ON SPLINE APPROXIMATION FOR BIVARIATE FUNCTIONS OF INCREASING CONVEX TYPE

. The motivation of the paper is to construct the largest and smallest families of functions that allow us to generate the bivariate continuous stochastic orderings of increasing convex type introduced recently in Denuit et al. (1999). The main step will consist in deriving a spline approximation for bivariate continuous increasing convex functions, which extends to the bivariate case a fundamental result obtained by Popoviciu (1941).


INTRODUCTION
The remarkable works of Tiberiu Popoviciu on the theory of convexity have deeply influenced certain research areas in numerical analysis, in theory of approximation and in functional analysis.A good idea of such developments can be found in the special issues 1-2 (vol. 26, 1997), of the Revue d'Analyse Numérique et de Théorie de l'Approximation, dedicated to the memory of Tiberiu Popoviciu.
Recently, another central role of the theory of convexity has also been pointed out in probability and statistics within the theory of stochastic orderings.This is not really surprising since the question of comparison is often encountered in the works of Tiberiu Popoviciu (as underlined, e.g., by E. Popoviciu [11]).
So, wide classes of stochastic orderings, univariate or bivariate, discrete or continuous, of (increasing) convex type have been introduced in order to compare random variables, univariate or bivariate, discrete or continuous.Roughly, a random variable is said to be smaller than another one in that sense that if the expectation of any (increasing) convex function of this random variable is smaller than for the other variable.
A problem of interest on its own and for certain applications is the derivation of the largest and smallest families of functions that allow us to generate the orderings.This question has been solved in Denuit et al. [4] for the univariate continuous case.Our purpose in the present paper is to determine the extremal generators for the bivariate continuous increasing convex stochastic orderings.The main step here will consist in constructing a spline approximation for bivariate continuous increasing convex functions.This extends to the bivariate case a fundamental result obtained by Popoviciu [14] and reexamined later by Bojanic and Roulier [2] and Dadu [3], inter alia.
For the notation in the sequel, the real line is denoted by R, the set of the non-negative integers by N and any point (x 1 , x 2 ) of the real space R 2 by an underlined small letter x.The vector of ones, that is (1, 1), is written as 1; similarly, 2 = (2, 2) and so on; x ± y stands for (x 1 ± y 1 , x 2 ± y 2 ).The space R 2 is endowed with the usual componentwise partial order, that is x ≤ y if + is equal to x k when x k > 0 and 0 otherwise (with the convention that x 0 + is equal to 1 when x > 0 and 0 otherwise).

BIVARIATE REAL FUNCTIONS AND STOCHASTIC ORDERINGS OF INCREASING CONVEX TYPE
A stochastic ordering is any binary relation defined on a set of probability measures and that allows us to compare any pair of these probability measures.Thus, it translates the notions of being greater or being more variable, for instance, to probability measures.Usually, the stochastic orderings under interest are partial orderings, i.e. binary relations satisfying the properties of reflexivity, transitivity and anti-symmetry.
In practice, it is often more convenient to work with random variables rather than with probability measures.A random variable is said to be smaller than another random variable in the sense when this ordering holds for their probability measures.Note that, as a consequence, the property of antisymmetry is lost.
A number of stochastic orderings have been introduced during the last two decades, mostly motivated by different areas of applications (statistics, queueing theory, reliability theory, economics, biomathematics, actuarial sciences, physics. ..).They gave rise to a rich and abundant literature; see, e.g., the books by Shaked and Shanthikumar [15] and Stoyan [16], and the classified bibliography by Mosler and Scarsini [9].
A rather general class of bivariate stochastic orderings is the class of integral orderings generated by some cones of bivariate functions (see, e.g., Marshall [8]).Let X and Y be a pair of bivariate random variables assumed to be continuous and valued in an interval Then, X is said to be smaller than Y for the integral stochastic ordering Eφ(X) ≤ Eφ(Y ), for all functions φ ∈ F, provided that the expectations exist.
In the present work, we will be concerned with a particular class of bivariate continuous integral orderings introduced in Denuit et al. [5].This class is generated by the cone of the bivariate continuous increasing convex functions on [a, b], hence its appellation of bivariate continuous of increasing convex type.
Specifically, given any s ∈ N 2 with s−cx be the family of functions φ defined as where is assumed to exist.For reasons given below (see ( 8)), s−icx denotes the subfamily of the regular s-increasing convex functions defined as Then, X is said to be smaller than Y in the s-increasing convex (resp.s convex) ordering, which is denoted by The notion of convex functions in the sense of Popoviciu [12] is more general than that defined in (2).Let us first recall the definition of the divided difference operator.In the univariate case, given a function φ : [a, b] → R and points x 0 < x 1 < . . .< x s ∈ [a, b], with s ∈ N, this operator is defined recursively by [x i ]φ = φ(x i ), i = 0, 1, . . ., s, and In the bivariate case, given a function φ : [a, b] → R and points As above, the family s−icx of the continuous s-increasing convex functions is defined by ( 7) s−cx has not necessarily a partial derivative φ (s 1 ,s 2 ) (although the s-convexity implies certain regularity properties); therefore, ( 9) s−icx , for some s ≥ 2, is continuous on [a, b]; this is not true, however, when s 1 or s 2 = 1.
We now introduce the following family of bivariate functions, denoted by Note that the functions in (10) have a very simple product form.It is easily seen that these functions are continuous s-increasing convex, i.e. (11) U Coming back to the notion of integral stochastic ordering F , it is interesting, from a theoretical point of view as well as for certain applications, to substitute for the generating cone F, either a dense subfamily of functions contained in F, or a larger family corresponding to the closure of F in some topology.The smallest and largest such classes, F and F say, are called the minimal and maximal generators (Müller [10]).Hereafter, we aim to establish that for the s−icx can be uniformly approximated by appropriate spline functions.

