In probability theory and statistics, the Weibull distribution/ˈwaɪbʊl/ is a continuous probability distribution. It models a broad range of random variables, largely in the nature of a time to failure or time between events. Examples are maximum one-day rainfalls and the time a user spends on a web page.
The distribution is named after Swedish mathematician Waloddi Weibull, who described it in detail in 1939,[1] although it was first identified by René Maurice Fréchet and first applied by Rosin & Rammler (1933) to describe a particle size distribution.
If the quantity, x, is a "time-to-failure", the Weibull distribution gives a distribution for which the failure rate is proportional to a power of time. The shape parameter, k, is that power plus one, and so this parameter can be interpreted directly as follows:[5]
A value of indicates that the failure rate decreases over time (like in case of the Lindy effect, which however corresponds to Pareto distributions[6] rather than Weibull distributions). This happens if there is significant "infant mortality", or defective items failing early and the failure rate decreasing over time as the defective items are weeded out of the population. In the context of the diffusion of innovations, this means negative word of mouth: the hazard function is a monotonically decreasing function of the proportion of adopters;
A value of indicates that the failure rate is constant over time. This might suggest random external events are causing mortality, or failure. The Weibull distribution reduces to an exponential distribution;
A value of indicates that the failure rate increases with time. This happens if there is an "aging" process, or parts that are more likely to fail as time goes on. In the context of the diffusion of innovations, this means positive word of mouth: the hazard function is a monotonically increasing function of the proportion of adopters. The function is first convex, then concave with an inflection point at .
In the field of materials science, the shape parameter k of a distribution of strengths is known as the Weibull modulus. In the context of diffusion of innovations, the Weibull distribution is a "pure" imitation/rejection model.
Alternative parameterizations
First alternative
Applications in medical statistics and econometrics often adopt a different parameterization.[7][8] The shape parameter k is the same as above, while the scale parameter is . In this case, for x ≥ 0, the probability density function is
the cumulative distribution function is
the quantile function is
the hazard function is
and the mean is
Second alternative
A second alternative parameterization can also be found.[9][10] The shape parameter k is the same as in the standard case, while the scale parameter λ is replaced with a rate parameter β = 1/λ. Then, for x ≥ 0, the probability density function is
the cumulative distribution function is
the quantile function is
and the hazard function is
In all three parameterizations, the hazard is decreasing for k < 1, increasing for k > 1 and constant for k = 1, in which case the Weibull distribution reduces to an exponential distribution.
Properties
Density function
The form of the density function of the Weibull distribution changes drastically with the value of k. For 0 < k < 1, the density function tends to ∞ as x approaches zero from above and is strictly decreasing. For k = 1, the density function tends to 1/λ as x approaches zero from above and is strictly decreasing. For k > 1, the density function tends to zero as x approaches zero from above, increases until its mode and decreases after it. The density function has infinite negative slope at x = 0 if 0 < k < 1, infinite positive slope at x = 0 if 1 < k < 2 and null slope at x = 0 if k > 2. For k = 1 the density has a finite negative slope at x = 0. For k = 2 the density has a finite positive slope at x = 0. As k goes to infinity, the Weibull distribution converges to a Dirac delta distribution centered at x = λ. Moreover, the skewness and coefficient of variation depend only on the shape parameter. A generalization of the Weibull distribution is the hyperbolastic distribution of type III.
where . The kurtosis excess may also be written as:
Moment generating function
A variety of expressions are available for the moment generating function of X itself. As a power series, since the raw moments are already known, one has
Alternatively, one can attempt to deal directly with the integral
If the parameter k is assumed to be a rational number, expressed as k = p/q where p and q are integers, then this integral can be evaluated analytically.[12] With t replaced by −t, one finds
The characteristic function has also been obtained by Muraleedharan et al. (2007). The characteristic function and moment generating function of 3-parameter Weibull distribution have also been derived by Muraleedharan & Soares (2014) harvtxt error: no target: CITEREFMuraleedharanSoares2014 (help) by a direct approach.
Minima
Let be independent and identically distributed Weibull random variables with scale parameter and shape parameter . If the minimum of these random variables is , then the cumulative probability distribution of is given by
That is, will also be Weibull distributed with scale parameter and with shape parameter .
Reparametrization tricks
Fix some . Let be nonnegative, and not all zero, and let be independent samples of , then[13]
The fit of a Weibull distribution to data can be visually assessed using a Weibull plot.[16] The Weibull plot is a plot of the empirical cumulative distribution function of data on special axes in a type of Q–Q plot. The axes are versus . The reason for this change of variables is the cumulative distribution function can be linearized:
which can be seen to be in the standard form of a straight line. Therefore, if the data came from a Weibull distribution then a straight line is expected on a Weibull plot.
There are various approaches to obtaining the empirical distribution function from data: one method is to obtain the vertical coordinate for each point using where is the rank of the data point and is the number of data points.[17][18]
Linear regression can also be used to numerically assess goodness of fit and estimate the parameters of the Weibull distribution. The gradient informs one directly about the shape parameter and the scale parameter can also be inferred.
