The Weibull family of probability distributions (see also Generalized Weibull Distributions) is one the most widely used parametric families of distributions for modeling failure times or lifetimes. This is especially true in engineering and science applications (as suggested originally by Weibull 1951) and is mainly due to the variety of shapes of its density function and the behaviors of its failure rate function. Literally thousands of references to the Weibull distribution can be found in the scientific literature. See Johnson et al. (1994) or a more recent treatment by Rinne (2008) for a detailed comprehensive overview of this family of distributions.
Let T denote a random variable (rv) representing the failure time or lifetime of an item under study. This rv has a Weibull distribution with shape parameter α > 0 and scale parameter β > 0 if its probability density function (pdf) is f(t) = αt α − 1 β − αexp[ − (t ∕ β)α] for t ≥ 0. The cumulative distribution function (cdf) is then F(...
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References and Further Reading
Black CM, Durham SD, Padgett WJ (1990) Parameter estimation for a new distribution for the strength of brittle fibers: a simulation. Comm Stat Simulat Comput 19:809–825
Burr IW (1942) Cumulative frequency functions. Ann Math Stat 13:215–222
Johnson NL, Kotz S, Balakrishnan N (1994) Continuous univariate distributions, vol 1, 2nd edn. Wiley Series in Probability and Mathematical Statistics, New York
Mann NR, Shafer RE, Singpurwalla ND (1974) Methods for statistical analysis of reliability and lifetime data. Wiley, New York
Mudholkar GS, Srivastava DK (1993) Exponentiated Weibull family for analyzing bathtub failure-rate data. IEEE Trans Reliab 42:299–302
Rinne H (2008) The Weibull distribution: a handbook. CRC Press, Boca Raton
Smith RJ (1991) Weibull regression models for reliability data. Reliab Eng Syst Saf 34:35–57
Weibull W (1951) A statistical distribution function of wide applicability. J Appl Mech 18:293–297
Wolstenholme LC (1995) A non-parametric test of the weakest-link property. Technometrics 37:169–175
Wolstenholme LC (1999) Reliability modelling: a statistical approach. Chapman & Hall/CRC, Boca Raton
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Padgett, W.J. (2011). Weibull Distribution. In: Lovric, M. (eds) International Encyclopedia of Statistical Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04898-2_611
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