An asymptotic distribution of bivariate extremes is studied to obtain the asymptotic probabilistic behaviour and to provide bivariate models of (asymptotic) extremes. The distribution of bivariate extremes and the limiting distribution of maxima are discussed with complementary basic, correlation and regression results. The classical correlation coefficient is linearly invariant. Orthogonal polynomials with respect to \(\mathrm{ \Lambda(X) }\) is used to improve the regression lines.
Bivariate asymptotic distributions of maxima are useful for the analysis of many concrete problems such as the greatest ages of death for men and women, each year, whose distribution, naturally, splits in the product of the margins, by independence; floods at two different places on the same river, each year; bivariate extreme meteorological data (pressures, temperature, wind velocity, etc.), each week; largest waves, each week, etc. The same can be said for the study of minima. Extensions to multivariate distributions, more complex, will be made later.
Evidently, the aim of study of asymptotic distributions of bivariate extremes is to obtain the asymptotic probabilistic behaviour, and also to provide bivariate models of (asymptotic) extremes that fit observed data. But, as will see in the next chapter, only some problems have been solved and the methods found so far cover a much smaller area than the theory for univariate extremes.
2 . The distribution of bivariate extremes
The theory of limiting bivariate (and multivariate) extremes, to a certain extent, follows the same lines of limiting univariate extremes. For simplicity of the exposé we now consider the bivariate case; the multivariate case will appear as an extension in a following chapter.
Let \(\mathrm{ ( X_{1},Y_{1} ) ,\dots, ( X_{k},Y_{k} ) ,\dots }\) be a sequence of i.i.d. random pairs with distribution function \(\mathrm{ F ( x,y ) =Prob \{ X \leq x,Y \leq y \} }\) and survival function \(\mathrm{ S ( x,y ) =Prob\{ { X>x,Y>y }\} =1+F ( x,y ) -F ( x,+ \infty ) -F ( + \infty,y ) }\); the inversion of the last relation expresses \(\mathrm{ F( x,y ) }\) in \(\mathrm{ S( x,y ) }\) by \(\mathrm{ F ( x,y ) =1+S ( x,y ) -S( x,- \infty ) -S ( - \infty,y ) }\)because, as known, we have \(\mathrm{ F ( x,+ \infty ) +S ( x,- \infty ) =1 }\) and \(\mathrm{ F ( + \infty,y ) +S ( - \infty,y ) =1 }\). The survival function plays the same rôle for minima as the distribution function played for maxima.
The random pair \((\begin{array}{c} \mathrm{ k } \\ \mathrm{ { max \,X_i} } \\ \mathrm{ { 1 } }\end{array} ,\begin{array}{c} \mathrm{ k } \\ \mathrm{ { max \,Y_i} } \\ \mathrm{ { 1 } }\end{array})\) has the distribution function
\(\mathrm{ F_{k} ( x,y ) =Prob \{ \begin{array}{c} \mathrm{ k } \\ \mathrm{ { max } } \\ \mathrm{ { 1 } }\end{array} X_{i} \leq x,\begin{array}{c} \mathrm{ k } \\ \mathrm{ { max } } \\ \mathrm{ { 1 } }\end{array}Y_{i} \leq y \} =F^{n} ( x,y ) }\).
The probability of \((\begin{array}{c} \mathrm{ k } \\ \mathrm{ { max \,X_i} } \\ \mathrm{ { 1 } }\end{array} ,\begin{array}{c} \mathrm{ k } \\ \mathrm{ { max \,Y_i} } \\ \mathrm{ { 1 } }\end{array})\) being an observed point is, evidently,
If \(\mathrm{ F( x,y ) }\) splits into its margins \(\mathrm{ F ( x,y ) =F ( x,+ \infty )\cdot\mathrm{ F ( + \infty,y ) } }\), we have \(\mathrm{ \pi _{k}=1/k }\). It is well known (see Fréchet, 1951) that \(\mathrm{ max ( 0,F ( x,+ \infty ) +F ( + \infty,y ) -1 ) \leq F ( x,y ) \leq min ( F ( x,+ \infty ) ,F ( + \infty,y ) ) }\)with the bounds attained; if \(\mathrm{ F( x,y ) }\) is equal to the LHS we have \(\mathrm{ \pi _{k}=0 }\), and for the RHS we get \(\mathrm{ \pi _{k}=1 }\) because in the first case we have \(\mathrm{ F ( x,+ \infty ) +F( + \infty,y) =1 }\)with probability 1, and in the second we have \(\mathrm{ F ( x,+ \infty ) =F( + \infty,y ) }\) with probability 1.
For the random pair \((\begin{array}{c} \mathrm{ k } \\ \mathrm{ { min \,X_i} } \\ \mathrm{ { 1 } }\end{array} ,\begin{array}{c} \mathrm{ k } \\ \mathrm{ { min \,Y_i} } \\ \mathrm{ { 1 } }\end{array})\) we have \(\mathrm{ S_{k} ( x,y ) =S^{k} ( x,y ) }\) in an analogous way, and the distribution function of the pair of minima is
The probability \(\mathrm{ \pi ’_{k} }\) of \((\begin{array}{c} \mathrm{ k } \\ \mathrm{ { min \,X_i} } \\ \mathrm{ { 1 } }\end{array} ,\begin{array}{c} \mathrm{ k } \\ \mathrm{ { min \,Y_i} } \\ \mathrm{ { 1 } }\end{array})\) being an observed point is also \(\mathrm{ 1/{k} }\) for the independence case and \(\mathrm{ 0 }\) and \(\mathrm{ 1 }\) if \(\mathrm{ F( x,y ) }\) is equal to the LHS or RHS of the corresponding Fréchet inequality.
As before we will stick essentially to the study of maxima, the conversion to minima being immediate by the relations \(\begin{array}{c} \mathrm{ k } \\ \mathrm{ { min} } \\ \mathrm{ { 1 } }\end{array}\,\mathrm{X_i } =-\begin{array}{c} \mathrm{ k } \\ \mathrm{ { max} } \\ \mathrm{ { 1 } }\end{array}\mathrm{(-X_i)}\) and \(\begin{array}{c} \mathrm{ k } \\ \mathrm{ { min } } \\ \mathrm{ { 1 } }\end{array}\,\mathrm{Y_i } =-\begin{array}{c} \mathrm{ k } \\ \mathrm{ { max } } \\ \mathrm{ { 1 } }\end{array}\mathrm{(-Y_i)}\); note that the distribution function of \(\mathrm{ ( -X,-Y ) }\) is \(\mathrm{ Prob \{ -X \leq x,-Y \leq y \} =Prob \{ X \geq -x,Y \geq -y \} =S( -x,-y ) }\) in the continuity points (with right-continuity continuation in the discontinuity set) and thus, here, all over the plane because the limiting distributions are continuous.
Note, as an hint for applications, if \(\mathrm{ ( X_{i},Y_{i} ) }\) are floods of the same river at two different points they must be dealt with as maxima; but if \(\mathrm{ ( X_{i},Y_{i} ) }\) are droughts at the same two points they must be dealt with as minima, i.e. \(\mathrm{ (-X_{i}\,,-Y_{i} ) }\) will be dealt with as maxima.
3 . The limiting distribution of maxima
An important question, both theoretical and applied, is to find out, analogously, as was done in the univariate case, if there exist positive linear transforms of \((\begin{array}{c} \mathrm{ k } \\ \mathrm{ { max \,X_i} } \\ \mathrm{ { 1 } }\end{array} ,\begin{array}{c} \mathrm{ k } \\ \mathrm{ { max \,Y_i} } \\ \mathrm{ { 1 } }\end{array})\) such that the reduced random pair
\(((\begin{array}{c} \mathrm{ k } \\ \mathrm{ { max } } \\ \mathrm{ { 1 } }\end{array}\,\mathrm{X_i-\lambda_k)\delta_k, }(\begin{array}{c} \mathrm{ k } \\ \mathrm{ { max } } \\ \mathrm{ { 1 } }\end{array}\,\mathrm{Y_i-\lambda_{k}^{'})\delta_{k}^{'} )}\),
whose distribution function is
\(\mathrm{ Prob \{ ( \begin{array}{c} \mathrm{ k } \\ \mathrm{ { max } } \\ \mathrm{ { 1 } }\end{array}X_{i}- \lambda _{k}) / \delta _{k} \leq x, ( \begin{array}{c} \mathrm{ k } \\ \mathrm{ { max } } \\ \mathrm{ { 1 } }\end{array}Y_{i}- \lambda ’_{k} ) / \delta ’_{k} \leq y \} =F^{k} ( \lambda _{k}+ \delta _{k}x, \lambda ’_{k}+ \delta ’_{k}y ) \mathrm{{\stackrel{\mathrm{w}}{\rightarrow}}}\mathrm{\ }L ( x,y ) }\), i.e., has a limiting, non-degenerate and proper distribution.
As a consequence we know that \(\mathrm{ F^{k}( \lambda _{k}+ \delta _{k}~x,+ \infty ) \mathrm{{\stackrel{\mathrm{w}}{\rightarrow}}}\mathrm{\ }L( x,+ \infty ) }\) and \(\mathrm{ F^{k}( + \infty,~ \lambda ’_{k}+ \delta ’_{k}~y ) \mathrm{{\stackrel{\mathrm{w}}{\rightarrow}}}\mathrm{\ }L ( + \infty,y ) }\) — see Deheuvels (1984).
