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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 (Lindy effect). 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 inflexion point at .
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.
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/? 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 = ?. 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 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.
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.
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 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. The interpretation of the cumulative distribution function is that is the mass fraction of particles with diameter smaller than , where is the mean particle size and is a measure of the spread of particle sizes.
In describing random point clouds (such as the positions of particles in an ideal gas): the probability to find the nearest-neighbor particle at a distance from a given particle is given by a Weibull distribution with and equal to the density of the particles.
for and for , where is the shape parameter, is the scale parameter and is the location parameter of the distribution. value sets an initial failure-free time before the regular Weibull process begins. When , this reduces to the 2-parameter distribution.
The Weibull distribution can be characterized as the distribution of a random variable such that the random variable
This implies that the Weibull distribution can also be characterized in terms of a uniform distribution: if is uniformly distributed on , then the random variable is Weibull distributed with parameters and . Note that here is equivalent to just above. This leads to an easily implemented numerical scheme for simulating a Weibull distribution.
The Weibull distribution interpolates between the exponential distribution with intensity when and a Rayleigh distribution of mode when .
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