Weibull distribution

Weibull (2-Parameter)
Probability density function

Cumulative distribution function

Parameters \lambda\in (0, +\infty)\, scale
k\in (0, +\infty)\, shape
Support x \in [0, +\infty)\,
PDF f(x)=\begin{cases} \frac{k}{\lambda}\left(\frac{x}{\lambda}\right)^{k-1}e^{-(x/\lambda)^{k}} & x\geq0\\ 0 & x<0\end{cases}
CDF \begin{cases}1- e^{-(x/\lambda)^k} & x\geq0\\ 0 & x<0\end{cases}
Mean \lambda \, \Gamma(1+1/k)\,
Median \lambda(\ln(2))^{1/k}\,
Mode \begin{cases} \lambda \left(\frac{k-1}{k} \right)^{\frac{1}{k}}\, &k>1\\ 0 &k=1\end{cases}
Variance \lambda^2\left[\Gamma\left(1+\frac{2}{k}\right) - \left(\Gamma\left(1+\frac{1}{k}\right)\right)^2\right]\,
Skewness \frac{\Gamma(1+3/k)\lambda^3-3\mu\sigma^2-\mu^3}{\sigma^3}
Ex. kurtosis (see text)
Entropy \gamma(1-1/k)+\ln(\lambda/k)+1 \,
MGF \sum_{n=0}^\infty \frac{t^n\lambda^n}{n!}\Gamma(1+n/k), \ k\geq1
CF \sum_{n=0}^\infty \frac{(it)^n\lambda^n}{n!}\Gamma(1+n/k)

In probability theory and statistics, the Weibull distribution /ˈveɪbʊl/ is a continuous probability distribution. It is named after 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

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

f(x;\lambda,k) = \begin{cases} \frac{k}{\lambda}\left(\frac{x}{\lambda}\right)^{k-1}e^{-(x/\lambda)^{k}} & x\geq0 ,\\ 0 & x<0, \end{cases}

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 \lambda = \frac{\sqrt{2}}{k} [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:

In the field of materials science, the shape parameter k of a distribution of strengths is known as the Weibull modulus.

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. It is interesting to note that 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 = 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.

Distribution function

The cumulative distribution function for the Weibull distribution is

F(x;k,\lambda) = 1- e^{-(x/\lambda)^k}\,

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

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

Q(p;k,\lambda) = \lambda {(-\ln(1-p))}^{1/k}

for 0 ≤ p < 1.

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

h(x;k,\lambda) = {k \over \lambda} \left({x \over \lambda}\right)^{k-1}.

Moments

The moment generating function of the logarithm of a Weibull distributed random variable is given by[3]

E\left[e^{t\log X}\right] = \lambda^t\Gamma\left(\frac{t}{k}+1\right)

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

E\left[e^{it\log X}\right] = \lambda^{it}\Gamma\left(\frac{it}{k}+1\right).

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

m_n = \lambda^n \Gamma\left(1+\frac{n}{k}\right).

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

\mathrm{E}(X) = \lambda \Gamma\left(1+\frac{1}{k}\right)\,

and

\textrm{var}(X) = \lambda^2\left[\Gamma\left(1+\frac{2}{k}\right) - \left(\Gamma\left(1+\frac{1}{k}\right)\right)^2\right]\,.

The skewness is given by

\gamma_1=\frac{\Gamma\left(1+\frac{3}{k}\right)\lambda^3-3\mu\sigma^2-\mu^3}{\sigma^3}

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

The excess kurtosis is given by

\gamma_2=\frac{-6\Gamma_1^4+12\Gamma_1^2\Gamma_2-3\Gamma_2^2 -4\Gamma_1\Gamma_3+\Gamma_4}{[\Gamma_2-\Gamma_1^2]^2}

where \Gamma_i=\Gamma(1+i/k). The kurtosis excess may also be written as:

\gamma_{2}=\frac{\lambda^4\Gamma(1+\frac{4}{k})-4\gamma_{1}\sigma^3\mu-6\mu^2\sigma^2-\mu^4}{\sigma^4}-3

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

E\left[e^{tX}\right] = \sum_{n=0}^\infty \frac{t^n\lambda^n}{n!}\Gamma\left(1+\frac{n}{k}\right).

Alternatively, one can attempt to deal directly with the integral

E\left[e^{tX}\right] = \int_0^\infty e^{tx} \frac{k}{\lambda}\left(\frac{x}{\lambda}\right)^{k-1}e^{-(x/\lambda)^k}\,dx.

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.[4] With t replaced by −t, one finds

E\left[e^{-tX}\right] = \frac1{ \lambda^k\, t^k} \, \frac{ p^k \, \sqrt{q/p}} {(\sqrt{2 \pi})^{q+p-2}} \, G_{p,q}^{\,q,p} \!\left( \left. \begin{matrix} \frac{1-k}{p}, \frac{2-k}{p}, \dots, \frac{p-k}{p} \\ \frac{0}{q}, \frac{1}{q}, \dots, \frac{q-1}{q} \end{matrix} \; \right| \, \frac {p^p} {\left( q \, \lambda^k \, t^k \right)^q} \right)

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.

