Abstract

In this paper, we propose a new class of Gumbel generated distributions called Gumbel-Marshall-Olkin family of distributions. The new family of distributions is represented as linear mixture of exponentiated-G distribution. Some of the sub-models are presented. We derived some characterizations such as the quantile, moments, moment generating function, entropy and order statistics of the proposed family of distributions. The estimation of the unknown parameters of the new class of distribution is through the maximum likelihood. The consistency of the MLEs of the sub-model is assessed by means of simulation. Furthermore, we derive the bivariate density function of the new class of distributions. Two real life data sets are used to illustrate the potential usefulness of the sub-models of the proposed class of distributions. The results of the applications clearly indicate that the sub-models of the proposed class of distribution provided better fit among the other competing models.

1. Introduction

In recent years, there have been growing interests in developing families of statistical distributions by extending already existing distributions through the addition of one or more parameters. The primary focus is to generate more flexible distributions that will provide better fits to many practical situations where ordinarily the classical distributions would not have provided adequate fits. ¹ proposed a method of generating new family of distributions for any baseline distribution with cumulative density function (cdf) and define the corresponding cdf as

(1)

where and is the survival function of the baseline distribution with vector of parameters . For, .

In the literatures, there are many other families of distributions such are exponentaited-G by ², beta–G by ³, transmuted-G by ⁴, gamma-G ⁵, Kumaraswamy-G by Cordeiro and de Castro (2011) ⁶, McDonald-G by ⁷.

Furthermore, ⁸ proposed a method of generating families of continuous distribution called the transformed-transformer (T-X) with the cdf of a class of continuous distributions for any given baseline distribution defined as

and the corresponding probability density function (pdf) is given by

where satisfies the following conditions: , is differentiable and monotonically non-decreasing, as and as .

⁹ proposed Gumbel-G family of distributions following ⁸. However, in this paper, we propose a new wider class of continuous distributions which generalizes the Gumbel-G family of distributions by taking

and

Here, is the cdf of eq. (1).

The new wider class of continuous distributions is called Gumbel Marshall Olkin family of distributions (GMO-G) having cdf and pdf are respectively given by

(2)

and

(3)

where , is the vector of parameters of the GMO-G. For in eqs. (2) and (3), GMO-G reduces to the cdf and pdf of Gumbel-G of ⁹, which is a special case of the newly proposed family of distributions..

The hazard rate function (hrf) and survival function (s.f) of GMO-G are respectively given by

and

The main motivations behind this paper are to generate more flexible distributions having bimodal, bathtub, symmetric, monotone increasing, increasing-decreasing-increasing, J and reverse-J shapes; hazard rates of constant, J, reverse-J, monotone increasing, increasing-decreasing-increasing shapes as shown in Figure 2, Figure 3, Figure 4 and Figure 5; construct heavy-tailed distributions which are not longer-tailed for modeling real-life data as shown in figure1; and generate models that will provide better fit even when compared with models having the same baseline distribution.

The remaining part of the paper is organized as follows. The linear representation of the distribution and density functions of the new class of distributions are presented in section 2. In section 3, we presented some special models of the proposed new family of distributions. The quantile function is presented in section 4. In section 5, the shapes of density and hazard rate functions are discussed. We derived the moments including the ordinary, incomplete and generating function in section 6. The Entropy and the distribution of the order statistics are presented in sections 7 and 8. In section 9, we presented the method of estimation of the unknown parameters of the new family of distributions. Simulation studies on the consistency of the MLEs are presented in section 10. Bivariate extension of the proposed family of distribution is presented in section 11. Finally in section 12, we provided the concluding remarks.

2. Linear Representation

Considering some useful series expansion

and

Using the series expansion above, equation (2) can re-written as mixture of exponentiated-G (exp-G) cumulative function given by

(4)

where

Here is the cdf of Exp-G of power parameter . See ^{2, 10, 11, 12}.

Some mathematical properties of Gumbel Marshall Olkin –G family of distributions can be derived from those properties of the EXP-G distribution. See ^{11, 12, 13}.

Differentiating eq. (4), eq. (3) can be re-written as

(5)

Here, is the density function of Exp-G with power parameter

3. Special Models

In this section, we consider two of the special models of GMO-G. However, equation (3) will be most tractable when the cdf and pdf of the baseline distribution have simply analytic expression.