Equating the sample quantities to , the moment estimate of the shape parameter can be read off either from a look up table or a graph of versus . A more accurate estimate of can be found using a root finding algorithm to solve
The moment estimate of the scale parameter can then be found using the first moment equation as
In describing the size of particles generated by grinding, milling and crushing operations, the 2-Parameter Weibull distribution is used, and in these applications it is sometimes known as the Rosin–Rammler distribution.[26] In this context it predicts fewer fine particles than the log-normal distribution and it is generally most accurate for narrow particle size distributions.[27] The interpretation of the cumulative distribution function is that is the mass fraction of particles with diameter smaller than , where is the mean particle size and is a measure of the spread of particle sizes.
In describing random point clouds (such as the positions of particles in an ideal gas): the probability to find the nearest-neighbor particle at a distance from a given particle is given by a Weibull distribution with and equal to the density of the particles.[28]
In calculating the rate of radiation-induced single event effects onboard spacecraft, a four-parameter Weibull distribution is used to fit experimentally measured device cross section probability data to a particle linear energy transfer spectrum.[29] The Weibull fit was originally used because of a belief that particle energy levels align to a statistical distribution, but this belief was later proven false[citation needed] and the Weibull fit continues to be used because of its many adjustable parameters, rather than a demonstrated physical basis.[30]
Related distributions
If , then the variable is Gumbel (minimum) distributed with location parameter and scale parameter . That is, .
The translated Weibull distribution (or 3-parameter Weibull) contains an additional parameter.[11] It has the probability density function
for and for , where is the shape parameter, is the scale parameter and is the location parameter of the distribution. value sets an initial failure-free time before the regular Weibull process begins. When , this reduces to the 2-parameter distribution.
The Weibull distribution can be characterized as the distribution of a random variable such that the random variable
This implies that the Weibull distribution can also be characterized in terms of a uniform distribution: if is uniformly distributed on , then the random variable is Weibull distributed with parameters and . Note that here is equivalent to just above. This leads to an easily implemented numerical scheme for simulating a Weibull distribution.
The Weibull distribution interpolates between the exponential distribution with intensity when and a Rayleigh distribution of mode when .
The distribution of a random variable that is defined as the minimum of several random variables, each having a different Weibull distribution, is a poly-Weibull distribution.
The Weibull distribution was first applied by Rosin & Rammler (1933) to describe particle size distributions. It is widely used in mineral processing to describe particle size distributions in comminution processes. In this context the cumulative distribution is given by
where
is the particle size
is the 80th percentile of the particle size distribution
is a parameter describing the spread of the distribution
Because of its availability in spreadsheets, it is also used where the underlying behavior is actually better modeled by an Erlang distribution.[33]
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^Therneau, T. (2020). "A Package for Survival Analysis in R." R package version 3.1.
^ a b cJohnson, Kotz & Balakrishnan 1994
^See (Cheng, Tellambura & Beaulieu 2004) harv error: no target: CITEREFChengTellamburaBeaulieu2004 (help) for the case when k is an integer, and (Sagias & Karagiannidis 2005) for the rational case.
^Balog, Matej; Tripuraneni, Nilesh; Ghahramani, Zoubin; Weller, Adrian (2017-07-17). "Lost Relatives of the Gumbel Trick". International Conference on Machine Learning. PMLR: 371–379.
^Cho, Youngseuk; Sun, Hokeun; Lee, Kyeongjun (5 January 2015). "Estimating the Entropy of a Weibull Distribution under Generalized Progressive Hybrid Censoring". Entropy. 17 (1): 102–122. doi:10.3390/e17010102. ISSN 1099-4300.
^Bauckhage, Christian (2013). "Computing the Kullback-Leibler Divergence between two Weibull Distributions". arXiv:1310.3713 [cs.IT].
^"1.3.3.30. Weibull Plot". www.itl.nist.gov.
^Wayne Nelson (2004) Applied Life Data Analysis. Wiley-Blackwell ISBN 0-471-64462-5
^Barnett, V. (1975). "Probability Plotting Methods and Order Statistics". Journal of the Royal Statistical Society. Series C (Applied Statistics). 24 (1): 95–108. doi:10.2307/2346708. ISSN 0035-9254. JSTOR 2346708.
^ a bCohen, A. Clifford (Nov 1965). "Maximum Likelihood Estimation in the Weibull Distribution Based on Complete and on Censored Samples" (PDF). Technometrics. 7 (4): 579–588. doi:10.1080/00401706.1965.10490300.
^ a bSornette, D. (2004). Critical Phenomena in Natural Science: Chaos, Fractals, Self-organization, and Disorder..
^"CumFreq, Distribution fitting of probability, free software, cumulative frequency".
^ a bLee, Se Yoon; Mallick, Bani (2021). "Bayesian Hierarchical Modeling: Application Towards Production Results in the Eagle Ford Shale of South Texas". Sankhya B. 84: 1–43. doi:10.1007/s13571-020-00245-8.
^"Wind Speed Distribution Weibull – REUK.co.uk". www.reuk.co.uk.
^Liu, Chao; White, Ryen W.; Dumais, Susan (2010-07-19). Understanding web browsing behaviors through Weibull analysis of dwell time. ACM. pp. 379–386. doi:10.1145/1835449.1835513. ISBN 9781450301534. S2CID 12186028.
^Sharif, M.Nawaz; Islam, M.Nazrul (1980). "The Weibull distribution as a general model for forecasting technological change". Technological Forecasting and Social Change. 18 (3): 247–56. doi:10.1016/0040-1625(80)90026-8.
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