Weibull distribution


In probability theory and statistics, the Weibull distribution is a continuous probability distribution. It is named after Swedish mathematician Waloddi Weibull, who described it in detail in 1951, although it was first identified by 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 and the Rayleigh distribution.
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

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 hazard function is
and the mean is
A third parameterization can also be found. The shape parameter k is the same as in the standard case, while the scale parameter is. Then, for x ≥ 0, the probability density function is
the cumulative distribution 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 = 0 for x < 0.
If x = λ then F = 1 − e−1 ≈ 0.632 for all values of k. Vice versa: at F = 0.632 the value of x ≈ λ.
The quantile function for the Weibull distribution is
for 0 ≤ p < 1.
The failure rate h 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 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 the mean is denoted by and the standard deviation is denoted by.
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.

Shannon entropy

The information entropy is given by
where 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 equal to ln − .

Parameter estimation

Maximum likelihood

The maximum likelihood estimator for the parameter given is
The maximum likelihood estimator for is the solution for k of the following equation
This equation defining only implicitly, one must generally solve for by numerical means.
When are the largest observed samples from a dataset of more than samples, then the maximum likelihood estimator for the parameter given is
Also given that condition, the maximum likelihood estimator for is
Again, this being an implicit function, one must generally solve for by numerical means.

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. 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 is the rank of the data point and 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 and the scale parameter can also be inferred.

[Kullback–Leibler divergence]

Applications

The Weibull distribution is used
, see also distribution fitting