In this article, we proposed a new distribution called exponentiated Gumbel exponential (EGuE) distribution. The new distribution is a member of the T-X family obtained through the logit transformation of the exponential random variable, using the exponentiated Gumbel distribution as the generator. The mathematical properties of the proposed distribution were studied. The maximum likelihood estimates of the parameters of the EGuE distribution were derived. The applicability of the new distribution is shown using a real data set.
The development of flexible distributions for modeling lifetime data has received widespread attention in recent times. This is largely due to the fact that some classical distributions are not flexible enough in modeling lifetime processes. Many methods have been proposed in the literature by statisticians for developing new distributions. Lee et al, 1 termed the recent methods of developing flexible distribution “method of Combination”. This is because these methods attempt to combine existing distributions to form a new one or adding parameters to existing distribution. Some examples of this method include but not limited to the beta-generated family of distributions Eugene et al, 2, Kumaraswamy family of distributions Jones 3, Cordeiro and de Castro 4, exponentiated-G family Gupta et al. 5, transmuted-G family Shaw and Buckley 6 and generalized transmuted family Alizadeh et al. 7.
Alzatreeh et al. 8 introduced the Transformed-Transformer family which addressed one of the limitations of the beta class by allowing for any continuous distribution to be used as a generator. Let
be a random variable in the interval
with probability density function
and cumulative density function
,
and
respectively. Let the function of the
of a random variable
be defined as
such that
meets the conditions stated below:
I.
II. is differentiable and monotonically non-decreasing.
III. as
and
as
The of the
family is given by
![]() | (1) |
The corresponding to (1) above is
![]() | (2) |
Let be the logit of
i.e
Hence (1) and (2) can be written in terms of the logit of as
![]() | (3) |
and
![]() | (4) |
respectively.
In this paper, we introduced the exponentiated Gumbel exponential distribution using the
approach introduced by Alzatreeh et al. 8. The generator used is exponentiated Gumbel distribution while the baseline distribution
is the exponential distribution. The mathematical properties of the new distribution are extensively explored. Maximum likelihood estimates of the new distribution are derived and the potentiality of the proposed model is illustrated using a lifetime data set.
The rest of this paper is organized as follows. Section 2 introduces the exponentiated Gumbel exponential distribution. Several mathematical properties of the proposed model such as the quantile function, shapes of the and
, moments, entropy, order statistics and inequality measures are discussed in Section 3. Estimation of the parameters of the model and application of the new distribution to real data set is done in Sections 4 and 5 respectively. Section 6 concludes the paper.
Suppose that is distributed as exponentiated Gumbel distribution with
and
respectively given by Nadarajah 9
![]() |
and
![]() |
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Let the random variable be exponentially distributed with
and
given respectively by
![]() |
![]() |
The logit of is
. Thus using (3), the
is given by
![]() | (5) |
where
The associated with (5) is obtained by taking the derivative of (5) with respect to the random variable
. Hence the
of
distribution is
![]() | (6) |
The survival function of is given by
![]() |
The hazard rate function, reversed - hazard rate function
and cumulative hazard rate function
of
distribution are respectively given by
![]() |
![]() |
and
![]() |
The plots of the ,
and
of
for selected parameter values are displayed in Figure 1, Figure 2 and Figure 3 respectively.
The quantile function of which follows the
distribution is given by
![]() | (7) |
Proof:
The quantile function of is obtained by inverting (5). Equating the random variable
to (5) we have
![]() |
![]() |
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![]() |
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Substituting in (7) gives the median of
distribution.
![]() | (8) |
The random variable in (7) is uniformly distributed in the interval
.
can be used to simulate a random sample of
distribution. The skewness and Kurtosis of the proposed distribution can be studied using measures based on quantiles. The Galton 10 skewness
and Moor 11 Kurtosis
are usually used for this purpose.
