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Orden de lorenz en la familia de distribuciones gamma triparametricas.

LORENZ ORDERING OF THREE PARAMETER GAMMA DISTRIBUTIONS

1. INTRODUCCION

El orden de Lorenz permite la comparacion en desigualdad de dos distribuciones de renta. Sea [F.sub.X](x) la funcion de distribucion de una variable aleatoria X no negativa y con media finita [[my].sub.X]. La curva de Lorenz, tambien llamada curva de concentracion, correspondiente a X se define (Gastwirth, 1971) como:

[L.sub.x] (p) = 1/[[my].sub.x] [[integral].sup.p.sub.0] [F.sup.-1.sub.x](t)dt

0 [menor que o igual a] p [menor que o igual a] 1 (1)

Donde por [F.sup.-1.sub.x] denotamos la inversa de [F.sub.X] definida por

[F.sup.-1.sub.x](p) = inf{x [F.sub.x](x) [mayor que o igual a] p}, p [elemento de] [0,1].

Si X representa ingresos anuales, entonces [L.sub.X] (p) es la proporcion del total de ingresos que corresponde a los individuos que se encuentran en el 100 p% de ingresos mas bajos. Un detallado estudio de la curva de Lorenz podemos encontrarlo en Gail y Gastwirth (1978), asi como una relacion precisa de sus propiedades en Dagum (1985). La curva de Lorenz nos permite definir el siguiente orden parcial ([[menor que o igual a].sub.L]) sobre la clase de variables aleatorias no negativas:

X [[menor que o igual a].sub.L] Y [??] [L.sub.X] (p) [mayor que o igual a] [L.sub.Y] (p) paratodo 0[menor que o igual a] p [menor que o igual a] 1. (2)

Si X [[mayor que o igual a].sub.L] Y, entonces diremos que X es menos desigual que Y en el sentido de Lorenz. Algunas referencias clasicas sobre el orden de Lorenz son Atkinson (1970), Dasgupta, Sen y Starrett (1973), Rothschild y Stiglitz (1973), Kakwani (1984) y Arnold (1987). De (1)y (2) se deduce facilmente que el orden de Lorenz es invariante frente a transformaciones de escala; es decir, X [[mayor que o igual a].sub.L] Y si y solo si aX [[mayor que o igual a].sub.L] [b.sub.Y] para todo a > 0, b > 0. En Arnold [1] puede verse que

X + a [[menor que o igual a].sub.L] X, para todo a > 0, (3)

para toda variable aleatoria Xno negativa con media finita.

El orden de Lorenz nos permite establecer una comparacion en desigualdad de forma absoluta, es decir, al margen de cualquier medida concreta de desigualdad que pudiera considerarse, las cuales, necesariamente, deben ser coherentes con dicho orden.

Desafortunadamente, no siempre dos variables aleatorias X e Y con distribuciones conocidas son susceptibles de ser comparadas en desigualdad en el sentido del orden de Lorenz. En este trabajo proponemos una condicion suficiente para el orden de Lorenz cuyo cumplimiento es facilmente contrastable. Esta condicion suficiente sera la que utilizaremos en la Seccion 3 para la ordenacion en desigualdad de la familia Gamma triparametrica que puede ser usada como modelo probabilistico para la distribucion de la renta.

Consideraremos los siguientes resultados previos (Shaked, 1982):

Definicion 1 Sea h(x) una funcion real definida en I [subconjunto] R. El numero de cambios de signo de h en I se define como sigue:

S(h) = sup S[h(x), h([x.sub.2]),. ..., h([x.sub.m])] (4)

donde S[h([x.sub.1]), h([x.sub.2]),. ..., h([x.sub.m])] es el numero de cambios de signo una vez eliminados los terminos iguales a cero, y el supremo en (4) se extiende a todos los conjuntos [x.sub.1] < [x.sub.2] < ... < [x.sub.m] ([x.sub.i] [elemento de] I), m < [infinito].

Necesitaremos el siguiente resultado.

Teorema 2 Sean X e Y variables aleatorias continuas con igual media

[[my].sub.X] = [[my].sub.Y] y sean F y G sus correspondientes funciones de densidad. Si S(F - G) = 1 yla secuencia de signos es -, +, entonces

[[integral].sup.u.sub.0] [F.sup.-1] (t) dt [mayor que o igual a] [[integral].sup.u.sub.0] [G.sup.-1] (t) dt, para todo 0 [menor que o igual a] u [menor que o igual a] 1. (5)

Demostracion. De acuerdo con lo asumido en el enunciado, tendremos que

S ([F.sub.-1] - [G.sub.-1]) = 1

siendo la secuencia de signos +,-.

