New properties of the orthant convex-type stochastic orders

Transcription

1 Noname manuscript No. (will be inserted by the editor) New properties of the orthant convex-type stochastic orders Fernández-Ponce, JM and Rodríguez-Griñolo, MR the date of receipt and acceptance should be inserted later SUPLEMENTARY MATERIAL Fernández-Ponce, JM Dpto. Estadística e Investigación Operativa Universidad de Sevilla 42 Sevilla, Spain ferpon@us.es Rodríguez-Griñolo, MR Dpto. de Economía, Métodos Cuantitativos e Historia Económica Universidad Pablo de Olavide 43 Sevilla, Spain mrrodgri@upo.es

2 2 Fernández-Ponce, JM and Rodríguez-Griñolo, MR A Measures of association The most widely known measures of association, which measure a form of dependence known as concordance, are Kendall s tau and Spearman s rho. If (x i,y i ) and (x j,y j ) denote two observations from the vector (X,Y), it is said that (x i,y i ) and (x j,y j ) are concordant if x i < x j and y i < y j or x i > x j and y i > y j. Similarly, it is said that (x i,y i ) and (x j,y j ) are discordant if x i < x j and y i > y j or x i > x j and y i < y j. Therefore, if (X,Y ) and (X 2,Y 2 ) are two independent and identically distributed random vectors distributed as (X, Y), then Kendall s tau coefficient for the vector (X,Y) is defined as the probability of concordance minus the probability of discordance. That is, τ = P[(X X 2 )(Y Y 2 ) > ] P[(X X 2 )(Y Y 2 ) < ]. Now,considerthree independent andidentically distributedrandomvectors (X,Y ),(X 2,Y 2 ) and (X 3,Y 3 ). Spearman s rho coefficient for the vector (X,Y) is defined as proportional to the probability of concordance minus the probability of discordance for the vectors (X,Y ) and (X 2,Y 3 ). That is, ρ = 3{P[(X X 2 )(Y Y 3 ) > ] P[(X X 2 )(Y Y 3 ) < ]}. Finally, one additional measure based on concordance is recalled. Blomqvist s coefficient for the vector (X,Y) is defined as β = P[(X η x)(y η y) > ] P[(X η x)(y η y) < ] where η x and η y are the medians of X and Y, respectively. Futher details about this concordance measure can be seen in Nelsen (26, page 82). B Example An example where two bivariate random vectors are ordered in the lo-cv sense but not in the PQD order sense is now explained. This example shows that the orthant convex orders cannot be considered as dependence stochastic orders in general. Example Let X = (X,X 2 ) and Y = (Y,Y 2 ) have the following joint probabilities (all probabilities are multiplied by 2) X 2 /X Y 2 /Y 2 3 After straightforward calculations, the distribution functions of X and Y are easily obtained. if x <, x 2 < /2 if x < 2, x 2 < 2 /2 if x < 2, 2 x 2 < 3 6/2 if x < 2, x 2 3 /2 if 2 x F(x,x 2 ) = < 3, x 2 < 2 3/2 if 2 x < 3, 2 x 2 < 3 4/2 if 2 x < 3, x 2 3 7/2 if x 3, x 2 < 2 /2 if x 3, 2 x 2 < 3 2/2 if x 3, x 2 3

3 New properties of the orthant convex-type stochastic orders 3 if x <, x 2 < /2 if x < 2, x 2 < 2 /2 if x < 2, 2 x 2 < 3 6/2 if x < 2, x 2 3 7/2 if 2 x G(x,x 2 ) = < 3, x 2 < 2 2/2 if 2 x < 3, 2 x 2 < 3 4/2 if 2 x < 3, x 2 3 7/2 if x 3, x 2 < 2 /2 if x 3, 2 x 2 < 3 2/2 if x 3, x 2 3 It is not difficult to see that F(x,x 2 )and G(x,x 2 ) are not ordered, given that F(2.,.) = /2 < 7/2 = G(2.,.) and F(2.,2.) = 3/2 > 2/2 = G(2.,2.). Therefore, X PQD Y and Y PQD X. Moreover, X PQD Y and Y PQD X. Consequently, if F = t it can be stated that: F = G = t2 F(x,x 2 )dx dx 2 and G = t t2 G(x,x 2 )dx dx 2, then if t <, t 2 < 2 (t )(t 2 ) if t < 2, t 2 < 2 2 (t )(t 2 ) if t < 2, 2 t 2 < 3 2 (t )( +6(t 2 3)) if t < 2, t (t )(t 2 ) if 2 t < 3, x 2 < (t 2 2)+ 2 (t 2)+ 3 2 (t 2 2)(t 2) if 2 t < 3, 2 t 2 < (t 2)+ 6 2 (t 2 3)+ 4 2 (t 2 3)(t 2) if 2 t < 3, t (t 2 )+ 7 2 (t 2 )(t 3) if t 3, t 2 < (t 3)+ 8 2 (t 2 2)+ 2 (t 2 2)(t 3) if t 3, 2 t 2 < (t 3)+(t 2 3)+(t 2 3)(t 3) if t 3, t 2 3 if t <, t 2 < 2 (t )(t 2 ) if t < 2, t 2 < 2 2 (t )(t 2 ) if t < 2, 2 t 2 < 3 2 (t )( +6(t 2 3)) if t < 2, t (t 2 )( +7(t 2)) if 2 t < 3, t 2 < (t 2)+ 2 (t 2 2)+ 2 2 (t 2 2)(t 2) if 2 t < 3, 2 < t 2 < (t 2)+ 6 2 (t 2 3)+ 4 2 (t 2)(t 2 3) if 2 t < 3, t (t 2 )+ 7 2 (t 3)(t 2 ) if t 3, t 2 < (t 2 2)+ 7 2 (t 3)+ 2 (t 3)(t 2 ) if t 3, 2 t 2 < (t 3)+(t 2 3)+(t 2 3)(t 3) if t 3, t 2 3

