Probability density function![]() | |||
Cumulative distribution function![]() | |||
Parameters |
λ
∈
(
0
,
+
∞
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{\displaystyle \lambda \in (0,+\infty )\,}
scale k ∈ ( 0 , + ∞ ) {\displaystyle k\in (0,+\infty )\,} shape | ||
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Support | x ∈ {\displaystyle \lambda ^{2}\left\,} | ||
Skewness | Γ ( 1 + 3 / k ) λ 3 − 3 μ σ 2 − μ 3 σ 3 {\displaystyle {\frac {\Gamma (1+3/k)\lambda ^{3}-3\mu \sigma ^{2}-\mu ^{3}}{\sigma ^{3}}}} | ||
Excess kurtosis | (see text) | ||
Entropy | γ ( 1 − 1 / k ) + ln ( λ / k ) + 1 {\displaystyle \gamma (1-1/k)+\ln(\lambda /k)+1\,} | ||
MGF | ∑ n = 0 ∞ t n λ n n ! Γ ( 1 + n / k ) , k ≥ 1 {\displaystyle \sum _{n=0}^{\infty }{\frac {t^{n}\lambda ^{n}}{n!}}\Gamma (1+n/k),\ k\geq 1} | ||
CF | ∑ n = 0 ∞ ( i t ) n λ n n ! Γ ( 1 + n / k ) {\displaystyle \sum _{n=0}^{\infty }{\frac {(it)^{n}\lambda ^{n}}{n!}}\Gamma (1+n/k)} | ||
Kullback–Leibler divergence | see below |
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, 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
f ( x ; λ , k ) = { k λ ( x λ ) k − 1 e − ( x / λ ) k , x ≥ 0 , 0 , x < 0 , {\displaystyle f(x;\lambda ,k)={\begin{cases}{\frac {k}{\lambda }}\left({\frac {x}{\lambda }}\right)^{k-1}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 λ = 2 σ {\displaystyle \lambda ={\sqrt {2}}\sigma } ).
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. 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. The shape parameter k is the same as above, while the scale parameter is b = λ − k {\displaystyle b=\lambda ^{-k}} . In this case, for x ≥ 0, the probability density function is
f ( x ; k , b ) = b k x k − 1 e − b x k , {\displaystyle f(x;k,b)=bkx^{k-1}e^{-bx^{k}},}the cumulative distribution function is
F ( x ; k , b ) = 1 − e − b x k , {\displaystyle F(x;k,b)=1-e^{-bx^{k}},}the hazard function is
h ( x ; k , b ) = b k x k − 1 , {\displaystyle h(x;k,b)=bkx^{k-1},}and the mean is
b − 1 / k Γ ( 1 + 1 / k ) . {\displaystyle b^{-1/k}\Gamma (1+1/k).} Second alternativeA second alternative parameterization can also be found. 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
f ( x ; k , β ) = β k ( β x ) k − 1 e − ( β x ) k {\displaystyle f(x;k,\beta )=\beta k({\beta x})^{k-1}e^{-(\beta x)^{k}}}the cumulative distribution function is
F ( x ; k , β ) = 1 − e − ( β x ) k , {\displaystyle F(x;k,\beta )=1-e^{-(\beta x)^{k}},}and the hazard function is
h ( x ; k , β ) = β k ( β x ) k − 1 . {\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
F ( x ; k , λ ) = 1 − e − ( x / λ ) k {\displaystyle F(x;k,\lambda )=1-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
Q ( p ; k , λ ) = λ ( − ln ( 1 − p ) ) 1 / k {\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
h ( x ; k , λ ) = k λ ( x λ ) k − 1 . {\displaystyle h(x;k,\lambda )={k \over \lambda }\left({x \over \lambda }\right)^{k-1}.}The Mean time between failures MTBF is
MTBF ( k , λ ) = λ Γ ( 1 + 1 / k ) . {\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
E = λ t Γ ( t k + 1 ) {\displaystyle \operatorname {E} \left=\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 = λ i t Γ ( i t k + 1 ) . {\displaystyle \operatorname {E} \left=\lambda ^{it}\Gamma \left({\frac {it}{k}}+1\right).}In particular, the nth raw moment of X is given by
m n = λ n Γ ( 1 + n k ) . {\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
E ( X ) = λ Γ ( 1 + 1 k ) {\displaystyle \operatorname {E} (X)=\lambda \Gamma \left(1+{\frac {1}{k}}\right)\,}and
var ( X ) = λ 2 . {\displaystyle \operatorname {var} (X)=\lambda ^{2}\left\,.}The skewness is given by
γ 1 = 2 Γ 1 3 − 3 Γ 1 Γ 2 + Γ 3 3 / 2 {\displaystyle \gamma _{1}={\frac {2\Gamma _{1}^{3}-3\Gamma _{1}\Gamma _{2}+\Gamma _{3}}{^{3/2}}}}where Γ i = Γ ( 1 + i / k ) {\displaystyle \Gamma _{i}=\Gamma (1+i/k)}
γ 1 = Γ ( 1 + 3 k ) λ 3 − 3 μ σ 2 − μ 3 σ 3 {\displaystyle \gamma _{1}={\frac {\Gamma \left(1+{\frac {3}{k}}\right)\lambda ^{3}-3\mu \sigma ^{2}-\mu ^{3}}{\sigma ^{3}}}} , which may also be written aswhere the mean is denoted by μ and the standard deviation is denoted by σ.
