A GENERALIZED LOGISTIC DISTRIBUTION
SARALEES NADARAJAH AND SAMUEL KOTZ
Received 11 October 2004
A generalized logistic distribution is proposed, based on the fact that the difference of two independent Gumbel-distributed random variables has the standard logistic distribution.
1. Introduction
If X1 and X2 are independent Gumbel-distributed random variables with the common cdf F(x) = exp − exp(−x) , (1.1) then it is well known that the difference Z = X1 − X2 has the standard logistic distribution with the pdf
exp(z) fZ(z) = (1.2) 1+exp(z) 2 for −∞
Copyright © 2005 Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences 2005:19 (2005) 3169–3174 DOI: 10.1155/IJMMS.2005.3169 3170 A generalized logistic distribution
A recent book by Kotz and Nadarajah [16], which describes this distribution, lists over fifty applications ranging from accelerated life testing through to earthquakes, floods, horse racing, rainfall, queues in supermarkets, sea currents, wind speeds, and track race records (to mention just a few).
2. The generalization The pdf corresponding to (1.3)is
1 x − µi x − µi fi(x) = exp − exp − exp − , (2.1) σi σi σi and thus the pdf of Z = X1 − X2 can be written as
∞ fZ(z) = f1(x) f2(x − z)dx −∞ ∞ x − µ x − µ = 1 − 1 − − 1 σ σ exp σ exp exp σ (2.2) 1 2 −∞ 1 1 x − z − µ x − z − µ × exp − 2 exp − exp − 2 dx. σ2 σ2
Setting y = exp(−x/σ1), (2.2) can be expressed as 1 µ1 µ2 + z fZ(z) = exp + I µ1,µ2,σ1,σ2 , (2.3) σ2 σ1 σ2 where I denotes the integral
∞ µ µ + z σ1/σ2 1 2 σ1/σ2 I µ1,µ2,σ1,σ2 = y exp − exp y +exp y dy. (2.4) 0 σ1 σ2
We refer to (2.3)asthegeneralized logistic distribution. The integral term in (2.4)isdif- ficult to calculate. However, for some particular choices of (µ1,µ2,σ1,σ2), one can obtain the following explicit expressions. (i) If σ1 = σ2 = σ, then by standard integration one can obtain
I = 1 . 2 (2.5) exp µ1/σ +exp µ2 + z /σ
If, in addition, µ1 = µ2 = µ, then the above reduces to
exp(−2µ/σ) I = . (2.6) 1+exp(z/σ) 2 S. Nadarajah and S. Kotz 3171
(ii) If σ1 = 2σ2, then one can show by [17, equation (2.3.15.7)] that
√ √ √ α πα αβ2 αβ αβ α2β I = exp 2erfc + αβ2 erfc − , (2.7) 8 4 2 2 4 where α = exp{(µ2 + z)/σ2}, β = exp{µ1/(2σ2)},anderfc(·), denotes the complementary error function defined by
2 x erfc(x) = 1 − √ exp − t2 dt. (2.8) π 0
(iii) If 0 <σ1/σ2 = p/q < 1(wherep ≥ 1andq ≥ 1 are co-prime integers), then one can show by [17, equation (2.3.1.13)] that
q− 1 −α j p j I = ( ) Γ (1 + ) β−(1+p(1+j)/q) j 1+ q j=0 ! (2.9) p j − q ppαq × F ∆ p (1 + ) ∆ q j ( 1) p+1 q 1, ,1+ q ; ( ,1+ ); qqβp ,
where α = exp{(µ2 + z)/σ2}, β = exp(µ1/σ1), ∆(k,a) denotes the sequence
a a a k − ∆ k a = +1 ... + 1 ( , ) k , k , , k , (2.10) mFn denotes the generalized hypergeometric function defined by
∞ α ··· α xk 1 k m k mFn α1,...,αm;β1,...,βn;x = , (2.11) β ··· βn k k=0 1 k k ! and (c)k = c(c +1)···(c + k − 1) denotes the ascending factorial. (iv) If σ1/σ2 = p/q > 1(wherep ≥ 1andq ≥ 1 are coprime integers), then one can show again by [17, equation (2.3.1.13)]that
p− 1 q −β j q j I = ( ) Γ (1 + ) α−(1+q(1+j)/p) pj 1+ p j=0 ! (2.12) q j − pqqβp × F ∆ q (1 + ) ∆ p j ( 1) q+1 p 1, ,1+ p ; ( ,1+ ); ppαq ,
where α = exp{(µ2 + z)/σ2} and β = exp(µ1/σ1). 3172 A generalized logistic distribution
0.3 )
x .
