↑ arvau

Wikipedia / 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 (1933) 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).

If the quantity, x, is a "time-to-failure", the Weibull distribution gives a distribution for which the hazard 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:

  • indicates that the hazard rate decreases over time (like in case of the Lindy effect, which however corresponds to Pareto distributions rather than Weibull distributions). This happens if there is significant "infant mortality", or defective items failing early and the hazard 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;

  • indicates that the hazard rate is constant over time. This might suggest random external events are causing mortality, or failure. The Weibull distribution reduces to an exponential distribution;

  • indicates that the hazard 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.

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. 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 is replaced with a rate parameter = 1/. 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 ofk. Vice versa: at F(x; k; λ) = 0.632 the value ofx≈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 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

, 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

. 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. (2007)

Minima

is given by

.

Reparametrization tricks

, then

  • .

Shannon entropy

The information entropy is given by

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 k and a fixed expected value of ln(xk) equal to ln(k)−.

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 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

  1. ,

is the number of data points. Another common estimator is

  1. .

can also be inferred.

Method of moments

The coefficient of variation of Weibull distribution depends only on the shape parameter:

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 is

is the solution for k of the following equation

by numerical means.

is

is

by numerical means.

Applications

The Weibull distribution is used

In communications systems engineering

  • In radar systems to model the dispersion of the received signals level produced by some types of clutters

  • To model fading channels in wireless communications, as the Weibull fading model seems to exhibit good fit to experimental fading channel measurements

  • In information retrieval to model dwell times on web pages.

  • In general insurance to model the size of reinsurance claims, and the cumulative development of asbestosis losses

  • In forecasting technological change (also known as the Sharif-Islam model)

  • In hydrology the Weibull distribution is applied to extreme events such as annual maximum one-day rainfalls and river discharges.

  • In decline curve analysis to model oil production rate curve of shale oil wells.

  • In describing the size of particles generated by grinding, milling and crushing operations, the 2-Parameter Weibull distribution is used, and in these applications it is sometimes known as the Rosin–Rammler distribution. In this context it predicts fewer fine particles than the log-normal distribution and it is generally most accurate for narrow particle size distributions. is the mass fraction is a measure of the spread of particle sizes.

  • equal to the density of the particles.

  • In calculating the rate of radiation-induced single event effects onboard spacecraft, a four-parameter Weibull distribution is used to fit experimentally measured device cross section probability data to a particle linear energy transfer spectrum. The Weibull fit was originally used because of a belief that particle energy levels align to a statistical distribution, but this belief was later proven false and the Weibull fit continues to be used because of its many adjustable parameters, rather than a demonstrated physical basis.

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 (known as the right Haar 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.https://cran.r-project.org/web/packages/fitdistcp/index.html

Related distributions

The Weibull distribution was first applied by Rosin & Rammler (1933) to describe particle size distributions. It is widely used in mineral processing to describe particle size distributions in comminution processes. In this context the cumulative distribution is given by where

  • is the particle size

  • is the 80th percentile of the particle size distribution

  • is a parameter describing the spread of the distribution

  • Because of its availability in spreadsheets, it is also used where the underlying behavior is actually better modeled by an Erlang distribution.

  • (Exponential distribution)

  • For the same values of k, the Gamma distribution takes on similar shapes, but the Weibull distribution is more platykurtic.

See also

Notes

References

Further reading

  • .

  • .