Max B. Mendel

De Finetti-type Representations for Life Distributions

Abstract Beginning with a finite population of units and the judgment of exchangeability for units with respect to lifetime, we argue that measures of similarity lead to the appropriate probabilistic models for aging. This in turn implies that Schur-concavity of the joint probability function (or, more generally, the joint survival distribution) provides the correct probabilistic description of aging. Following this argument and using the principle of indifference, we argue that the appropriate probability models for life distributions conditional on average life are in a family of distributions that we call the generalized gamma distributions. If, on the other hand, we are interested in probabilistic models for aging conditional on average lifetime maintenance cost, it follows from our development that generalized Weibull distributions are appropriate.

An engineering basis for statistical lifetime models with an application to tribology

The authors propose a method for incorporating engineering information about failure mechanisms into a statistical lifetime model. The central idea is that wear, stress, and strain are more directly related to failure than is component age. With an application to tribology, they use the method to derive a lifetime model which explicitly uses information about wear. The model is contrasted with the more common technique of fitting the parameters of a statistical distribution. A primary benefit of an engineering-based model is its interpretability for connecting lifetime data for similar components used under differing operating conditions, such as during accelerated lifetime testing.

Predicting dynamic imbalance in rotors

This paper derives probability models which predict dynamic imbalance of rotors. The models are based on physical laws and specific knowledge of how the rotors are made. Several sources of imbalance are considered. It is shown that wider tolerances in the orientation of a rotor lead to more likely worst-case imbalances. It is shown that the expected dynamic imbalance of a cylinder is proportional to the square root of the feed rate of the cutting tool over the speed. Thus, such models show how various parameters in the manufacturing process affect the distribution of imbalance. In this way, manufacturing decisions on how to effectively reduce imbalance can be made prior to observing any data.

A model for the frequency of extreme river levels based on river dynamics

A new model for predicting the frequency of extreme river levels is proposed which encapsulates physical knowledge about river dynamics. The central idea is the use of continuous time stochastic processes that use hydrological equations and ergodic theory to model extreme events, rather than relying on statistical fits of classical models to local maximum data. A simple example shows how changes in discharge characteristics change the extreme river level frequencies. Solutions are provided for special cases, and directions for more general techniques are provided.

Operational parameters in Bayesian models

SummaryOperational parameters are parameters that are defined in terms of the data that are being modelled. This paper shows under what conditions they can be used in place of the usual formal parameters and discusses the advantages of doing this. Example applications include the normal, exponential and uniform model, the multivariate-normal and other multivariate models, finitepopulation versions, and also several new models. Also presented is their relationship to de Finetti’s work on exchangeability and other symmetrybased approaches.

Using wear curves to predict the cost of changes in cutting conditions

A recently-proposed probability model for cutting tool lifetimes based on tool wear curves is presented. Optimal tool replacement strategies and cutting conditions are calculated using this wear-based model. The models use of wear curves permits extrapolation of cost calculations to cutting conditions for which little statistical data has been collected, assuming data is available for other cutting conditions which have the same failure mechanism. This is particularly important during process design changes, when data is expensive to obtain or limited in quantity. Analysis indicates that an important factor in tool replacement is run-in wear.

Deriving Probability Models for Stress Analysis

This paper presents an approach to derive probability models for use in structural reliability studies. Two main points are made. First, that it is possible to translate engineering and physics knowledge into a requirement on the form of a probability model. And second, that making assumptions about a probability model for structural failure implies either explicit or hidden assumptions about material and structural properties. The work is foundational in nature, but is developed with explicit examples taken from planar and general stress problems, the von Mises failure criterion, and a modified Weibull distribution.

A generalized framework for probabilistic design with application to rotors

This paper generalizes an existing method for deriving probability models of manufacturing quality metrics. We specifically consider the problem of deriving probability models for the inertia tensor of a rotor. The inertia tensor is a 3 /spl times/ 3 matrix that determines various dynamical properties of the rotor as it spins, affecting its reliability. The key contribution of this paper is that the quality metric of interest is a matrix or a second-order tensor, and the various manufacturing imperfections that cause deviations in the inertia tensor may be vectors. Existing methods, by contrast, assume that the quality metric, as well as the manufacturing imperfections, are scalar quantities. By using rotational properties of matrices & vectors, we show that the relationship between the inertia tensor, and the manufacturing errors must have a specific form, when the errors are small. This structure significantly restricts the class of allowable distributions for the inertia tensor. For example, we show that the multivariate s-normal distribution is not a physically appropriate distribution for the inertia tensor. The results in this paper, while applied specifically to the inertia tensor, are general, and depend only on transformational properties of vectors & matrices. Thus, the framework is applicable to modeling other engineering systems involving second-order tensors, such as the stress tensor, or strain tensor.

A new method of drawing probability densities

This paper proposes a new method of drawing probability densities. The method has the following advantages in visualizing probability: It is easy to recover event probabilities from the plots, bivariate plots show marginal and conditional densities, the plots require one less dimension than density function plots, the plots can show in a single picture the dependence of a density on some parameter, and the plots are coordinate invariant.

Deriving fatigue life models from the physical failure mechanism

We derive two fatigue life models for structural components in a varial>le-amplitude loading environment. The failure meclianisnl can be interpretccl in two distinct ways, each leading to a different prolxhility nlotlel. One of the models is related to the familiar Gaus-