IIT Kanpur

Generalized ExponentialDistribution

Debasis Kundu

Department of Mathematics and Statistics

Indian Institute of Technology, Kanpur

India

Joint Work: R.D. Gupta, Univ. of New Brunswick & A.

Manglick, Univ. of Umea.

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Outline of the Talk

• Genesis of the Model

• Model Description

• Common Properties with Weibull and

Gamma Distribution

• Moments of the GED

• Inference

• Closeness with Other Distributions

• Generation of Gamma and Normal Random

Variables Using GED

• References

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Genesis of the Model

Gompertz (1825, Phil. Trans. Roy. Soc. Lon.)

used the following distribution function to

represent mortality growth

G(t) =(

1− ρe−λt)α

t >1

λln ρ.

Ahuja and Nash (1967, Sankhya A) also used

this model and some related model for growth

curve mortality

Gupta and Kundu (1999, ANZJS) consider a spe-

cial case of this model

F (x;α, λ) =(

1− e−λx)α; x > 0.

This is a special case of the exponentiated

Weibull model proposed by Mudholkar and his

co-workers (1995, Technometrics)

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Model Description

Here α is the shape and λ is the scale param-

eter. It has different shapes of the density

function

f (x;α, λ) = αλ(

1− e−xλ)α−1

e−xλ; x > 0.

It has different shapes of the hazard functions

h(x;α, λ) =f (x;α, λ)

1− F (x;α, λ)=αλ

(

1− e−λx)α−1

e−λx

1− (1− e−λx)α ; x > 0.

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Physical Interpretations

A parallel system is a system where the system

works if at least one of the components works

If the shape parameter α is an integer it repre-

sents the life time of a parallel system when

each component follows exponential distribu-

tion

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Shapes of the DifferentDensity Functions

Since λ is the scale parameter we take λ = 1

α = 0.50

α = 1.0

α = 2.0 α = 10.0

α = 50.0

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 2 4 6 8 10

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Shapes of the Different HazardFunctions

Since λ is the scale parameter we take λ = 1

α = 0.4

α = 0.5

α = 1.0

α = 2.0

α = 5.0

0

0.5

1

1.5

2

0 2 4 6 8 10

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Comparisons with Gamma andWeibull Distributions

Density Functions

• Shape parameter = 1 all are equal to ex-

ponential distribution

• Shape parameter > 1, all are unimodal

• Shape parameter < 1, all are decreasing

functions

Hazard Functions

• Shape parameter = 1 all have constant haz-

ard functions

• Shape parameter > 1, all are increasing

functions

• Shape parameter < 1, all are decreasing

functions

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Closer look at the hazardfunctions

Shape-Parameter Gamma Weibull GE

1 1 1 1

> 1 0 ↑ 1 0 ↑ ∞ 0 ↑ 1

< 1 ∞ ↓ 1 ∞ ↓ 0 ∞ ↓ 1

GE is more closer to the gamma distribution

rather than the Weibull distribution in this

respect

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Ordering Relations in Terms ofthe Shape Parameter

LR HAZ ST

Gamma Y Y Y

Weibull N N N

GE Y Y Y

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Moments

Moment generating Function (λ = 1).

M(t) =Γ(α + 1)Γ(1− t)

Γ(α− t + 1)(t < 1).

E(X) = ψ(α + 1)− ψ(1), V (X) = ψ′(1)− ψ′(α + 1).

Stochastic representation

Xd=

[α]∑

j=1

Yj

j+ < α >+ Z

Here [α] is the integer part of α and < α > is

the fractional part of α. Yj’s are i.i.d. exponen-

tial with mean 1 and Z follows GE with shape

parameter < α >.

E(X) =n∑

i=1

1

i, V (X) =

n∑

i=1

1

i2.

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Sum of n i.i.d GE Distribution

If X1, . . . , Xn are n i.i.d. GE(α), then the density

function of X =n∑

i=1Xi is

fX(x) =∞∑

j=0cjfGE(x;nα + j)

Here

cj > 0n∑

i=1cj = 1.

If we approximate it by M terms, i.e.

fX(x) ≈M−1∑

j=0cjfGE(x;nα + j),

then the error due to approximations is

bounded by

1−M−1∑

j=0cj

g(x),

Explicit expression of g is available.

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Estimation

The problem is to estimate the unknown param-

eters from a random sample of size n.