Evidently, as we dealing with maxima, the marginal distribution functions \(\mathrm{ L( x,+ \infty ) }\) and \(\mathrm{ L ( + \infty,y ) }\) must be of any of the three forms — Weibull, Gumbel or Fréchet. As they are easily translated from one to another, as seen before, we will suppose that the margins have a reduced Gumbel distribution function, i.e., \(\mathrm{ L( x,+ \infty ) = \Lambda ( x ) ,L ( + \infty,y ) = \Lambda ( y ) }\). As the margins are continuous, so is \(\mathrm{ L ( x,y ) }\) and, thus, the convergence \(\mathrm{ F^{k} ( \lambda _{k}+ \delta _{k}~x, \lambda _{k}^{’}+ \delta _{k}^{’}~y ) {{\stackrel{\mathrm{w}}{\rightarrow}}}\mathrm{\ }L ( x,y )}\) is also uniform. For notational convenience, we will denote by \(\mathrm{ \Lambda ( x,y ) }\) the limiting distribution functions \(\mathrm{ L ( x,y ) }\) when the margins have a reduced Gumbel distribution function.
The special case where we have minima with an reduced exponential distribution will also be considered, sometimes by duality, but the kernel of the Chapter, as well as the next ones, is the case of maxima with reduced Gumbel margins.
Let us show that \(\mathrm{ \Lambda ( x,y ) =e^{- ( e^{-x}+e^{-y} )\, k\, ( y-x ) } }\) where \(\mathrm{ k(.) }\), called the dependence function must satisfy some conditions to be seen later.
as said before. The dependence function expresses the probabilistic inter-relation, or association, between the margins.
An actual characterization of the distribution function \(\mathrm{ \Lambda ( x,y ) }\) can be made in the following way. It is immediate that a random pair \(\mathrm{( X,Y ) }\) with distribution function \(\mathrm{ \Lambda ( x,y ) }\) is such that \(\mathrm{ max ( X+a,Y+b ) }\)has a Gumbel distribution function with a location parameter. But, also, the converse is true. In fact putting \(\mathrm{ z-a=x,z-b=y }\) we have \(\mathrm{ L( x,y ) = \Lambda ( z- \varphi ( z-x,z-y ) ) }\)independent of \(\mathrm{ z }\), from which we get \(\mathrm{ \varphi ( \xi, \eta ) = \xi +\varphi( 0, \eta - \xi ) }\), and then \(\mathrm{( 1+e^{-w} ) \,k \,( w ) =e^{ \varphi ( 0,-w ) } }\). Thus:
\(\Lambda ( x,y ) \)is the only distribution function \(L( x,y )\)such that\(Prob \{ max ( X+a,Y+b ) \leq z \} =L ( z-a,z-b ) = \Lambda ( z-\varphi ( a,b ) ) \).
Another approach may be the following: let us call the structure function of a bivariate distribution function \(\mathrm{ F( x,y ) }\) with continuous margins \(\mathrm{ A( x) =F ( x,+ \infty ) }\) and \(\mathrm{ B ( y) =F( + \infty,y ) }\) the function \(\mathrm{ \bar{S} ( \xi , \eta ) }\) such that
\(\mathrm{ \bar{S} \left( A \left( x \right) ,B \left( y \right) \right) =F \left( x,y \right) }\) ;
\(\mathrm{ \bar{S} ( \xi , \eta ) }\) is a bivariate distribution function with uniform margins, defined in the unit square \(\mathrm{[ 0,1] \times [ 0,1 ] }\) and is continuous.
Then any structure function of the limiting distribution functions of bivariate extremes verifies the functional equation \(\mathrm{ \left( 0< \omega <+ \infty \right) }\)
Let \(\mathrm{ S_0 }\) denote the (initial) structure function of the distribution function with margins \(\mathrm{ A(x) }\) and \(\mathrm{ B(y) }\) and suppose, as before, that the marginal limiting distributions are \(\mathrm{ \Lambda( x ) }\) and \(\mathrm{ \Lambda ( y) }\), with \(\mathrm{ \lambda _{k}+ \delta _{k}>0, \lambda ’_{k}+ \delta ’_{k}>0 }\), as a system of attraction coefficients. We then have
and \(\mathrm{ F^{k} ( \lambda _{k}+ \delta _{k} \,x, \lambda ’_{k}+ \delta ’_{k} \,y ) =\bar{S}_{0}^{k} ( \xi _{k}^{1/k}, \eta _{k}^{1/k} ) }\). Let us now show that \(\mathrm{ \bar{S}_{0}^{k} ( \xi _{k}^{1/k}, \eta _{k}^{1/k} ) }\) has a limit \(\mathrm{ \bar{S} ( \xi , \eta ) }\) when, \(\mathrm{ \xi _{k} \rightarrow \xi , \eta _{k} \rightarrow \eta }\) if and only if \(\mathrm{ \bar{S}_{0}^{k} ( \xi ^{{1}/{k}}, \eta ^{{1}/{k}} ) \rightarrow \bar{S} ( \xi, \eta ) }\). The ‘if’ part is obvious; let us show the converse.
As \(\mathrm{ \xi _{k} \rightarrow \xi , \eta _{k} \rightarrow \eta }\) , for \(\mathrm{ \eta >N ( \varepsilon ) }\), we have
Taking, as stated, the reduced margins to be \(\mathrm{ \Lambda( x ) =exp ( -e^{-x} ) }\) and \(\mathrm{ \Lambda ( y ) =exp ( -e^{-y} ) }\), we have the following fundamental result:
Any limiting distribution function of bivariate pairs with reduced Gumbel margins is of the form
When the asymptotic extremal pair \(\mathrm{( \tilde{X} ,\tilde{Y} ) }\) does not have reduced margins, that is \(\mathrm{ \tilde{X} }\) and \(\mathrm{ \tilde{Y} }\) have location parameters \(\mathrm{ \lambda _{x}, \lambda _{y} }\) and dispersion parameters \(\mathrm{ \delta _{x}>0, \delta _{y}>0 }\), the distribution function of \(\mathrm{( \tilde{X} ,\tilde{Y} ) }\), as \(\mathrm{ ( ( \tilde{X} - \lambda _{x} ) / \delta _{x}, ( \tilde{Y} - \lambda _{y})/ \delta _{y} ) }\)has reduced margins, is \(\mathrm{ \Lambda ( ( \tilde{x} - \lambda _{x} ) / \delta _{x}, ( \tilde{y} - \lambda _{y}/ \delta _{y} ) }\).
We have shown in two ways that the limiting and the stable distribution functions of extremal pairs form the same class.
Let us now deal briefly with minima as \(\mathrm{ X̰_{k}=min ( X_{1},\dots,X_{k} ) =-max ( -X_{1},\dots,-X_{k} ) }\) and \(\mathrm{ Y ̰_k=min ( Y_{1},\dots,Y_{k} ) =-max ( -Y_{1},\dots,-Y_{k} ) }\), the asymptotic distribution function of minima with reduced Gumbel margins \(\mathrm{ 1-exp ( -e^{x} ) }\) and \(\mathrm{ 1-exp( -e^{y}) }\), if they exist, and correspondence with \(\mathrm{ \{ \Lambda ( x ) \cdot \Lambda ( y ) \} ^{k ( y-x ) } }\), is
Using the fact that \(\begin{array}{c} \mathrm{ k } \\ \mathrm{ { min } } \\ \mathrm{ { 1 } }\end{array}\mathrm{(Y_i )} =-\begin{array}{c} \mathrm{ k } \\ \mathrm{ { max } } \\ \mathrm{ { 1 } }\end{array}\mathrm{(-Y_i)}\) we can also prove easily that the limiting distribution of \((\begin{array}{c} \mathrm{ k } \\ \mathrm{ { min } } \\ \mathrm{ { 1 } }\end{array}\,\mathrm{X_i } ,\begin{array}{c} \mathrm{ k } \\ \mathrm{ { max } } \\ \mathrm{ { 1 } }\end{array}\mathrm{Y_i})\) , if it exists, is
Let us now obtain some inequalities for \(\mathrm{ \Lambda ( x,y ) }\) or \(\mathrm{ k ( w ) }\) which will simplify some of the derivations below. Although \(\mathrm{ \Lambda ( x,y ) }\) is a continuous function, it does not necessarily have derivatives, and consequently we cannot expect all bivariate maxima random pairs with distribution function \(\mathrm{ \Lambda(x,y ) = [ \Lambda ( x ) \Lambda ( y ) ] ^{k ( y-x ) } }\) to have a planar density. From the Boole-Fréchet inequality — Fréchet (1951) —
\(\mathrm{ max ( 0, \Lambda ( x ) + \Lambda ( y ) -1 ) \leq \Lambda ( x,y ) \leq min( \Lambda ( x ) \Lambda( y ) ) }\)
we have, replacing \(\mathrm{ x }\) and \(\mathrm{ y }\) by \(\mathrm{ x+log~k }\) and \(\mathrm{ y+log~k }\), raising to the power \(\mathrm{ k }\) and letting \(\mathrm{ k \rightarrow \infty }\), the limit inequality
\(\mathrm{ \Lambda ( x ) \Lambda ( y ) \leq \Lambda ( x,y ) \leq min( \Lambda ( x ) \Lambda ( y ) ) }\)
The LHS inequality shows that we have positive association, that is \(\mathrm{ Prob \{{ X \leq x,Y \leq y } \}+Prob \{ X>x,Y>y \} }\)is larger or equal to the corresponding values the case of for independence for all \(\mathrm{ x }\) and \(\mathrm{ y }\), which means that if we have a large (small) value of \(\mathrm{ X }\), the value of \(\mathrm{ Y }\), tends also to be large (small). \(\mathrm{ \Lambda ( x )\cdot \Lambda ( y ) }\) is , evidently, the independence situation; \(\mathrm{ min ( \Lambda ( x ) , \Lambda ( y ) ) }\) is called the diagonal or complete dependence case where we have, for reduce values, \(\mathrm{ Prob \{ Y=X \} =1 }\) . As regards \(\mathrm{ k ( w ) }\), we have
\(\mathrm{ ( 1/2 \leq) \frac{max ( 1,e^{w} ) }{1+e} \leq k ( w ) \leq 1 }\) ;
\(\mathrm{ k_{D} ( w ) =\frac{max ( 1,e^{w} ) }{1+e^{w}}=\frac{1}{1+e^{- \vert w \vert }} }\) is the dependence function corresponding to the diagonal case and \(\mathrm{k_{1}( w ) =1 }\) is the dependence function for the independence situation. Positive association, shown by Sibuya (1960), is a result that could be expected.
