Journal of Statistical Theory and Applications

Volume 17, Issue 3, September 2018, Pages 537 - 553

Estimating the Parameters of a Generalized Exponential Distribution

Authors
Syed Afzal Hossain
Department of Mathematics and Statistics, University of Nebraska at Kearney
Received 22 March 2017, Accepted 13 September 2018, Available Online 30 September 2018.
DOI
10.2991/jsta.2018.17.3.9How to use a DOI?
Keywords
Generalized exponential distribution; Maximum Likelihood Estimation; Method of Moments; Moment generating function; Sufficient condition
Abstract

A three-parameter generalized exponential distribution was suggested by Hossain and Ahsanullah [5]. Some important aspects of this distribution in the area of estimation remain unexplored in the earlier works. We discuss here the maximum likelihood (ML) method and the method of moments to estimate the parameters. The sufficient condition for the existence of a unique solution of the parameters obtained by the method of moments is derived. Finally, a popular real life numerical example is illustrated to investigate the application of both methods of estimation and the results obtained therein are compared with four similar well known three-parameter distributions.

Copyright
© 2018, the Authors. Published by Atlantis Press.
Open Access
This is an open access article under the CC BY-NC license (http://creativecommons.org/licences/by-nc/4.0/).

1. Introduction

The estimation of parameters and drawing conclusions based on the estimated parameters is one of the important aspects of inferential statistics. The three-parameter gamma and Weibull distributions are commonly used in life time data analysis. These distributions have many desirable statistical properties. For more information on these distributions see Alexander [1] and Jackson [6].

Mudholkar [9] considered a three-parameter exponentiated Weibull distribution. This new family is suitable for modeling data that indicate non-monotone hazard rates and can be adopted for testing goodness of fit of Weibull as a submodel. It is a right skewed unimodal density function. The usefulness and flexibility of the family is illustrated by reanalyzing five classical data sets on bus-motor failures.

Gupta [3] proposed special cases of the exponentiated Weibull and exponentiated exponential models and compared their performances with the two-parameter gamma family and two-parameter Weibull family, mainly through data analysis and computer simulations. For more on exponentiated Weibull, beta Gumbel, and beta exponentiated distribution see Nadarajah [10], Nadarajah and Kotz [11], Nadarajah and Kotz [12], Nassar and Eissa [13], and Raqab and Ahsanullah [14].

A three-parameter generalized exponential distribution was suggested by Gupta and Kundu [4]. This distribution is a particular case of the exponentiated Weibull distribution originally proposed by Mudholkar, Srivastava, and Freimer [9]. Since its distribution function has closed form the inference based on the censored data can be handled more easily than gamma family. One of its drawbacks being it is less flexible than the other two families for graduating tail thickness.

Recently, Kundu and Gupta [7] introduced the bivariate generalized exponential distribution so that the marginal have generalized exponential distributions. They reanalyzed one set of data and the results have shown that the bivariate generalized exponential distribution provides a better fit than the bivariate exponential distribution.

Hossain and Ahsanullah [5] introduced a new three-parameter generalized exponential distribution that represents a different type of generalization than Gupta and Kundu [4]. This distribution approaches a two parameter exponential distribution when the shape parameter approaches zero, whereas the distribution in Gupta and Kundu [4] approaches two-parameter exponential if the shape parameter approaches one.

In this article we have considered the estimation of parameters of the three-parameter generalized exponential distribution introduced by Hossain and Ahsanullah [5] by using the maximum likelihood estimation and the method of moments. A sufficient condition for the existence of unique solution for the parameters estimated by the method of moments is derived. A real time numerical example is analyzed and compared with four other similar distributions.

The rest of the article is organized as follows: Section 2 introduces the generalized exponential distribution and some its important properties. In section 3 we described the parameter estimation procedure using maximum likelihood method and method of moments. In section 4 we showed the applications by analyzing a set of real time data and compared the results with similar three-parameter distributions.

2. Generalized Exponential Distribution

2.1 Distribution and density functions

A random variable X is said to have generalized exponential distribution (GE2) if it has the following distribution function

F(x)={0ifx<δ1(1αxδσ)1αifδxδ+σα1ifx>δ+σαwherex>δ>0,α>0,andσ>0.
The corresponding probability density function is given by
f(x)={1σ(1αxδσ)1α1ifδ<x<δ+σα0Otherwise
Hossain and Ahsanullah [5] has showed that f(x) is approaches a two-parameter exponential distribution when α → 0+. Therefore we will limit our discussions on α close to zero. However, later discussions will justify that 0 < α < 1.

The generalized exponential distribution introduced by Gupta and Kundu [4] is given by

F(x)={0ifx<δ(1eαxδσ)αifx>δ
The distribution is a two-parameter exponential distribution when α → 1.

To keep a clear distinction between GE2 and the Gupta and Kundu [4], the generalized exponential distribution in Gupta and Kundu [4] will be denoted as GE1.

