Abstract
We introduce a generalized version of the standard Gumble type-2 distribution. The new lifetime distribution is called the Exponentiated Gumbel (EG) type-2 distribution. The EG type-2 distribution has three nested submodels, namely, the Gumbel type-2 distribution, the Exponentiated Fréchet (EF) distribution, and the Fréchet distribution. Some statistical and reliability properties of the new distribution were given and the method of maximum likelihood estimates was proposed for estimating the model parameters. The usefulness and flexibility of the Exponentiated Gumbel (EG) type-2 distribution were illustrated with a real lifetime data set. Results based on the log-likelihood and information statistics values showed that the EG type-2 distribution provides a better fit to the data than the other competing distributions. Also, the consistency of the parameters of the new distribution was demonstrated through a simulation study. The EG type-2 distribution is therefore recommended for effective modelling of lifetime data.
1. Introduction
The Gumbel distribution, also known as the type-1 extreme value distribution, has received significant research attention, over the years particularly, in extreme value analysis of extreme events. For a review of the recent developments and applications of the Gumbel distribution, see Pinheiro and Ferrari [1]. There is no question that, before now, the Gumbel type-2 distribution is not popularly used in statistical modelling and the reason may not be far from its lack of fits in data modelling. Generally, standard probability distributions are well known for their lack of fits in modelling complex data sets. On this note, users of this distributions across various fields in general and statistics and mathematics in particular have been fantastically motivated to developing sophisticated probability distributions from the existing ones. Exponentiated distributions have been introduced to solve the problem of lack of fits that is commonly encountered when using the standard probability distributions for modelling complex data sets. Results from this advancement have frequently been proven more reasonable than the one based on the standard distributions. Exponentiating distributions are indeed a powerful technique in statistical modelling that offers an effective way of introducing an additional shape parameter to the standard distribution to achieve robustness and flexibility. This method of generalizing probability distributions is traceable to the work of Gupta et al. [2] who introduced the exponentiated exponential (EE) distribution as a generalized form of the standard exponential distribution by simply raising the cumulative density function (cdf) to a positive constant power. Ever since the introduction of the EE distribution, exponentiated distributions have achieved reasonable feats in modelling data sets from various complex phenomena. A good number of standard probability distributions have their corresponding exponentiated versions. Gupta et al. [2] introduced the Exponentiated Weibull distribution as a generalization of the standard Weibull distribution. Nadarajah and Kotz [3] modified the method by Gupta et al. [2] and introduced the Exponentiated Fréchet distribution as a generalization of the standard Fréchet distribution. Using the same method in Nadarajah and Kotz [3], Nadarajah [4] introduced the Exponentiated Gumbel distribution as a generalization of the standard Gumbel distribution. Mudholkar and Srivastava [5] introduced the Exponentiated Weibull family distribution as a generalization of the Weibull family distribution. Ashour and Eltehiwy [6] developed the exponentiated power Lindley distribution generalizing the power Lindley distribution and so forth. Therefore, this paper is aimed at generalizing the standard Gumbel type-2 distribution to a wider class of distribution so as to improve its performance and encourage its applicability, in modelling varieties of complex data sets.
The cumulative density function of the exponentiated family of distributions according to Nadarajah and Kotz [3] is defined bydifferentiating (1) with respect to gives the corresponding probability density function aswhere and are the vector of parameters and parameter space of the baseline distribution , respectively.
The remaining part of this paper is organized as follows; Section 2 introduces the Gumbel type-2 distribution, its exponentiated version, and special cases (submodels); Section 3 presents some important reliability characteristics of the new distribution and their asymptotic properties; Section 4 presents an explicit derivation of the moments, variance, and moment generating function of the new model; Section 5 presents the th quantile function of the new distribution; Section 6 presents the Rényi’s entropy of the new distribution; Section 7 presents the th order statistics of the new distribution; Section 8 proposes the maximum likelihood estimation method for estimating the parameters of the new distribution; Section 9 presents the application of the new distribution to a real data set and a simulation study; Section 10 is the discussion of results and Section 11 contains the conclusion of the study.
2. Exponentiated Gumbel Type-2 Distribution
Definition 1. According to Gumbel [7–9], a random variable is said to follow the Gumbel type-2 distribution if its cumulative density function (cdf) is given bywhile the corresponding probability density function (pdf) is given byUsing (3), we obtain the cdf of the Exponentiated Gumbel (EG) type-2 distribution aswhile the corresponding pdf is given bywhere and are the shape parameters and is the scale parameter.
