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.
The probability density function of a Weibull random variable is[2][3]
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 λ λ --> = 2 σ σ --> {\displaystyle \lambda ={\sqrt {2}}\sigma } [4]).
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]
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.
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 b = λ λ --> − − --> k {\displaystyle b=\lambda ^{-k}} . 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
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
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.
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.
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
The moment generating function of the logarithm of a Weibull distributed random variable is given by[11]
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 Γ Γ --> i = Γ Γ --> ( 1 + i / k ) {\displaystyle \Gamma _{i}=\Gamma (1+i/k)} , which may also be written as
where the mean is denoted by μ and the standard deviation is denoted by σ.
The excess kurtosis is given by
where Γ Γ --> i = Γ Γ --> ( 1 + i / k ) {\displaystyle \Gamma _{i}=\Gamma (1+i/k)} . The kurtosis excess may also be written as:
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
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) harvtxt error: no target: CITEREFMuraleedharanSoares2014 (help) by a direct approach.
Let X 1 , X 2 , … … --> , X n {\displaystyle X_{1},X_{2},\ldots ,X_{n}} be independent and identically distributed Weibull random variables with scale parameter λ λ --> {\displaystyle \lambda } and shape parameter k {\displaystyle k} . If the minimum of these n {\displaystyle n} random variables is Z = min ( X 1 , X 2 , … … --> , X n ) {\displaystyle Z=\min(X_{1},X_{2},\ldots ,X_{n})} , then the cumulative probability distribution of Z {\displaystyle Z} given by
That is, Z {\displaystyle Z} will also be Weibull distributed with scale parameter n − − --> 1 / k λ λ --> {\displaystyle n^{-1/k}\lambda } and with shape parameter k {\displaystyle k} .
Fix some α α --> > 0 {\displaystyle \alpha >0} . Let ( π π --> 1 , . . . , π π --> n ) {\displaystyle (\pi _{1},...,\pi _{n})} be nonnegative, and not all zero, and let g 1 , . . . , g n {\displaystyle g_{1},...,g_{n}} be independent samples of Weibull ( 1 , α α --> − − --> 1 ) {\displaystyle {\text{Weibull}}(1,\alpha ^{-1})} , then[13]
The information entropy is given by
where γ γ --> {\displaystyle \gamma } 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) − γ γ --> {\displaystyle \gamma } .
The Kullback–Leibler divergence between two Weibulll distributions is given by[14]
The fit of a Weibull distribution to data can be visually assessed using a Weibull plot.[15] The Weibull plot is a plot of the empirical cumulative distribution function F ^ ^ --> ( x ) {\displaystyle {\widehat {F}}(x)} of data on special axes in a type of Q–Q plot. The axes are ln --> ( − − --> ln --> ( 1 − − --> F ^ ^ --> ( x ) ) ) {\displaystyle \ln(-\ln(1-{\widehat {F}}(x)))} versus ln --> ( x ) {\displaystyle \ln(x)} . 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 F ^ ^ --> = i − − --> 0.3 n + 0.4 {\displaystyle {\widehat {F}}={\frac {i-0.3}{n+0.4}}} where i {\displaystyle i} is the rank of the data point and n {\displaystyle n} is the number of data points.[16][17]
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 {\displaystyle k} and the scale parameter λ λ --> {\displaystyle \lambda } can also be inferred.
The coefficient of variation of Weibull distribution depends only on the shape parameter:[18]
Equating the sample quantities s 2 / x ¯ ¯ --> 2 {\displaystyle s^{2}/{\bar {x}}^{2}} to σ σ --> 2 / μ μ --> 2 {\displaystyle \sigma ^{2}/\mu ^{2}} , the moment estimate of the shape parameter k {\displaystyle k} can be read off either from a look up table or a graph of C V 2 {\displaystyle CV^{2}} versus k {\displaystyle k} . A more accurate estimate of k ^ ^ --> {\displaystyle {\hat {k}}} 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
The maximum likelihood estimator for the λ λ --> {\displaystyle \lambda } parameter given k {\displaystyle k} is[18]
The maximum likelihood estimator for k {\displaystyle k} is the solution for k of the following equation[19]
This equation defines k ^ ^ --> {\displaystyle {\widehat {k}}} only implicitly, one must generally solve for k {\displaystyle k} by numerical means.
When x 1 > x 2 > ⋯ ⋯ --> > x N {\displaystyle x_{1}>x_{2}>\cdots >x_{N}} are the N {\displaystyle N} largest observed samples from a dataset of more than N {\displaystyle N} samples, then the maximum likelihood estimator for the λ λ --> {\displaystyle \lambda } parameter given k {\displaystyle k} is[19]
Also given that condition, the maximum likelihood estimator for k {\displaystyle k} is[citation needed]
Again, this being an implicit function, one must generally solve for k {\displaystyle k} by numerical means.
The Weibull distribution is used[citation needed]
f ( x ; k , λ λ --> , θ θ --> ) = k λ λ --> ( x − − --> θ θ --> λ λ --> ) k − − --> 1 e − − --> ( x − − --> θ θ --> λ λ --> ) k {\displaystyle f(x;k,\lambda ,\theta )={k \over \lambda }\left({x-\theta \over \lambda }\right)^{k-1}e^{-\left({x-\theta \over \lambda }\right)^{k}}\,}
X = ( W λ λ --> ) k {\displaystyle X=\left({\frac {W}{\lambda }}\right)^{k}}
f F r e c h e t ( x ; k , λ λ --> ) = k λ λ --> ( x λ λ --> ) − − --> 1 − − --> k e − − --> ( x / λ λ --> ) − − --> k = f W e i b u l l ( x ; − − --> k , λ λ --> ) . {\displaystyle 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 80 , m ) = { 1 − − --> e ln --> ( 0.2 ) ( x P 80 ) m x ≥ ≥ --> 0 , 0 x < 0 , {\displaystyle f(x;P_{\rm {80}},m)={\begin{cases}1-e^{\ln \left(0.2\right)\left({\frac {x}{P_{\rm {80}}}}\right)^{m}}&x\geq 0,\\0&x<0,\end{cases}}}
F ( x ; k , λ λ --> ) = { ∫ ∫ --> 0 ∞ ∞ --> 1 ν ν --> F ( x ; 1 , λ λ --> ν ν --> ) ( Γ Γ --> ( 1 k + 1 ) N k ( ν ν --> ) ) d ν ν --> , 1 ≥ ≥ --> k > 0 ; or ∫ ∫ --> 0 ∞ ∞ --> 1 s F ( x ; 2 , 2 λ λ --> s ) ( 2 π π --> Γ Γ --> ( 1 k + 1 ) V k ( s ) ) d s , 2 ≥ ≥ --> k > 0 ; {\displaystyle F(x;k,\lambda )={\begin{cases}\displaystyle \int _{0}^{\infty }{\frac {1}{\nu }}\,F(x;1,\lambda \nu )\left(\Gamma \left({\frac {1}{k}}+1\right){\mathfrak {N}}_{k}(\nu )\right)\,d\nu ,&1\geq k>0;{\text{or }}\\\displaystyle \int _{0}^{\infty }{\frac {1}{s}}\,F(x;2,{\sqrt {2}}\lambda s)\left({\sqrt {\frac {2}{\pi }}}\,\Gamma \left({\frac {1}{k}}+1\right)V_{k}(s)\right)\,ds,&2\geq k>0;\end{cases}}}
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