Remark that for \(\mathrm{ \bar{S} ( \xi , \eta ) }\) defined above we have \(\mathrm{ \xi \eta \leq \bar{S} ( \xi, \eta ) \leq min ( \xi , \eta ) }\).
If there exists a planar density almost everywhere, i.e., if \(\mathrm{ k" ( w ) }\) exists almost everywhere, \(\mathrm{ k ( w ) }\) must satisfy the relations:
Note that as \(\mathrm{ ( 1+e^{w} ) \,k\,( w) }\) and \(\mathrm{ ( 1+e^{-w} )\, k\,( w ) }\) are monotonic (non-decreasing and non-increasing), we know that \(\mathrm{ k’ ( w ) }\) exists everywhere except for a denumerable set of points of \(\mathrm{ 1R}\).
As \(\mathrm{ \Lambda( x,y )}\) is a continuous function, \(\mathrm{ k ( w ) }\) is also a continuous function. Condition 1) is easily seen to be equivalent to
II) and Ill) are obtained taking, respectively, \(\mathrm{ y \rightarrow - \infty }\) and \(\mathrm{ x \rightarrow + \infty }\). To show IV), let us put
Letting \(\mathrm{ \xi }\) vary with \(\mathrm{ x,y, \eta }\) but such that \(\mathrm{ x- \xi ,y- \xi , \eta - \xi }\)are fixed, \(\mathrm{ A,B,C,D }\) are also fixed. We then have to prove the full equivalence of 2) and II) under the new form for fixed \(\mathrm{ A,B,C,D }\) and \(\mathrm{0 \leq Z \leq 1 }\).
Let us then consider the function \(\mathrm{ f ( Z ) =Z^{-A}+Z^{-B}-Z^{-C}-Z^{-D} }\). As \(\mathrm{ f ( Z ) \geq 0 }\) by 2) and \(\mathrm{ f( 1 ) =0 }\) we have \(\mathrm{ f’ ( 1 ) \leq 0 }\) so that IV) is true.
Let us now prove the converse, i.e., that II), III) and IV) imply 2). Conditions II) and III) give \(\mathrm{ A \geq D,C \geq B }\).
Let us prove now 2). Supposing IV) to be true, we have
\(\mathrm{ Z^{-A}\cdot\,Z^{-B} \geq Z^{-C}\cdot\,Z^{-D} }\) and thus
\(\mathrm{\begin {array}{c}\\ \\\mathrm{ max} \\ \mathrm{ { A+B \leq C+D\\B \leq C,D \leq A } }\end {array}( Z^{-C}+Z^{-D} ) \leq\,\begin{array}{c} \\\mathrm{ max} \\ \mathrm{ { B \leq C \leq A } } \end{array} ( Z^{-C}+\frac{Z^{-A}~Z^{-B}}{Z^{-C}} ) }\).
The RHS function of \(\mathrm{ C \in [ B,A ] }\) has a minimum at \(\mathrm{ C=\frac{A+B}{2} }\) and the maximum value in the interval is attained at \(\mathrm{ C=A }\) and \(\mathrm{ C=B }\) with the value \(\mathrm{ Z^{-B}+Z^{-A} }\). We have then proved \(\mathrm{ Z^{-A}+Z^{-B} \geq Z^{-C}+Z^{-D} }\) as desired.
If we are dealing with minima with exponential margins, corresponding to the transformation \(\mathrm{ \xi =e^{-x}, \eta =e^{-y} }\) , the asymptotic distribution for \(\mathrm{ \xi , \eta \geq 0 }\) takes the form \(\mathrm{ \Psi ( x,y ) =1-e^{- \xi }-e^{- \eta }+e^{- ( \xi + \eta ) A( \eta / ( \xi + \eta ) ) } }\)or the survival function \(\mathrm{ S ( \xi , \eta ) =e^{- ( \xi + \eta ) A ( \eta /( \xi + \eta ) ) } }\). It can be shown that the conditions for \(F(.)\)are equivalent to
\(A ( 0) =A ( 1 ) =1 \),
\(0 \leq A’ ( 0 ) ,A’ ( 1 ) \leq 1 \)
and \(A ( u ) convex~in~ [ 0,1 ] \).
The functions \(\mathrm{ k( w) }\) and \(\mathrm{ A ( u ) }\) are related by \(\mathrm{ k ( w ) =A( \frac{1}{1+e^{w}}) }\) or \(\mathrm{ A ( u ) =k ( log\frac{1-u}{u} ) }\). We have \(\mathrm{ A( u ) =1 }\) for independence and \(\mathrm{ A ( u ) =max( u,1-u ) }\)in the diagonal case, with, also, \(\mathrm{ max ( u,1-u ) A ( u ) \leq 1 }\).
Another formulation is
\(A ( 0) =A ( 1 ) =1 \) ,
\(max( u,1-u ) \leq A( u) \leq 1\)
and \(A ( u ) convex~in [ 0,1] \).
4 . Complementary basic results
Some properties can be ascribed to the set of \(\mathrm{ k( w) }\). The first one is a symmetry property, i.e., if \(\mathrm{ k( w) }\) is a dependence function, then \(\mathrm{ k( -w ) }\) is also a dependence function. The proof is immediate if we consider the conditions in the differentiable case (where a planar density does exist) and slightly longer in the general case. If \(\mathrm{ k ( w ) =k ( -w ) }\) then \(\mathrm{( X,Y ) }\) is an exchangeable pair and \(\mathrm{ \Lambda ( x,y ) = \Lambda ( y,x ) }\).
Also it is immediate that if \(\mathrm{ k_1 \left( w \right) }\) and \(\mathrm{ k_2 \left( w \right) }\) are dependence functions, any mixture \(\mathrm{ \theta\, k_{1} ( w ) + ( 1- \theta )\, k_{1} ( w ) ,0 \leq \theta \leq 1 }\)is also a dependence function. The set of dependence functions is, then, convex. And this convexity property
is very useful for obtaining models: the mixed model, as well as the Gumbel model, are such examples. We will call this way of modelling the mix-technique.
As a generalization, it can be observed that if \(\mathrm{ G( u ) }\) is a distribution function in \(\mathrm{ [ 0,1 ] }\), then \(\mathrm{ k( w) }\) given by \(\mathrm{ ( 1+e^{-w} ) ~k ( w) = \int _{0}^{1}( 1+e^{-w/u} ) ^{u}~d~G ( u ) }\)is also a dependence function.
Another method of generating models is the following max-technique. Let \(\mathrm{( X,Y ) }\) be an extreme random pair, with dependence function \(\mathrm{ k( w) }\) and reduced Gumbel margins, and consider the new random pair \(\mathrm{ ( \tilde{X},\tilde{Y}) \ }\)with \(\mathrm{ \tilde{X}=max ( X+a,Y+b ) ,\tilde{Y}=max ( X+c,Y+d ) }\). To have reduced Gumbel margins we must have \(\mathrm{ ( e^{a}+e^{b} ) \,k \,( a-b ) =1 }\) and \(\mathrm{ ( e^{c}+e^{d} ) \,k \, ( c-d ) =1 }\). Then we have
\(\mathrm{\tilde{ k}( w ) =\frac{ [ e^{max ( a+w,c t) }+e^{max ( b+w,d ) } ] k [ max ( a+w,c ) -max ( b+w,d ) ] }{1+e^{w}} }\)
with \(\mathrm{ ( a,b ) }\) and \(\mathrm{( c,d ) }\) satisfying the conditions given before. This max-technique will be used towards the end of the paper to generate the biextremal and natural models.
Let us stress that independence has a very important position as a limiting situation. If we denote by \(\mathrm{ P ( u,v ) }\) the function defined by \(\mathrm{ Prob \{ X>x,Y>y ) =P ( F ( x,+ \infty ) ,F ( + \infty,y ) ) }\), Sibuya (1960) has shown that the necessary and sufficient condition to have limiting independence is that \(\mathrm{ P( 1-s,1-s ) /s \rightarrow 0~as~s \rightarrow 0 }\). He also showed that the necessary and sufficient condition for having the diagonal case as a limiting situation is that \(\mathrm{ P( 1-s,1-s ) /s \rightarrow 0~as~s \rightarrow 0 }\). With the first result we can easily show that the maxima of the binormal distribution has independence as a limiting distribution if \(\mathrm{ \vert \rho \vert <1 }\).
Also Geffroy (1958/59) showed that a sufficient condition for limiting independence is that
\(\mathrm{ \frac{1+F( x,y ) -F( x,w_{y} ) -F ( w_{x},y ) }{1-F ( x,y ) } \rightarrow 0 }\) as \(\mathrm{ x \rightarrow \bar{w}_{x} }\) and \(\mathrm{y \rightarrow \bar{w}_{y} }\),
\(\mathrm{ \bar{w}_{x} }\) and \(\mathrm{ \bar{w}_{y} }\) being the right end-points of the supports of \(\mathrm{ X }\) and \(\mathrm{ Y }\).
Sibuya conditions (and Geffroy sufficient condition) are easy to interpret: we have limiting independence if \(\mathrm{ Prob \{ X>x,Y>y \} }\) is a vanishing summand of \(\mathrm{ Prob~ \{ X>x,or~Y>y \} }\) and the diagonal case as limit if \(\mathrm{ Prob \{ X>x,Y>y \} }\) is the leading summand of \(\mathrm{ Prob~ \{ X>x,or~Y>y \} }\).
