Weibull distribution

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

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

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 ${\displaystyle \lambda ={\sqrt {2}}\sigma }$[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]

• A value of ${\displaystyle k<1\,}$ indicates that the failure rate decreases over time (Lindy effect). 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 ${\displaystyle k=1\,}$ 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 ${\displaystyle k>1\,}$ 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 inflexion point at ${\displaystyle (\mathrm {e} ^{1/k}-1)/\mathrm {e} ^{1/k},\,k>1\,}$.

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.[4][5] The shape parameter k is the same as above, while the scale parameter is ${\displaystyle b=\lambda ^{-k}}$. In this case, for x ≥ 0, the probability density function is

${\displaystyle f(x;k,b)=bkx^{k-1}\mathrm {e} ^{-bx^{k}},}$

the cumulative distribution function is

${\displaystyle F(x;k,b)=1-\mathrm {e} ^{-bx^{k}},}$

the hazard function is

${\displaystyle h(x;k,b)=bkx^{k-1},}$

and the mean is

${\displaystyle b^{-1/k}\Gamma (1+1/k).}$

A third parameterization can also be found.[6][7] The shape parameter k is the same as in the standard case, while the scale parameter is ${\displaystyle \beta =1/\lambda }$. Then, for x ≥ 0, the probability density function is

${\displaystyle f(x;k,\beta )=\beta k({\beta x})^{k-1}\mathrm {e} ^{-(\beta x)^{k}}}$

the cumulative distribution function is

${\displaystyle F(x;k,\beta )=1-\mathrm {e} ^{-(\beta x)^{k}},}$

and the hazard function is

${\displaystyle h(x;k,\beta )=\beta k({\beta x})^{k-1}.}$

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

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

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

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

for 0 ≤ p < 1.

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

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

The Mean time between failures MTBF is

${\displaystyle {\text{MTBF}}(k,\lambda )=\lambda \Gamma (1+1/k).}$

The moment generating function of the logarithm of a Weibull distributed random variable is given by[8]

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

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

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

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

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

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

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

and

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

The skewness is given by

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

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

The excess kurtosis is given by

${\displaystyle \gamma _{2}={\frac {-6\Gamma _{1}^{4}+12\Gamma _{1}^{2}\Gamma _{2}-3\Gamma _{2}^{2}-4\Gamma _{1}\Gamma _{3}+\Gamma _{4}}{[\Gamma _{2}-\Gamma _{1}^{2}]^{2}}}}$

where ${\displaystyle \Gamma _{i}=\Gamma (1+i/k)}$. The kurtosis excess may also be written as:

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

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

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

Alternatively, one can attempt to deal directly with the integral

${\displaystyle \operatorname {E} \left[e^{tX}\right]=\int _{0}^{\infty }e^{tx}{\frac {k}{\lambda }}\left({\frac {x}{\lambda }}\right)^{k-1}e^{-(x/\lambda )^{k}}\,dx.}$

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.[9] With t replaced by −t, one finds

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

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) harvtxt error: no target: CITEREFMuraleedharanSoares2014 (help) by a direct approach.

The information entropy is given by

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

where ${\displaystyle \gamma }$ 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 x^k equal to λ^k and a fixed expected value of ln(x^k) equal to ln(λ^k) − ${\displaystyle \gamma }$.

The maximum likelihood estimator for the ${\displaystyle \lambda }$ parameter given ${\displaystyle k}$ is

${\displaystyle {\widehat {\lambda }}^{k}={\frac {1}{n}}\sum _{i=1}^{n}x_{i}^{k}}$

The maximum likelihood estimator for ${\displaystyle k}$ is the solution for k of the following equation[10]

${\displaystyle 0={\frac {\sum _{i=1}^{n}x_{i}^{k}\ln x_{i}}{\sum _{i=1}^{n}x_{i}^{k}}}-{\frac {1}{k}}-{\frac {1}{n}}\sum _{i=1}^{n}\ln x_{i}}$

This equation defining ${\displaystyle {\widehat {k}}}$ only implicitly, one must generally solve for ${\displaystyle k}$ by numerical means.

When ${\displaystyle x_{1}>x_{2}>\cdots >x_{N}}$ are the ${\displaystyle N}$ largest observed samples from a dataset of more than ${\displaystyle N}$ samples, then the maximum likelihood estimator for the ${\displaystyle \lambda }$ parameter given ${\displaystyle k}$ is[10]

${\displaystyle {\widehat {\lambda }}^{k}={\frac {1}{N}}\sum _{i=1}^{N}(x_{i}^{k}-x_{N}^{k})}$

Also given that condition, the maximum likelihood estimator for ${\displaystyle k}$ is

${\displaystyle 0={\frac {\sum _{i=1}^{N}(x_{i}^{k}\ln x_{i}-x_{N}^{k}\ln x_{N})}{\sum _{i=1}^{N}(x_{i}^{k}-x_{N}^{k})}}-{\frac {1}{N}}\sum _{i=1}^{N}\ln x_{i}}$

Again, this being an implicit function, one must generally solve for ${\displaystyle k}$ by numerical means.