The graphs of probability density function and distribution function for various values of parameters α, δ, and σ are shown in Figure 1 and Figure 2 respectively.

Figure 1.

Plots of probability density function for various values of parameters

Figure 2.

Plots of distribution function for various values of parameters

3. Estimation of the Parameters

3.1 Maximum Likelihood Method

In this section we briefly discuss the maximum likelihood estimators, see Bickel and Docksum [2]. Let x1, x2,...,xn be a random sample of size n from GE2(α,σ,δ); then the likelihood function, L(α,σ,δ), is given by

L(α,δ,σ)=1σni=1n(1αxiδσ)1α1.
Clearly, the maximum likelihood estimate δ^ of δ is
δ^=Min{x1,x2,,xn}if0<α<1.
We estimate the parameters α and σ by solving the following normal equations
dln(L(α,δ,σ))dσ=nσ+(1α1)i=1nαxiδ^σ21αxiδ^σ=0
dln(L(α,δ,σ))dα=1α2i=1nln(1αxiδ^σ)(1α1)i=1nxiδ^σ1αxiδ^σ=0
It may be arduous to determine the necessary and sufficient conditions for the existence of unique solution to the parameters for nonlinear equations. However, two equations in two variables should be solvable with modern technology unless solution does not exist for the parameters. Note that the MLEs exist only if 0 < α < 1.

3.2 Method of Moments

The r-th raw moment of the generalized exponential distribution, Hossain and Ahsanullah [5], is given by j=0rr!(rj)!σj(α+1)(2α+1)(jα+1)δrj, and will be used to determine the raw moments of the distribution. If a parametric family has r parameters, the moment equations are 1ni=1nxij=μj,j=1,2,,r where μ j=E(Xj|θ) a function of the unknown parameter vector θ. The method of moments estimator, see Bickel and Docksum [2], is the solution to these equations.

Theorem 3.1.

The parameters obtained by the method of moments for a (α, δ, σ) generalized exponential distribution to have a unique solution, it is sufficient to show that the sample skewness is bounded between 0 and 2, that is,

0<1n(xx¯)3(1n(xx¯)2)32<2

Proof.

Equating the sample raw moments with the corresponding population raw moments (Hossain and Ahsanullah [5]) we obtain the following three equations in terms of the parameters α, δ, and σ.

x¯=δ+σα+1
1nx2=δ2+2δσα+1+2σ2(α+1)(2α+1)
1nx3=δ3+3δ2σα+1+6δσ2(α+1)(2α+1)+6σ3(α+1)(2α+1)(3α+1)
Solving equations (4) and (5) simultaneously for δ and σ, we obtain
σ=(α+1)(2α+1)12(1nx2x¯2)12=(α+1)(2α+1)12s
and
δ=x¯(2α+1)12s
where s=(1nx2x¯2)12.

Substituting σ and δ from the equations (7) and (8) into the equation (6), we obtain

1nx3=(x¯(2α+1)12s)3+3(x¯(2α+1)12s)2((α+1)(2α+1)12s)α+1+6(x¯(2α+1)12s)((α+1)(2α+1)12s)2(α+1)(2α+1)+6((α+1)(2α+1)12s)3(α+1)(2α+1)(3α+1).
On further simplification, we obtain
(2α+1)12(1α)(3α+1)K=0
where K=(1nx3x¯33s2x¯2s3).

Now it comes down to solving a non-linear equation in only one unknown parameter α. Let f(α)=(2α+1)12(1α)(3α+1)K. Therefore,

f(α)=3α(2α+1)123K.
Equation (10) shows that f(α) is decreasing in α if K > 0. It can be easily shown that f(0) = 1 − K and f(∞) < 0. Thus, f(α) has a unique solution in α if 1− K > 0. Therefore, unique solution for α exists and can be obtained by solving equation (9) if 0 < K < 1, that is,
0<1nx3x¯33s2x¯2s3<1.0<1n(xx¯)3(1n(xx¯)2)32<20<SampleSkewness<2.
The unique solution for σ and δ can be obtained by solving equations (7) and (8) respectively. This completes the proof.

4. Application

4.1 Data Analysis

In this section we analyze a set of data that arose in tests on the endurance of deep grove ball bearings. They were discussed by Lieblein and Zelen [8], Gupta and Kundu [3], and Gupta and Kundu [4]. The data are the number of revolutions in millions before failure for each of the 23 ball bearings in the life test and they are –

17.88 28.92 33.00 41.52 42.12 45.60 48.40 51.84 51.96
54.12 55.56 67.80 68.64 68.64 68.88 84.12 93.12 98.64
105.12 105.84 127.92 128.04 173.40

We have fitted four distributions, namely three-parameter Weibull, three-parameter gamma, and three parameter GE1, and three-parameter GE2 to this data set.

