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On the Families of Generalized Exponentiated Weibull Distributions: Properties and Applications

Anyiam Kizito Ebere*, Onyeagu Sydney Iheanyi, Njoku Modestus Onyekachi

Department of Statistics, School of Physical Sciences, Federal University of Technology, Owerri, Imo State, Nigeria

Abstract: Some new families of exponentiated Weibull (EW) distribution named T-Exponentiated Weibull{Y} using the T-R{Y} context are presented in this article. The quantile functions of five notable distributions, namely, Logistic, Log-Logistic, Rayleigh, Exponential and Lomax were used to develop five sub-families T-EW{Logistic}, T-EW{Log-Logistic}, T-EW{Rayleigh}, T-EW{Exponential} and T-EW{Lomax} and some general properties such as the quantile functions, Moments, mean deviations from the mean and median, Shannon entropies are obtained. The shapes of the exponentiated Weibull family densities shows can be unimodal, bimodal, monotonically decreasing, skewed to the left, skewed to the right and almost symmetric curves while the hazard function can be bathtub shaped, up-side down bathtub shaped, and increasing-deceasing-increasing. To demonstrate the flexibility and usefulness of the model, three real-life datasets analyzed and the results compared with some competing models.

Keywords: T-R{Y} family, quantile function, moments, Shannon entropy

Received November 08, 2021; Revised December 11, 2021; Accepted December 20, 2021

Citation: Anyiam Kizito Ebere, Onyeagu Sydney Iheanyi, Njoku Modestus Onyekachi, On the Families of Generalized Exponentiated Weibull Distributions: Properties and Applications. Turkish Journal of Analysis and Number Theory, 9 (3). 52-64. January 2021. doi:10.12691/tjant-9-3-4

1. Introduction

Weibull distribution is one of the distributions widely studied in statistical literature because of its structural properties and flexibility in modeling lifetime datasets. Some of the areas of application include insurance, metrology and environmental sciences, informatics, forestry, medicine, engineering and reliability analysis, economics and social sciences. In the past few decades, complex nature of lifetime datasets has made researchers to constantly seek for ways of making the existing distributions more flexible and capable of dealing with these emerging complexities. These involves transformation of existing distributions [1], adding skewness to distribution [2], compounding of continuous univariate distributions and discrete distributions [3,4], and transmutation of distributions, addition of scale, shape or location parameters which help in reflection of the distributions. [5] proposed the exponentiated Weibull distribution by adding extra shape parameter to the Weibull distribution by raising the cumulative distribution function (cdf) to a positive index parameter. The introduction of the new positive index parameter to Weibull distribution extended the distribution and made it more applicable in modeling data in several areas.

Studies of some new generalized families utilizing the T-R{Y} structure include the works of [6,7,8,9,10,11]. In each case, the quantile function of Y random variable were used to realizing generalizations of the expoenentiated Weibull distributions.

The remaining part of the paper is organized as follows: In section 2, the T-R{Y} structure which forms the backbone of the work, the quantile function and their support and different generalizations of the exponentiated Weibull distributions are given. In section 3, several properties of the T-exponentiated Weibull family of distributions are derived. In Section 4, some new generalized exponentiated Weibull distributions are developed.

Given a random variable X, the 3-parameter Exponentiated Weibull (EW) distribution cumulative distribution function (cdf) and probability density function (pdf) are respectively given as

[picture] | (1)

and

[picture] | (2)

where is the rate parameter and are the shape parameters.

Suppose we have and random variables with cumulative distribution functions (cdfs) and respectively and corresponding probability density function (pdfs) and The quantile functions of the T, R and Y random variables are and respectively. Since T and Y must be on the same support, if and then and Utilizing the T-R{Y}structure by [12] and [13], the cumulative distribution function (cdf), the probability density function (pdf) and hazard rate function (hrf) of the T-R{Y} family is presented respectively as

[picture] | (3)

[picture] | (4)

and

[picture] | (5)

For any non-uniform T and Y random variables, and given the random variable R, T-R{Y} generalizes the R distribution.

Five different families of generalized exponentiated Weibull (GEW) distribution is provided using the quantile function of the Y random variable.

