Multivariate Pareto Distributions Author(s): K. V. Mardia Source: The Annals of Mathematical Statistics, Vol. 33, No. 3 (Sep., 1962), pp. 1008-1015 Published by: Institute of Mathematical Statistics Stable URL: http://www.jstor.org/stable/2237876 Accessed: 20-03-2017 13:37 UTC REFERENCES Linked references are available on JSTOR for this article: http://www.jstor.org/stable/2237876?seq=1&cid=pdf-reference#references_tab_contents You may need to log in to JSTOR to access the linked references. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact
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MULTIVARIATE PARETO DISTRIBUTIONS
BY K. V. MARDIA University of Rajasthan, Jaipur
1. Introduction and summary. It is well known that the family of Pareto distributions with densities
f(x; a, p) = pa'/x', x> a>O, = O, x _ a,p > O, provides reasonably good fits to many empirical distributions, e.g., to distributions of income and of property values. In most of these cases, ancillary informa-
tion is present, which could be utilized if an appropriate multivariate Pareto distribution were available.
The objects of this note are (i) to suggest two families of bivariate Pareto distributions with the property that both marginal distributions are of univariate
Pareto form; (ii) to extend these to multivariate forms; and (iii) to discuss estimation of the parameters in the bivariate distributions.
2. The bivariate Pareto distribution of type 1. A simple bivariate density function satisfying the marginal property is
fi(x, y; a, b, p) = [p (p + 1) (ab) '+1]/(bx + ay -ab) (2.1)
x
>
a =
> 0,
O, x
_
y
>
a,y
_
b
>
b,p
0, >
0.
We shall call it a bivariate Pareto distribution of type 1.
The density for this distribution is constant on every line a7'x + b-'y = c. The marginal density functions of x and y are f(x; a, p) and f(y; b, p) respectively. Also, the conditional distribution of x given y is
(2.2) fi(x I y) = [b(p+ 1)(ay)P+I]/(bx + ay-ab) +2, x>a >O,y >b>O, = 0 otherwise, which is again of Pareto form but with displaced origin. Further, we have
(2.3) E(x) = ap/(p - 1), p> 1; V(x) = a2p/{ (p -1) 2 (p - 2) }, p > 2, (2.4) Cor(x, y) = 1/p, p > 2, (2.5) E(x y) = a + { (ay) / (bp)},
(2.6) V(x i Y) = a2y2(p + 1) /{ b2(p _-1) p2J.
Similarly E(y), V(y), fi(y I x) and V(y I x) can be immediately Received December 14, 1959; revised March 10, 1962. 1008
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MULTIVARIATE PARETO DISTRIBUTIONS 1009
3. A bivariate distribution of type 2. The density function of u = p log(x/a), where x has the density function f(x; a, p) is e-u, u > O,
(3.1)
0, u? 0,v > 0, 0,
u
4qpl.
Incidentally we can obtain the probability density function of R. On applica-
tion of the inverse theorem to the characteristic function (10.5), we find that the density function of z is
1 c'd+ d)I*-Kh< [2 z(c + d)l, f(z) 1 e [zl(c + d - z(c + d)( Z (10.6)1 1 c d e-z(c-d)[r/c+dY~K~ )
c(d [ Z(c dIn2z(c +) z U2, p) then the same problem, and
its solution for large n, can be written for a random sample from Pareto type 2,
on replacing xi, yi, x, y7, mln, M2, 0-i, a02, and p by the respective quantities log xi, log y , log G,, log Gy , log v, log 6, l/p, l/q and a. THEOREM 11.2. y is a more efficient estimate than log Gy , at least for large values of n.
PROOF. Application of Lemma 11.1 to the variance of the corresponding regression estimate for (x, y) of bivariate normal population (Sukhatme [10], pp. 203) gives for large values of n
V(y) (1 - a2) /(nq2) < 1/(nq2) = V(logG,). 12. Regression estimate (double sampling). Let (xl, ** , x') be a random
sample on variate x from Pareto type 2, (xi, * , xn) be a sub-sample of it (m < n), and (yi, *** , yn) are corresponding observed values of variate y.
Suppose G, is the geometric mean of (xl, * * , xI). We define
= log Gy + 0(logG. - log G-). THEOREM 12.1. pi is an unbiased estimate and, at least for large n, more efficient estimate of log 6 than log Gy,. PROOF. Unbiasedness may be established as in Theorem 11.1. Further, application of Lemma 11.1 in the solution to a similar problem for (x, y) of the bivariate normal population (Cochran [9], pp. 278) gives for large n,
V( ) _ l{a2/(mq2)} + {(1 - a 2) /(nq2)}; n< m,