4.1.1 ML estimates

From equation (1) we obtain the ML estimate for location parameter δ^=17.88. By solving the equations (2) and (3) simultaneously we obtain ML estimates α^=0.4341 and σ^=77.33.

4.1.2 Method of moments estimates

The sufficient condition for the existence of unique solution is given by 1n(xx¯)3(1n(xx¯)2)32=0.94. Therefore, satisfies the condition of Theorem 1 for unique solution of all three parameters. Solving equations (4) and (5) for δ and σ in terms of α, and substituting these in equation (6), we obtain

(2α+1)12(1α)(3α+1)(1nx3x¯33s2x¯2s3)=0.
Solving equation (11) for α and substituting back into equations (7) and (8) we obtain the estimated parameters: α^=0.2957, σ^=59.9330, and δ^=25.9650.

The following TABLE 1 appends the ML estimates of parameters, chi-squared statistics, and the Kolmogorov-Smirnov (K-S) statistics of four three-parameter distributions and the moments estimate of parameters for GE2. The TABLE 2 shows the observed and the expected frequencies. All the distributions fit well to this popular data set. The moment estimates of parameters of GE2 performed better than the other three-parameter distributions with ML estimated parameters except for GE1. The GE1 distribution fitted marginally better than GE2 for this data set.

α^ σ^ δ^ χ2 K − S
Gamma (MLE) 2.7316 0.0441 10.2583 0.950 0.107
Weibull (MLE) 1.5979 0.0156 14.8479 1.321 0.118
GE1 (MLE) 4.1658 0.0314 4.7476 0.675 0.103
GE2(MLE) 0.4341 77.3300 17.8800 4.289 0.112
GE2(Moments) 0.2957 59.9330 25.9650 1.557 0.185
TABLE 1

Estimates of parameters for similar three-parameter distributions

Intervals Observed Gamma Weibull GE1 GE2(Moments) GE2(ML)
0 – 40 3 4.493 4.738 4.322 4.956 6. 053
40 – 80 12 10.658 10.016 10.913 9.984 8. 382
80 – 120 5 5.423 5.627 5.303 5.268 5. 331
120 – 160 2 1.805 1.992 1.739 2.201 2. 655
160 – 200 1 0.621 0.627 0.723 0.560 0.381
TABLE 2

Observed and expected frequencies

Figure 3.

Plots of (a) estimated probability density function, (b) estimated distribution function, and (c) expected values under different models and observed values.

5. Conclusion

This paper offers a new family of three-parameter generalized exponential distribution. The distribution can be used as an alternative to analyzing skewed data. Since the distribution function is in closed form, the inference based on the censored data can be handled by this model more easily than the gamma family distributions.

The model is bounded on both sides by positive values and has an increasing hazard rate function. This makes the model suitable for demographers and actuaries to determine the force of mortality for older age groups.

The Generalized exponential distribution (GE2) is a right skewed unimodal density function. Its hazard function is monotonically increasing for 0 < α < 1. The one real life example used in this article shows that this distribution can quite effectively be used to analyze lifetime data in place of generalized gamma, generalized Weibull and generalized exponential (GE1) distributions. This three-parameter GE2 distribution is always as flexible as the three-parameter GE1, Weibull or gamma distributions.

Acknowledgements

The author would like to thank the referees for their valuable comments and suggestions which contributed to improving the write-up of the manuscript.

References

[1]GN Alexander, “The use of the gamma distribution in estimating the regulated output from the storage”, Trans. Civil Engineering, Inst. of Engineers, Australia, Vol. 4, 1962, pp. 29-34.
[2]PJ Bickel and KA Docksum, Mathematical Statistics: Basic Ideas and Selected Topics, 2nd ed., Prentice Hall, New Jersey, Vol. 1, 2001.
[5]SA Hossain and M Ahsanullah, “On Generalized Exponential Distributions”, Advances and Applications in Statistical Sciences, Vol. 4, No. 1, 2010, pp. 1-22.
Journal
Journal of Statistical Theory and Applications
Volume-Issue
17 - 3
Pages
537 - 553
Publication Date
2018/09/30
ISSN (Online)
2214-1766
ISSN (Print)
1538-7887
DOI
10.2991/jsta.2018.17.3.9How to use a DOI?
Copyright
© 2018, the Authors. Published by Atlantis Press.
Open Access
This is an open access article under the CC BY-NC license (http://creativecommons.org/licences/by-nc/4.0/).

Cite this article

TY  - JOUR
AU  - Syed Afzal Hossain
PY  - 2018
DA  - 2018/09/30
TI  - Estimating the Parameters of a Generalized Exponential Distribution
JO  - Journal of Statistical Theory and Applications
SP  - 537
EP  - 553
VL  - 17
IS  - 3
SN  - 2214-1766
UR  - https://doi.org/10.2991/jsta.2018.17.3.9
DO  - 10.2991/jsta.2018.17.3.9
ID  - Hossain2018
ER  -