# Muth distribution

The Muth distribution is a non-negative continuous random variable model. Although this distribution is not typically covered in introductory probability and statistics courses, this model is often encountered in reliability theory where it examines the ability of a system to function under specific conditions over a period of time. Despite its historical lack of attention in statistical literature [1], the Muth distribution plays an important role in this field.

## Muth distribution properties

The Muth distribution, which is well-suited for a broad range of lifetime datasets, has several notable characteristics. It incorporates the bathtub hazard function with a vertical asymptote [2] and closely corresponds to the exponential distribution as β approaches zero. Additionally, its tail is lighter than other commonly used lifetime distributions, such as the exponential, lognormal, and Weibull.

The probability density function (PDF) for the standard Muth distribution is [3]:

Where x > 0 and κ is a shape parameter.

The cumulative distribution function (CDF) is

for x >0.

The inverse distribution function, median, moment generating function and characteristic function of X are not tractable.

## History of the Muth distribution

The Muth distribution was first introduced by French biologist Teissier in 1934 [4]. Teissier considered the mortality of several domestic animal species resulting from pure ageing. Through empirical analysis, Teissier identified that animal mortality does not follow the human mortality pattern, as supported by Gompertz law, frequently used in actuarial practice [5].

Laurent [6] later introduced a location version of Teissier’s distribution, characterizing it with life expectancy and exploring applications to demographic studies. Muth [7] observed that the Muth distribution displays a heavier tail, compared to well-known lifetime distributions such as the gamma distribution, lognormal distribution, and Weibull distribution. Rinne [8] used this model to estimate the lifetime distribution, expressed in kilometers, for a German data set based on used car prices. However, following Rinne’s work, Teissier’s distribution and its location version were forgotten and not referenced in available literature.

Leemis and McQueston [9] acknowledged and named the Muth distribution in their study of schematic representation in various univariate distributional relationships. In 2015, Pedro et al. reintroduced this distribution and its scaled version [10], studying its important properties through the exponential integral function.

## References

[1] Jodrá, P. & Arshad, M. (2021). An intermediate Muth distribution with increasing failure rate. Communications in Statistics – Theory and Methods. Retrieved November 12, 2021 from: https://www.tandfonline.com/doi/abs/10.1080/03610926.2021.1892133?journalCode=lsta20

[2] Kosznik-Biernacka, S. (2007). Makeham’s Generalised Distribution. Computational Methods in Science and Technology 13(2), 113-120 from: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.552.4465&rep=rep1&type=pdf

[3] Muth distribution. Retrieved May 18, 2023 from: http://www.math.wm.edu/~leemis/chart/UDR/PDFs/Muth.pdf

[4] G. TEISSIER (1934). Recherches sur le vieillissement et sur les lois de la mortalité. II. essai
d’interprétation genérale des courbes de survie. Annales de Physiologie et de Physicochimie Biologique, 10, pp. 260–284.

[5] B. GOMPERTZ (1825). On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies. Philosophical Transactions of the Royal Society of London, 115, pp. 513–583.

[6] A. G. LAURENT (1975). Failure and mortality from wear and ageing – The Teissier model. In G. PATIL, S. KOTZ, J. ORD (eds.), A Modern Course on Statistical Distributions in Scientific Work, Springer, Heidelberg, vol. 17, pp. 301–320.

[7] J.E. Muth. (1977). Reliability models with positive memory derived from the mean residual life function. In C.P. Tsokos and I. Shimi (Eds.), The Theory and Applications of Reliability, volume 2, pp. 401–435. Academic Press, Inc., New York.

[8] H. RINNE (1981). Estimating the lifetime distribution of private motor-cars using prices of
used cars-the Teissier model. Statistiks Zwischen Theorie und Praxis, pp. 172–184.

[9] L. M. LEEMIS, J. T. MCQUESTON (2008). Univariate distribution relationships. The
American Statistician, 62, no. 1, pp. 45–53.

[10] J. PEDRO, M. D. JIMÉNEZ GAMERO, V. ALBA-FERNÁNDEZ (2015). On the Muth distribution. Mathematical Modelling and Analysis, 20, pp. 291–310.

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