SPLINE APPROXIMATION FOR FUNCTIONS IN U [a,b] s−icx
For simplicity, we consider the particular interval [0, 1]; the general case [a, b] follows by straightforward substitutions.
We are going to establish that the family of functions For that, we will mainly apply to the bivariate case an argument of Bojanic and Roulier [2], and which is based on an intermediate uniform approximation of any function φ in U In the sequel, any partial derivative which will be used is assumed to exist.

Given any continuous function
In the next lemma, we show that the Bernstein polynomial of any s-increasing convex function is also a (regular) s-increasing convex function.This property is the bivariate extension of a classical result by Popoviciu [13].
Let us return to the original problem of the extremal generators for the bivariate stochastic orderings of increasing convex type.The main result is Corollary 3.6 below which deals with the maximal generator F. s−icx .By Proposition 3.3, φ is the uniform limit of some sequence of functions φ (n) , implying that Eφ (n) (X) → Eφ(X) and Eφ (n) (Y ) → Eφ(Y ) as n → +∞.Moreover, these φ (n) 's are non-negative linear combinations of functions in U s−icx , so that by Corollary 3.5, Eφ (n) (X) ≤ Eφ (n) (Y ) for all n.Therefore, we deduce that Eφ(X) ≤ Eφ(Y ).
[a,b] s−icx ordering, s ≥ 1, the extremal generators are F = U [a,b] s−icx and F = U [a,b]s−icx .The proof for F will be immediate from a known approximation property.For F, the result will follow directly by showing that any function φ in U[a,b] [a,b]s−icx , s ≥ 2, by Bernstein polynomials.

[0, 1 ]Proposition 3 . 3 .
s−icx , s ≥ 1, we are going to build a sequence of spline functions ψ (n) m (φ, .)that are non-negative linear combinations of functions in U [a,b] s−icx , and that converge uniformly to the polynomial B m (φ, .)as n → ∞.Combining both approximations will then provide a uniform approximation of φ ∈ U [0,1] s−icx by the functions ψ (n) m (φ, .).This will show also that U [a,b]s−icx is dense in U [a,b] s−icx , s ≥ 1.For clarity, the precise statement is given with respect to a general interval [a, b] (instead of [0, 1]).Every function φ ∈ U [a,b] s−icx , s ≥ 2, can be approximated uniformly on [a, b], as n → ∞, by spline functions ψ (n) m (φ, .),m ≥ s and n ≥ 2, which are of the form

Remark 3 . 4 .
The result of Proposition 3.3 holds for every function φ ∈ U [a,b]