Por lo tanto, la integral

[[integral].sup.u.sub.0] [[F.sup.-1] (t) - [G.sup.-1] (t)]dt

alcanza su menor valor cuando u = 1.

Por otra parte, de la igual de medias se sigue que

[[integral].sup.u.sub.0] [[F.sup.-1] (t) - [G.sup.-1] (t)]dt [mayor que o igual a] [[integral].sup.1.sub.0] [[F.sup.-1] (t) - [G.sup.-1] (t)]dt = 0, y, en consecuencia, se verifica (5).

2. CONDICIONES SUFICIENTES PARA EL ORDEN DE LORENZ

Utilizaremos el siguiente resultado de

Arnold (1987).

Teorema 3 Sean X and Y variables aleatorias no negativas con medias finitas [[my].sub.X] y [[my].sub.Y], respectivamente, y sean F y G sus correspondientes funciones de distribucion. Si S(F(x [[my].sub.X])- G(x [[my].sub.Y])) = 1 y la secuencia de signos es -,+, entonces X [[menor que o igual a].sub.L] Y.

Demostracion. Sean [EXPRESION MATEMATICA IRREPRODUCIBLE EN ASCII] funciones de distribucion de X/[[my].sub.X] y Y/[[my].sub.Y] respectivamente. Puesto que

[EXPRESION MATEMATICA IRREPRODUCIBLE EN ASCII]

y la secuencia de signos es -,+, del Teorema 2 se sigue que X/[[my].sub.X] y Y/[[my].sub.Y] ya que el orden de Lorenz es invariante frente a transformaciones de escala, tendremos que X [[menor que o igual a].sub.L] Y.

Basandonos en el Teorema 3, obtenemos en el siguiente corolario una condicion suficiente para la comparacion en desigualdad en el sentido de Lorenz de dos variables aleatorias absolutamente continuas.

Corolario 4 Sean X e Y variables aleatorias absolutamente continuas y no negativas, con medias finitas [[my].sub.X] y Y/[[my].sub.Y], y soportes supp(X) y supp(Y), respectivamente, y sean f y g sus correspondientes funciones de densidad. Asumiremos que X/[[my].sub.X] [??] supp Y/[[my].sub.Y].

En estas condiciones, si f([[my].sub.X]x)/g ([[my].sub.Y]x) es unimodal para valores de x restringidos al supp (Y/[[my].sub.Y]), donde la moda es un supremo, entonces X [[menor que o igual a].sub.L] Y.

Demostracion. Puesto que

[EXPRESION MATEMATICA IRREPRODUCIBLE EN ASCII] (6)

y f ([[my].sub.X]x)/g ([[my].sub.Y]x) es unimodal en supp(Y/[[my].sub.Y]), entonces tambien lo sera [EXPRESION MATEMATICA IRREPRODUCIBLE EN ASCII], siendo la moda un supremo. Por lo tanto,

[EXPRESION MATEMATICA IRREPRODUCIBLE EN ASCII]

Como el orden estocastico no puede darse ya que X/[[my].sub.X] y Y/[[my].sub.Y] tienen la misma media, tendremos que [EXPRESION MATEMATICA IRREPRODUCIBLE EN ASCII] siendo la secuencia de signos -,+,- y, en consecuencia, [EXPRESION MATEMATICA IRREPRODUCIBLE EN ASCII] siendo la secuencia de signos -,+. Finalmente, del Teorema 3 se sigue que X [[menor que o igual a].sub.L] Y.

Observacion 5 Una condicion suficiente para que f/g sea unimodal es que f/g sea log-concava (Keilson y Gerber, 1971).

Corolario 6 Sean X e Y variables aleatorias absolutamente continuas con medias finitas y soportes respectivos supp(X) = (a, [infinito]) y supp(Y) = (b, [infinito]), a > 0, b [mayor que o igual a] 0, y sean f y g sus correspondientes funciones de densidad. Si f ([[my].sub.X]x)/g ([[my].sub.Y]x) decrece en X para

x [mayor que o igual a] a/[[my].sub.X], entonces X [[menor que o igual a].sub.L] Y.

