Abstract

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.

1. Introduction

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. asand as

The of the family is given by

(1)

The corresponding to (1) above is

(2)

Let be the logit ofi.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.

2. Exponentiated Gumbel Exponential Distribution

Suppose that is distributed as exponentiated Gumbel distribution with and respectively given by Nadarajah ⁹

and

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.

3. Mathematical Properties of the (EGuE) Distribution

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

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)

4. Estimation

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.

5. Application

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.

6. Conclusion

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.

References