The excess kurtosis is given by
γ 2 = − 6 Γ 1 4 + 12 Γ 1 2 Γ 2 − 3 Γ 2 2 − 4 Γ 1 Γ 3 + Γ 4 2 {\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}}{^{2}}}}where Γ i = Γ ( 1 + i / k ) {\displaystyle \Gamma _{i}=\Gamma (1+i/k)}
γ 2 = λ 4 Γ ( 1 + 4 k ) − 4 γ 1 σ 3 μ − 6 μ 2 σ 2 − μ 4 σ 4 − 3. {\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.} . 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
E = ∑ n = 0 ∞ t n λ n n ! Γ ( 1 + n k ) . {\displaystyle \operatorname {E} \left=\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 = ∫ 0 ∞ e t x k λ ( x λ ) k − 1 e − ( x / λ ) k d x . {\displaystyle \operatorname {E} \left=\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. With t replaced by −t, one finds
E = 1 λ k t k p k q / p ( 2 π ) q + p − 2 G p , q q , p ( 1 − k p , 2 − k p , … , p − k p 0 q , 1 q , … , q − 1 q | p p ( q λ k t k ) q ) {\displaystyle \operatorname {E} \left={\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.
Let X 1 , X 2 , … , X n {\displaystyle X_{1},X_{2},\ldots ,X_{n}}
F ( z ) = 1 − e − n ( z / λ ) k . {\displaystyle F(z)=1-e^{-n(z/\lambda )^{k}}.} be independent and identically distributed Weibull random variables with scale parameter 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 byThat is, Z {\displaystyle Z}
will also be Weibull distributed with scale parameter n − 1 / k λ {\displaystyle n^{-1/k}\lambda } and with shape parameter k {\displaystyle k} .Fix some α > 0 {\displaystyle \alpha >0}
. Let ( π 1 , . . . , 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 − 1 ) {\displaystyle {\text{Weibull}}(1,\alpha ^{-1})} , thenThe information entropy is given by
H ( λ , k ) = γ ( 1 − 1 k ) + ln ( λ k ) + 1 {\displaystyle H(\lambda ,k)=\gamma \left(1-{\frac {1}{k}}\right)+\ln \left({\frac {\lambda }{k}}\right)+1}where γ {\displaystyle \gamma } 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 {\displaystyle \gamma } .
is theThe Kullback–Leibler divergence between two Weibulll distributions is given by
D KL ( W e i b 1 ∥ W e i b 2 ) = log k 1 λ 1 k 1 − log k 2 λ 2 k 2 + ( k 1 − k 2 ) + ( λ 1 λ 2 ) k 2 Γ ( k 2 k 1 + 1 ) − 1 {\displaystyle D_{\text{KL}}(\mathrm {Weib} _{1}\parallel \mathrm {Weib} _{2})=\log {\frac {k_{1}}{\lambda _{1}^{k_{1}}}}-\log {\frac {k_{2}}{\lambda _{2}^{k_{2}}}}+(k_{1}-k_{2})\left+\left({\frac {\lambda _{1}}{\lambda _{2}}}\right)^{k_{2}}\Gamma \left({\frac {k_{2}}{k_{1}}}+1\right)-1}The fit of a Weibull distribution to data can be visually assessed using a Weibull plot. 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:
F ( x ) = 1 − e − ( x / λ ) k − ln ( 1 − F ( x ) ) = ( x / λ ) k ln ( − ln ( 1 − F ( x ) ) ) ⏟ 'y' = k ln x ⏟ 'mx' − k ln λ ⏟ 'c' {\displaystyle {\begin{aligned}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{aligned}}}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.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 {\displaystyle \lambda } can also be inferred.The coefficient of variation of Weibull distribution depends only on the shape parameter:
C V 2 = σ 2 μ 2 = Γ ( 1 + 2 k ) − ( Γ ( 1 + 1 k ) ) 2 ( Γ ( 1 + 1 k ) ) 2 . {\displaystyle CV^{2}={\frac {\sigma ^{2}}{\mu ^{2}}}={\frac {\Gamma \left(1+{\frac {2}{k}}\right)-\left(\Gamma \left(1+{\frac {1}{k}}\right)\right)^{2}}{\left(\Gamma \left(1+{\frac {1}{k}}\right)\right)^{2}}}.}Equating the sample quantities s 2 / x ¯ 2 {\displaystyle s^{2}/{\bar {x}}^{2}}
