Remaining useful life estimation - A review on the statistical data driven approaches

https://rss.onlinelibrary.wiley.com/doi/pdf/10.1111/j.2517-6161.1978.tb01039.x

The Inverse Gaussian Distribution and its Statistical Application—A Review

This article systematically investigates the inverse Gaussian (IG) process as an effective degradation model. The IG process is shown to be a limiting compound Poisson process, which gives it a meaningful physical interpretation for modeling degradation of products deteriorating in random environments. Treated as the first passage process of a Wiener process, the IG process is flexible in incorporating random effects and explanatory variables that account for heterogeneities commonly observed in degradation problems. This flexibility makes the class of IG process models much more attractive compared with the Gamma process, which has been thoroughly investigated in the literature of degradation modeling. The article also discusses statistical inference for three random effects models and model selection. It concludes with a real world example to demonstrate the applicability of the IG process in degradation analysis. Supplementary materials for this article are available online.

The Inverse Gaussian Process as a Degradation Model

Supplemental Material for the article "The Inverse Gaussian Process as a Degradation Model," published in Technometrics Volume 56 Issue 3.

Supplemental Material: The Inverse Gaussian Process as a Degradation Model

Pursuing the classification initiated by Morris (1982), we describe all the natural exponential families on the real line such that the variance is a polynomial function of the mean with degree less than or equal to 3. We get twelve different types; the first six appear in the fundamental paper by Morris (1982); most of the other six appear as distributions of first passage times in the literature, the inverse Gaussian type being the most famous example. An explanation of this occurrence of stopping times is provided by the introduction of the notion of reciprocity between two measures or between two natural exponential families, and by classical fluctuation theory.

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Natural Real Exponential Families with Cubic Variance Functions

HALDANE1 has recently explained a method of using a haemacytometer in which counting is stopped when a fixed number of cells has been recorded. This sampling technique, with suitable modifications, can be applied to other populations and the theory reveals some interesting relationships.

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M. C. K. Tweedie

Papers

Inverse Statistical Variates