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Mean residual life processes and associated submartingales

Antoine-Marie Bogso

TL;DR

The paper extends the Madan-Yor link between mean residual life (MRL) ordering and associated submartingales to two-parameter processes, using the Cox-Hobson/Azéma-Yor framework to construct submartingales when the MRL order holds. It establishes a sharp equivalence: the integrated survival function $C_mbda$ is $MTP_2$ iff it is $TP_2$ in each argument pair with the remaining variable fixed, enabling propagation of total positivity through a wide class of two-parameter constructions. The authors provide explicit two-parameter MRL process examples (diatomic, censoring, subordination, convolution) and show how $MTP_2$ properties are preserved under natural transformations, thereby generating many new MRL processes. They also develop methods to obtain associated submartingales for certain non-MRL processes by decomposing into MRL components and applying convex combinations, broadening the set of processes that admit explicit submartingale representations. Overall, the work enriches the theoretical framework connecting multivariate MRL structure with submartingale constructions and total positivity, with potential applications in stochastic ordering and Skorokhod embedding contexts.

Abstract

We use Madan-Yor's argument to construct associated submartingales to a class of two-parameter processes that are ordered by the increasing convex dominance. This class includes processes which have MTP$_2$ integrated survival functions. We prove that the integrated survival function of an integrable two-parameter process is MTP$_2$ if and only if it is TP$_2$ in each pair of arguments when the remaining argument is fixed. This result can not be deduced from known results since there are several two-parameter processes whose integrated survival functions do not have interval support. The MTP$_2$ property of certain MRL processes is useful to exhibit numerous other processes having the same property.

Mean residual life processes and associated submartingales

TL;DR

The paper extends the Madan-Yor link between mean residual life (MRL) ordering and associated submartingales to two-parameter processes, using the Cox-Hobson/Azéma-Yor framework to construct submartingales when the MRL order holds. It establishes a sharp equivalence: the integrated survival function is iff it is in each argument pair with the remaining variable fixed, enabling propagation of total positivity through a wide class of two-parameter constructions. The authors provide explicit two-parameter MRL process examples (diatomic, censoring, subordination, convolution) and show how properties are preserved under natural transformations, thereby generating many new MRL processes. They also develop methods to obtain associated submartingales for certain non-MRL processes by decomposing into MRL components and applying convex combinations, broadening the set of processes that admit explicit submartingale representations. Overall, the work enriches the theoretical framework connecting multivariate MRL structure with submartingale constructions and total positivity, with potential applications in stochastic ordering and Skorokhod embedding contexts.

Abstract

We use Madan-Yor's argument to construct associated submartingales to a class of two-parameter processes that are ordered by the increasing convex dominance. This class includes processes which have MTP integrated survival functions. We prove that the integrated survival function of an integrable two-parameter process is MTP if and only if it is TP in each pair of arguments when the remaining argument is fixed. This result can not be deduced from known results since there are several two-parameter processes whose integrated survival functions do not have interval support. The MTP property of certain MRL processes is useful to exhibit numerous other processes having the same property.

Paper Structure

This paper contains 16 sections, 16 theorems, 122 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Two-parameter mean residual life processes
  3. The Cox-Hobson algorithm and the mean residual life ordering
  4. Organization of the paper
  5. Total positivity properties of MRL processes
  6. Multivariate totally positive functions
  7. Integrated survival functions of MRL processes
  8. Some examples of two-parameter MRL processes
  9. A family of diatomic MRL processes
  10. MRL processes obtained by censoring transformations
  11. MRL processes obtained by subordination
  12. MRL processes obtained by convolution
  13. Construction of associated submartingales to a class of non-MRL processes
  14. Censoring transformed processes
  15. Processes obtained by subordination
  16. ...and 1 more sections

Key Result

Proposition 2.3

KaR80. Let $n$, $m$, $l$ be positive integers. Let $\mathcal{I}=\prod\limits_{i=1}^n\mathcal{I}_i$, $\mathcal{J}=\prod\limits_{i=1}^m\mathcal{J}_i$, $\mathcal{K}=\prod\limits_{i=1}^l\mathcal{K}_i$, where $\mathcal{I}_i$, $\mathcal{J}_i$ and $\mathcal{K}_i$ are totally ordered sets. Let $f$ be MTP$_2 where $\varrho=\varrho_1\times\cdots\times\varrho_m$ and $\varrho_i$ is a $\sigma$-finite positive

Figures (1)

  • Figure 1: $\pi_{\mu_\mathbf{t}}$ for a $\mu_\mathbf{t}$ with positive mean and such that $\text{supp}(\mu_\mathbf{t})=\mathbb{R}$.

Theorems & Definitions (44)

  • Definition 1.1
  • Remark 1.2
  • Definition 1.3
  • Remark 1.4
  • Remark 1.5
  • Definition 2.1
  • Remark 2.2
  • Proposition 2.3
  • Remark 2.4
  • Proposition 2.5
  • ...and 34 more