Figure 1 shows the plots of the pdf (a) and cdf (b) of the EG type-2 distribution for certain parameter values.
(a)
(b)
2.1. Special Cases of the EG Type-2 Distribution
The EG type-2 distribution is developed for the purpose of modelling data sets that arise from complex phenomena. It generalizes some standard distributions; for instance, the EG type-2 distribution reduces to the Gumbel type-2 distribution, Exponentiated Fréchet (EF) distribution, and Fréchet distribution when , , and , respectively.
Theorem 2. If and is distributed according to the EG type-2 distribution then, is distributed according to the exponentiated exponential (EE) distribution due to Gupta et al. [2].
Proof. The transformation of a random variable to a random variable is defined by , where is known as the Jacobian of transformation. Thus, and
Corollary 3. When (8) reduces to the exponential distribution with parameter ; that is, .
3. Some Reliability Properties of the EG Type-2 Distribution
Reliability theory is generally concerned with the estimation of the probability of longevity or failure of a system.
3.1. Reliability Function
Definition 4. The reliability function or the survival function of a random variable is defined by . It could be interpreted as the probability of a system not failing before some specified time , Lee and Wang [10]. The reliability function of the EG type-2 distribution is given by
3.2. Hazard Rate Function
Definition 5. The hazard rate function or the instantaneous failure rate of a random variable is the probability that a system fails given that it has survived up to time and is given by (Lee and Wang [10]). Hence, we define the hazard rate function of the EG type-2 distribution as follows:Figure 2 shows plots of the reliability function (a) and hazard rate function (b) of the EG type-2 distribution for selected parameter values.
(a)
(b)
3.3. Asymptotics
pdf and cdf of the EG type-2 distribution is unimodal and monotonically increasing, respectively, with increasing values of The reliability function of the EG type-2 distribution is 0 as and 1 as . Also, is a monotonic decreasing function of . For example, when , Hence, is strictly a monotonic decreasing function of .
The hazard rate function of the EG type-2 distribution is 0 for both and and its shape appears increasingly upside-down bathtub with decreasing values of .
4. The th Crude Moment of the EG Type-2 Distribution
In probability theory, the moments of a random variable are one of the most important properties of a distribution that could be used to derive other essential properties such as mean, variance, skewness, and kurtosis statistics that describes a probability distribution. The th crude moment of a continuous random variable is defined by ; then the th crude moment of the EG type-2 distribution follows asSubstituting into (13) we haveand thus,Since can only take values on the positive real line we can introduce the exponential integral defined by (see Chapter 5 of Abramowitz and Stegun [11] and Equation (6.2.6) of Olver et al. [12]):Substituting in (16) we haveThus, evaluating (18) at and yields the mean and second crude moment then we can obtain the variance of the EG type-2 distribution as . Denoting by the coefficient of variation (cv), skewness , and kurtosis statistics of the EG type-2 distribution can be obtained by evaluatingrespectively.
4.1. The Moment Generating Function of the EG Type-2 Distribution
Generally, the moment generating function (mgf) denoted by of a random variable is defined asIf a random variable is distributed according to the EG Type-2 distribution, then its mgf is given by
5. The th Quantile Function of the EG Type-2 Distribution
The th quantile function of the EG Type-2 distribution is the inverse of (5) and it is obtained asWe can simulate random variables from the EG type-2 distribution through the inversion of the cdf method by simply replacing in (24) with variates. Also, evaluating (24) at gives the median of the distribution.
6. The Rényi Entropy
The Rényi entropy is used to measure uncertainty or variation in a random variable . The Rényi’s entropy measure has been shown to be effective in comparing the tails and shapes of various standard distributions, Song [13]. The Rényi entropy measure for a continuous random variable is given byThen the Rényi entropy measure for the EG type-2 distribution could be obtained as follows:Setting in (26) we haveSubstituting in (27) we haveusing the expression for the function defined in Section 4; we haveand substituting in (29) we haveand thus,
7. The Order Statistics of the EG Type-2 Distribution
The distribution of the th order statistics of a random sample of size is generally given asHence, the density of the th order statistics of the EG Type-2 distribution is given byThe density of the th smallest order statistics of the EG Type-2 distribution is given byThe density of the th largest order statistics of the EG Type-2 distribution is given by
8. Parameter Estimation of the EG Type-2 Distribution
In this section, we propose the method of maximum likelihood estimates (MLE) for the estimation of the parameters of the EG type-2 distribution. Suppose a random variable of size has the EG type-2 distribution then, its MLE are obtained as follows:
The likelihood function is given bywith the corresponding log-likelihood functionTaking the partial derivatives of the log-likelihood function with respect to , , and , respectively, and equating to 0 giveEquations (38) can only be solved by some numerical optimization methods such as Newton Raphson’s algorithm to obtain the MLE of , , and .
