Column
Weibull Distribution
The Weibull Distribution
Parameter Estimates
The probability density function of a Weibull random variable is:[1]
f(x;,k) =
\[\begin{cases}
\frac{k}{\lambda}\left(\frac{x}{\lambda}\right)^{k-1}e^{-(x/\lambda)^{k}} & x\geq0 ,\\
0 & x<0,
\end{cases}\]
where k > 0 is the shape parameter and > 0 is the scale parameter of the distribution.
The cumulative distribution function for the Weibull distribution is \[F(x;k,\lambda) = 1- e^{-(x/\lambda)^k}\,
for x ≥ 0, and F(x; k; \lambda ) = 0 for x < 0.\] The quantile (inverse cumulative distribution) function for the Weibull distribution is \[Q(p;k,\lambda) = \lambda {(-\ln(1-p))}^{1/k}
for 0 ≤ p < 1.\] The failure rate h (or hazard rate) is given by \[ h(x;k,\lambda) = {k \over \lambda} \left({x \over \lambda}\right)^{k-1}.\]
Failure rate is the frequency with which an engineered system or component fails, expressed, for example, in failures per hour. It is often denoted by the Greek letter (lambda) and is important in reliability engineering.
Weibull distributions
\[ \lambda(t) = \alpha \beta t^{\beta -1} \] \((\lambda , \alpha, \beta > 0)\)
Gompertz Distribution
\[ \lambda(t) = \alpha e^{\beta t} (\lambda , \alpha > 0), ( \beta \mbox{ is real})\]
GTDL \[ \lambda(t) = \frac{\gamma e^{\alpha+\beta t}}{1+e^{\alpha+\beta t}} \] \[ (\lambda > 0),(\alpha , \beta \mbox{is real})\]
When given the hazard function, the properties of the corresponding survival distribution can be deduced
\[
\LARGE
f(t)=\lambda(t)S(t) = \lambda(t) exp(-\int_0^t \lambda(u)d(u))
\]
These models are called parametric models. \(\lambda(t)\) involves parameters = unknown constants in these models. The parameters
control the shape and scale variation. They are not unknown functions, i.e., functions are wholly specified in parametric models.
Applications
The Weibull distribution is used
- In survival analysis[8]
- In reliability engineering and failure analysis
- In industrial engineering to represent manufacturing and delivery times
- In extreme value theory
- In weather forecasting
- To describe wind speed distributions, as the natural distribution often matches the Weibull shape[9]
- 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
Fitted cumulative Weibull distribution to maximum one-day rainfalls using CumFreq, see also distribution fitting 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)[10] In hydrology the Weibull distribution is applied to extreme events such as annual maximum one-day rainfalls and river discharges. The blue picture illustrates an example of fitting the Weibull distribution to ranked annually maximum one-day rainfalls showing also the 90% confidence belt based on the binomial distribution. The rainfall data are represented by plotting positions as part of the cumulative frequency analysis.
Time to Failure
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 k < 1 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.
A value of k = 1 indicates that the failure rate is constant over time. This might suggest random external events are causing mortality, or failure.
A value of k > 1 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.