We have asymptotic independence when \(\mathrm{ \bar{S} \left( \xi , \eta \right) = \xi , \eta }\), that is, when \(\mathrm{ \bar{S}_{0}^{k}( \xi ^{1/k}, \eta ^{1/k} ) \rightarrow \xi , \eta }\). The sufficient condition for independence given by Geffroy can be proved in a simpler way, as follows. It is equivalent to:
\(\mathrm{ h ( \xi ^{1/k}, \eta ^{1/k} ) \} ^{k} \rightarrow 1 }\) when \(\mathrm{ \eta \rightarrow \infty }\).
Its meaning is straightforward, and, in general, we can expect extremal pairs to be asymptotically independent. Mardia (1964) proved that the sample extremes of a bivariate pair are independent under very general conditions.
A simple example now shows that we can have non-independence cases. Let \(\mathrm{ ( X,Y,Z ) }\) be three independent random variables with the same distribution function \(\mathrm{ F }\) . The distribution function of the pair
It can be remarked that \(\mathrm{ P ( x,y) }\), as well as the limit, does not have a planar density.
Sometimes it is usual to study the equidistribution and equidensity curves (if they exist). The latter are very difficult to deal with so that we will restrict ourselves to the former.
As \(\mathrm{ k ( - \infty ) =k ( + \infty ) =1 }\), the asymptotes to the median curves are
\(\mathrm{ x=-log\,log\,2 }\) and \(\mathrm{ y=-log\,log\,2 }\).
As these asymptotes are independent of the dependence function, the equidistribution curves behave similarly for large values of \(\mathrm{ x }\) or \(\mathrm{ y }\), as could be expected, and, then, like the median curve for independence. The relations deduced from Fréchet inequalities yield the double inequality
showing that any median curve is, necessarily, between the median curves corresponding to the independence
\(\mathrm{ e^{-x}+e^{-y}=log~2 }\)
and to the diagonal case
\(\mathrm{ min\,( x,y ) =-log\,log\,2 }\)
which degenerates in part of the asymptotes.
Consider now the random variable \(\mathrm{ W=Y-X }\), the difference of reduced extremes. We have, evidently, \(\mathrm{ M \left( W \right) =0 }\) and \(\mathrm{ V ( W ) =\frac{ \pi ^{2}}{3} ( 1- \rho ) }\) . Recall that as \(\mathrm{ V ( X) =V ( Y ) = \pi ^{2}/6 }\) the covariance \(\mathrm{ C ( X,Y ) =\frac{ \pi ^{2}}{6} \rho }\) exists as well as the correlation coefficients \(\mathrm{ \rho }\).
When \(\mathrm{ \rho =1 }\) we have \(\mathrm{ V ( W ) =0 }\); \(\mathrm{ W }\) is a random variable almost surely equal to zero and we have \(\mathrm{ Y=X }\) almost surely. We then have the diagonal case.
Let us denote by \(\mathrm{ D ( w ) =Prob \{{ Y-X \leq w } \} }\); we have
\(\mathrm{ D \left( w \right) = \int _{- \infty}^{+ \infty}d~x~e^{-e^{-x} \left( 1+e^{-w} \right) k \left( w \right) }e^{-x}~ \{ k \left( w \right) + \left( 1+e^{-w} \right) k^{’} \left( w \right) \} =\frac{1}{1+e^{-w}}+\frac{k^{’} \left( w \right) }{k \left( w \right) } }\)
as is obvious.
The relations between \(\mathrm{ D( w ) }\) and \(\mathrm{ k ( w ) }\) are immediate. Integration between \(\mathrm{ 0 }\) and \(\mathrm{ w }\) of the differential relation gives
\(\mathrm{ ( 1+e^{w}) ~k ( w ) =2~k ( 0 ) exp \int _{0}^{w}D ( w ) d~w }\),
and letting \(\mathrm{ w \rightarrow - \infty ( k ( - \infty ) =1) }\) we obtain
\(\mathrm{ k ( w) =\frac{exp~ \int _{- \infty}^{w}D ( w ) d~w}{ ( 1+e^{w} ) } }\).
Note that \(\mathrm{ \int _{- \infty}^{0}D( w )\, d~w }\) and \(\mathrm{ \int _{0}^{+ \infty} ( 1-D ( w ) ) d~w }\) exist and are equal because \(\mathrm{ W }\) has mean value zero.
The conditions for \(\mathrm{ k ( w ) }\) are immediately translated into conditions for \(\mathrm{ D( w ) }\) so that
\(\mathrm{D ( w ) =Prob \{ Y-X \leq w \} }\)
verifies the following relations (apart from being a distribution function):
\(\mathrm{\int _{- \infty}^{0}D ( w) d~w= \int _{0}^{+ \infty} [ 1-D ( w ) ] d~w }\) or \(\mathrm{ \int _{- \infty}^{+ \infty}d~w~D \left( w \right) =0 }\)
and
\(\mathrm{ D’ ( w ) \geq D ( w ) [ 1-D ( w ) ] }\) .
We can also write
\(\mathrm{ D ( w ) =k’ ( w ) /k ( w ) +L ( w ) }\),
where \(\mathrm{ L \left( w \right) = \left( 1+e^{-w} \right) ^{-1} }\) is the standard logistic distribution function.
We have \(\mathrm{ D \left( w \right) =L \left( w \right) }\) in the independence case and \(\mathrm{ D \left( w \right) =H \left( w \right) }\) where \(\mathrm{ H \left( w \right) }\) is the Heaviside jump function at \(\mathrm{ 0, H \left( w \right) =0 }\) if \(\mathrm{ w<0 }\) and \(\mathrm{ H \left( w \right) =1 }\) if \(\mathrm{ w\geq0 }\) — for the diagonal case. In the case of exchangeability \(\mathrm{ \left( k \left( w \right) =k \left( -w \right) \right) }\) we have \(\mathrm{ D \left( w \right) +D \left( -w \right) =1 }\), and \(\mathrm{ W }\) has a symmetric distribution.
As said before, \(\mathrm{ k’ \left( w \right) }\) exists everywhere except for a denumerable set of points and thus \(\mathrm{ D( w ) }\) is defined everywhere: directly almost everywhere and at the discontinuity set by the right-continuity of the distribution function \(\mathrm{ D\left(. \right) }\).
Let us denote by \(\mathrm{ ID \left( w \right) = \int _{- \infty}^{w}D \left( t \right) ~d~t }\); we have seen that
\(\mathrm{ k ( w ) =\frac{e^{ID ( w) }}{1+e^{w}} }\).
The conditions on the distribution function \(\mathrm{ D( w ) }\), in the case of the existence of planar density, are then
\(\mathrm{ \int _{- \infty}^{+ \infty}d~w~D \left( w \right) =0 }\)
or equivalently
\(\mathrm{ \int _{- \infty}^{+ \infty} ( H ( w ) -D ( w ) ) d~w=0 }\)
and
\(\mathrm{ D’ \left( w \right) \geq D \left( w \right) \left( 1-D \left( w \right) \right) }\) .
More generally, the last condition is substituted by
Note that \(\mathrm{ ID ( - \infty ) =0,ID ( w ) -w \rightarrow + \infty,ID ( w+ \beta) -ID ( w ) \leq \beta ( \beta >0 ) }\),\(\mathrm{ e^{ID \left( w \right) }-e^{ID \left( w- \alpha \right) } \leq e^{w}-e^{w- \alpha } \left( \alpha \geq 0 \right) ,max \left( 0,w \right) \leq ID \left( w \right) \leq log \left( 1+e^{w} \right) }\)(from Boole-Fréchet inequalities), \(\mathrm{ \int _{- \infty}^{0}D ( t ) ~d~t= \int _{0}^{+ \infty} ( 1-D ( t ) ) ~d~t }\) (from the null mean value), \(\mathrm{ D \left( w \right) e^{ID \left( w \right) } \leq e^{w} }\) and \(\mathrm{ \left( 1-D \left( w \right) \right)e^{ID \left( w \right) } \leq 1 }\) (by convenient integration of \(\mathrm{ D’ \geq D \left( 1-D \right) ) }\) .
Let \(\mathrm{ \underline{ w } }\) and \(\mathrm{\bar{w}}\) ( \(\mathrm{ \underline{w} \leq 0 \leq \bar{w},\bar{w}=\underline{w}=0 }\) in the diagonal case and \(\mathrm{ \underline{w} <0< \bar{w} }\) in the other cases) be the left and right end-points of \(\mathrm{ D( w ) }\).
For \(\mathrm{w>\bar{w},}\) if \({\mathrm {\bar{w}< \infty}} \), we have
\(\mathrm{ID \left( w \right) = \int _{- \infty}^{w}D \left( t \right) d~t= \int _{+ \infty}^{\bar{w}}D \left( t \right) d~t+w-\bar{w} }\);
as
\(\mathrm{ ID \left( w \right) -w= \int _{- \infty}^{\bar{w}}D \left( t \right) d~t-\bar{w} \rightarrow 0 }\)
we have
\(\mathrm{ \bar{w}= \int _{- \infty}^{\bar{w}}D \left( t \right) d~t }\),
a result that is also true \(\mathrm{ \bar{w}=+ \infty }\).
Consider now, for \(\mathrm{ \underline{w} >- \infty }\) ,
\(\mathrm{ -\underline{w}- \int _{\underline{w}}^{0}D \left( t \right) d~t+ \int _{0}^{+ \infty}{ \left( 1-D \left( t \right) \right) d~t=-\underline{w}} }\)
which is also true \(\mathrm{ \underline{w}=- \infty }\) .