Demostracion. En primer lugar probaremos que a/[[my].sub.X] > b/[[my].sub.Y] Consideremos, por reduccion al absurdo, que supp(Y/[[my].sub.Y]) [??] supp(X/[[my].sub.X]), es decir que b/[[my].sub.Y] [mayor que o igual a] a/[[my].sub.X] Entonces, teniendo en cuenta (6), definiremos

[EXPRESION MATEMATICA IRREPRODUCIBLE EN ASCII]

Puesto que f([[my].sub.X]x)/g([[my].sub.Y]x) es decreciente para x [mayor que o igual a] a/[[my].sub.X], tendremos que g([[my].sub.Y]x)/f([[my].sub.X]x) es creciente para x > b/[[my].sub.Y] En consecuencia,

[EXPRESION MATEMATICA IRREPRODUCIBLE EN ASCII]

lo que no es posible ya que Y/[[my].sub.Y] y X/[[my].sub.X] tienen la misma media. Por tanto, se cumple que supp(X/[[my].sub.X]) [subconjunto] supp (Y/[[my].sub.Y]), es decir, a/[[my].sub.X] > b/[[my].sub.Y]. Finalmente, puesto que f ([[my].sub.X]x)/g([[my].sub.Y]x) es decreciente para x [mayor que o igual a] a/[[my].sub.X], tenemos que f([[my].sub.X]x)/g([[my].sub.Y]x) es unimodal para x restringido a supp (Y/[[my].sub.Y]). Finalmente, aplicando el Corolario 4, el resultado queda demostrado.

3. APLICACION

En esta seccion aplicaremos nuestro anterior resultado (Corolario 6) para la ordenacion de la familia de distribuciones Gamma triparametricas.

3.1. Orden de Lorenz en la familia de distribuciones Gamma triparametricas

Sea X la distribucion Gamma triparametrica cuya funcion de densidad es

[f.sub.x](x) = [[beta].sup.-[alfa]] [(x - [theta]).sup.[alfa]-1][e.sup.-(x-[theta]/[beta])]/[GAMMA]([alfa]), [theta] [mayor que o igual a] 0, [alfa] > 0, [beta] > 0, x > [theta]. (7)

Puesto que el orden de Lorenz es invariante frente a transformaciones de escala, podemos considerar, sin perdida alguna de generalidad, que el parametro de escala [beta] es igual a 1 por lo que

[f.sub.x](x) = [(x - [theta]).sup.[alfa]-1] [e.sup.-(x-[theta])]/[GAMMA]([alfa]), [alfa] > 0, [theta] [mayor que o igual a] [theta]. (8)

Llamaremos G([theta], [alfa]). a la correspondiente funcion de distribucion. Arnold, Brockett, Robertson y Shu (1987) consideran [X.sub.1] ~ G([theta], [[alfa].sub.1]) y [X.sub.2] ~ G([theta], [[alfa].sub.2]) ([theta] fijo) y demuestran que [X.sub.2] [[menor que o igual a].sub.L] [X.sub.1] para [[alfa].sub.1] [menor que o igual a] [[alfa].sub.2]. Por otra parte, si consideramos [X.sub.1] ~ G([[theta].sub.1], [alfa]) y [X.sub.2] ~G([[theta].sub.2], [alfa]) [alfa] (fijo) puede verse que [X.sub.2] [[menor que o igual a].sub.L] [X.sub.1] para [[theta].sub.1] [menor que o igual a] [[theta].sub.2]. En efecto, puesto que [X.sub.2] = [X.sub.1] +([[theta].sub.2] - [[theta].sub.1]), con [[theta].sub.2] - [[theta].sub.2] > 0, se sigue de (3) que [X.sub.2] [[menor que o igual a].sub.L] [X.sub.1]. Estos resultados pueden ser extendidos para [X.sub.1] ~ G([[theta].sub.1], [[alfa].sub.1]) y [X.sub.2] ~G([[theta].sub.2], [[alfa].sub.2]). Aplicando la transitividad del orden de Lorenz y los anteriores resultados, es facil probar que si [[alfa].sub.1] [menor que o igual a] [[alfa].sub.2] y [[theta].sub.1] [menor que o igual a] [[theta].sub.2], entonces [X.sub.2] [[menor que o igual a].sub.L] [X.sub.1]. Sin embargo, no sabemos que ocurre cuando [[alfa].sub.1] < [[alfa].sub.2] y [[theta].sub.1] > [[theta].sub.2]. Esta cuestion nos conduce al siguiente resultado.