Suppose and denotes the cdf and pdf of standard normal distribution where; , , then the pdf of GMO-N distribution is given by

Here, is the parameter vector of GMO-N distribution. Figure 2 shows some possible shapes which are not limited to bathtub, symmetric, J-shape and monotone increasing shapes for some selected parameter values. Figure 3 also shows some possible shapes of the hazard rate function for some selected parameter values. These shapes indicate the flexibility of GMO-N and its potential to model real-life data.

A random variable with

and

where and are the shape and scale parameters is said to follow Weibull distribution. By substituting and in equation (3), the pdf of GMO-W is defined as

The plots of the pdf and hrf of GMO-W distribution for selected parameter values are shown in Figure 4 and Figure 5 respectively. Shapes such as bimodal, j and reverse-J, symmetric and left-skewed are for the density function while constant, S-shape, monotone increasing, increasing-decreasing-increasing shapes are for hazard rate function.

4. Quantile Function

The quantile function (qf) of GMO-G family is obtained by inverting equation (1.2) and it is given by

(6)

Here, and is the qf of the baseline distribution.

The effect of the parameters GMO-G on skewness and kurtosis are determined by its quantile measures using the Bowley’s skewness and Moor’s kurtosis measures. These measures are respectively given by

and

However, these measures are less sensitive to outliers and they do exist for distribution with moments. We considered the effect of parameters and on the skewness and kurtosis of GMO-Normal. The plots are presented in Figure 1. Both measures equal zero for the normal distribution.

Proposition 1: If follows Gumbel distribution, then for any baseline distribution with cdf the qf can be expressed as

(7)

where is the qf of Marshall Olkin and the qf of the transformed distribution.

PROOF

and are obtained by inverting (1) and cdf of and are respectively given by

and

Substituting and in (7) and simplifying, (6) is obtained.

5. Shape of GMO-G pdf and Hazard Rate Function

The shapes of the density and hazard rate function of GMO-G can be described analytically. The critical points of the GMO-G density function are the roots of the equation below

(8)

By equation (8), there may be more than one root. Suppose is a root of (8), then , and correspond to the local maximum, local minimum, and point of reflection respectively. Here is given by

Figure 2 and Figure 4 show the graphical pdf plots of two GMO-G sub-models (GMO-N and GMO-W) for selected parameter values indicating different shapes.

Furthermore, the shape of the GMO-G hazard rate function can be described analytically. The critical points of the GMO-G hazard rate function are the roots of the eq. (9).

(9)

There may be more than one root of equation (9). Thus, if is a root of (9), then ,, and correspond to the local maximum, local minimum, and point of reflection respectively. Here is given by

Figure 3 and Figure 5 show the graphical hrf plots of two GMO-G sub-models (GMO-N and GMO-W) for selected parameter values indicating different shapes.

6. Moments

In this section, we obtained some moments associated with GMO-G family of distribution.

Suppose is a random variable from GMO-G distribution, then the moment about the origin can be expressed by two formulas.

Firstly, let a random variable having pdf with power parameter, then the moment of is given by

Secondly,

where

The incomplete moment plays important role in determining measures of inequality such as Lorenz, Bonferroni and Gini measures of inequality ¹⁴. For a random variable having GMO-G density function, the incomplete moment is given by

The moment generating function (mgf) of a random variable X having GMO-G defined as can be obtained firstly by

where is the mgf of Y~Exp^m+1-G distribution.

Secondly, can be obtained by

where , and are as defined in equation (6).

7. Entropy

The measure of uncertainty of a random variable is through entropy. The Renyi and Shannon entropies are the two popular entropies. However, the Renyi entropy is a generalization of Shannon entropy. For a random variable with pdf , according to ¹⁵, the Renyi entropy is defined as

However, suppose X~GMO-G, the Renyi entropy is given by

8. Order Statistics

Let be a random sample from a population having pdf and the order statistics. The pdf of the order statistics is given by

Following ¹⁶, the pdf of ordered statistics of sample of size n from GMO-G population is given by

(10)

9. Estimation

We consider the maximum likelihood method for the estimation of unknown parameters of GMO-G distribution. For a random sample of size from eq. (2), the log-likelihood function is given by

The score functions for the parameters of the distribution are given by

The MLEs say, of can be obtained as the solution to the non-linear equations and Iterative techniques such as a Newton-Raphson type algorithm can be employed to evaluate Here, the optim function in R statistical software is used to obtain the numerical solution. Taking the minus expectation of the partial derivative of the score function with respect to the parameters of the distribution, we obtain the Fisher’s information matrix ; , where is the number of parameters of the baseline distribution. However, under general regularity conditions, is asymptotically multivariate normal distributed . By the asymptotic confidence interval for the parameters of the distribution appropriately constructed.