Table 1. Domain of T and Quantile Function of Y

Following equations (3), (4) and (5), and using the quantile functions in Table 1, the cdf, pdf and hrf of each of the family of distributions; (i) T-exponentiated Weibull{Logistic}, (ii) T-exponentiated Weibull{Log-Logistic}, (iii) T-exponentiated Weibull{Rayleigh}, (vi) T-exponentiated Weibull{Exponential} and (v) T-exponentiated Weibull{Lomax}are given by

i) T-Exponentiated Weibull{Logistic} (T-EW{L})

Suppose the quantile function for Y random variable follows the Logistic distribution, then the distribution function density function and hazard function of T-EW{L} are presented respectively in (6), (7), (8) below

[picture] | (6)

[picture] | (7)

[picture] | (8)

where and are the probability density function, cumulative density function and failure rate function of the exponentiated Weibull distribution, and is the failure rate function of the random variable.

ii) T-Exponentiated Weibull {Log-Logistic} (T-EW{LL})

Assuming the quantile function for Y random variable follows the Log-Logistic distribution, then the distribution function density function and hazard function of T-EW{LL} are presented respectively in (9), (10), (11) below

[picture] | (9)

[picture] | (10)

[picture] | (11)

iii) T-Exponentiated Weibull {Rayleigh} (T-EW{R})

Let the quantile function for the Y random variable follow the Rayleigh distribution, then the distribution function density function and hazard function of T-EW{R} are presented in (12), (13) and (14) below

[picture] | (12)

[picture] | (13)

[picture] | (14)

iv) T-Exponentiated Weibull {Exponential} (T-EW{E})

Suppose the quantile function for Y random variable follows exponential function, then the distribution function density function and hazard function of T-EW{E} are presented in (15), (16) and (17) below

[picture] | (15)

[picture] | (16)

where is the cumulative hazard function of R random variable. T-EW{Ex} is

[picture] | (17)

where and is the survival function and hazard rate function of R random variable respectively.

[picture] | (18)

v) T-Exponentiated Weibull{Lomax}(T-EW{Lx})

Suppose the quantile function for Y random variable follows Lomax function, then the distribution function density function () and hazard function () of T-EW{Lx} are presented in (19), (20) and (21) below

[picture] | (19)

[picture] | (20)

[picture] | (21)

2. Properties of the T-exponentiated Weibull Family of Distributions

Here, some general properties of the T-exponentiated Weibull family of distributions are discussed.

Relationships between the Random Variable T, and the Random Variables, X.

Lemma 1 (Transformations).

Given that then and using the relation connection between the T random variable and X random variable can be established by transformation.

i)

ii)

iii)

iv)

v)

Lemma 2 (Quantiles). Suppose the random variable X has quantile function with The quantile functions for the (i) T-exponentiated Weibull {Logistic}, (ii) T-exponentiated Weibull{Log-Logistic}, (iii) T-exponentiated Weibull {Rayliegh}, (iv) T-exponentiated Weibull{Exponential} and (v) T-exponentiated Weibull{Lomax} distributions are respectively given as

i)

ii)

iii)

iv)

v)

Proof: By solving for directly from the results follow.

Theorem 1: The Shannon entropies of the T-Exponentiated Weibull {Y} family of distributions for (i) T-exponentiated Weibull{Logistic}, (ii) T-exponentiated Weibull{Log-Logistic}, (iii) T-exponentiated Weibull{Rayleigh}, (iv) T-exponentiated Weibull{Pareto}, (v) T-exponentiated Weibull{Exponential} and (vi) T-exponentiated Weibull{Lomax}

[picture] | (22)

[picture] | (23)

[picture] | (24)

[picture] | (25)

[picture] | (26)

where denotes the Shannon entropy of the T random variable.

Proof: The prove of the result in equation (22) for the T-exponentiated Weibull {Logistic}family shall be provided here. Given the expression for the Shannon entropy of the random variable X as

But, and

Given the random variable for the T-exponentiated Weibull {Logistic} family, then

[picture] | (27)

Also,

It follows from Lemma1 (i), that follows the T-exponentiated Weibull{Logistic} family, thus

[picture] | (28)

Result from equation (22) follows from equations (27) and (28). Following the same approach results in equations (23) – (26) are obtained.

Theorem 2: The non-central moments for the (i) T-exponentiated Weibull {Logistic}, (ii) T-exponentiated Weibull{Log-Logistic}, (iii) T-exponentiated Weibull {Rayliegh} and (iv) T-exponentiated Weibull {Pareto}, (v) T-exponentiated Weibull {exponential} and (vi) T-exponentiated Weibull{Lomax} distributions are respectively given by

[picture] | (29)

[picture] | (30)

[picture] | (31)

[picture] | (32)

[picture] | (33)

Here, is the moment generation function of the random variable X.