If both \(\mathrm{ \underline{w} }\) and \(\mathrm{ \bar{w} }\) are finite we have
\(\mathrm{ \bar{w}= \int _{\underline{w}}^{w}D \left( t \right) d~t }\) and \(\mathrm{ \underline{w}=- \int _{\underline{w}}^{\bar{w}} \left( 1-D \left( t \right) \right) d~t }\).
The symmetry condition (exchangeability) \(\mathrm{ k \left( w \right) =k \left( -w \right) }\) is equivalent to \(\mathrm{ D \left( w \right) +D \left( -w \right) =1 }\) or \(\mathrm{ ID \left( w \right) =w+ID \left( -w \right) }\) .
From the conditions on \(\mathrm{ D( w ) }\) we can give new methods of generating bivariate models of maxima, as follows, essentially for absolutely continuous distribution functions. The most natural is the one that follows.
In \(\mathrm{ [ \underline{w},\bar{w}] }\), the support of \(\mathrm{ D(.) }\), we can define the function \(\mathrm{ \Psi \left( w \right) =\frac{D’ \left( w \right) }{D \left( w \right) \left( 1-D \left( w \right) \right)} \geq 1 }\). Notice that as a consequence every point of \(\mathrm{ [ \underline{w},\bar{w}] }\) is a point of increase of \(\mathrm{ D(.) }\), and thus the quantiles are uniquely defined. From the absolute continuity we have \(\mathrm{ D \left(\underline{w}\right) =0 }\) and \(\mathrm{ D \left( \bar{w} \right) =1 }\). Note that \(\mathrm{\underline{w}<0<\bar{w} }\), as the mean value is zero.
Let us fix \(\mathrm{ v }\) as the median. Integrating the differential equation \(\mathrm{ D’=D \left( 1-D \right) \Psi }\)we get with \(\mathrm{ I \Psi \left( w \right) = \int _{- \infty}^{w} \Psi \left( t \right) d~t }\),
\(\mathrm{ D \left( w \vert v \right) =\frac{1}{1+e^{- \left( I \Psi \left( w \right) -I \Psi \left( v \right) \right) }} }\),
and thus the relation between \(\mathrm{ D \left( w \vert v \right) }\) and \(\mathrm{ D \left( w \vert v' \right) }\) is
\(\mathrm{ D \left( w \vert v’ \right) =\frac{D \left( w\vert v \right) }{D \left( w\vert v \right) +e^{-K} \left( 1-D \left( w \vert v \right) \right) } }\)
with \(\mathrm{ K= \int _{v’}^{v} \Psi \left( t \right) d~t }\) and the same support.
As \(\mathrm{ \Psi \left( w \right) \geq 1 }\) it is immediate that \(\mathrm{ \int _{v}^{w} \Psi \left( t \right) d~t \geq \left( w-v \right) }\) for \(\mathrm{ w>v }\). We get, thus
\(\mathrm{ D \left( w \vert v \right) \geq \frac{1}{1+e^{- \left( w-v \right) }} }\)
and for \(\mathrm{ w<v }\), analogously, we obtain
\(\mathrm{ D \left( w \vert v \right) \leq \frac{1}{1+e^{- \left( w-v \right) }} }\),
results not known previously.
Having fixed, temporarily, the median at \(\mathrm{ v \left( \underline{w}<V<\bar{w} \right) }\), the condition
\(\mathrm{ \int _{- \infty}^{+ \infty}w~d~D \left( w \vert v \right) = \int _{- \infty}^{+ \infty} \left[ H \left( w \right) -D \left( w \vert v \right) \right] d~w=0 }\) reads as
\(\mathrm{ I \left( K \right) =0 }\) where
\(\mathrm{ I \left( K \right) = \int _{\underline{w}}^{\bar{w}} [ H \left( w \right) -\frac{D \left( w \vert v \right) }{D \left( w \vert v \right) +e^{-K} \left( 1-D \left( w \vert v \right) \right) } ] d~w }\).
As \(\mathrm{ I \left( K \right) =\underline{w}}\) when \(\mathrm{ K \rightarrow + \infty,I \left( K \right) \rightarrow \bar{w} }\) when \(\mathrm{ K \rightarrow +- \infty }\) and \(\mathrm{ I \left( K \right) }\) is increasing, we see that the solution of \(\mathrm{ I \left( K \right) =0 }\) is unique. The unique solution \(\mathrm{ K_0 }\) of \(\mathrm{ I \left( K \right) =0 }\) leads to
\(\mathrm{ \int _{v’}^{v} \Psi \left( t \right) d~t=K_{0} }\)
and, as \(\mathrm{ \Psi \left( w \right) \geq 1 }\), the only solution for \(\mathrm{ D(.) }\) is immediately obtained. It is thus sufficient to give an integrable function \(\mathrm{ \Psi \left( w \right) \geq 1 }\) in every finite interval \(\mathrm{ [ \underline{w},\bar{w}] }\) to obtain the unique \(\mathrm{ D(.) }\) related to it. For simplicity we can take \(\mathrm{ v=0}\).
5 . Correlation results
The results concerning correlation that follow and the regression results to be given later, to a certain extent, illuminate some other features of the situation.
Let us now compute some widely used correlation coefficients.
The classical correlation coefficient always exists as shown before and is \(\mathrm{ \rho =\frac{6}{ \pi ^{2}}~C \left( X,Y \right) }\).
A known expression for \(\mathrm{ C \left( X,Y \right) }\) is
\(\mathrm{ C \left( X,Y \right) = \int _{- \infty}^{+ \infty} \int _{}^{} \left[ F \left( x,y \right) -F \left( x,+ \infty \right) F \left( + \infty,y \right) \right] d~x~d~y }\)
which, in our case, takes the form
\(\mathrm{ C \left( X,Y \right) = \int _{- \infty}^{+ \infty} \int _{}^{} \left[ \Lambda \left( x,y \right) - \Lambda \left( x \right) \Lambda \left( y \right) \right] d~x~d~y }\),
and because of the positive association we have \(\mathrm{ C \left( X,Y \right) \geq 0 }\), substituting the expressions of \(\mathrm{ \Lambda \left( x,y \right) , }\)\( \mathrm{\Lambda \left( x \right)}\) and \(\mathrm{ \Lambda \left( y \right) }\) and using the change of variables
The first two integrals, being an expression for \(\mathrm{ M \left( X \right) }\), add to \(\mathrm{ \gamma }\). Changing \(\mathrm{v }\) to \(\mathrm{ v=z+log~k \left( w \right) }\) in the two last integrals, we have after simple computations for the inner integral
when needed we will write \(\mathrm{ \rho \left( k \right) }\) to indicate the dependence function.
From the expression of \(\mathrm{ D(w) }\) we can obtain, once more in the differentiable case, also the expression \(\mathrm{ \rho =-\frac{6}{ \pi ^{2}} \int _{- \infty}^{+ \infty}log~k \left( w \right) d~w }\). Let us give the supplementary proof.
Let us recall first that the existence of the variance \(\mathrm{ V(W) }\) implies that \(\mathrm{ {lim}_{w \rightarrow - \infty}w^{2}D \left( w \right) =0~and~{lim}_{w \rightarrow \infty}w^{2} \left( 1-D \left( w \right) \right) =0 }\), and, as the same happens for the logistic \(\mathrm{ 1/ \left( 1+e^{-w} \right)) }\), those conditions are equivalent to \(\mathrm{ lim_{w \rightarrow + \infty}w^{2}\frac{k’ \left( w \right) }{k \left( w \right) }=0 }\). As the variance of the logistic \(\mathrm{ \left( \rho =0 \right) }\) is \(\mathrm{ \pi ^{2}/3 }\), we have the general relation ( \(\mathrm{ M \left( W \right) }\) being zero)
\(\mathrm{ V ( W ) =\frac{ \pi ^{2}}{3} ( 1- \rho ) = \int _{- \infty}^{+ \infty}w^{2}~d~D( w ) = \int _{- \infty}^{+ \infty}w^{2}d ( \frac{1}{1+e^{-w}} ) + \int _{- \infty}^{+ \infty}w^{2}d ( \frac{k’ ( w ) }{k ( w ) } ) }\)
\(\mathrm{ \frac{ \pi ^{2}}{3}~ \rho =2 \int _{- \infty}^{+ \infty}w~\frac{k’ \left( w \right) }{k \left( w \right) }~, }\)
the integrated parts being zero as a consequence of
\(\mathrm{ lim_{w \rightarrow \pm \infty}w^{2}~\frac{k’ \left( w \right) }{k \left( w \right) }=0. }\)
The last integral is equal, by integration by parts, to
\(\mathrm{ \int _{- \infty}^{+ \infty}logk \left( w \right) ~d~w }\),
with the integrated part \(\mathrm{ w~log~k \left( w \right) \rightarrow 0 }\) when \(\mathrm{ w \rightarrow \pm \infty }\), as follows from the Fréchet derived inequalities.
We can summarize the results by giving the equivalent expressions for \(\mathrm{ \rho }\) :
\(\mathrm{ \rho =1-\frac{3}{ \pi ^{2}} \int _{- \infty}^{+ \infty}w^{2}d~D \left( w \right) =\frac{6}{ \pi ^{2}} \int _{- \infty}^{+ \infty}w \{ D \left( w \right) -L \left( w \right) \} d~w }\)
\(\mathrm{ =\frac{6}{ \pi ^{2}}\cdot \int _{- \infty}^{+ \infty} \left( IL \left( t \right) -ID \left( t \right) \right) d~t=-\frac{6}{ \pi ^{2}} \int _{- \infty}^{+ \infty}log\,k \left( w \right) d~w }\).