Teorema 7 Si [X.sub.1] ~ G([[theta].sub.1], [[alfa].sub.1])([[alfa].sub.1] [menor que o igual a] 1) y [X.sub.2] ~G([[theta].sub.2], [[alfa].sub.2]), con [[theta].sub.1] - [[theta].sub.2] > [[alfa].sub.2] - [[alfa].sub.1] > 0, entonces [X.sub.1] [[menor que o igual a].sub.L] [X.sub.2].

Demostracion. En primer lugar observemos que la relacion

[[theta].sub.1] - [[theta].sub.2] > [[alfa].sub.2] - [[alfa].sub.1] > 0 (9)

es posible unicamente si [[alfa].sub.1] < [[alfa].sub.2] y [[theta].sub.1] > [[theta].sub.2]. Teniendo en cuenta que la funcion de densidad de X~ G([theta], [alfa]) es (8) y que [micron] = E[X]= [alfa] + [theta], tendremos que

[EXPRESION MATEMATICA IRREPRODUCIBLE EN ASCII]

Donde, k = [GAMMA]([[alfa].sub.2])/[GAMMA]([[alfa].sub.1]) * exp [[theta].sub.1] - [[theta].sub.2] > 0.

De (9) se sigue claramente que exp[([[alfa].sub.2] - [[alfa].sub.1] + [[theta].sub.2] - [[theta].sub.1])x] es decreciente en x. Supongamos ahora que x [mayor que o igual a] [[theta].sub.1]/[[my].sub.1] = 91/[a1 +91 ] Puesto que [[alfa].sub.1] < [[alfa].sub.2], se deduce que

[EXPRESION MATEMATICA IRREPRODUCIBLE EN ASCII]

es tambien decreciente en x para x [mayor que o igual a] [[theta].sub.2]/[[[alfa].sub.2] + [[theta].sub.1]]. De (9) se sigue que

[[theta].sub.1]/[[alfa].sub.1] + [[theta].sub.1] > [[theta].sub.2]/[[alfa].sub.2] + [[theta].sub.2].

Por lo tanto, (10) es decreciente en x para x [mayor que o igual a] [[theta].sub.1]/[[my].sub.1]. Finalmente, llamando

[EXPRESION MATEMATICA IRREPRODUCIBLE EN ASCII],

se demuestra a partir de (9) que h'(x) [menor que o igual a] 0 si y solo si [[alfa].sub.2] [[theta].sub.1] > [[alfa].sub.1] + [[theta].sub.2]. Por consiguiente, la razon [f.sub.1]([[my].sub.1]x)/[f.sub.2 ([[my].sub.2]x) es decreciente para x [mayor que o igual a] [[theta].sub.1]/[[my].sub.1]. Por tanto, aplicando el Corolario 6, el teorema queda demostrado.

1. INTRODUCTION

Lorenz ordering is used to compare the amounts of inequality in two income distributions. Let [F.sub.X] (x) be the distribution function of a non-negative random variable X with finite mean [[mu].sub.x], then the Lorenz curve, also called curve of concentration, corresponding to X can be defined (Gastwirth, 1971) as:

[L.sub.X](p) = 1/[[mu].sub.X] [[integral].sup.p.sub.X] (t)dt 0 [less than or equal to] p [less than or equal to] 1 (1)

where we denote by [F.sup.-1.sub.X] the inverse of [F.sub.X] defined by

[[F.sub.-1.sub.X] (p) = inf{x:[F.sub.x] (x) [greater than or equal to] p}, p [[member of][0,1].