These measures of skewness and kurtosis exist even when the moments of the distribution do not exist and they are not sensitive to outliers.Alizadeh 12. These are some of the advantages of these measures over the ones based on moments. The expression for obtaining the Galton skewness and Moor’s kurtosis
respectively are given by
![]() | (9) |
![]() | (10) |
The shapes of the and
can be described analytically. This can be done by taking their log, differentiating with respect to
and equating to zero. The shape of the
of
can be described by
![]() | (11) |
(11) may have more than one root. If is a root of (11), then it corresponds to a local maximum, local minimum or point of inflexion depending on whether
or
where
The shape of the of
distribution is described by
![]() | (12) |
The roots of (12) may be more than one. If is a root of (12), then it corresponds to a local maximum, local minimum or point of inflexion depending on whether
,
or
where
A representation of the and
of
distribution is made in this sub-section. The
of
can be written as
![]() | (13) |
Applying general binomial expansion (14) to A in (13)
![]() | (14) |
we have
![]() |
and
![]() |
Applying power series expansion for exponential functions to
![]() |
Substituting for in (13)
![]() |
Applying the general binomial expansion for negative powers to we have
![]() |
Thus the density can be represented as infinite linear combination of exponential distribution. Therefore
![]() | (15) |
where
![]() |
The expansion of cumulative density distribution is obtained as follows
![]() |
where is a positive integer.
![]() |
![]() |
![]() | (16) |
Applying general binomial expansion for negative powers to in (16) reduces to
![]() |
![]() | (17) |
where
![]() |
If is distributed as (6), then its
noncentral moment is obtained as follows
![]() | (18) |
Substituting (15) in (18)
![]() |
![]() | (19) |
where is as defined in (15) and
is a gamma function.
The moment generating function, of
distribution is given by
![]() | (20) |
Substituting (19) into (20) we have
![]() | (21) |
The Renyi entropy of a random variable is the measure of variation of uncertainty. The Renyi entropy is given by
![]() | (22) |
To obtain the Renyi entropy of thedistribution, we substitute (6) in (22) and apply the general binomial expansion and power series expansion for exponential function. Hence,
![]() |
Applying the general binomial expansion again
![]() |
![]() |
where
![]() |
![]() |
![]() |
Hence the Renyi entropy of distribution is given by
![]() |
The q-entropy is given by
![]() |
![]() |
Let be a random sample from the
distribution. Then the
of the
order statistic can be expressed as
![]() | (23) |
where is the beta function. Substituting (15) and (17) into (23) and replacing
with
we have
![]() |
where
![]() |
Hence, the order statistic of
distribution may be expressed as a mixture of exponential density with parameter
The raw moment of the order statistic of the
distribution is
![]() |
![]() |
![]() |
The incomplete moment
is given by
![]() | (24) |
Substituting (15) in (24), the incomplete moment of
distribution becomes
![]() |
![]() | (25) |
where is the lower incomplete gamma function.
The following relation may be used to obtain the probability weighted moments of a random variable.
![]() | (26) |
Substituting (15) and (17) in (26) and replacing with
we have the PWM of
as
![]() |
The spread from the center of a population can be measured using the deviation from mean or deviation from the median. Letting the mean deviation from the mean, and mean deviation from the median be and
respectively. The mean deviation about mean is given by
![]() | (27) |
Using the result of the lower incomplete moment in (25) we have
![]() |
Letting . Hence (27) can be written as
![]() |
where
![]() |
For the mean deviation from the median, we have
![]() |
![]() |
where
![]() |
The moment of residual life of
is given by
![]() | (28) |
where is the survival function Substituting (15) in (28) we have
![]() |
Thus the moment of the residual life function is given by
![]() |
where is the upper incomplete gamma function. The mean residual life function is obtained from (28) by substituting
.
The Lorenz and Bonferroni curves are very important inequality measures in income and wealth distribution. The Lorenz curve for distribution is given by
![]() | (29) |
substituting (19) and (25) for in (29) we have
![]() |
while and Bonferroni curve is given by
![]() | (30) |
![]() |
Given a random sample of size
, the log-likelihood function of
distribution is given by
![]() | (31) |
where
Let be the vector of unknown parameters, the score function associated with it is given by
![]() |
where,
,
, and
are the partial derivatives of
with respect to
,
,
, and
. The score function’s elements are:
![]() |
![]() |
![]() |
![]() |
![]() |
The maximum likelihood estimates are now obtained by numerically solving which is a system of non-linear equations. Numerical optimization methods are normally used in solving such systems of equations.
In this section, we presented an application of distribution to a real data set. The fit of
is compared to the fits of other competing models with the same baseline distribution; exponentiated Weibull exponential
Elgarhy et al. 13, Weibull exponential
Oguntunde et al. 14, Kumaraswamy exponential
Cordeiro and de Castro 4 and exponential
distributions.
Estimates of parameters of the distribution and other competing distributions were obtained using the method of maximum likelihood. The Kolmogorov-Smirnov (K-S), Cramer-von Mises (W*), Anderson-Darling (A*) statistics, Akaike information criterion (AIC) and Bayesian information criterion (BIC) are the goodness of fit criteria used to compare the fits of the distributions to the data set.