( 0 2 f
0.1
0 −3 −2 −10 1 2 3 x
σ1/σ2 = 0.2 σ1/σ2 = 5 σ1/σ2 = 1 σ1/σ2 = 10 σ1/σ2 = 2
Figure 2.1. The generalized logistic pdf (2.3)forσ1/σ2 = 0.2,1,2,5,10, σ2 = 1, µ1 = 0, and µ2 = 1.
0.25
0.2
0.15 ) x ( f 0.1
0.05
0 −3 −2 −10 1 2 3 x
σ1/σ2 = 0.2 σ1/σ2 = 5 σ1/σ2 = 1 σ1/σ2 = 10 σ1/σ2 = 2
Figure 2.2. The generalized logistic pdf (2.3)forσ1/σ2 = 0.2,1,2,5,10, σ2 = 1, µ1 = 1, and µ2 = 0.
Figures 2.1 and 2.2 illustrate possible shapes of the pdf (2.3) for selected values of (µ1,µ2,σ1,σ2). The magnitude of σ1/σ2 clearly controls the shape of the pdf. In fact, if µ1 = 0, then 1 µ2 + z µ2 + z fZ(z) −→ exp exp − exp (2.13) σ2 σ2 σ2 S. Nadarajah and S. Kotz 3173 as σ1/σ2 → 0. Also, f (z) → 0foreveryz ∈ (0,∞)asσ1/σ2 →∞. On the other hand, if µ1 = 0, then f (z) → 0foreveryz ∈ (0,∞) irrespective of whether σ1/σ2 → 0orσ1/σ2 →∞.
3. Applications
The standard logistic distribution given by (1.2) has important uses in describing growth and as a substitute for the normal distribution. It has also attracted interesting applica- tions in the modeling of the dependence of chronic obstructive respiratory disease preva- lence on smoking and age, degrees of pneumoconiosis in coal miners, geological issues, hemolytic uremic syndrome data for children, physiochemical phenomenon, psycholog- ical issues, survival time of diagnosed leukemia patients, and weight gain data. The main feature of the generalized logistic distribution in (2.3)isthatnewparametersareintro- duced to control both location and scale. Thus, (2.3)allowsforagreaterdegreeofflexi- bility and we can expect this to be useful in many more practical situations.
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Saralees Nadarajah: Department of Statistics, University of Nebraska, Lincoln, NE 68583, USA E-mail address: [email protected]
Samuel Kotz: Department of Engineering Management and Systems Engineering, The George Washington University, Washington, DC 20052, USA E-mail address: [email protected] Mathematical Problems in Engineering
Special Issue on Time-Dependent Billiards
Call for Papers This subject has been extensively studied in the past years Guest Editors for one-, two-, and three-dimensional space. Additionally, Edson Denis Leonel, Departamento de Estatística, such dynamical systems can exhibit a very important and still Matemática Aplicada e Computação, Instituto de unexplained phenomenon, called as the Fermi acceleration Geociências e Ciências Exatas, Universidade Estadual phenomenon. Basically, the phenomenon of Fermi accelera- Paulista, Avenida 24A, 1515 Bela Vista, 13506-700 Rio Claro, tion (FA) is a process in which a classical particle can acquire SP, Brazil ; [email protected] unbounded energy from collisions with a heavy moving wall. This phenomenon was originally proposed by Enrico Fermi Alexander Loskutov, Physics Faculty, Moscow State in 1949 as a possible explanation of the origin of the large University, Vorob’evy Gory, Moscow 119992, Russia; energies of the cosmic particles. His original model was [email protected] then modified and considered under different approaches and using many versions. Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos). We intend to publish in this special issue papers reporting research on time-dependent billiards. The topic includes both conservative and dissipative dynamics. Papers dis- cussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome. To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned. Mathematical papers regarding the topics above are also welcome. Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/. Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/ according to the following timetable:
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