Kaniadakis Weibull Distribution
   HOME

TheInfoList



OR:

The Kaniadakis Weibull distribution (or ''κ''-Weibull distribution) is a
probability distribution In probability theory and statistics, a probability distribution is a Function (mathematics), function that gives the probabilities of occurrence of possible events for an Experiment (probability theory), experiment. It is a mathematical descri ...
arising as a generalization of the
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 on ...
. It is one example of a Kaniadakis ''κ''-distribution. The κ-Weibull distribution has been adopted successfully for describing a wide variety of
complex system A complex system is a system composed of many components that may interact with one another. Examples of complex systems are Earth's global climate, organisms, the human brain, infrastructure such as power grid, transportation or communication sy ...
s in seismology, economy, epidemiology, among many others.


Definitions


Probability density function

The Kaniadakis ''κ''-Weibull distribution is exhibits power-law right tails, and it has the following
probability density function In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a Function (mathematics), function whose value at any given sample (or point) in the sample space (the s ...
: : f_(x) = \frac \exp_\kappa(-\beta x^\alpha) valid for x \geq 0, where , \kappa, < 1 is the entropic index associated with the Kaniadakis entropy, \beta > 0 is the scale parameter, and \alpha > 0 is the shape parameter or
Weibull modulus The Weibull modulus is a Dimensionless quantity, dimensionless parameter of the Weibull distribution. It represents the width of a probability density function (PDF) in which a higher modulus is a characteristic of a narrower distribution of values ...
. The
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 on ...
is recovered as \kappa \rightarrow 0.


Cumulative distribution function

The
cumulative distribution function In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Ever ...
of ''κ''-Weibull distribution is given by
F_\kappa(x) = 1 - \exp_\kappa(-\beta x^\alpha)
valid for x \geq 0. The cumulative
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 on ...
is recovered in the classical limit \kappa \rightarrow 0.


Survival distribution and hazard functions

The survival distribution function of ''κ''-Weibull distribution is given by :S_\kappa(x) = \exp_\kappa(-\beta x^\alpha) valid for x \geq 0. The survival
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 on ...
is recovered in the classical limit \kappa \rightarrow 0. The hazard function of the ''κ''-Weibull distribution is obtained through the solution of the ''κ''-rate equation:
\frac = -h_\kappa S_\kappa(x)
with S_\kappa(0) = 1, where h_\kappa is the hazard function: :h_\kappa = \frac The cumulative ''κ''-Weibull distribution is related to the ''κ''-hazard function by the following expression: :S_\kappa = e^ where :H_\kappa (x) = \int_0^x h_\kappa(z) dz :H_\kappa (x) = \frac \textrm\left(\kappa \beta x^\alpha \right) is the cumulative ''κ''-hazard function. The cumulative hazard function of the
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 on ...
is recovered in the classical limit \kappa \rightarrow 0: H(x) = \beta x^\alpha .


Properties


Moments, median and mode

The ''κ''-Weibull distribution has moment of order m \in \mathbb given by :\operatorname ^m= \frac \frac \Gamma\Big(1+\frac\Big) The median and the mode are: : x_ (F_\kappa) = \beta^ \Bigg(\ln_\kappa (2)\Bigg)^ : x_ = \beta^ \Bigg( \frac\Bigg)^ \Bigg( \sqrt - 1 \Bigg)^ \quad (\alpha > 1)


Quantiles

The
quantile In statistics and probability, quantiles are cut points dividing the range of a probability distribution into continuous intervals with equal probabilities or dividing the observations in a sample in the same way. There is one fewer quantile t ...
s are given by the following expression
x_ (F_\kappa) = \beta^ \Bigg \ln_\kappa \Bigg(\frac \Bigg) \Bigg
with 0 \leq F_\kappa \leq 1.


Gini coefficient

The
Gini coefficient In economics, the Gini coefficient ( ), also known as the Gini index or Gini ratio, is a measure of statistical dispersion intended to represent the income distribution, income inequality, the wealth distribution, wealth inequality, or the ...
is:
\operatorname_\kappa = 1 - \frac \frac \frac


Asymptotic behavior

The ''κ''-Weibull distribution II behaves asymptotically as follows: : \lim_ f_\kappa (x) \sim \frac (2 \kappa \beta)^ x^ : \lim_ f_\kappa (x) = \alpha \beta x^


Related distributions

*The ''κ''-Weibull distribution is a generalization of: ** ''κ''-Exponential distribution of type II, when \alpha = 1; **
Exponential distribution In probability theory and statistics, the exponential distribution or negative exponential distribution is the probability distribution of the distance between events in a Poisson point process, i.e., a process in which events occur continuousl ...
when \kappa = 0 and \alpha = 1. *A ''κ''-Weibull distribution corresponds to a ''κ''-deformed Rayleigh distribution when \alpha = 2 and a
Rayleigh distribution In probability theory and statistics, the Rayleigh distribution is a continuous probability distribution for nonnegative-valued random variables. Up to rescaling, it coincides with the chi distribution with two degrees of freedom. The distributi ...
when \kappa = 0 and \alpha = 2.


Applications

The ''κ''-Weibull distribution has been applied in several areas, such as: * In
economy An economy is an area of the Production (economics), production, Distribution (economics), distribution and trade, as well as Consumption (economics), consumption of Goods (economics), goods and Service (economics), services. In general, it is ...
, for analyzing personal income models, in order to accurately describing simultaneously the income distribution among the richest part and the great majority of the population. * In
seismology Seismology (; from Ancient Greek σεισμός (''seismós'') meaning "earthquake" and -λογία (''-logía'') meaning "study of") is the scientific study of earthquakes (or generally, quakes) and the generation and propagation of elastic ...
, the κ-Weibull represents the statistical distribution of magnitude of the earthquakes distributed across the Earth, generalizing the
Gutenberg–Richter law In seismology, the Gutenberg–Richter law (GR law) expresses the relationship between the Richter magnitude scale, magnitude and total number of earthquakes in any given region and time period of ''at least'' that magnitude. : \log_ N = a - b M ...
, and the interval distributions of seismic data, modeling extreme-event return intervals. * In
epidemiology Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and Risk factor (epidemiology), determinants of health and disease conditions in a defined population, and application of this knowledge to prevent dise ...
, the κ-Weibull distribution presents a universal feature for epidemiological analysis.


See also

* Giorgio Kaniadakis * Kaniadakis statistics * Kaniadakis distribution * Kaniadakis κ-Exponential distribution * Kaniadakis κ-Gaussian distribution * Kaniadakis κ-Gamma distribution * Kaniadakis κ-Logistic distribution * Kaniadakis κ-Erlang distribution


References

{{Reflist


External links


Kaniadakis Statistics on arXiv.org
Continuous distributions Exponential family distributions Survival analysis