 4, y = b b µ X µ X y µ Y µ Y +c log, a, b >, we obtain F 4, y = a b+c log y since c log y = y c log, we have a and the theorem is completely proved. The parameter values a, b > is required for the eistence of the means.

8 282 P. G. Sankaran, N. Unnikrishnan Nair and Preethi John Setting c = 0 in F 4, y and working similarly, we have the following result. Theorem 5. For the random vector X, Y in Theorem 4, the following statements are equivalent a F, y = a y b ;, y >, a, b > b X, Y satisfies BDP-2. c m, y, m 2, y = µ X, yµ Y Remark 6. It may be noticed that BDP is stronger than BDP 2. Remark 7. The properties BDP and BDP 2 can be interpreted in income analysis as follows. Let X and Y be the incomes from two different sources of a unit in a population. Assume that the incomes of X and Y are at least and y respectively. The average under-reporting error is proportional to the amount by which the income eceeds the ta eemption level. The under-reporting error in XY is a linear function of the reported income if and only if the incomes X, Y follow bivariate Pareto law. In the case of BDP, the proportionality is independent of and y, while in BDP 2, it is independent of in the case of X and independent of y in the case of Y. Remark 8. Hanagal 996 has considered another version of the dullness property defined as P X > t, Y > yt = P X >, Y > yp X > t, Y > t which characterized a Marshall-Olkin type bivariate Pareto model. Note that this distribution does not belongs to the family. Theorem 9. Let X, Y be a non-negative echangeable random vector with absolutely continuous survival function and µ = EX = EY <. Then m, y, m 2, y = µ + µ py, µy + µ p 20 for some non-negative function p. with p = 0 iff the survival function of X, Y is µ + y cy µ F 0, y = ;, y > 2 c where c is a real constant different from unity. Proof. Assume that X, Y has the distribution 2. Then m, y = + F t, ydt F, y = + + y cy cy = µ + y µ cy µ

9 Characterizations of a family of bivariate Pareto distributions 283 which is of the form 20 with py = y cy and p = 0. The proof for m 2, y is similar. Conversely, if the relation 20 holds, from 4, F, y = ep = y + p + p From 4 and 20, we obtain µ y µ t dt F, y = µ Equating 22 and 23 and simplifying, Since 24 holds for all, y >, one should have µ µ t + µ p dt µ. 22 µ y + py µ py p + p = ypy + py y. 24 p + p = c, a constant independent of and y. Solving the above, we get p = c. Substituting p in 22, we have 2. This completes the proof. Remark 0. The distribution specified by 2 is a bivariate distribution with Pareto I marginals. It contains some members of our family. When c = 0 and m, y = µ + µ y m 2, y = µy + µ characterized the bivariate Pareto distribution F 3, y = + y a ;, y >, the well known Mardia s962 type I bivariate Pareto model. Similarly when c =, we have m, y, m 2, y = µ + µ y, µy + µ y + +

10 284 P. G. Sankaran, N. Unnikrishnan Nair and Preethi John characterized the bivariate Pareto distribution F, y = [ 2 + y + y ] a,, y >, a > which is a special case of F5, y in Table when α = β = so that σ = a. Finally c = q, q > 0 gives and that characterizes m, y = µ + m 2, y = µy + µ qy q y µ q q F 2, y = q + p qy q a a special case of F 9, y obtained by taking λa = λb =. Remark. It is easy to see that all the bivariate distributions discussed in Remark 0, including our models F 3, y, F 5, y and F 9, y do not satisfy the dullness properties BDP and BDP 2. The etent to which they depart from BDP is accounted for by the terms µp and µpy. Remark 2. A closely related function to m, y, m 2, y used etensively in reliability analysis is the bivariate mean residual life function see Nair and Nair, 988 defined as EX X >, Y > y, EY y X >, Y > y = m, y, m 2, y y 25 It follows that Theorems 4 through 9 provide useful characterizations of the concerned distributions by the form of the bivariate mean residual life function that can be easily deduced from the relationship Bivariate income gap ratio In the contet of applications in economics, in the univariate case, two functions that are closely related to m are the income gap ratio and the left proportional residual income. For a continuous non-negative random variable Z which represents the income of a population, those with income eceeding are deemed to be affluent or rich. We call Z = to be the affluence line. Then Ḡz = P Z > z represents the proportion of rich in the population. The proportion of rich, their average income and the measures of income inequality are important indices discussed in connection with income analysis and also for comparison between the rich and poor. Of these, Sen 988 defines the income gap ratio among the affluent as i = 26 EX X >

