Supriya Ghosh (Editor)

Q Weibull distribution

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Mean
  
(see article)

Q-Weibull distribution

Parameters
  
q < 2 {displaystyle q<2} shape (real) λ > 0 {displaystyle lambda >0} rate (real) κ > 0 {displaystyle kappa >0,} shape (real)

Support
  
x ∈ [ 0 ; + ∞ )  for  q ≥ 1 {displaystyle xin [0;+infty )!{ ext{ for }}qgeq 1} x ∈ [ 0 ; λ ( 1 − q ) 1 / κ )  for  q < 1 {displaystyle xin [0;{lambda over {(1-q)^{1/kappa }}}){ ext{ for }}q<1}

PDF
  
{ ( 2 − q ) κ λ ( x λ ) κ − 1 e q − ( x / λ ) κ x ≥ 0 0 x < 0 {displaystyle {egin{cases}(2-q){ rac {kappa }{lambda }}left({ rac {x}{lambda }} ight)^{kappa -1}e_{q}^{-(x/lambda )^{kappa }}&xgeq 00&x<0end{cases}}}

CDF
  
{ 1 − e q ′ − ( x / λ ′ ) κ x ≥ 0 0 x < 0 {displaystyle {egin{cases}1-e_{q'}^{-(x/lambda ')^{kappa }}&xgeq 00&x<0end{cases}}}

In statistics, the q-Weibull distribution is a probability distribution that generalizes the Weibull distribution and the Lomax distribution (Pareto Type II). It is one example of a Tsallis distribution.

Contents

Probability density function

The probability density function of a q-Weibull random variable is:

f ( x ; q , λ , κ ) = { ( 2 − q ) κ λ ( x λ ) κ − 1 e q ( − ( x / λ ) κ ) x ≥ 0 , 0 x < 0 ,

where q < 2, κ > 0 are shape parameters and λ > 0 is the scale parameter of the distribution and

e q ( x ) = { exp ⁡ ( x ) if  q = 1 , [ 1 + ( 1 − q ) x ] 1 / ( 1 − q ) if  q ≠ 1  and  1 + ( 1 − q ) x > 0 , 0 1 / ( 1 − q ) if  q ≠ 1  and  1 + ( 1 − q ) x ≤ 0 ,

is the q-exponential

Cumulative distribution function

The cumulative distribution function of a q-Weibull random variable is:

{ 1 − e q ′ − ( x / λ ′ ) κ x ≥ 0 0 x < 0

where

λ ′ = λ ( 2 − q ) 1 κ q ′ = 1 ( 2 − q )

Mean

The mean of the q-Weibull distribution is

μ ( q , κ , λ ) = { λ ( 2 + 1 1 − q + 1 κ ) ( 1 − q ) − 1 κ B [ 1 + 1 κ , 2 + 1 1 − q ] q < 1 λ Γ ( 1 + 1 κ ) q = 1 λ ( 2 − q ) ( q − 1 ) − 1 + κ κ B [ 1 + 1 κ , − ( 1 + 1 q − 1 + 1 κ ) ] 1 < q < 1 + 1 + 2 κ 1 + κ ∞ 1 + κ κ + 1 ≤ q < 2

where B ( ) is the Beta function and Γ ( ) is the Gamma function. 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 κ = 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 distributions ( q ≥ 1 + κ κ + 1 ) .

The q-Weibull is a generalization of the Lomax distribution (Pareto Type II), as it extends this distribution to the cases of finite support and adds the κ parameter. The Lomax parameters are:

α = 2 − q q − 1   ,   λ Lomax = 1 λ ( q − 1 )

As the Lomax distribution is a shifted version of the Pareto distribution, the q-Weibull for κ = 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:

If  X ∼ q-Weibull ( q , λ , κ = 1 )  and  Y ∼ [ Pareto ( x m = 1 λ ( q − 1 ) , α = 2 − q q − 1 ) − x m ] ,  then  X ∼ Y

References

Q-Weibull distribution Wikipedia
(Text) CC BY-SA