The real data set used for illustration is the survival times (in days) of 72 virulent tubercle bacilli infected guinea pigs. The data is obtained from Bjerkedal 15 and has been used in SElgarhy et al. 13. It is right-skewed and unimodal data. The data is as shown below.
0.1, 0.33, 0.44, 0.56, 0.59, 0.72, 0.74, 0.77, 0.92, 0.93, 0.96, 1, 1, 1.02, 1.05, 1.07, 07, .08, 1.08,1.08, 1.09, 1.12, 1.13, 1.15, 1.16, 1.2, 1.21, 1.22, 1.22, 1.24, 1.3, 1.34, 1.36, 1.39, 1.44, 1.46,1.53, 1.59, 1.6, 1.63, 1.63, 1.68, 1.71, 1.72, 1.76, 1.83, 1.95, 1.96, 1.97, 2.02, 2.13, 2.15, 2.16,2.22, 2.3, 2.31, 2.4, 2.45, 2.51, 2.53, 2.54, 2.54, 2.78, 2.93, 3.27, 3.42, 3.47, 3.61, 4.02, 4.32,4.58, 5.55
The maximum likelihood estimates, standard errors of the estimates and log-likelihood values of and other competing models are shown in Table 1. The values of the K-S, W*,A* statistics, AIC and BIC are presented in Table 2.
We observe that the four-parameter distribution provides a better fit to the data set than the other distributions with the same baseline given that it has the lowest value in all the goodness of fit criteria considered.
The plot of the histogram and estimated of
,
,
,
and
distributions are displayed in Figure 4 while the empirical and estimated
are shown in Figure 5. Both plots affirm the results of the goodness of fit criteria that
distribution provides a better fit to the data set than the other competing models.
In this paper, we introduced a four-parameter distribution called the exponentiated Gumbel exponential distribution using the transformed-transformer method introduced by Alzatreeh et al. 7. We expressed the density of the new model as an infinite linear combination of exponential distribution. Several mathematical properties of the new model were derived. Estimates of the parameters of the proposed model were obtained using the method of maximum likelihood. The importance of the new model was demonstrated using a real data set.
[1] | Lee, C., Famoye, F. and Alzaatreh, A, “Methods of generating families of univariate continuous distributions in the recent decades”. WIRESs Computational Statistics, 5. 219-238. 2013. | ||
In article | View Article | ||
[2] | Eugene, N., Lee, C. and Famoye F, “A beta Beta-normal distribution and its applications”. Communications in Statistics. Theory and Methods 31 (4), 497-512. 2002. | ||
In article | View Article | ||
[3] | Jones, M.C, “Kumaraswamy’s distribution: a beta-type distribution with tractability advantages”. Statistical Methodology 6, 70-81. 2009. | ||
In article | View Article | ||
[4] | Cordeiro, G. M., and de Castro, M “A new family of generalized distributions.” Journal of Statistical Computation and Simulations 81 (7). 833-898. 2011. | ||
In article | View Article | ||
[5] | Gupta, R.C., Gupta, P.L. and Gupta, R.D, “Modeling failure time data by Lehman alternatives”. Communications in Statistics- Theory and Methods 27(4) .887-904. 1998. | ||
In article | View Article | ||
[6] | Shaw, W, I., and Buckley, I, R “The alchemy of probability distributions: beyond Gram-charlier expansion, and a skew-kurtitic-normal distribution from a rank transmutation map” arxiv print, arXiv: 0901.0434. 2009. | ||
In article | |||
[7] | Alizadeh , M, Merovci F. and Hamedani , G G , “Generalized Transmuted Family of Distributions: Properties and Applications”. Hacettepe journal of Mathematics and Satatistics. 46 (4), 2017. | ||
In article | View Article | ||
[8] | Alzaatreh, A., Lee, C. and Famoye, F, “A new method for generating families of continuous distributions”, Metron 71(1), 63-79. 2013. | ||
In article | View Article | ||
[9] | Nadarajah, S, “The exponentiated Gumbel distribution with climate application”. Environmetrics 17. 13-23. 2006. | ||
In article | View Article | ||
[10] | Galton, F. Enquires into human faculty and its development. Macmillan and company London. 1983. | ||
In article | |||
[11] | Moor J.J “A quantile alternative for kuutosis”. The Statistician 37, 25-32 .1988. | ||
In article | View Article | ||