11 Characterizations of a family of bivariate Pareto distributions 285 Distribution F, y F 2, y F 3, y F, y F 2, y TABLE 2 Bivariate income gap ratios i, y, i 2, y µ X µ X, µ Y µ Y µx y µ X, µ Y y µ Y µ +y 2, µ +y 2 µ+µ y µ+µ y µ +y+y, µ +y+y µy++µ y µy++µ y µ q y+qy, µ yq +q µq y+qµ y µq y+qµ y The measure i is used in defining indices of affluence in Sen 988. On the other hand, Belzunce et al. 998 defined the mean left proportional residual incomemlp RI as X l = E X > = i. 27 We propose bivariate generalization of these concepts. For a non-negative random vector X, Y, the bivariate income gap ratio is defined by the vector i, y, i 2, y = EX X >, Y > y, y EY X >, Y > y = m, y, y. 28 m 2, y Equation 28 shows that there is one-to-one relationship between i, y, i 2, y and m, y, m 2, y, so that each determine other and the corresponding distribution uniquely. The functional forms of i, y, i 2, y characterizing some members of our family are given in Table 2. The bivariate generalization of MLP RI is proposed as the vector l, y, l 2, y = E = m, y X X >, Y > y, E Y X >, Y > y y, m 2, y 29 y The calculation of l, y, l 2, y is easily facilitated from those of m, y, m 2, y. Thus the characterizations established in Section 3 using m, y, m 2, y can be translated in terms of l, y, l 2, y. 5. Bivariate generalized failure rate For a non-negative random variable Z, the generalized failure rate is given by r = dlogḡ d 30

12 286 P. G. Sankaran, N. Unnikrishnan Nair and Preethi John Lariviere and Porteus 200 and Lariviere 2006 discussed the properties of r and its applications in operations management. The well known income model derived by Singh and Maddala 976 is based on a relationship between r and Ḡ as r = α β Ḡγ, α, β, γ > 0. For a non-negative random vector X, Y, the bivariate generalized failure rate vector is defined by r, y, r 2, y = log F, y, y log F, y. 3 y There eists an identity connecting r, y, r 2, y and m, y, m 2, y. Differentiating m, y = + F t, y dt F, y with respect to and rearranging terms, This gives and similarly F, y m, y r, y = r 2, y = = m, y + F, y. m,y m, y 32 y m 2,y y m 2, y y. 33 A redeeming feature of r, y, r 2, y is that it allows simple analytically tractable epression for various distributions in the bivariate Pareto family, while the other functions can be epressed in terms of special functions only for many members. See Table 3 for epressions of r, y, r 2, y. It may be noted that the characterizations developed in Section 3 can be transformed in terms of r, y, r 2, y. From Theorems 4 and 5 and the above deliberations, the following result is apparent. Theorem 3. The following statements are equivalent: a. X, Y possesses BDP BDP 2 b. i, y, i 2, y is constantlocally constant c. l, y, l 2, y is constantlocally constant d. r, y, r 2, y is constantlocally constant e. X, Y is distributed as F, y F 4, y Local constancy of a vector means that the first component is independent of and the second component is independent of y.

13 Distribution F, y F 2, y F 3, y F 4, y TABLE 3 Bivariate generalized failure rates r, y, r 2, y a = µ X, b = µ Y µ X µ Y a aα, by bα aα +y bα aα +y bα a, by +y +y µx y, µ Y µ X y µ Y σα α +y β, σαy β + α α +y β + α y β α +y β + α y β F 5, y F 6, y ca c log c b log y c, cb c log y c a log c F 7, y + 2 a y b a y b a a 2y b, by b 2 a F 8, y λ α {λ α a α log α + λ α b α log y α } α a α log α, b α log y α [ F 9, y pq + λa qy λb q ] ap λa y λb q, y λb λa q F, y a + y + y + y, y + F 2, y ap [pq + qy q] y q, y q Characterizations of a family of bivariate Pareto distributions 287

