Authors' Rejoinder

Abstract

First of all, we would like to express our sincere gratitude to Professor Fabrizio Ruggeri, the editor-in-chief, for his kind invitation for us to prepare this discussion paper and then making painstaking efforts for its review as well as to have numerous discussions on it done by various experts active on different aspects of Birnbaum-Saunders (BS) model. Next, we would like to express our great thanks and deep appreciation to all the discussants for their many incisive comments and suggestions on our expository article on BS and related distributions. Below, we present our specific detailed responses to the comments made by each of the discussants.Professor Anthony Desmond: It was an interesting read to see how Professor Desmond's interest in BS model originated from the work of Cramér and Leadbetter 1 on properties of sample paths of stochastic processes including level crossings and maxima. He then went on to derive extensions of the famous Miner's rule, in which the point process of local maxima played a key role. His approach in this direction led to an interesting finding that the damage process D(t) could be modelled as a compound Poisson process. Of course, he was able to achieve this by exploiting the approximate Poisson nature of local maxima; then, without the Poisson approximation, by using only the mean and variance of the damage process D(t), he was able to provide BS distribution as an approximation for the fatigue lifetime distribution.His idea of probability intervals for fatigue life is quite interesting and practically useful. His idea of allowing for some uncertainty in the material parameters, either in a Bayesian or frequentist approach, and then evaluating the sensitivity and robustness characteristics of these probability intervals will be very interesting, of course! With regard to his comment about the potential use of estimating functions, many of the point estimators proposed in the literature (and also detailed in our article) are indeed solutions of some forms of estimating equations. Though the maximum-likelihood estimate (MLE) score-based estimating equations are known to be optimal, use of estimating equations to come up with robust (to departures from the model assumptions) and efficient estimators will be an interesting and challenging problem to undertake! In this regard, three different approaches have been taken in the literature, in our view, concerning robust estimation of model parameters of BS distribution. One is basing the estimation on censored samples, so that the effect of extreme order statistics is minimized in the estimation process. Second is the traditional influence function approach put forward by Dupuis and Mills, 2 and the third one is by basing the inference on generalized BS distribution based on a heavy-tailed kernel (instead of the traditional normal one). It is true, as correctly pointed out by Professor Desmond, that such a generalized model may lack a motivation from the underlying fatigue process itself; it does provide great flexibility while model...

Top‐Down Fabrication of Atomic Patterns in Twisted Bilayer Graphene

“…The hazard function has a distinct peak at the beginning and converges asymptotically to a constant value determined by the shape, α , and scale, β , parameters. [ 44 ] This value is expressed as 1/(2 α 2 β ) and evaluates to 0.05 s −1 for the fit shown.…”