[12] | Alizadeh, M., Cordeiro, G. M., de Brito, E. and Demetrio, C .B, “The beta Marshall-Olkin family of distributions”, Journal of Statistical distributions and Applications 2 (4). 2015. | ||
In article | View Article | ||
[13] | Elgarhy, M., Shakil, M. and Kibria, B.M., “Exponentiated Weibull-exponential distribution with applications”, Applications and Applied Mathematics: An International journal, 12. 710-725. 2017. | ||
In article | |||
[14] | Oguntunde, P.E., Balogun, O.S., Okagbue, H.I. and Bishop, S.A., “The Weibull-exponential distribution: its properties and applications”, Journal of Applied Statistics. 15. 1305-1311. 2015 | ||
In article | View Article | ||
[15] | Bjerkedal, T., “Acquisition of resistance in guinea pigs infected with different doses of virulent tubercle bacilli”, American Journal of Epidemilology, 72(1). 130-148.1960. | ||
In article | View Article PubMed | ||
Published with license by Science and Education Publishing, Copyright © 2019 Uchenna U Uwadi, Emmanuel W Okereke and Chukwuemeka O Omekara
This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit
http://creativecommons.org/licenses/by/4.0/
[1] | Lee, C., Famoye, F. and Alzaatreh, A, “Methods of generating families of univariate continuous distributions in the recent decades”. WIRESs Computational Statistics, 5. 219-238. 2013. | ||
In article | View Article | ||
[2] | Eugene, N., Lee, C. and Famoye F, “A beta Beta-normal distribution and its applications”. Communications in Statistics. Theory and Methods 31 (4), 497-512. 2002. | ||
In article | View Article | ||
[3] | Jones, M.C, “Kumaraswamy’s distribution: a beta-type distribution with tractability advantages”. Statistical Methodology 6, 70-81. 2009. | ||
In article | View Article | ||
[4] | Cordeiro, G. M., and de Castro, M “A new family of generalized distributions.” Journal of Statistical Computation and Simulations 81 (7). 833-898. 2011. | ||
In article | View Article | ||
[5] | Gupta, R.C., Gupta, P.L. and Gupta, R.D, “Modeling failure time data by Lehman alternatives”. Communications in Statistics- Theory and Methods 27(4) .887-904. 1998. | ||
In article | View Article | ||
[6] | Shaw, W, I., and Buckley, I, R “The alchemy of probability distributions: beyond Gram-charlier expansion, and a skew-kurtitic-normal distribution from a rank transmutation map” arxiv print, arXiv: 0901.0434. 2009. | ||
In article | |||
[7] | Alizadeh , M, Merovci F. and Hamedani , G G , “Generalized Transmuted Family of Distributions: Properties and Applications”. Hacettepe journal of Mathematics and Satatistics. 46 (4), 2017. | ||
In article | View Article | ||
[8] | Alzaatreh, A., Lee, C. and Famoye, F, “A new method for generating families of continuous distributions”, Metron 71(1), 63-79. 2013. | ||
In article | View Article | ||
[9] | Nadarajah, S, “The exponentiated Gumbel distribution with climate application”. Environmetrics 17. 13-23. 2006. | ||
In article | View Article | ||
[10] | Galton, F. Enquires into human faculty and its development. Macmillan and company London. 1983. | ||
In article | |||
[11] | Moor J.J “A quantile alternative for kuutosis”. The Statistician 37, 25-32 .1988. | ||
In article | View Article | ||
[12] | Alizadeh, M., Cordeiro, G. M., de Brito, E. and Demetrio, C .B, “The beta Marshall-Olkin family of distributions”, Journal of Statistical distributions and Applications 2 (4). 2015. | ||
In article | View Article | ||
[13] | Elgarhy, M., Shakil, M. and Kibria, B.M., “Exponentiated Weibull-exponential distribution with applications”, Applications and Applied Mathematics: An International journal, 12. 710-725. 2017. | ||
In article | |||
[14] | Oguntunde, P.E., Balogun, O.S., Okagbue, H.I. and Bishop, S.A., “The Weibull-exponential distribution: its properties and applications”, Journal of Applied Statistics. 15. 1305-1311. 2015 | ||
In article | View Article | ||
[15] | Bjerkedal, T., “Acquisition of resistance in guinea pigs infected with different doses of virulent tubercle bacilli”, American Journal of Epidemilology, 72(1). 130-148.1960. | ||
In article | View Article PubMed | ||