14 288 P. G. Sankaran, N. Unnikrishnan Nair and Preethi John 6. Conclusion In this paper, we have developed characterizations of a family of bivariate Pareto distributions. The well known dullness property was etended to the bivariate set up and characterizations of bivariate Pareto distributions using this property were derived. The measures of income inequality such as income gap ratio and mean left proportional residual income were proposed and studied in the bivariate case. The generalized failure rate was etended to the bivariate set up and characterizations using this concept were derived. The properties of the family of bivariate Pareto distribtions using copula theory are yet to be studied. The work in this direction will be reported elsewhere. Acknowledgements We thank the referee and the editor for the constructive comments. The third author thanks Department of Science and Technology, Government of India for providing financial support. References B. C. Arnold 985. Pareto Distribution. Wiley, New York. B. C. Arnold 990. A fleible family of Multivariate Pareto distributions. Journal of Statistical Planning and Inference, 24, pp T. Artikis, A. Voudouri, M. Malliaris 994. Certain selecting and underreporting processes. Mathematical and Computer Modelling, 20, pp N. Balakrishnan, C. D. Lai Continuous Bivariate Distributions. Springer, New York. F. Belzunce, J. Candel, J. M. Ruiz 998. Ordering and asymptotic properties of residual income distributions. Sankhyā, Series B, pp D. D. Hanagal 996. A multivariate Pareto distribution. Communications in Statistics-Theory and Methods, 25, pp T. P. Hutchinson, C. D. Lai 990. Continuous Bivariate Distributions, Emphasising Applications. Rumsby Scientific Publishing, Adelaide. R. Korwar 985. On the joint characterization of distributions of true income and the proportion of reported income to the true income through a model of underreported incomes. Statistics and Probability Letters, 3, pp H. Langseth Bayesian Networks with Applications in Reliability analysis. Ph.D. thesis, Norwegian University of Science and Technology, Guntur, India. M. A. Lariviere A note on probability distributions with increasing generalized failure rates. Operations Research, 54, pp

15 Characterizations of a family of bivariate Pareto distributions 289 M. A. Lariviere, E. L. Porteus 200. Selling to the newsvendor: An analysis of price-only contracts. Manufacturing & Service Operations Management, 3, pp D. V. Lindley, N. D. Singpurwalla 986. Multivariate distributions for the life lengths of components of a system sharing a common environment. Journal of Applied Probability, pp K. V. Mardia 962. Multivariate Pareto distributions. The Annals of Mathematical Statistics, 33, pp S. Nadarajah Models for over reported income. Applied Economics Letters, 6, pp K. R. M. Nair, N. U. Nair 989. Bivariate mean residual life. IEEE Transactions on Reliability, 38, pp T. Pham-Gia, N. Turkkan 997. Change in income distribution in the presence of reporting errors. Mathematical and Computer Modelling, 25, pp P. G. Sankaran, D. Kundu 204. A bivariate Pareto model, Statistics. p. DOI:0.080/ P. G. Sankaran, N. U. Nair 993. A bivariate Pareto model and its applications to reliability. Naval Research Logistics, 40, pp P. G. Sankaran, N. U. Nair, J. Preethi 204. A family of bivariate pareto distributions. Statistica, 74, pp P. K. Sen 988. The harmonic Gini coefficient and affluence indees. Mathematical Social Sciences, 6, pp S. K. Singh, G. S. Maddala 976. A function for size distribution of incomes. Econometrica, 44, pp S. Talwalker 980. On the property of dullness of Pareto distribution. Metrika, 27, pp W. Wong 982. On the property of dullness of Pareto distribution. Tech. rep., Purdue University. E. Xekalaki 983. A property of the Yule distribution and its applications. Communications in Statistics-Theory and Methods, 2, pp Summary In the present paper, we study properties of a family of bivariate Pareto distributions. The well known dullness property of the univariate Pareto model is etended to the bivariate setup. Two measures of income inequality viz. income gap ratio and mean left proportional residual income are defined in the bivariate case. We also introduce

16 290 P. G. Sankaran, N. Unnikrishnan Nair and Preethi John bivariate generalized failure rate useful in reliability analysis. Characterizations, using the above concepts, for various members of the family of bivariate Pareto distributions are derived. Keywords: Bivariate Pareto distributions; characterization; dullness property; income gap ratio; generalized failure rate.