If X represents annual income, [L.sub.X] (p) is the proportion of total income that accrues to individuals having the 100p% lowest incomes. There is extensive discussion of the Lorenz curve in Gail and Gastwirth (1978) and a concise account of its properties in Dagum (1985). The Lorenz curve can be used to define a partial ordering (denoted [[less than or equal to].sub.L]) on the class of non-negative random variables, as follows:

X [[less than or equal to].sub.L] Y [??] [L.sub.X] (p) [greater than or equal to] [L.sub.Y] (p) for every 0 [less than or equal to] p [less than or equal to] 1. (2)

If X [[less than or equal to].sub.L] Y, then X is said to exhibit less inequality in the Lorenz or relative Lorenz sense than Y. Some standard references for the Lorenz order are Atkinson (1970), Dasgupta, Sen and Starrett (1973), Rothschild and Stiglitz (1973), Kakwani (1984) and Arnold (1987). It is obvious from (1) and (2) that the Lorenz order is scale-invariant, that is, X [[less than or equal to].sub.L] Y if and only if aX [[less than or equal to].sub.L] bY for all a > 0, b > 0. Arnold [1] shows that

X + a [[less than or equal to].sub.L] X, for all a > 0, (3)

for every non-negative random variable X with finite mean.

Lorenz ordering may be viewed as the maximal ranking generated by relative inequality measures, that is, if X [[less than or equal to].sub.L] Y then, whatever the measures of relative inequality one may choose, Y must not be judged as less unequal that X and conversely.

Unfortunately, for the two random variables X and Y with known distributions, it is sometimes not clear how to verify the relation X [[less than or equal to].sub.L] Y. In this paper, we give a simple condition that ensure these orderings. This condition will be used in Section 3 for the ordering of the three parameter Gamma income distribution model. As in Shaked (1982), the following notation is used:

Definition 1 Let h(x) be a real function defined in I [subset] R. The number of sign changes of h in I is defined by

S(h) = sup S[h([x.sub.1]), h([x.sub.2]),. ..., h([x.sub.m])] (4)

where S[h([x.sub.1]), h([x.sub.2]),. ..., h([x.sub.m])] is the number of sign changes of the indicated sequence, zero terms being discarded, and the supremum in (4) is extended over all sets [x.sub.1] < [x.sub.2] < ... < [x.sub.m] ([x.sub.i] [member of] I), m < [infinity].

We require the following well known result.

Theorem 2 Let X and Y be continuous random variables with equal means [[mu].sub.X] = [[mu].sub.Y] and let F and G the corresponding densities. If S(F - G) = 1 and the sign sequence is -,+, then

[[integral].sup.u.sub.0] [F.sup.-1] (t)dt [greater than or equal to] [[integral].sup.u.sub.0] [G.sup.-1] (t)dt, for all 0 [less than or equal to] u [less than or equal to] 1. (5)

Proof. By the assumptions on F and G we have that

S ([F.sub.-1] - [G.sub.-1]) = 1 with the sequence +,-.

Therefore, the integral

[[integral].sup.u.sub.0] [[F.sup.-1] (t) - [[G.sup.-1] (t)]dt

assumes its smallest value for u = 1. From the equality of the means it follows that

[[integral].sup.u.sub.0] [[F.sup.-1] (t) - [[G.sup.-1] (t)]dt [greater than or equal to] [[integral].sup.1.sub.0][[F.sup.-1] (t) - [G.sup.-1] (t)]dt = 0, and consequently (5) holds.

2. SUFFICIENT CONDITIONS FOR LORENZ ORDERING

The next theorem, due to Arnold (1987), will be used below.

Theorem 3 Let X and Y be nonnegative random variables with finite means [[mu].sub.X] and [[mu].sub.Y], respectively, and let F and G be the corresponding distribution functions. If S(F(x[[mu].sub.X]) - G(x [[mu].sub.Y])) = 1 and the sign sequence is -,+, then X [[less than or equal to].sub.L] Y.

Proof. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be the distribution functions of X/[[mu].sub.X] and Y/[[mu].sub.Y], respectively. Since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and the sign sequence is -,+, it follows from Theorem 2 that X/[[mu].sub.X] [[less than or equal to].sub.L] Y/[[mu].sub.Y]. Since the Lorenz order is invariant under scale transformations we have that X [[less than or equal to].sub.L] Y.

Based on Theorem 3 the next corollary gives a sufficient condition for the Lorenz comparison of two absolutely continuous random variables.

