Ergodicity means that cross-section probabilities equal longitudinal lifetime probabilities. (“Ergos” is Greek for “work.” Think of “ergonomics”.) Ergodicity means that we can estimate age-specific field reliability functions from cross-section data: ships (installed base) and returns (complaints, failures, service parts’ sales, etc.). Ships and returns provide information about lifetimes. Returns are the superpositions of failures of products or their parts started at different times. What does ergodicity have to do with toilet paper? [Read more…]

Andrew Vázsonyi led an interesting life. He collaborated with mathematician Paul Erdös, he was co-founder of The Institute of Management Sciences, and he wrote “Which Door has the Cadillac: Adventures of a Real-Life Mathematician”. Around 1970, Andrew Vázsonyi interviewed for a teaching job in Sauder School of Business, University of British Columbia. During the job interview, he taught us Gozinto Theory. [Read more…]

Myron Tribus’ UCLA Statistical Thermodynamics class introduced me to entropy, -Sp(t)ln(p(t)). (p(t) is the probability of state t of a system.) Professor Tribus later advocated maximum-entropy reliability estimation, because that “…best represents the current state of knowledge about a system…” [Principle of maximum entropy – Wikipedia] Caution! This article contain statistical neurohazards.

Claude Shannon wrote that entropy (log base 2) represents information bits, “…an absolute mathematical limit on how well data from the source can be losslessly compressed onto a perfectly noiseless channel.”  [Beirlant et al.]

Maximum likelihood estimation is one way to estimate reliability from data. It maximizes the probability density function of observed data, Pp(t), e.g., for observed failures at ages t. It is equivalent to maximize -Sln(p(t). Maximum entropy reliability estimation maximizes entropy -Sp(t)ln(p(t). That’s same as maximizing the expected value, -Sp(t)ln(p(t), of the log likelihood -ln(p(t). Fine, if you have life data, ages at failures t censored or not. [Read more…]

Ralph Evans was editor of the IEEE Transactions on Reliability from 1969 until 2004. He was a very good editor for my 1977 article, and he used me as a reviewer, because I was critical of BS and academic exercises. Ralph moved to University Retirement Community, Davis, CA. He died in 2013, https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=6587564. I wish I’d known he lived nearby so I could have visited and argued with him.

Ralph’s editorials [1 and 2] pled, “Data, Data, Oh Where Art Thou Data?” He wrote, “Field-data are largely garbage. I believe they deserve all the negative thinking possible.” “True field-data are wonderful-much better than fancy equations. Unfortunately, they are very difficult to get. Thus data from the field are largely garbage because they do not represent what really happened.” [Read more…]