9. Application
In this section we would fit the EG type-2 distribution to a real and uncensored data set to demonstrate its applicability and flexibility. The goodness of fit of the new distribution would be compared with the three submodels, namely, the Gumbel type-2 distribution, Exponentiated Fréchet distribution, and Fréchet distribution and two other related heavy tail distributions: Weibull distribution , , , and , and log-normal (LN) distribution, , , and . The model comparison would be based on the minimized log-likelihood estimate and the following information statistics: AIC by Akaike [14], AICC by Sugiura [15], CAIC by Bozdogan [16], HQC by Schwarz [17], and BIC by Hannan and Quinn [18]. The model with the smallest minimized log-likelihood and information statistics value is the best. The data set in Table 1 shows the survival times in months of 20 acute myeloid leukemia patients reported in Afify et al. [19].
9.1. Monte-Carlo Simulation
In this section we present a Monte-Carlo simulation study to investigate the effect of sample size on the maximum likelihood estimates of the parameters of the EG type-2 distribution and further to assess the stability of these parameters. Different sample sizes were drawn from the EG type-2 distribution with parameters , , and using the inverse transformation method with (24) where each sample was replicated 5000 times. Using the simulated random variables we estimate the parameters of the EG type-2 distribution through the method of maximum likelihood estimation and the procedure was repeated 5000 times for each sample size. The mean (parameter estimate) and standard deviation (standard error (se)) of the 5000 parameters each for α, θ, and ϕ for each sample size were computed and the result is presented in Table 3. Furthermore, the corresponding bias and mean square errors (mse) of each of the parameter estimates are tabulated in Table 4. Analogously, Tables 5 and 6 show simulation results for the EG type-2 distribution with parameters , , and .
10. Discussion of Results
From the pdf and cdf plots in Figure 1, the pdf of the EG type-2 distribution is unimodal and increasingly unimodal for increasing values of (shape parameter) while its cdf is monotonic increasing and more monotonically increasing for increasing values of . Also, the plots of the reliability function and hazard rate function in Figure 2 show that is monotonic decreasing and more monotonically decreasing for increasing values of while is upside-down bathtub and becomes more upside-down bathtub for decreasing values of . Results from the model fittings as tabulated in Table 2 indicate that the EG type-2 distribution provides the best fit to the data based on its smallest minimized log-likelihood and information statistics values. Figure 3 depicts the cdf’s of all the estimated distributions in Table 2 superimposed on the empirical cdf of the data, where the cdf of the EG type-2 distribution is closely aligned to the empirical one than the other distributions. From the simulation results in Tables 3, 4, 5, and 6 it is clear that the parameters of the EG type-2 distribution approach the true value as the sample size increases, while the standard error, bias, and the mse decrease down the column with increasing sample size.
11. Conclusion
This paper introduces a new lifetime distribution, the Exponentiated Gumbel (EG) type-2 distribution. The new distribution generalizes the standard Gumbel type-2 distribution and has the following distributions as special cases: Gumbel type-2 distribution, Exponentiated Fréchet distribution, and Fréchet distribution. We have provided explicit mathematical expressions for some of its basic statistical properties such as the probability density function, cumulative density function, th crude moment, variance, coefficient of variation, skewness, kurtosis, moment generating function, and th quantile function and some reliability characteristics like the reliability and hazard rate functions. Estimation of the model parameters was approached through the method of maximum likelihood estimates. The flexibility and applicability of the new lifetime distribution were illustrated with a real data set and the results obtained revealed that the EG type-2 distribution provides the best fit among all the compared related distributions. We recommend the EG type-2 distribution for modelling complex data sets and hope that it would receive significant applications in the future.
Competing Interests
The authors declare that there are no competing interests regarding the publication of this paper.