As \(\mathrm{ k \left( w \right) \leq 1 \left( -log~k \left( w \right) \geq 0 \right) }\) we have \(\mathrm{ 0 \leq \rho }\) , as could be expected from the positive association. It is very easy to show that for the diagonal case
\(\mathrm{k_{D} \left( w \right) =\frac{max \left( 1,e^{w} \right) }{1+e^{w}} }\)
we have \(\mathrm{ \rho=1 }\). The value of \(\mathrm{ \rho }\) does not identify the dependence function (or the distribution): \(\mathrm{ \rho }\) is the same for \(\mathrm{ k(w) }\) and \(\mathrm{ k(-w) }\) . But \(\mathrm{ \rho=0 }\), as \(\mathrm{ k(w)\leq1 }\), implies \(\mathrm{ k(w)=1 }\), or independence. Now writing \(\mathrm{ \rho }\) under the form
\(\mathrm{ \rho =1-\frac{6}{ \pi ^{2}} \int _{- \infty}^{+ \infty}log\frac{k \left( w \right) }{k_{D} \left( w \right) }d~w }\),
we see, analogously, that \(\mathrm{ \rho=1 }\) as \(\mathrm{ k \left( w \right) \geq k_{D} \left( w \right) }\) implies \(\mathrm{ k \left( w \right) =k_{D} \left( w \right) }\), or the diagonal case.
For the non-parametric correlation coefficients used, see see Fraser (1957) and Konijn (1950) for details.
in the case of differentiability, we show that the expression for the difference-sign correlation is
\(\mathrm{ \tau= \int _{- \infty}^{+ \infty}\frac{k’ \left( w \right) ^{2}}{k \left( w \right) }d~w-2 \int _{- \infty}^{+ \infty}\frac{e^{w}}{ \left( 1+e^{w} \right) ^{2}}log\,k \left( w \right) d~w }\),
which can also take the form
\(\mathrm{ \tau=1- \int _{- \infty}^{+ \infty}D \left( w \right) \left( 1-D \left( w \right) \right) d~w= \int _{- \infty}^{+ \infty} \left( D^{2} \left( w \right) -L^{2} \left( w \right) \right) d~w }\).
By integration by parts of \(\mathrm{ \int _{- \infty}^{+ \infty}D \left( w \right) \left( 1-D \left( w \right) \right) d~w~as~ \int _{- \infty}^{+ \infty}w~d~D \left( w \right) =0 }\), we get the simple expression
\(\mathrm{ \tau=1- \int _{- \infty}^{+ \infty}w~d~D^{2} \left( w \right) }\).
\(\mathrm{ \tilde{ \mu} =-~log~log~2 }\) being the median, we see that the medial correlation coefficient has the expression \(\mathrm{ v=4^{1-k \left( 0 \right) }-1 \left( \geq 0 \right) }\).
The probability of concordance, that is, the probability that the two pairs \(\mathrm{ \left( X_{1},Y_{1} \right) }\) and \(\mathrm{\left( X_{2},Y_{2} \right) }\) are such that \(\mathrm{X_{1}-X_{2} }\) and \(\mathrm{ Y_{1}-Y_{2} }\) have the same sign, being given by \(\mathrm{ \left( 1+ \tau \right) /2 }\), is always greater than \(\mathrm{ 1/2 }\).
Computations show easily that if \(\mathrm{ k \left( w \right) }\) is differentiable and \(\mathrm{ w_0 }\) is such that \(\mathrm{k’ \left( w_{0} \right) =0 }\) the value of \(\mathrm{ \Delta \left( k \right) }\) is, with \(\mathrm{ k_{0}=k \left( w_{0} \right) }\),
for \(\mathrm{ k \left( w \right) =k \left( -~w \right) }\) we have \(\mathrm{ w_0=0 }\). It is very easy to see from the expression of \(\mathrm{ k '\left( w \right) }\) has only an interval of solutions so that \(\mathrm{ k \left( w \right) }\) has the same value for all \(\mathrm{ w_0 }\) in the interval (a minimum). The same result an be obtained directly without recourse to the hypothesis of differentiability, as follows.
Denoting by \(\mathrm{ Z= \Lambda \left( x \right) ~ \Lambda \left( y \right) }\), the expression of \(\mathrm{ \Delta \left( k \right) }\) can be written \(\mathrm{ \left( as~k \left( w \right) \leq 1 \right) }\)
which for fixed \(\mathrm{ Z \left( \leq 1 \right) }\) has the maximum for the minimum \(\mathrm{ k_{0}=k \left( w_{0} \right) }\) of \(\mathrm{ k \left( w \right) }\). The maximum of \(\mathrm{ Z^{k \left( w_{0} \right) }-Z }\) is given by the previous value. Then the index of dependence
has the value \(\mathrm{ \Lambda \left( k \right) =k_{0}^{k_{0}/ \left( 1-k_{0} \right) }-k_{0}^{1/ \left( 1-k_{0} \right) },k_{0} }\) being the minimum of \(\mathrm{ k }\).
It can be recalled that the classical correlation coefficient is linearly invariant (independent of the margin parameters) and that all the others are transformation invariant (non-parametric).
Consider, finally, the case for bivariate minima with standard exponential margins. Then the mean values and variances are equal to 1; the covariance (evidently equal to the correlation coefficient) is, using the same formula as the one at the beginning, transformed to survival functions, \(\mathrm{ \left( \xi ~, \eta \geq 0 \right) }\)
where \(\mathrm{ S_{0} \left( \xi ~, \eta \right) =e^{- \xi }~e^{- \eta } }\) is the survival function for independence. By the change of variables \(\mathrm{ s= \xi + \eta ,u= \eta / \left( \xi + \eta \right) }\) we get
\(\mathrm{ = \int _{0}^{1}A^{-2} \left( u \right) d~u-1 \geq 0 }\) as \(\mathrm{ \left( {1}/{2} \leq \right) max \left( u,1-u \right) \leq A \left( u \right) \leq 1 }\)
with the value \(\mathrm{ \rho =0 }\) for independence \(\mathrm{ \left( A \left( u \right) =1 \right) }\) and \(\mathrm{ \rho =1 }\) for the diagonal case \(\mathrm{ \left( A \left( u \right) =max \left( u,1-u \right) \right) }\) as should be expected. Recall that \(\mathrm{ \rho }\) is linearly invariant although in this case only scale invariance matters.
The other dependence indicators, being non-parametric, have the same expression, after transformation from \(\mathrm{ k \left( w \right) }\) to \(\mathrm{ A \left( u \right) }\).
We have then:
for the grade correlation \(\mathrm{ \chi =12 \int _{0}^{1}\frac{d~u}{ \left( 1+A \left( u \right) \right) ^{2}}-3 }\)
for the medial correlation \(\mathrm{ v=4^{1-A \left( 1/2 \right) }-1 }\) ;
for the index of dependence \(\mathrm{ \Lambda \left( A \right) =A_{0}^{A_{0}/ \left( 1-A_{0} \right) }-A_{0}^{1/ \left( 1-A_{0} \right) } }\) where \(\mathrm{ A_{0} \left( \geq 1/2 \right) }\) denotes the minimum of \(\mathrm{ A \left( u \right) }\).
As \(\mathrm{ \xi =e^{-X}, \eta =e^{-Y},u= \left( 1+e^{w} \right) ^{-1} }\), we see that \(\mathrm{ B \left( u \right) =Prob \{{ \frac{ \eta }{ \xi + \eta } \leq u } \}=Prob \{{ Y-X \geq log\frac{1-u}{u} }\} = }\)\(\mathrm{ 1-D( log\frac{1-u}{u} ) =u+\frac{u \left( 1-u \right) A’ \left( u \right) }{A \left( u \right) } }\) and \(\mathrm{ \tau=1- \int _{0}^{1}\frac{B \left( u \right) \left( 1-B \left( u \right) \right) }{u \left( 1-u \right) }d~u~=1- \int _{0}^{1}log\frac{1-u}{u}d~B^{2} \left( u \right) because \int _{0}^{1}log\frac{1-u}{u}d~B \left( u \right) =0 }\)
Let us now discuss some results on regression in the case of reduced Gumbel margins. For convenience, we will only consider the regression of \(\mathrm{ Y }\) on \(\mathrm{ X }\), the regression of \(\mathrm{ X }\) on \(\mathrm{ Y }\) being dealt with in the same way with the substitution of \(\mathrm{ k \left( w \right) by~k \left( -w \right) }\) as said before. The linear regression line for reduced margins is evidently \(\mathrm{ L_{y} ( x ) = \gamma + \rho \left( k \right) \left( x- \gamma \right) , \rho \left( k \right) }\)being given before.
As \(\mathrm{ Prob \{ Y \leq y \vert X=x \} = \Lambda \left( y \vert x \right) }\) is given, in the case of existence of a density, by
\(\mathrm{ \bar{y} ( x \vert k ) = \int _{- \infty}^{+ \infty}y~d~ \Lambda ( y \vert x ) }\)
and as \(\mathrm{ \gamma = \int _{- \infty}^{+ \infty}y~d~ \Lambda \left( y \right) }\) we get, by integration by parts,
\(\mathrm{ \bar{y} \left( x \vert k \right) = \gamma + \int _{- \infty}^{+ \infty} \left[ \Lambda \left( y \right) - \Lambda \left( y \vert x \right) \right]d~y }\)
\(\mathrm{ = \gamma + \int _{- \infty}^{+ \infty} {exp( e^{-x}\, {e^{-w}})- [ k \left( w \right)+ \left( 1+e^{-w} \right) k' \left( w \right)] } }\).
