Weibull distribution


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 Rosin & Rammler 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 Cumulative distribution function#Complementary [cumulative distribution function (tail distribution)|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:
  • A value of indicates that the failure rate decreases over time. 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 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 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 inflection point at.
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.

Optional parameterizations

First option

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 option

A second parameterization option 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 = 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, which may also be written as
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 Muraleedharan et al.

Minima

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

Reparametrization tricks

Fix some. Let be nonnegative, and not all zero, and let be independent samples of, then
  • .

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 − .

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. 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. 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 and the scale parameter 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 can be read off either from a look up table or a graph of versus. A more accurate estimate of 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 parameter given is
The maximum likelihood estimator for is the solution for k of the following equation
This equation defines 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.

Applications

The Weibull distribution is used

Prediction

  • It is often of interest to predict probabilities of out-of-sample data under the assumption that both the training data and the out-of-sample data follow a Weibull distribution.
  • Predictions generated by substituting the method of moments or maximum likelihood estimates of the Weibull parameters given above into the cumulative distribution function ignore parameter uncertainty. As a result, the probabilities are not well calibrated, do not reflect the frequencies of out-of-sample events, and, in particular, underestimate the probabilities of out-of-sample tail events.
  • Predictions generated using the objective Bayesian approach of calibrating prior prediction completely eliminate this underestimation. The Weibull distribution is one of a number of statistical distributions with group structure. As a result of the group structure, the Weibull has associated left and right Haar measures. The use of the right Haar measure as the prior in a Bayesian prediction gives probabilities that are perfectly calibrated, for any underlying true parameter values. Calibrating prior prediction for the Weibull using the appropriate right Haar prior is implemented in the R software package fitdistcp.

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