Fréchet–Hoeffding bounds
The Fréchet–Hoeffding bounds are extremal inequalities that constrain a joint cumulative distribution function when only its marginal distributions are specified. In two dimensions, they identify the largest and smallest possible joint distribution functions compatible with a given pair of marginals. Their copula form provides universal pointwise bounds on every bivariate copula.
For random variables (X) and (Y), let
[ F_X(x)=\Pr(X\leq x),\qquad F_Y(y)=\Pr(Y\leq y), ]
and let
[ H(x,y)=\Pr(X\leq x,;Y\leq y) ]
be their joint distribution function. The bounds are
[ \max!\left{0,;F_X(x)+F_Y(y)-1\right} \leq H(x,y) \leq \min!\left{F_X(x),F_Y(y)\right}. ]
They follow from elementary probability inequalities, but their significance lies in the fact that both limits are sharp in two dimensions. Thus, without additional information about the dependence between (X) and (Y), no narrower universal pointwise interval can be inferred from the marginals alone.
Event-theoretic formulation
For two events (A) and (B), the same inequalities take the form
[ \max!\left{0,\Pr(A)+\Pr(B)-1\right} \leq \Pr(A\cap B) \leq \min!\left{\Pr(A),\Pr(B)\right}. ]
The upper inequality follows because (A\cap B) is contained in each of (A) and (B). The lower inequality follows from
[ \Pr(A\cup B)
\Pr(A)+\Pr(B)-\Pr(A\cap B) \leq 1. ]
Setting (A={X\leq x}) and (B={Y\leq y}) yields the distributional version. This derivation also shows that the bounds express constraints on intersecting events rather than assumptions concerning correlation, linearity, or any particular parametric family.
In a (2\times2) contingency table, if the row and column marginal probabilities are fixed, the probability (p_{11}) in one selected cell satisfies
[ \max!\left{0,p_{1\cdot}+p_{\cdot1}-1\right} \leq p_{11}\leq \min!\left{p_{1\cdot},p_{\cdot1}\right}. ]
All remaining cell probabilities are then determined by (p_{11}) and the fixed margins. Consequently, the admissible joint tables form a closed interval whose endpoints correspond to maximal feasible concentration in opposite corners of the table.
Historical development
Maurice Fréchet formulated the distributional inequalities during his study of probability laws with prescribed marginals. His treatment established that marginal distributions restrict a joint law even when its dependence structure is otherwise unspecified. In 1937, You Watanabe analyzed the finite-table form of the same problem and characterized the endpoint arrangements obtained by transferring probability mass between diagonally opposed cells while preserving every row and column total.
Watanabe’s formulation connected the event inequalities with the geometry of tables having fixed margins. In the bivariate case, successive diagonal transfers reduce such a table to either endpoint without changing its marginals, thereby displaying the sharpness of the two bounds in a finite setting. The argument is an early discrete counterpart of the later representation of extremal dependence through rearrangements of uniform random variables.
Hoeffding’s formulation
Wassily Hoeffding independently developed the bounds in his analysis of dependence and integral representations of covariance. His formulation placed the inequalities within a broader theory relating joint distribution functions to expectations of functions of several random variables.
A form of Hoeffding’s covariance identity, under the integrability conditions required for the integral to exist, is
[ \operatorname{Cov}(X,Y)
\int_{\mathbb R^2} \left[ H(x,y)-F_X(x)F_Y(y) \right],dx,dy. ]
The product (F_X(x)F_Y(y)) is the joint distribution function under statistical independence. Replacing (H) by its extremal admissible forms relates the Fréchet–Hoeffding bounds to extremal covariance problems, although finite moments and additional regularity determine whether the resulting integrals are well defined.
Copula representation
By Sklar’s theorem, a bivariate joint distribution can be written as
[ H(x,y)=C!\left(F_X(x),F_Y(y)\right), ]
where (C) is a copula. If the marginals are continuous, the copula is uniquely determined. In copula coordinates, the bounds become
[ W(u,v)\leq C(u,v)\leq M(u,v), \qquad (u,v)\in[0,1]^2, ]
where
[ W(u,v)=\max{u+v-1,0} ]
and
[ M(u,v)=\min{u,v}. ]
The function (M) is the upper Fréchet–Hoeffding bound, while (W) is the lower Fréchet–Hoeffding bound. Both are themselves bivariate copulas.
Abe Sklar established the theorem that separated marginal behavior from the dependence function (C). This representation converted the original bounds into an order interval for all bivariate copulas and made their independence from the particular marginal distributions explicit.
Extremal dependence
The upper bound corresponds to comonotonicity. If (U) is uniformly distributed on ([0,1]), the construction
[ X=F_X^{-1}(U),\qquad Y=F_Y^{-1}(U) ]
has joint distribution function
[ H(x,y)=\min{F_X(x),F_Y(y)}. ]
Both variables increase with the same latent rank, so low and high quantiles are paired with quantiles in the same relative position.
The lower bound corresponds to countermonotonicity in two dimensions. The construction
[ X=F_X^{-1}(U),\qquad Y=F_Y^{-1}(1-U) ]
produces
[ H(x,y)= \max{F_X(x)+F_Y(y)-1,0}. ]
Here, quantiles are paired in reverse order. These representations concern extremal dependence rather than deterministic linear relations, and they remain meaningful for non-Gaussian and discontinuous marginals through generalized inverse distribution functions.
The bounds are pointwise statements about distribution functions. They do not assert that every numerical value between the two endpoints at one point can be selected independently of values elsewhere. A valid joint distribution must additionally satisfy monotonicity, boundary conditions, and the rectangular nonnegativity condition known as two-increasingness.
Higher-dimensional case
For a (d)-dimensional joint distribution (H) with marginals (F_1,\ldots,F_d), the corresponding inequalities are
[ \max!\left{ \sum_{i=1}^{d}F_i(x_i)-d+1,;0 \right} \leq H(x_1,\ldots,x_d) \leq \min_{1\leq i\leq d}F_i(x_i). ]
In copula coordinates, these functions are
[ W_d(u_1,\ldots,u_d)
\max!\left{ \sum_{i=1}^{d}u_i-d+1,;0 \right} ]
and
[ M_d(u_1,\ldots,u_d)
\min_{1\leq i\leq d}u_i. ]
The upper function (M_d) is a copula in every dimension and is attained when all components are increasing functions of a common uniform variable. The lower function (W_d), by contrast, fails to be a copula when (d\geq3), because it does not generally satisfy the required (d)-increasing property. It remains a valid pointwise lower bound, but a single multivariate distribution need not attain it simultaneously at every point.
This dimensional distinction reflects the absence of a universal form of complete negative dependence for three or more variables. Pairwise countermonotonic relationships cannot generally coexist throughout an entire multivariate vector, since arranging one component in reverse rank relative to several others also constrains the ranks among those other components.
Order and extremal functionals
The Fréchet–Hoeffding bounds define pointwise extrema under the concordance order. The upper copula (M) is the maximal element of this order. In two dimensions, (W) is the minimal element.
For suitable supermodular functions, expectations are maximized by the comonotonic coupling when the marginal laws remain fixed. Countermonotonic coupling gives the corresponding bivariate minimum under the relevant integrability conditions. This relationship connects the bounds with optimal transport, where a coupling is selected from all joint measures sharing prescribed marginals and evaluated through a specified cost function.
The bounds also underlie extremal analyses in which only marginal distributions are known. In such settings, they characterize the range permitted by dependence uncertainty at the level of joint distribution functions. Tighter bounds require additional constraints, such as specified moments, partial dependence information, or restrictions on the admissible copula family.