Corollary 4 Let X and Y be nonnegative and absolutely continuous random variables with finite means and supports supp(X) and supp(Y), respectively, and let f and g be the corresponding densities. Assume that supp (X/[[mu].sub.X]) [??] supp (Y/[[mu].sub.Y]). If f([[mu].sub.X]x)/g([[mu].sub.Y]x) is unimodal for x restricted to supp (Y/[[mu].sub.Y], where the mode is a supremum, then X [[less than or equal to].sub.L] Y. Proof. Since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

and f([[mu].sub.X]x)/g([[mu].sub.Y]x) is unimodal on supp(Y/[[mu].sub.Y]), so is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], with the mode yielding a supremum. Hence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since X/[[mu].sub.X]) and (Y/[[mu].sub.Y] have the same mean, ordinary stochastic order is not possible, so that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and the sign sequence is -,+,-. Then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and the sign sequence is -,+. From theorem 3 it follows that X [[less than or equal to].sub.L] Y.

Remark 5 A sufficient condition for f/g to be unimodal is for f/g to be log-concave (Keilson and Gerber, 1971).

Corollary 6 Let X and Y be absolutely continuous random variables with finite means and supports supp (X)= (a, [infinity]) and supp(Y)= (b, [infinity]) a > 0, b [greater than or equal to] 0, and let f and g be the corresponding densities. If f([[mu].sub.X]x)/g([[mu].sub.Y]x) decreases in x for x [greater than or equal to] a/[[mu].sub.X], then X [[less than or equal to].sub.L] Y.

Proof. First, we prove that a/[[mu].sub.X] > b/[[mu].sub.y].

Assume, by the way of contradiction, that supp(Y/[[mu].sub.Y]) [??] supp(X/[[mu].sub.X]x), i.e. that b/[[mu].sub.Y] [greater than or equal to] a/[[mu].sub.X]. Then, using (6) we define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since f ([[mu].sub.X]x)/g([[mu].sub.Y]x) decreases in x for

x [greater than or equal to] a/[[mu].sub.X], it follows that g([[mu].sub.Y]x)/f ([[mu].sub.X]x) increases in x for x > b/[[mu].sub.Y] Hence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which is not possible because Y/[[mu].sub.Y] and X/[[mu].sub.X] have the same mean. Thus, it follows that supp(X/[[mu].sub.X]) [subset] supp (Y/[[mu].sub.Y]), i.e. a/[[mu].sub.X] > b/[[mu].sub.Y] Again, since f ([[mu].sub.X]x)/g ([[mu].sub.Y]x) decreases in x for x [greater than or equal to] a/[[mu].sub.X], we have that f ([[mu].sub.X]x)/g([[mu].sub.Y]x) is unimodal for x restricted to supp (Y/[[mu].sub.Y]). The result follows by applying Corollary 4.

3. APPLICATION

3.1. Lorenz ordering of three parameter Gamma distributions

Let X be the three-parameter Gamma distribution with density function

Since the Lorenz order is invariant under scale changes, the scale parameter [beta] can be set equal to 1 without loss of generality. This gives

The corresponding distribution will be denoted by G([theta], [alpha]). Arnold, Brockett, Robertson and Shu (1987) consider [X.sub.1] ~ G([theta], [[alpha].sub.1]) and [X.sub.2] ~ G([theta], [[alpha].sub.2]) ([theta] fixed) and prove that [X.sub.2] [[less than or equal to].sub.L] [X.sub.1] for [[alpha].sub.1] < [[alpha].sub.2]. On the other hand, if we consider [X.sub.1] ~ G([[theta].sub.1],[alpha]), and [X.sub.2] ~G([[theta].sub.2],[alpha]) ([alpha] fixed) it can be shown that [X.sub.2] [[less than or equal to].sub.L] [X.sub.1] for [[theta].sub.1] [less than or equal to] [[theta].sub.2]. In fact, since [X.sub.2] = [X.sub.1] +([[theta].sub.2] - [[theta].sub.1]), with [[theta].sub.2] - [[theta].sub.1] > 0, from (3) it follows that [X.sub.2] [[less than or equal to].sub.L] [X.sub.1] These results can be completed considering [X.sub.1] ~G([[theta].sub.1], [[alpha].sub.1]) and [X.sub.2] ~G([[theta].sub.2], [[alpha].sub.2]). Using the transitivity of the Lorenz order and the above results, it is easy to prove that if [[alpha].sub.1] [less than or equal to] [[alpha].sub.2] and [[theta].sub.1] < [less than or equal to] [[theta].sub.2] then [X.sub.2] [[less than or equal to].sub.L] X . However, what happen if [[alpha].sub.1] < [[alpha].sub.2] and [[theta].sub.1] > [[theta].sub.2]? This leads to the following result.

