Pointwise Comparison Research Articles

Lorenz and Excess Wealth Orders, with Applications in Reinsurance Theory

The purpose of this paper is to study the stochastic orderings defined by means of pointwise comparison of the Lorenz curves, or of the excess wealth transforms, of two risks. The resulting order relations are presented in an actuarial context and put in relation with classical stochastic orderings, namely the stochastic dominance, the stop-loss order, the convex order and the dispersive order. Several relevant applications in reinsurance theory are provided.

Operational Monitoring of Rainfall over the Arno River Basin Using Dual-Polarized Radar and Rain Gauges

Abstract Reflectivity (ZH) and differential reflectivity (ZDR) measurements collected by Polar 55C over the Arno River basin in Italy are presented. The applicability of dual-polarization (ZDR)-based rainfall algorithms at C band in an operational setting is studied in conjunction with a network of rain gauges. Conventional pointwise comparison of radar and rain gauge estimates of rainfall, as well as statistical comparison of dual-polarization radar and rain gauge data via probability matching procedure, are presented. Error structure of reflectivity rainfall Z-R relation, as well as ZDR-based algorithms, is evaluated as a function of spatial and temporal averaging. Pointwise comparison, as well as statistical evaluation based on cumulative distribution function (CDF) matching, are used to show that in an operational environment with excessive ground-clutter contamination and attenuation problems the dual-polarization-based rainfall algorithm performs better than any arbitrary Z-R relation. In addition, ...

Some Negative Results Concerning the Process of Fejér Type Trigonometric Convolution

Abstract The process of Fejér type trigonometric convolution and its discrete analogue have equivalent uniform residues. The situation changes under pointwise comparison. In this direction negative results have been obtained by different authors. One result of such a type is given in the present paper. In particular, a counter-example is constructed for which both comparisons diverge on the set of complete measure. The smoothness of the counter-example as well as some other problems are investigated.

Some negative results concerning the process of Fejér type trigonometric convolution

The process of Fejer type trigonometric convolution and its discrete analogue have equivalent uniform residues. The situation changes under pointwise comparison. In this direction negative results have been obtained by different authors. One result of such a type is given in the present paper. In particular, a counter-example is constructed for which both comparisons diverge on the set of complete measure. The smoothness of the counter-example as well as some other problems are investigated.

Convolution processes of Fejér type and the divergence almost everywhere of a pointwise comparison

Convolution processes of Fejér type and the divergence almost everywhere of a pointwise comparison

Divergence almost everywhere of a pointwise comparison of trigonometric convolution processes with their discrete analogues

Continuing previous investigations this paper deals with negative results concerning a pointwise comparison of trigonometric convolution processes with their discrete analogues. Though the uniform errors of those processes are equivalent (under suitable conditions), application of an appropriate extension of a familiar lemma of A. P. Calderón in connection with a general quantitative resonance principle establishes that corresponding pointwise interpretations may fail almost everywhere.

Comparison of Bernstein polynomials with the metrical means of Kantorovitch

In 1934 Kantorovitch modified the Bernstein polynomials B n by means of metrical means to yield a nonlinear polynomial process B n ∗ which approximates measurable functions almost everywhere. The present paper is concerned with the pointwise comparison of B n and B n ∗ on C[0, 1] (the space of continuous functions on [0, 1]). We establish direct estimates of the form ¦(B nf − f)(x)¦ ⩽ ¦(B n ∗f − f)(x)¦ + ω(3n − 1 3 , f) with the first modulus of continuity ω. On the other hand, it is the main purpose of this paper to show that this inequality can not be strengthened to ¦(B nf − f)(x)¦ ⩽ C x ¦(B nf − f)(x)¦ so that B n ∗ is not a pointwise extension of B n on C[0, 1]. To this end, a previous condensation principle is applied concerning nonlinear functionals which assures the nonvalidity of this last inequality for x on a dense, in fact, residual, set of [0, 1].

The Divergence almost Everywhere of a Pointwise Comparison of Fejér and Abel-Poisson means

On etablit l'existence de fonctions lipschitziennes pour lesquelles la comparaison ponctuelle n'a pas lieu

A duality theorem on a pair of simultaneous functional equations

Given P and Q convex compact sets in R k and R s , respectively, and u a continuous real valued function on P × Q, we consider the following pair of dual problems: Problem I—Minimize ƒ so that ƒ: P × Q → R and ƒ ⩾ Cav p Vex q × max(u, ƒ) . Problem II—Maximize g so that g: P × Q → R and g ⩽ Vex q × Cav p min( u, g). Here Cav p is the operation of concavification of a function with respect to the variable p ϵ P (for each fixed q ϵ Q). Similarly, Vex q is the operation of convexification with respect to q ϵ Q. Maximum and minimum are taken here in the partial ordering of pointwise comparison: ƒ ⩽ g means ƒ(p, q) ⩽ g(p, q) ∀(p, q) ϵ P × Q . It is proved here that both problems have the same solution which is also the unique simultaneous solution of the following pair of functional equations: (i) ƒ = Vex q max(u, ƒ) . (ii) ƒ = Cav p min(u, ƒ) . The problem arises in game theory, but the proof here is purely analytical and makes no use of game-theoretical concepts.

Some point-wise estimates for solutions of a class of nonlinear functional-integral inequalities

The celebrated Gronwall-Bellman Lemma provides explicit bounds on solutions of a class of linear integral inequalities. The aim of this paper is to prove sufficiently general results analogous to this Lemma for functional-integral inequalities. These results are useful for obtaining point-wise estimates or comparison theorems for solutions of functional differential equations and functional-integral equations of Volterra type.