Γ ( 1 + 2 k ) − ( Γ ( 1 + 1 k ) ) 2 ( Γ ( 1 + 1 k ) ) 2 = s 2 x ¯ 2 . {\displaystyle {\frac {\Gamma \left(1+{\frac {2}{k}}\right)-\left(\Gamma \left(1+{\frac {1}{k}}\right)\right)^{2}}{\left(\Gamma \left(1+{\frac {1}{k}}\right)\right)^{2}}}={\frac {s^{2}}{{\bar {x}}^{2}}}.} to σ 2 / 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 solveThe moment estimate of the scale parameter can then be found using the first moment equation as
λ ^ = x ¯ Γ ( 1 + 1 k ^ ) . {\displaystyle {\hat {\lambda }}={\frac {\bar {x}}{\Gamma \left(1+{\frac {1}{\hat {k}}}\right)}}.}The maximum likelihood estimator for the λ {\displaystyle \lambda } parameter given k {\displaystyle k} is
λ ^ = ( 1 n ∑ i = 1 n x i k ) 1 k {\displaystyle {\widehat {\lambda }}=\left({\frac {1}{n}}\sum _{i=1}^{n}x_{i}^{k}\right)^{\frac {1}{k}}}The maximum likelihood estimator for k {\displaystyle k}
0 = ∑ i = 1 n x i k ln x i ∑ i = 1 n x i k − 1 k − 1 n ∑ i = 1 n ln x i {\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}} is the solution for k of the following equationThis 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}}
λ ^ k = 1 N ∑ i = 1 N ( x i k − x N k ) {\displaystyle {\widehat {\lambda }}^{k}={\frac {1}{N}}\sum _{i=1}^{N}(x_{i}^{k}-x_{N}^{k})} 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 {\displaystyle \lambda } parameter given k {\displaystyle k} isAlso given that condition, the maximum likelihood estimator for k {\displaystyle k}
0 = ∑ i = 1 N ( x i k ln x i − x N k ln x N ) ∑ i = 1 N ( x i k − x N k ) − 1 N ∑ i = 1 N ln x i {\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}} isAgain, this being an implicit function, one must generally solve for k {\displaystyle k}
by numerical means.The Weibull distribution is used
f ( x ; k , λ , θ ) = k λ ( x − θ λ ) k − 1 e − ( x − θ 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}}\,}
for x ≥ θ {\displaystyle x\geq \theta } and f ( x ; k , λ , θ ) = 0 {\displaystyle f(x;k,\lambda ,\theta )=0} for x < θ {\displaystyle x<\theta } , where k > 0 {\displaystyle k>0} is the shape parameter, λ > 0 {\displaystyle \lambda >0} is the scale parameter and θ {\displaystyle \theta } is the location parameter of the distribution. θ {\displaystyle \theta } value sets an initial failure-free time before the regular Weibull process begins. When θ = 0 {\displaystyle \theta =0} , this reduces to the 2-parameter distribution.X = ( W λ ) k {\displaystyle X=\left({\frac {W}{\lambda }}\right)^{k}}
is the standard exponential distribution with intensity 1.f F r e c h e t ( x ; k , λ ) = k λ ( x λ ) − 1 − k e − ( x / λ ) − k = f W e i b u l l ( x ; − k , 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}}}
whereF ( x ; k , λ ) = { ∫ 0 ∞ 1 ν F ( x ; 1 , λ ν ) ( Γ ( 1 k + 1 ) N k ( ν ) ) d ν , 1 ≥ k > 0 ; or ∫ 0 ∞ 1 s F ( x ; 2 , 2 λ s ) ( 2 π 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}}}
where N k ( ν ) {\displaystyle {\mathfrak {N}}_{k}(\nu )} is the Stable count distribution and V k ( s ) {\displaystyle V_{k}(s)} is the Stable vol distribution.Authority control databases: National |
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