Information entropy

The information entropy is given by

H(\lambda,k) = \gamma\left(1\!-\!\frac{1}{k}\right) + \ln\left(\frac{\lambda}{k}\right) + 1

where \gamma is the Euler–Mascheroni constant.

Parameter estimation

Maximum likelihood

The maximum likelihood estimator for the \lambda parameter given k is,

\hat \lambda^k = \frac{1}{n} \sum_{i=1}^n x_i^k

The maximum likelihood estimator for k is,

\hat k^{-1} = \frac{\sum_{i=1}^n x_i^k \ln x_i } {\sum_{i=1}^n x_i^k } - \frac{1}{n} \sum_{i=1}^n \ln x_i

This being an implicit function, one must generally solve for k by numerical means.

When x_1 > x_2 > ... > x_N are the N largest observed samples from a dataset of more than N samples, then the maximum likelihood estimator for the \lambda parameter given k is,[5]

\hat \lambda^k = \frac{1}{N} \sum_{i=1}^N (x_i^k - x_N^k)


Also given that condition, the maximum likelihood estimator for k is,

\hat k^{-1} = \frac{\sum_{i=1}^N (x_i^k \ln x_i - x_N^k \ln x_N)} {\sum_{i=1}^N (x_i^k - x_N^k)} - \frac{1}{N} \sum_{i=1}^N \ln x_i

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

Weibull plot

The fit of data to a Weibull distribution can be visually assessed using a Weibull Plot.[6] The Weibull Plot is a plot of the empirical cumulative distribution function \hat F(x) of data on special axes in a type of Q-Q plot. The axes are \ln(-\ln(1-\hat F(x))) versus \ln(x). The reason for this change of variables is the cumulative distribution function can be linearized:

\begin{align} F(x) &= 1-e^{-(x/\lambda)^k}\\ -\ln(1-F(x)) &= (x/\lambda)^k\\ \underbrace{\ln(-\ln(1-F(x)))}_{\textrm{'y'}} &= \underbrace{k\ln x}_{\textrm{'mx'}} - \underbrace{k\ln \lambda}_{\textrm{'c'}} \end{align}

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 \hat F = \frac{i-0.3}{n+0.4} where i is the rank of the data point and n is the number of data points.[7]

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 k and the scale parameter \lambda can also be inferred.

The Weibull distribution is used

Fitted cumulative Weibull distribution to maximum one-day rainfalls using CumFreq, see also distribution fitting

Related distributions

f(x;k,\lambda, \theta)={k \over \lambda} \left({x - \theta \over \lambda}\right)^{k-1} e^{-({x-\theta \over \lambda})^k}\,

for x \geq \theta and f(x; k, λ, θ) = 0 for x < θ, where k >0 is the shape parameter, \lambda >0 is the scale parameter and \theta is the location parameter of the distribution. When θ=0, this reduces to the 2-parameter distribution.

X = \left(\frac{W}{\lambda}\right)^k

is the standard exponential distribution with intensity 1.[3]

f_{\rm{Frechet}}(x;k,\lambda)=\frac{k}{\lambda} \left(\frac{x}{\lambda}\right)^{-1-k} e^{-(x/\lambda)^{-k}} = *f_{\rm{Weibull}}(x;-k,\lambda).
f(x;P_{\rm{80}},m) = \begin{cases} 1-e^{ln\left(0.2\right)\left(\frac{x}{P_{\rm{80}}}\right)^m} & x\geq0 ,\\ 0 & x<0 ,\end{cases}

where

x: Particle size
P_{\rm{80}}: 80th percentile of the particle size distribution
m: Parameter describing the spread of the distribution

See also

References

  1. ↑ Papoulis, Pillai, "Probability, Random Variables, and Stochastic Processes, 4th Edition
  2. ↑ http://www.mathworks.com.au/help/stats/rayleigh-distribution.html
  3. ↑ 3.0 3.1 3.2 Johnson, Kotz & Balakrishnan 1994
  4. ↑ See (Cheng, Tellambura & Beaulieu 2004) for the case when k is an integer, and (Sagias & Karagiannidis 2005) for the rational case.
  5. ↑ Sornette, D. (2004). Critical Phenomena in Natural Science: Chaos, Fractals, Self-organization, and Disorder..
  6. ↑ The Weibull plot
  7. ↑ Wayne Nelson (2004) Applied Life Data Analysis. Wiley-Blackwell ISBN 0-471-64462-5
  8. ↑ Survival/Failure Time Analysis
  9. ↑ Wind Speed Distribution Weibull
  10. ↑ "The Weibull distribution as a general model for forecasting technological change". Sciencedirect.com. Retrieved 2013-09-05.
  11. ↑ "System evolution and reliability of systems". Sysev (Belgium). 2010-01-01.
  12. ↑ Montgomery, Douglas. Introduction to statistical quality control. [S.l.]: John Wiley. p. 95. ISBN 9781118146811.
  13. ↑ C. Chatfield and G.J. Goodhardt: “A Consumer Purchasing Model with Erlang Interpurchase Times”; Journal of the American Statistical Association, Dec. 1973, Vol.68, pp.828-835

Bibliography

External links