Theorem 7 For [X.sub.1] ~G([[theta].sub.1], [[alpha].sub.1])([[alpha].sub.1] [less than or equal to] 1) and [X.sub.2] ~ G([[theta].sub.2], [[alpha].sub.2]), with [[theta].sub.1] - [[theta].sub.2] > [[alpha].sub.2] - [[alpha].sub.1] > 0, we have [X.sub.1] [[less than or equal to].sub.L] [X.sub.2].

Proof. First note that the relationship

[[theta].sub.1] - [[theta].sub.2] > [[alpha].sub.2] - [[alpha].sub.1] > 0 (9)

is only possible if [[alpha].sub.1] < [[alpha].sub.2] and [[theta].sub.1] > [[theta].sub.2]. Now, taking into account the fact that the density of X~ G([theta], [alpha])is (8) and [micro] = E[X] = [alpha] + [theta], we have that

From (9) we clearly have that exp[([[alpha].sub.2] - [[alpha].sub.1] + [[theta].sub.2] - [[theta].sub.1])x] decreases in x. Now, suppose x [greater than or equal to] [[theta].sub.1]/[[mu].sub.1] = [[theta].sub.1]/[[[alpha].sub.1] + [[theta].sub.1]. Since [[alpha].sub.1] < [[alpha].sub.2], it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is also decreasing in x for x [greater than or equal to] [[theta].sub.2]/[ [[alpha].sub.2] + [[theta].sub.2]] From (9) it follows that

[[theta].sub.1]/[[alpha].sub.1] + [[theta].sub.1] > [[theta].sub.2]/[[alpha].sub.2] + [[theta].sub.2]

Thus, (10) decreases in x for x [greater than or equal to] [[theta].sub.1]/[[mu].sub.1]. Finally, denoting

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it can be proven that h'(x) [less than or equal to] 0 if and only if [[alpha].sub.2][[theta].sub.1] > [[alpha].sub.1] + [[theta].sub.2], which follows from (9). Therefore, the assumption implies that the ratio [f.sub.1] ([[mu].sub.1]x)/[f.sub.2] ([[mu].sub.2]x) is decreasing for x [greater than or equal to] [[theta].sub.1]/[[mu].sub.1]. Thus, applying Corollary 6, the proof is complete.

BIBLIOGRAFIA/REFERENCES

Arnold, B.C. (1987). Majorization and the Lorenz Order: A brief introduction. Berlin: Springer-Verlag.

Arnold, B.C., Brockett, P.L., Robertson, C.A. y Shu, B. (1987). Generating ordered families of Lorenz curves by strongly unimodal distributions. Journal of Business and Economic Statistics, 5, 305-308.

Atkinson, B. (1970). On the measurement of inequality. Journal of Economic Theory, 2, 244-263.

Dagum, C. (1985). Lorenz Curve, encyclopedia of statistical sciencies (5, pp. 156-161). S. Kotz, N.L. Johnson & C.B. Read (Eds.). New York: Wiley.

Dasgupta, P., Sen, A.K. y Starrett, D. (1973). Notes on the measurement of inequality. Journal of Economic Theory, 6, 180-187.

Gail, M.H. y Gastwirth, J.L. (1978). A scale-free goodness-of-fit test for the exponential distribution based on the Lorenz curve. Journal of the American Statistical Association, 73, 786-793.

Gastwirth, J.L. (1971). A general definition of the Lorenz curve. Econometrica, 39, 1037-1039.

Kakwani, N. (1984). Welfare ranking of income distributions. Advances in Econometrics, 3, 191213.

Keilson, J. y Gerber, H. (1971). Some result for discrete unimodality. Journal of the American Statistical Association, 66, 386-389.

Rothschild, M. y Stiglitz, J.E. (1973). Some further results on the measurement of inequality. Journal of Economic Theory, 6, 188-204.

Shaked, M. (1982). Dispersive ordering of distributions. Journal of Applied Probability, 19, 310-320.

Hector M. Ramos (1)

hector.ramos@uca.es

Miguel A. Sordo (1)

mangel.sordo@uca.es

Universidad de Cadiz

(1) Departamento de Estadistica e I.O., Facultad de CC. Economicas y Empresariales, Universidad de Cadiz, Duque de Najera 8, 11002 Cadiz (Espana).
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