• The family of the GED satisfy all the reg-

ularity conditions

• The MLEs work quite well if α is not very

close to 0

• Fixed point type iterative process can be

used to solve the non-linear equation

• If α is close to 0, the iterative process takes

longer time to converge

• Other estimators like Moment Estimators,

L-Moment estimators, Percentile Estima-

tors, Least Squares Estimators, BLUE have

been tried

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If one of the parameters is known

• If the scale parameter is knownthen the MLE of the shape parame-ter can be obtained in explicit form

• If the shape parameter is knownthen the MLEs of the scale parame-ter can be obtained by solving a non-linear equation.

• Other estimators also can be ob-tained accordingly

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Testing and ConfidenceIntervals

• LRT can be used for testing purposes if

both are unknown

• If λ is known it is an exponential family,

then the UMP or UMPU test exists for

testing the shape parameter

• If α is known then testing the scale param-

eter LRT can be used

• If both the parameters are unknown then

asymptotic confidence intervals can be used

for constructing confidence intervals

• If λ is known then exact confidence inter-

vals based on χ2 distribution is available

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Closeness with OtherDistributions

For certain ranges of the shape and scale pa-

rameters the distribution function of the GE

distribution can be very close to the cor-

responding distribution functions of Weibull,

gamma and log-normal distributions.

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The distribution function of GE(12.9)and LN(0.3807482, 2.9508672)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2 4 6 8 10

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Disadvantage: It is very difficult todistinguish between the two models.Selecting the correct model for smallsample sizes becomes almost impossible.

Advantage: Log-normal distributionfunction or gamma distribution func-tion can be approximated very well byGE

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We can generate approximate randomsamples of log-normal ( implies normalalso) and gamma distributions using GE

A very convenient approximation ofthe standard normal distribution func-tion can be used as

Φ(z) ≈(

1− e−ezσ+µ)12.9

where

σ = 0.3807482 µ = 1.0820991

In this case the error of approximationis less than 0.0003.

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Generation of N(0, 1)

Using the approximation, approximateN(0, 1) can be easily generated

Generation of Gamma(α)

Consider the Gamma distribution withthe following density function

fGA(x;α) =1

Γαxα−1e−x; x > 0

It can be shown that for 0 < α < 1

fGA(x;α) ≤2α

Γ(α + 1)fGE(x;α,

1

2)

Using the inequality and byAcceptance-Rejection method gammarandom numbers can be generated

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References

• Ahuja, J. C. and Nash, S. W. (1967), “The

generalized Gompertz-Verhulst family of

distributions”, Sankhya, Ser. A., vol. 29, 141

- 156.

• Gompertz, B. (1825), “On the nature of the

function expressive of the law of human

mortality, and on a new mode of determin-

ing the value of life contingencies”, Philo-

sophical Transactions of the Royal Society London,

vol. 115, 513 - 585.

• Gupta, R. D. and Kundu, D. (1999). “Gener-

alized exponential distributions”, Australian

and New Zealand Journal of Statistics, vol. 41, 173

- 188.

• Gupta, R. D. and Kundu, D. (2003), “Discrim-

inating between the Weibull and the GE

distributions”, Computational Statistics and Data

Analysis, vol. 43, 179 - 196.

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References (cont.)

• Gupta, R. D. and Kundu, D. (2005), “Compar-

ison of the Fisher information between the

Weibull and generalized exponential dis-

tribution”, (to appear in the Journal of Sta-

tistical Planning and Inference)

• Kundu, D., Gupta, R D. and Manglick,

A. (2005),“Discriminating between the log-

normal and generalized exponential distri-

bution”, Journal of the Statistical Planning and In-

ference, vol. 127, 213 - 227.

• Raqab, M. Z. (2002), “Inferences for gen-

eralized exponential distribution based on

record statistics”, Journal of Statistical Plan-

ning and Inference, vol. 104, 339 - 350.

• Raqab, M. Z. and Ahsanullah, M. (2001),

“Estimation of the location and scale pa-

rameters of generalized exponential distri-

bution based on order statistics”, Journal of

Statistical Computation and Simulation, vol. 69,

109 - 124.

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References (cont.)

• Zheng, G. (2002), “Fisher information ma-

trix in type -II censored data from exponen-

tiated exponential family”, Biometrical Jour-

nal, vol. 44, 353 - 357.