10. Simulation Studies

In this section, we assess the performance of the MLEs of GMO-Normal distribution, a sub model of GMO-G through simulation studies of different starting parameter values and Simulated samples data of sizes n=50, 100, 150 generated by inverting the GMO-N distribution function given by

where is the distribution function of normal distribution with and The MLEs, say =,,, , are determined and the process is repeated for N=1000 times. We obtain the average estimates (Mean value), average baises and the mean square error (MSE) given respectively by

and

The results of the simulations for the three different starting parameter values are respectively shown in Table 1, Table 2, and Table 3. The values in the tables indicate that as the sample size increases the mean values tend to starting parameter values and the MSE decreases which conforms to the first order asymptotic theory.

11. Bivariate Extension of GMO-G

Let X and Y be two random variables from GMO-G family of distribution, the bivariate cdf is given by

(11)

By taking the partial derivative of (11) with respect to and, the bivariate pdf of GMO-G is given by

where

and

The marginal distributions are respectively given by

and

The conditional density functions are respectively given by

and

and

12. Applications

In this section, we considered the applications of two sub models of GMO-G (GMO-Weibull and GMO-Lomax) to real life data sets to show the potentials of the new class of distributions. Comparison with other models having the same baseline distribution is made based on goodness-of-statistics Cramer Von Mises (W^*), Anderson Darling, Koromogrov-Smirnov, Akaike Information Criterion, and Bayesian Information Criterion. The model with the least value of goodness-of-fit statistics provides the best fit ¹⁷.

First data set: The first real data set is on the observed survival times (weeks) for AG positive reported by ¹⁸. The data set is as follow: 65, 156, 100, 134, 16, 108, 121, 4, 39, 143, 56, 26, 22, 1, 1, 5, 65. The GMO-Weibull, Exponentiated-Weibull(ExpW) due to ², Beta-Weibull (BW) due to ¹⁹, Gumbel-Weibull(GuW) due to ⁹, logistic-Weibull(LW) and Weibull distribution are fitted to the data set.

Second data set: The second real data set refers to 30 devices failure times reported by Meeker and Escobar ²⁰ and Tahir et al. ²¹. The data set are as follow: 275,13, 147, 23, 181, 30, 65, 10, 300, 173, 106, 300, 300, 212, 300, 300, 300, 2, 261, 293, 88,247, 28, 143, 300, 23, 300, 80, 245, 266. The GMO-Lomax, Beta-Lomax (BL) due to ²², Gumbel-Lomax(GuL) due to ²³, Exponentiated-Lomax(ExpL) due to ²⁴, logistic-Lomax (LL) due to ²⁵ and Lomax distribution are fitted to the data set.

The results of the two applications to the real data sets are shown in Table 4, Table 5 and Table 6, Table 7 for first and second data set respectively. The parameter estimates with standard errors in parenthesis are shown in Table 4 and Table 6 while the goodness-of-fit statistics are contained in Table 5 and Table 7 for the first and second data set respectively. The plots of the estimated pdfs and empirical cdf with cdfs of some of the competing models in the first and second application are shown respectively in Figure 6 and Figure 7. The plots in Figure 6 and Figure 7 indicate that the sub models of the newly proposed class of distributions provide better fits among the competing models in agreement with the Table 5 and Table 7.

13. Conclusion

We proposed a new class of Gumbel generated family of distributions called Gumbel-Marshall Olkin-G family of distribution which has Gumbel-G as a special model. The cdf and pdf of the new class of distributions are represented as linear combination of exponentiated-G family of distribution. Some sub models of the new class of distributions are presented and several continuous distributions can be obtained for any baseline distribution. Furthermore, some statistical properties such as the quantile function. Ordinary and incomplete moments, generating function, entropy, and density function of the order statistics are derived. The unknown parameters of the new family of distributions are estimated through maximum likelihood method. The consistency of the MLEs is considered using GMO-N by means of simulations studies. We further derived the bivariate extension of the new class of distributions. Finally, the potentials of the new class of the distributions are illustrated by means of comparing the GMO-W and GMO-Lomax distributions with other competing distributions in two real life data sets. The goodness-of-statistics indicate that the two sub models of the new class of distributions provide the best fit among other competing models.

References