In probability theory and statistics, the Weibull distribution 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 to describe a particle size distribution.

Definition

Standard parameterization

The probability density function of a Weibull random variable is 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 ). 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.

Alternative parameterizations

First alternative

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. 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

Second alternative

A 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 the cumulative distribution function is the quantile 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 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, 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. 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. With t replaced by −t, one finds where G is the Meijer G-function. The characteristic function has also been obtained by. The characteristic function and moment generating function of 3-parameter Weibull distribution have also been derived by by a direct approach.

Minima

Let be independent and identically distributed Weibull random variables with scale parameter \lambda and shape parameter k. If the minimum of these n random variables is, then the cumulative probability distribution of Z is given by That is, Z will also be Weibull distributed with scale parameter and with shape parameter k.

Reparametrization tricks

Fix some \alpha > 0. Let be nonnegative, and not all zero, and let be independent samples of, then

Shannon entropy

The information entropy is given by where \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) − \gamma.

Kullback–Leibler divergence

The Kullback–Leibler divergence between two Weibull distributions is given by

Parameter estimation

Ordinary least square using Weibull plot

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 of data on special axes in a type of Q–Q plot. The axes are versus \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 where i is the rank of the data point and n is the number of data points. Another common estimator is 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.

Method of moments

The coefficient of variation of Weibull distribution depends only on the shape parameter: Equating the sample quantities to, the moment estimate of the shape parameter k can be read off either from a look up table or a graph of CV^2 versus k. A more accurate estimate of \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

Maximum likelihood

The maximum likelihood estimator for the \lambda parameter given k is The maximum likelihood estimator for k is the solution for k of the following equation This equation defines \widehat k only implicitly, one must generally solve for k by numerical means. When 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 Also given that condition, the maximum likelihood estimator for k is Again, this being an implicit function, one must generally solve for k by numerical means.

Applications

The Weibull distribution is used

Related distributions