4 4 Fernández-Ponce, JM and Rodríguez-Griñolo, MR It can been seen that G (t,t 2 ) F (t,t 2 ), for all (t,t 2 ). That is, Y lo cv X. Moreover, from Corollary of our paper, it follows that ( X) uo cx ( Y). However, + + [ F(x,x 2 ) Ḡ(x,x 2 ) ] dx dx 2 = /2, and + + [ F(x,x 2 ) Ḡ(x,x 2 )]dx dx 2 = 2/2, 2.. thatis,x uo cx Y.Thus,( X) uo cx ( Y)[Y lo cv X]but X PQD Y[Y PQD X]. C Example 2 Example 2 Assume C X and C Y are the corresponding copulas of X and Y, respectively, defined as: C X (u,v) =.4max{u+v,}+.min{u,v} and C Y (u,v) = uv. We denote (u,v) as the function defined by (u,v) = C X (u,v) C Y (u,v). It is easily obtained that u(. v) if (u,v) A, ( v)(u.4) if (u,v) B, (u,v) = ( u)(v.4) if (u,v) C, v(. u) if (u,v) D. () where the zones A,B,C, and D are given in Figure (a). The aim is now to show that different values for (u,v) can be obtained depending on the zone where it is evaluated (see the function given in ()). It is easily observed that C X (u,v) and C Y (u,v) are not ordered in the PQD sense since (.,.2) > and (.,.66) <. To obtain the order lo cv between these copulas, the zones defined in Figure (b) should be taken into account. Note that the function (u,v) is negative in the A,B,C, and D zones (see Figure (b)). Nevertheless, the following symmetry property is verified: t z (u, v)dvdu = t2 z2 (u, v)dvdu when (t,z ) A and (t 2,z 2 ) D. Similarly, if (t,z ) B and (t 2,z 2 ) C, then the symmetry property is also verified. For the sake of simplicity, we denote g(t,z) = t z u (u,v)dvdu for (t,z) A. It can be deduced that: t z t u g(t,z) = u(. v)dvdu + v(. u)dvdu u t u t u 2 = (z u)(. z u)du+ (. u)du. (2) 2 2

5 New properties of the orthant convex-type stochastic orders Note that for a fixed z in A, the function g(t,z) is non-decreasing in t since it is the sum of integrals of non-negative functions. By taking into account that g(,z) =, for all z (.,), and, for a fixed z, that the function g(t,z) is non-decreasing, it is immediately obtained that g(t,z), for all (t,z) in A. Now, denote f(t,z) = t z (u,v)dvdu when (t,z) is in B. It follows that: z z t u f(t,z) = u(. v)dvdu + u(. v)dvdu u z u t z t u + ( v)(u.4)dvdu + v(. u)dvdu. (3) z u By straightforward computation, z z t f(t,z) = u(. v)dvdu + u(..u)du u z t t +.(u.4)(2z z 2 +u 2 )du +.u 2 (. u)du. (4) z It can therefore be obtained that f(,) = and f(t, t) = g(t, t). Consequently, f(t, t) for all t in (,.4). It is necessary to show, for a fixed value z in (.,), that the function f(t,z) is non-decreasing for all t in ( z,.4). By using the expression of f(t,z) given in (4), it is sufficient to show that the integrand function is non-negative, f(t,z)/ t. That is, h(t) = t(..2t)+(t.4)(2z z 2 +t 2 )+t 2 (. t) is non-negative subject to t in ( z,.4) where z is a fixed value in (.,). It can be immediately obtained that h(t) =.t 2 +t [. ( z) 2] +.4( z) 2. The function h(t) is concave with a maximum at the point t =. ( z) 2. Therefore, if the function h(t) is non-negative in the extremes of the interval ( z,.4) then h(t) holds. It can easily shown that h( z) = ( z)[( z)(z.6)+.], and h(.4) =.247. Thus, h(t) for all t in ( z,.4). Consequently, it is obtained that the minimum of f(t,z), when (t,z) is in B, is equal to for (t,z ) = (,). Therefore, it is shown that C X lo cv C Y. Our aim now is to show that two bivariate random variables, X and Y with copulas C X and C Y respectively, are not ordered in the lo-cv sense. To this end, Chi-squared distributions with and 4 degrees of freedom are selected for the corresponding marginal distributions. That is, F(x,x 2 ) = C X (F (x ),F 2 (x 2 )) and G(x,x 2 ) = C Y (F (x ),F 2 (x 2 )), where F is the Chi-squared distribution with degree of freedom, and F 2 is the Chi-squared distribution with 4 degrees of freedom. Clearly, the random variable X and Y are not ordered in the PQD sense since the corresponding copulas are not ordered in this sense. Moreover, if t z h(t,z) = [F(x,x 2 ) G(x,x 2 )]dx 2 dx then it can been proved, by using Monte Carlo integration, that h(,2.) =.429 and h(.,) =.3. That is, the distribution function F and G are not ordered in the lo-cv sense.

6 6 Fernández-Ponce, JM and Rodríguez-Griñolo, MR a) b) v A D B C..4 A B D C u.4. Fig. : Zones for (u,v). References Nelsen, R. (26). An Introduction to Copulas. Springer-Verlag, New York.