Load-sharing survival models in the analysis of minima within subsets of random variables

Spizzichino Fabio, Sapienza University

\begin{document} \setlength{\marginparwidth}{5pc} \title{Load-sharing survival models in the analysis of minima within subsets of random variables\\ } % \author{Fabio L. Spizzichino\\ % University La Sapienza, Rome\\ \texttt{fabio.spizzichino@fondazione.uniroma1.it} } \date{}%% DO NOT INPUT ANY DATE \maketitle \section*{\noindent Abstract } Starting from a vector $\mathbf{X}\equiv(X_{1},...,X_{m})$ of, generally interdependent, non-negative random variables, we focus attention on the family \[ \mathcal{A}\equiv\{a_{j}(A)|A\subseteq\lbrack m],j\in A\}, \] where $[m]:=\{1,...,m\}$ and $a_{j}(A):=$ $\mathbb{P}\left( \min_{i\in a}X_{i}=A\right) $. In different fields, the quantities $a_{j}(A)$ can be seen as "winning probabilities". Assuming the "no-tie" condition% \[ \mathbb{P}\left( X_{1}\neq...\neq X_{m}\right) =1 \] and denoting by $X_{1:m}...,X_{m:m}$ the order statistics of $X_{1},...,X_{m}% $, we consider the $[m]$-valued random variables $J_{1},...,J_{m}$ defined by% \[ J_{h}=i\Leftrightarrow X_{h:m}=X_{i}. \] $\mathbf{J\equiv}\left( J_{1},...,J_{m}\right) $ is obviously a random permutation of the elements of $[m]$ and we denote by $\mathbf{P}_{\mathbf{J}% }^{\left[ \mathbf{X}\right] }$ the corresponding joint probability distribution. By means of elementary formulas, the quantities $a_{j}(A)$'s can be obtained from the knowledge of $\mathbf{P}_{\mathbf{J}}^{\left[ \mathbf{X}\right] }$ (see \cite{DsS23a}). As it will be first recalled in the talk, in the absolutely continuous case a possible method to describe dependence among non-negative random variables is based on the "multivariate conditional hazard rate" functions. This method is specially convenient in the analysis of minima (see \cite{DsMS20}). Under such a description of dependence, multivariate survival models of the type "Time-Homogeneous Load-Sharing" arise from a natural, and very strong, condition of invariance imposed on the form of the multivariate conditional hazard rates. The talk will then present a discussion centered on the claim that these models can be of special interest, in particular, in the study of different features of the family $\mathcal{A}$. Substantially, such a claim hinges on the following two circumstances (see \cite{DsS23a}): i) In the THLS case, the computation of the quantities $a_{j}(A)$ can easily be developed in terms of the parameters of the model ii) For an arbitrary choice of $\left( X_{1},...,X_{m}\right) $ one can determine a corresponding THLS-distributed vector $\left( \widetilde{X}% _{1},...,\widetilde{X}_{m}\right) $ such that% \[ \mathbf{P}_{\mathbf{J}}^{\left[ \mathbf{X}\right] }=\mathbf{P}_{\mathbf{J}% }^{\left[ \widetilde{\mathbf{X}}\right] }. \] Concerning with the validity of the above property ii), we point out that it is necessary to introduce the class of the "order-dependent" load-sharing models (see \cite{DsS23a}). This class emerges as a completely natural extension of the one of the standard load-sharing models, that have be considered so far in the applied fields of engineering reliability, medicine, or financial risk. Joint work with Emilio De Santis at Sapienza University, Rome. \bigskip \begin{thebibliography}{99} \bibitem{DsMS20} E. De Santis, Y. Malinovsky, F. L. Spizzichino. Stochastic Precedence and Minima among Dependent Variables. Meth. Comp. Appl. Probab., \textbf{23} (2020), 187--205. \bibitem{DsS23a} E. De Santis, F. L. Spizzichino. Construction of Aggregation Paradoxes through Load-sharing Models. Adv. Appl. Probab., \textbf{55} (2023), 223--244. \end{thebibliography} \end{document}

Area: CS53 - Connections between stochastic models and game and voting theory (Emilio De Santis)

Keywords: Winning probabilities, Multivariate conditional hazard rate functions, Load-sharing models