Deviation from the Mean D(µ) and from the Median D(M)

The absolute mean deviation from the Mean and absolute mean deviation from the median for (i) T-exponentiated Weibull{Logistic}, (ii) T-exponentiated Weibull{Log-Logistic} (iii) T-exponentiated Weibull{Rayliegh}, (iv) T-exponentiated Weibull{Pareto}, (v) T-exponentiated Weibull{exponential} and T-exponentiated Weibull{Lomax} distributions, are given respectively as

[picture] | (34)

[picture] | (35)

Proof: Using the deviation from the mean and deviation from the median for the T-exponentiated Weibull{Logistic} distribution. Suppose the expression for the mean deviation from the mean and from the median are given as is given by

and respectively and

Then

Taking then

But then

Therefore,

Using logarithmic series representation (see formula 1.511 in [14]) we have that

Substituting, we have

Which can be simplified as

3. New Generalized Exponentiated Weibull Distributions

Some new generalized Exponentiated Weibull distributions in the T-Exponentiated Weibull {Y} families are presented in this section. The additional parameter(s) from the T and Y usually deepens the flexibility of the distributions and also improves the tail behaviour. The five distributions are Gumbel-Exponentiated Weibull {Logistic} (Gu-EW{L}) [15], Frechet-Exponentiated Weibull {Log-Logistic} (F-EW{LL}), Chen-exponentiated Weibull{Rayleigh} (Chen-EW{R}), Burr(III)-Exponentiated Weibull{Exponential} (Burr(III)-EW{E}) and Exponential-Exponentiated Weibull {Lomax} (E-EW{Lx}).

3.1. Gumbel-Exponentiated Weibull{Logistic}(Gu-EW{L}) Distribution

Let the random variable T follow the Gumbel distribution with location parameter and scale parameter and cdf given by

Then the cdf of the random variable X following the Gu-EW{L} distribution is obtained as

and the pdf of the random variable following the Gu-EW{L} obtained as

where, presents the shape parameters, is the rate parameter and is the location parameter. is the cdf of exponentiated Weibull distribution and is the pdf of exponentiated Weibull distribution given in equations (1) and (2) respectively. If the shape parameter Gu-EW{L} distribution reduces to Gumbel-Weibull (GW) distribution.

Figure 1. Plots of the Gu-EW{L}densities for different values of α, β, λ, b and s

The graph shows different shapes of the distribution for different choice of the parameters (skewed to the left, bimodal, skewed to the right and monotonically decreasing). The parameters b and s are from the Gumbel distribution.

3.2. Frėchet-Exponentiated Weibull{Log-Logistic} (F-EW{LL}) Distribution

Let the random variable T follow the Frechet distribution with parameter cdf of Frechet distribution is given by The cdf for the Frechet-Exponentiated Weibull{Log-Logistic}(F-EW{LL}) distribution is given by

where, represents shape parameters and λ is the rate parameter. is the cdf of exponentiated Weibull distribution and is the pdf of exponentiated Weibull distribution given in equations (1) and (2) respectively. The parameter is a shape parameter from log-logistic distribution and the parameter is shape parameter from the Frėchet distribution.

Figure 2. Plots of the pdf of F-EW{LL}distribution for various values of α, β, λ, δ and π

The plots in Figure 2 give a reversed-J, right-skewed, left-skewed unimodal densities and almost bimodal density shapes.

3.3. Chen-Exponentiated Weibull{Rayleigh} (Chen-EW{R}) Distribution

Let T be Chen random variable with distribution function given by then the cdf and pdf of the random variable X following the Chen-EW{R} distribution given by

and

respectively.

where, and represents shape parameters and and λ are the rate parameters. is the cdf of exponentiated Weibull distribution and is the pdf of exponentiated Weibull distribution given in equations (1) and (2) respectively. The parameter is from the Rayleigh distribution and the parameters are the scale and shape parameters from the Chen distribution respectively.

Figure 3. Plots of the densities of Chen-EW{R} distribution for various values of α, β, θ, γ, η and λ

The plots in Figure 3 shows a J-shaped, monotonically decreasing (reversed -J shaped), density shapes. The pdf displays curves which are highly heterogeneous in the shapes which supports the fact that the six parameter model is suitable for modelling lifetime for several industrial and economic products.

