Weibull (2-parameter)
Probability density function
Cumulative distribution function
Parameters scale
shape
Support
PDF
CDF
Mean
Median
Mode
Variance
Skewness
Ex. kurtosis (see text)
Entropy
MGF
CF
Kullback-Leibler divergence see below

In probability theory and statistics, the Weibull distribution /ˈwbʊl/ is a continuous probability distribution. It is named after Swedish mathematician Waloddi Weibull, who described it in detail in 1951, although it was first identified by Fréchet (1927) and first applied by Rosin & Rammler (1933) to describe a particle size distribution.

Definition

Standard parameterization

The probability density function of a Weibull random variable is:[1]

where k > 0 is the shape parameter and λ > 0 is the scale parameter of the distribution. Its complementary cumulative distribution function is a stretched exponential function. The Weibull distribution is related to a number of other probability distributions; in particular, it interpolates between the exponential distribution (k = 1) and the Rayleigh distribution (k = 2 and [2]).

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:[3]

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

Applications in medical statistics and econometrics often adopt a different parameterization.[5][6] 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 hazard function is

and the mean is

A third parameterization can also be found.[7][8] 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

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.

Cumulative distribution function

The cumulative distribution function for the Weibull distribution is

for x ≥ 0, and F(x; k; λ) = 0 for x < 0.

If x = λ then F(x; k; λ) = 1 − e−1 ≈ 0.632 for all values of k. Vice versa: at F(x; k; λ) = 0.632 the value of x ≈ λ.

The quantile (inverse cumulative distribution) function for the Weibull distribution is

for 0 ≤ p < 1.

The failure rate h (or hazard function) is given by

The Mean time between failures MTBF is

Moments

The moment generating function of the logarithm of a Weibull distributed random variable is given by[9]

where Γ is the gamma function. Similarly, the characteristic function of log X is given by

In particular, the nth raw moment of X is given by

The mean and variance of a Weibull random variable can be expressed as

and

The skewness is given by

where the mean is denoted by μ and the standard deviation is denoted by σ.

The excess kurtosis is given by

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.[10] With t replaced by −t, one finds

where G is the Meijer G-function.

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) by a direct approach.

Shannon entropy

The information entropy is given by

where is the Euler–Mascheroni constant. The Weibull distribution is the maximum entropy distribution for a non-negative real random variate with a fixed expected value of xk equal to λk and a fixed expected value of ln(xk) equal to ln(λk) − .

Parameter estimation

Maximum likelihood

The maximum likelihood estimator for the parameter given is

The maximum likelihood estimator for is the solution for k of the following equation[11]

This equation defining only implicitly, one must generally solve for by numerical means.

When are the largest observed samples from a dataset of more than samples, then the maximum likelihood estimator for the parameter given is[11]

Also given that condition, the maximum likelihood estimator for is[citation needed]

Again, this being an implicit function, one must generally solve for by numerical means.

Weibull plot

The fit of a Weibull distribution to data can be visually assessed using a Weibull plot.[12] 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.[13]

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.

Kullback–Leibler divergence

[14]

Applications

The Weibull distribution is used[citation needed]

Fitted cumulative Weibull distribution to maximum one-day rainfalls using CumFreq, see also distribution fitting[15]
Fitted cumulative Weibull distribution to maximum one-day rainfalls using CumFreq, see also distribution fitting[15]
Fitted curves for oil production time series data [16]
Fitted curves for oil production time series data [16]

Related distributions

See also

References

  1. ^ Papoulis, Athanasios Papoulis; Pillai, S. Unnikrishna (2002). Probability, Random Variables, and Stochastic Processes (4th ed.). Boston: McGraw-Hill. ISBN 0-07-366011-6.
  2. ^ "Rayleigh Distribution – MATLAB & Simulink – MathWorks Australia". www.mathworks.com.au.
  3. ^ Jiang, R.; Murthy, D.N.P. (2011). "A study of Weibull shape parameter: Properties and significance". Reliability Engineering & System Safety. 96 (12): 1619–26. doi:10.1016/j.ress.2011.09.003.
  4. ^ Eliazar, Iddo (November 2017). "Lindy's Law". Physica A: Statistical Mechanics and Its Applications. 486: 797–805. Bibcode:2017PhyA..486..797E. doi:10.1016/j.physa.2017.05.077.
  5. ^ Collett, David (2015). Modelling survival data in medical research (3rd ed.). Boca Raton: Chapman and Hall / CRC. ISBN 978-1439856789.
  6. ^ Cameron, A. C.; Trivedi, P. K. (2005). Microeconometrics : methods and applications. p. 584. ISBN 978-0-521-84805-3.
  7. ^ Kalbfleisch, J. D.; Prentice, R. L. (2002). The statistical analysis of failure time data (2nd ed.). Hoboken, N.J.: J. Wiley. ISBN 978-0-471-36357-6. OCLC 50124320.
  8. ^ Therneau, T. (2020). "A Package for Survival Analysis in R." R package version 3.1.
  9. ^ a b c Johnson, Kotz & Balakrishnan 1994
  10. ^ See (Cheng, Tellambura & Beaulieu 2004) for the case when k is an integer, and (Sagias & Karagiannidis 2005) for the rational case.
  11. ^ a b Sornette, D. (2004). Critical Phenomena in Natural Science: Chaos, Fractals, Self-organization, and Disorder..
  12. ^ "1.3.3.30. Weibull Plot". www.itl.nist.gov.
  13. ^ Wayne Nelson (2004) Applied Life Data Analysis. Wiley-Blackwell ISBN 0-471-64462-5
  14. ^ Bauckhage, Christian (2013). "Computing the Kullback-Leibler Divergence between two Weibull Distributions". arXiv:1310.3713 [cs.IT].
  15. ^ "CumFreq, Distribution fitting of probability, free software, cumulative frequency".
  16. ^ a b Lee, Se Yoon; Mallick, Bani (2021). "Bayesian Hierarchical Modeling: Application Towards Production Results in the Eagle Ford Shale of South Texas". Sankhya B. doi:10.1007/s13571-020-00245-8.
  17. ^ "Wind Speed Distribution Weibull – REUK.co.uk". www.reuk.co.uk.
  18. ^ 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.
  19. ^ 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.
  20. ^ Austin, L. G.; Klimpel, R. R.; Luckie, P. T. (1984). Process Engineering of Size Reduction. Hoboken, NJ: Guinn Printing Inc. ISBN 0-89520-421-5.
  21. ^ Chandrashekar, S. (1943). "Stochastic Problems in Physics and Astronomy". Reviews of Modern Physics. 15 (1): 86.
  22. ^ "System evolution and reliability of systems". Sysev (Belgium). 2010-01-01.
  23. ^ Montgomery, Douglas (2012-06-19). Introduction to statistical quality control. [S.l.]: John Wiley. p. 95. ISBN 9781118146811.
  24. ^ Chatfield, C.; Goodhardt, G.J. (1973). "A Consumer Purchasing Model with Erlang Interpurchase Times". Journal of the American Statistical Association. 68 (344): 828–835. doi:10.1080/01621459.1973.10481432.

Bibliography