\(\mathrm{ exp \{ e^{-x} \left[ \left( 1+e^{-w} \right) k \left( w \right) -1 \right] \} d~w }\)
\(\mathrm{ = \gamma +e^{x}+ \int _{- \infty}^{+ \infty} [ e^{-e^{-x}\,e^{-w}}- \left( 1+e^{-w} \right) k \left( w \right) e^{-e^{-x} [ \left( 1+e^{-w} \right) k \left( w \right) -1] }d~w }\)
The correlation ratio is given by
\(\mathrm{ R^{2} \left( y \vert x;k \right) =\frac{6}{ \pi ^{2}} \int _{- \infty}^{+ \infty} \left( \bar{y} \left( x \vert k \right) - \gamma \right) ^{2}d~ \Lambda \left( x \right) }\)
\(\mathrm{ =\frac{6}{ \pi ^{2}} \int _{- \infty}^{+ \infty} \int _{}^{}d~v~d~w [ \frac{e^{-v}~e^{-w}}{1+e^{-v}+e^{-w}}-2\frac{e^{-v} \left( \left( 1+e^{-w} \right) k \left( w \right) -1 \right) }{e^{-v}+ \left( 1+e^{-w} \right) k \left( w \right) }+ }\)
\(\mathrm{ +\frac{ \left( \left( 1+e^{-v} \right) k \left( v \right) -1 \right) \left( \left( 1+e^{-w} \right) k \left( w \right) -1 \right) }{ \left( 1+e^{-w} \right) k \left( v \right) + \left( 1+e^{-w} \right) k \left( w \right) -1} ] }\).
Median regression can be defined as the solution of the equation \(\mathrm{ \Lambda \left( \tilde{y} \vert x \right) =1/2 }\); it takes the form \(\mathrm{ \tilde{y}(x \vert k ) =x+ \varphi \left( x \right) }\) where \(\mathrm{ \varphi \left( x \right) }\) is given by \(\mathrm{ e^{-x} \left( 1+e^{-\varphi} \right) k \left( \varphi \right) -log \left[ k \left( \varphi \right) + \left( 1+e^{- \varphi } \right) k’ \left( \varphi \right) \right] = \left( log~2 \right) e^{-x} }\). In two of the models (logistic and mixed) described below the curves are approximately linear—see Gumbel and Mustafi (1968).
Evidently where we have margin parameters we must substitute \(\mathrm{ x }\) and \(\mathrm{ y }\) by \(\mathrm{ \left( x- \lambda _{x} \right) / \delta _{x} }\) and \(\mathrm{ \left( y- \lambda _{y} \right) / \delta _{y} }\) in all the three cases above. For instance, for linear regression we have \(\mathrm{ \frac{L_{y} \left( x \right) – \lambda _{y}}{ \delta _{y}}= \gamma + \rho ( \frac{x- \lambda _{x}}{ \delta _{x}}- \gamma ) ~or~L_{y} \left( x \right) = \lambda _{y}+ \gamma ~ \delta _{y}+ \rho ~ \delta _{y}\frac{x- \lambda _{x}- \gamma ~ \delta _{x}}{ \delta _{x}} }\), which can take the usual form \(\mathrm{ L_{y} ( x ) = \mu _{y}+ \rho \frac{ \sigma _{y}}{ \sigma _{x}} ( x- \mu _{x} ) }\).
Let us return, from now on, to the study of regression for reduced values of \(\mathrm{ \left( X,Y \right) }\).
As is well known, see Cramér (1946), the mean-square error of the linear regression for reduced Gumbel margins is
it can be considered the index of non-linearity, evaluating the improvement in using non-linear regression — see Tiago de Oliveira (1974).
It can observe that in all the cases studied so far, \(\mathrm{ NL }\) is very small and so the improvement is negligible. This is easily understandable because \(\mathrm{ L_{y}( X) }\) and \(\mathrm{ \bar{y} \left( x \right) }\) are, in fact, very different for \(\mathrm{x<-2~or~x>5 }\) whose total probability is very low (about .007 or less).
The bounds for \(\mathrm{ k \left( w \right) }\) can give bounds for \(\mathrm{ \bar{y} \left( x \right) }\), using the expression of \(\mathrm{ \bar{y}(x \vert k ) - \gamma }\) given before, but the bounds are so large that they are not useful. In fact we have
Let us see how we can, if needed, improve the regression lines, through the use of orthogonal polynomials with respect to \(\mathrm{ \Lambda \left( x \right) }\), that we are going to construct.
Let \(\mathrm{ \{ \Psi _{p} ( x ) \} }\) denote the complete set of orthogonal polynomials with respect to \(\mathrm{ \Lambda \left( x \right) }\), i.e.
\(\mathrm{ \int _{- \infty}^{+ \infty} \Psi _{p} \left( x \right) \Psi _{q} \left( x \right) d~ \Lambda \left( x \right) = \delta _{pq} }\)
where \(\mathrm{ \delta _{pq} }\) is the Kroneker symbol \(\mathrm{ \left( \delta _{pp}=1, \delta _{pq}=0~if~p \neq q \right) , \Psi _{p} \left( x \right) }\)having the degree \(\mathrm{ p }\). Recall that \(\mathrm{ \{ \left( x- \gamma \right) ^{p} \} }\) is a complete set of polynomials (not orthogonal) in any bounded interval.
Let us take \(\mathrm{ \Psi _{0} \left( x \right) =1~and~ \Psi _{k+1} \left( x \right) = \sum _{0}^{k}a_{kj} \Psi _{j} \left( x \right) +b_{k+1} \left( x- \gamma \right) ^{k+1} }\). The orthonormality conditions give for \(\mathrm{ t=0,1,\dots,k+1 }\)
\(\mathrm{ \int _{- \infty}^{+ \infty} \Psi _{k+1} \left( x \right) \Psi _{t} \left( x \right) d~ \Lambda \left( x \right) = \delta _{k+1,t} }\)
The results contained in this section were, in part, given in Tiago de Oliveira (1962/63) and (1964). They refer to bivariate pairs with reduced Gumbel margins.
Let us prove that
If\( \left( X,Y \right)\)is an extremal pair, the distribution of \(\left( Y \vert X \right) \)is extremal if and only if\( \left( X,Y \right)\)is an independent pair.
It is evident that if \(\mathrm{ \left( X,Y \right) }\) is an independent pair \(\mathrm{ \left( Y \vert X \right) =Y }\) is an extremal variate. Let us prove the converse.
Denoting by (as in the previous consideration of regression)\(\mathrm{ \Lambda \left( y \vert x \right) =e^{-( e^{-x}+e^{-y }) k ( y-x ) +e^{-x}} \{ k ( y-x ) + ( 1+e^{-y+x ) }k’ ( y-x ) \} }\)
the distribution function of \(\mathrm{ \left( Y \vert X \right) }\), we must have
\(\mathrm{ \Lambda ( y \vert x ) = \Lambda ( \frac{y-v ( x ) }{ \tau ( x ) } ) }\).
so that \(\mathrm{ \beta =1, \alpha =1 }\). Consequently we have
\(\mathrm{ k \left( w \right) =1- \alpha ’\frac{e^{- \beta ’w}}{1+e^{-w}} }\) , \(\mathrm{ \left( \beta ’ \leq 1 \right) }\),
Condition II) gives \(\mathrm{ ~ \alpha ’\leq0 }\) so that and \(\mathrm{ ~ \alpha ’=0 }\) and \(\mathrm{ k \left( w \right) =1 }\). The case \(\mathrm{ \beta' \leq \beta }\) is dealt with in the same way.
From the expression \(\mathrm{ \rho =-\frac{6}{ \pi ^{2}} \int _{- \infty}^{+ \infty}log~k \left( w \right) d~w }\), we see that \(\mathrm{ \rho =0 }\) if and only if \(\mathrm{ k \left( w \right) =1 }\), that is, if the extremal pair is independent.
An extremal pair \( \left( X,Y \right)\)is independent if and only if any pair \(\left( X_{ \theta },Y_{ \theta } \right) ,X_{ \theta }=cos~ \theta X+sen~ \theta Y+a,Y_{ \theta }=-sen~ \theta X+cos~ \theta Y+b~for~any~ \theta \neq 0, \pi /2, \pi ,3 \pi /2 \)have equal variances.
\(Z=a+a'X+a"Y \)has an extremal distribution when \( \left( X,Y \right)\) is an extremal independent pair if and only if \(a’=0~or~a"=0 \).
One half of the proof is straightforward. Let us prove the converse. We can suppose that \(\mathrm{ X,Y,and\,Z }\) have reduced extremal distributions, only having to transform linearly \(\mathrm{ ~Z }\) if this is not the case (the coefficients of \(\mathrm{ ~X }\) and \(\mathrm{ ~Y }\) being proportional to \(\mathrm{ ~a' }\) and \(\mathrm{ ~a'' }\) ). As \(\mathrm{ M \left( X \right) =M \left( Y \right) =M \left( Z \right) = \gamma }\) we obtain
\(\mathrm{ \left( 6-2 \beta _{2} \right) a’^{2}~a"^{2}=0;as~ \beta _{2}=5.4 }\) we have \(\mathrm{ a’=0 }\) or \(\mathrm{ a’'=0 }\).
As an immediate consequence the pair \(\mathrm{ ( X,Y ) ( X=a+a’X_{0}+a"Y_{0},Y=b+b’X_{0}+b"Y_{0} ) }\)is an extremal pair, \(\mathrm{ \left( X_{0},Y_{0} \right) }\) being an independent extremal pair, if and only if \(\mathrm{ a"=b’=0~or~a’=b"=0 }\).
Another result, corresponding to the characterization of \(\Lambda \left( x,y \right) \), is that \(Z=max \left( \alpha\, X+a, \beta\, Y+b \right) \left( \alpha , \beta >0 \right) \)is extremal, \( \left( X,Y \right)\)being a reduced extremal pair, if and only if \(\alpha=\beta\).
As \(\mathrm{\Lambda \left( z \right) =Prob \{{ max \left( \alpha \,X+a, \beta \,Y+b \right) \leq z }\} = \Lambda ( \frac{z-a}{ \alpha },\frac{z-b}{ \beta } ) }\)we must seek conditions such that
Multiplying this equation by \(\mathrm{ e^{v} }\) and letting \(\mathrm{ v \rightarrow \pm \infty }\) we have a limit if and only if \(\mathrm{ \alpha ’= \beta ’=1~or~ \alpha = \beta = \delta }\) and then
which relates \(\mathrm{ \lambda }\) and \(\mathrm{ \delta }\).
