In statistics, the ''q''-Weibull distribution is a

probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomeno ...

that generalizes the

Weibull distribution In probability theory and statistics, the Weibull distribution is a continuous probability distribution. It is named after Swedish mathematician Waloddi Weibull, who described it in detail in 1951, although it was first identified by Maurice R ...

and the

Lomax distribution The Lomax distribution, conditionally also called the Pareto Type II distribution, is a heavy-tail probability distribution used in business, economics, actuarial science, queueing theory and Internet traffic modeling. It is named after K. ...

(Pareto Type II). It is one example of a

Tsallis distribution In statistics, a Tsallis distribution is a probability distribution derived from the maximization of the Tsallis entropy under appropriate constraints. There are several different families of Tsallis distributions, yet different sources may refere ...

Characterization

Probability density function

The

probability density function In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) c ...

of a ''q''-Weibull

random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the p ...

is: :

f(x;q,\lambda,\kappa) =
\begin
(2-q)\frac\left(\frac\right)^ e_q(-(x/\lambda)^)& x\geq0 ,\\
0 & x<0,
\end

where ''q'' < 2,

\kappa

> 0 are ''

shape parameter In probability theory and statistics, a shape parameter (also known as form parameter) is a kind of numerical parameter of a parametric family of probability distributionsEveritt B.S. (2002) Cambridge Dictionary of Statistics. 2nd Edition. CUP. t ...

s'' and λ > 0 is the ''

scale parameter In probability theory and statistics, a scale parameter is a special kind of numerical parameter of a parametric family of probability distributions. The larger the scale parameter, the more spread out the distribution. Definition If a family o ...

'' of the distribution and :

\end

is the ''q''-exponential

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. Ev ...

of a ''q''-Weibull

is: :

\begin1- e_^ & x\geq0\\ 0 & x<0\end

where :

\lambda' =

q' =

Mean

The mean of the ''q''-Weibull distribution is :

\mu(q,\kappa,\lambda) =
\begin
\lambda\,\left(2+\frac+\frac\right)(1-q)^\,B\left +\frac,2+\frac\right q<1 \\
\lambda\,\Gamma(1+\frac) & q=1\\
\lambda\,(2 - q) (q-1)^\,B\left +\frac, -\left(1+\frac+\frac\right)\right & 1 where B() is the

Beta function In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral : \Beta(z_1,z_2) = \int_0^1 t^ ...

and

\Gamma()

is the

Gamma function In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except th ...

. The expression for the mean is a continuous function of ''q'' over the range of definition for which it is finite.

Relationship to other distributions

The ''q''-Weibull is equivalent to the Weibull distribution when ''q'' = 1 and equivalent to the ''q''-exponential when

\kappa=1

The ''q''-Weibull is a generalization of the Weibull, as it extends this distribution to the cases of finite support (''q'' < 1) and to include

heavy-tailed distribution In probability theory, heavy-tailed distributions are probability distributions whose tails are not exponentially bounded: that is, they have heavier tails than the exponential distribution. In many applications it is the right tail of the distrib ...

(q \ge 1+\frac)

. The ''q''-Weibull is a generalization of the

(Pareto Type II), as it extends this distribution to the cases of finite support and adds the

\kappa

parameter. The Lomax parameters are: :

\alpha =  ~,~ \lambda_\text =

As the Lomax distribution is a shifted version of the

Pareto distribution The Pareto distribution, named after the Italian civil engineer, economist, and sociologist Vilfredo Pareto ( ), is a power-law probability distribution that is used in description of social, quality control, scientific, geophysical, actu ...

, the ''q''-Weibull for

\kappa=1

is a shifted reparameterized generalization of the Pareto. When ''q'' > 1, the ''q''-exponential is equivalent to the Pareto shifted to have support starting at zero. Specifically: :

\text X \sim Y \,

References