\(\Lambda \left( x,y \vert \theta \right) =exp \left( - \left( e^{-x}+e^{-y} \right) k \left( y-x \vert \theta \right) \right) \)cannot have sufficient statistics (of rank 1) if for some \(\theta = \theta _{0} \)we have independence.
As is well known, if \(\mathrm{ \Lambda \left( x,y \vert \theta \right) }\) does have a sufficient statistic for \(\mathrm{ \theta }\) (of rank 1), its density
For \(\mathrm{ \theta = \theta _{0}=0 }\) we will suppose independence and thus \(\mathrm{ P \left( w \vert 0 \right) =0 }\) and \(\mathrm{ Q \left( w \vert 0 \right) =e^{-w} }\); we can suppose \(\mathrm{ a \left( 0 \right) =d \left( 0 \right) =0 }\) by a simple transformation and, as \(\mathrm{ a’ \left( 0 \right) }\) is a factor of the variance , we can suppose \(\mathrm{ a’ \left( 0 \right) =1 }\).
Then, putting \(\mathrm{ \theta = 0 }\), we obtain \(\mathrm{ c \left( x,y \right) =x+y+e^{-x}+e^{-y} }\), so that the condition for sufficiency can be written
Denoting by \(\mathrm{ h \left( w \right) =\frac{ \partial ~k \left( w \vert \theta \right) }{ \partial ~ \theta } \vert _{ \theta =0} }\) and deriving the relation above in order to \(\theta \), we obtain
Putting \(\mathrm{ y=w+x }\) and letting \(\mathrm{ x \rightarrow + \infty }\) we see that we must have \(\mathrm{ P \left( w \vert 0 \right) =0 }\) for any \(\mathrm{ \theta }\), which implies independence.
As regards correlation, we can also give some results.
The first one is that symmetrization reduces correlation.
If \(\mathrm{ \Lambda _{1} ( x,y ) }\) and \(\mathrm{ \Lambda _{2} ( x,y ) }\) are two distribution functions of bivariate reduced extremes with dependence functions \(\mathrm{ k_{1} ( w ) }\) and \(\mathrm{ k_{2} \left( w \right) }\) and correlation coefficients \(\mathrm{ \rho _{i}=-\frac{6}{ \pi ^{2}} \int _{- \infty}^{+ \infty}log~k_{i} \left( w \right) d~w }\), consider now the dependence function \(\mathrm{ \bar{k} \left( w \right) = \left( k_{i} \left( w \right) +k_{2} \left( w \right) \right) /2 }\) — also a dependence function by the mix-technique — and as \(\mathrm{ \frac{k_{1}+k_{2}}{2} \geq \sqrt[]{k_{1}~k_{2}} }\) we obtain
\(\mathrm{ \varphi \left( k_{0} \right) }\) is a decreasing function of \(\mathrm{ k_{0} }\), and the relation for \(\mathrm{ k_{0} }\) can be rewritten as \(\mathrm{ 1/2 \leq k_{0} \leq \varphi^{-1} \left( \rho \right) \leq 1 }\); and as the index of dependence \(\mathrm{ \Delta \left( k_{0} \right) }\) is also a decreasing function of \(\mathrm{ k_{0} }\), we get
and so the knowledge of \(\mathrm{ \rho }\) gives a lower bound for \(\mathrm{ \Delta \left( k_{0} \right) }\).
An inequality with the reverse direction was not obtained although such a result seems possible as we know that for \(\mathrm{ \rho =0 }\) we have \(\mathrm{ k_{0}=1 ( k \left( w \right) =1 , independence )}\) and for \(\mathrm{ \rho =1 }\) we have \(\mathrm{ k_{0}=1/2 ( k \left( w \right) =\frac{max \left( 1,e^{w} \right) }{1+e^{w}} }\), diagonal case) against the fact that by the previous inequality for \(\mathrm{ \rho =1 }\) we have \(\mathrm{ k_{0}=1/2~ }\) and \(\mathrm{ \Delta \left( 1/2 \right) =1 }\) but for \(\mathrm{ \rho =0 }\) we have \(\mathrm{ k_{0}=1 }\) and \(\mathrm{ 0 \leq \Delta \left( k_{0} \right) \leq 1 }\).
An important and unexpected relation between \(\mathrm{ \rho }\) and \(\mathrm{ \tau }\) is
In fact, as we have \(\mathrm{ \rho =1-\frac{3}{ \pi ^{2}} \int _{- \infty}^{+ \infty}w^{2}d~D \left( w \right) ~and~ \tau=1- \int _{- \infty}^{+ \infty}wd~D^{2} \left( w \right) }\), the Schwarz inequality gives the desired result.
The relation can be written as \(\mathrm{ 1- \tau \leq \frac{2 \pi }{3}~\sqrt[]{1- \rho }~or~1-\frac{2 \pi }{3}\sqrt[]{1- \rho } \leq \tau }\). Then we get
Also, given \(\mathrm{ \tau }\) , we get \(\mathrm{ 0 \leq \rho \leq 1-\frac{9}{4 \pi ^{2}} \left( 1- \tau \right) ^{2} }\).
Let \(\mathrm{ k_{1} \left( w \right) }\) and \(\mathrm{ k_{2} \left( w \right) }\) be the dependence functions and \(\mathrm{ D_{1} \left( w \right) }\) and \(\mathrm{ D_{2} \left( w \right) }\) the distribution functions of the reduced extremes of the distribution functions \(\mathrm{ \Lambda _{1} \left( x,y \right) and \,\Lambda _{2} \left( x,y \right)}\).
The difference between the correlation coefficients is
\(\mathrm{ \rho _{2}- \rho _{1}=-\frac{6}{ \pi ^{2}} \int _{- \infty}^{+ \infty}log\frac{k_{2} \left( w \right) }{k_{1} \left( w \right) }d~w }\).
Let us denote by \(\mathrm{ Q_{1}=inf\,\frac{k_{2} \left( w \right) }{k_{1} \left( w \right) } }\) and \(\mathrm{ Q_{2}=sup\,\frac{k_{2} \left( w \right) }{k_{1} \left( w \right) } }\); as for \(\mathrm{ w \rightarrow \pm \infty }\) we know that \(\mathrm{ k_{2} \left( w \right) /k_{1} \left( w \right) \rightarrow 1 }\) and by the relation \(\mathrm{ k_{D} \left( w \right) ={1}/{1}+e^{- \vert w \vert } \leq k_{1} \left( w \right),k_{2} \left( w \right) \leq 1=k_{1} \left( w \right) }\), we get the inequalities \(\mathrm{ 1/2 \leq Q_{1} \leq 1 \leq Q_{2} \leq 2 }\).
As we have \(\mathrm{ Q_{1}\,k_{1} ( w ) \leq k_{2} ( w ) \leq Q_{2}\,k_{1} ( w ) }\) and by the previous relations we get \(\mathrm{ max ( k_{D} ( w ) ,Q_{1}\,k_{1} ( w ) ) \leq k_{2} ( w ) \leq min ( 1,Q_{2}\,k_{1}( w ) ) }\), then
with \(\mathrm{ k_{D} \left( w \right) \leq k_{1} \left( w \right) ,k_{2} \left( w \right) \leq 1 }\).
Then \(\mathrm{ 1/k_{1} \left( w \right) \leq 1/k_{D} \left( w \right) }\) and, thus, \(\mathrm{ min \left( Q_{2},1/k_{1} \left( w \right) \right) \leq min \left( Q_{2},1/k_{D} \left( w \right) \right) }\)and so
\(\mathrm{ 1 \leq 1/k_{1} \left( w \right) }\) so \(\mathrm{ max ( \frac{k_{D} \left( w \right) }{k_{1} \left( w \right) },Q_{1} ) \geq max \left( k_{D} \left( w \right) ,Q_{1} \right) }\)
Also, the exchange of \(\mathrm{ k_1 }\) and \(\mathrm{ k_2 }\) (exchanging \(\mathrm{ Q_{1} }\) and \(\mathrm{ Q_{2} }\) for \(\mathrm{ Q^{-1}_2 }\) and \(\mathrm{ Q^{-1}_1 }\) ) gives the same result and, so, \(\mathrm{ \vert \rho _{2}- \rho _{1} \vert \leq max ( \varphi ( Q_{2}^{-1} ) ,\varphi ( Q_{1} ) ) }\) and, as \(\mathrm{ \varphi (Q) }\) is an increasing function for \(\mathrm{ Q^{-1}>1 }\), we get
where \(\mathrm{ Q_{1} }\) and \(\mathrm{ Q^{-1}_2 }\) are analogous to \(\mathrm{ k_0}\) in the beginning, and the technique of computation is the same. As \(\mathrm{ ( Q_{2}-1 ) e^{-{Q_{2}}/{ ( Q_{2}}-1 ) } \leq Q_{2}-1~and~ \left( 1-Q_{1} \right) Q_{1}^{{Q_{1}}/{ \left( Q_{1}-1 \right) }} \leq 1-Q_{1} }\)are rough approximations, a very rough approximation is \(\mathrm{ sup \vert \Lambda _{1}- \Lambda _{2} \vert \leq max ( Q_{2}-1,1-Q_{1} ) }\). If we consider the two non-differentiable models, biextremal with
this distance is an increasing function of \(\mathrm{ Q_{2}=1+ \theta \left( 1- \theta \right) }\)whose maximum value is \(\mathrm{ Q_{2}=1+1/4 }\). So, with equal correlation coefficients for the biextremal model and its dual, we have the maximum distance equal to
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