3.4. Burr(III)-Exponentiated Weibull{Exponential} (Burr(III)-EW{E}) Distribution

Suppose T random variables are taking from Burr(III) distribution with distribution function given by where c and k are shape parameters of Burr(III) distribution. The distribution function of Burr(III)-EW{E} distribution is obtained as

and

where, are the shape parameters and is the scale parameter. is the cdf of exponentiated Weibull distribution and is the pdf of exponentiated Weibull distribution given in equations (1) and (2) respectively.

Figure 4. Plots of the pdf of Burr (III)-EW{E} for different combination of parameters

Figure 4 depicts the behavour of the pdf of Burr(III)-EW{E} for different selections of the parameters. The pdf plots shows reversed-J shaped, right-skewed, left-skewed and unimodal densities.

3.5. Exponential-Exponentiated Weibull{Lomax} (E-EW{Lx}) Distribution

Let the random variable follow the exponential distribution with parameter 1. The pdf of T random variable is Then the cdf of exponential-exponentiated Weibull{Lomax} (E-EW{Lx}) distribution is given by

and pdf is provided as

where, are the shape parameters and the scale parameters are and λ, is the cdf of exponentiated Weibull distribution and is the pdf of exponentiated Weibull distribution given in equations (1) and (2) respectively.

Figure 5. Plots of the pdf of E-EW{Lx} distribution for various parameter values α, β, λ, τ and ν

The E-EW{Lx} densities take various shapes for selected parameter values such as unimodal, reversed-J shaped, right-skewed and left skewed and almost symmetric shapes.

4. Simulation and Numerical Computations

Suppose are independent random sample from a Gumbel-exponentiated Weibull{Logistic} distribution, the likelihood function, L, based on observed values for the Gu -EW{L} distribution is given by

By maximizing the likelihood function, the maximum likelihood estimate of is obtained. The log-likelihood function for Gu-EW{L} distribution is given by

where,

[picture] | (36)

The maximum likelihood estimates of say can be derived by taking the partial derivatives of the log-likelihood function and then solving iteratively,

[picture] | (37)

[picture] | (38)

[picture] | (39)

[picture] | (40)

[picture] | (41)

A simulation study to investigate the finite sample properties of the Maximum likelihood estimates of the parameters was conducted. We generated 5000 samples of size and 700 and the fixed choice of parameter values, Random samples are simulated by using the quantile function of the G-EW{L} distribution and Average Estimates (AEs), Root Mean Square Error (RMSE), Average Width (AW) and Coverage Probability were computed to examine the estimation accuracies. It can be observed from the results that as the sample sizes increases the average estimates of the parameters moves rapidly to the true values. In addition, the coverage probabilities tend to the nominal level of 95% as the samples sizes increases for each of the parameters. For expediency, the results for RMSE and AW are presented graphically in Figure 6.

Figure 6. Plots of the Monte Carlo simulations results for the Gu-EW{L}; RMSE and AW

The graphs of RMSE and AW as a function of n in Figure 6, shows clearly that as the sample size increases the values of RMSE decays to zero and AW decreases reasonably well.

Four applications to real data to establish the usefulness of the Gu-EW{L} distribution since the pdf has different shapes and also to demonstrate that the model can be applied in many situations are presented in this section.

The first dataset relates to the symmetric, multimodal and platykurtic Kevlar 49/epoxy strands failure times data (pressure at 70%) with skewness = 0.0938 and excess kurtosis = 0.9154 extracted from [16]. We fit the density of the Gumbel-exponentiated Weibull{Logistic} (Gu-EW{L}) distribution using MLE to the data set and compare with three other distributions namely; Exponentiated Weibull distribution (EWD), Gumbel-Weibull distribution (GWD) and beta-Normal distribution (BND). The result of the parameters estimates, log-likelihood, AIC, K-S and p-value.

Table 2. Model Parameter estimates for the Kevlar 49/epoxy strands failure times data (pressure at 70%)

Figure 7. Histogram and fitted densities functions of the Gu-EW{L}D, EWD, GWD and BND

The above results in Table 2 clearly show that the proposed Gu-EW{L}distribution provided a good fit to the dataset and also performed better than other distribution in fitting the dataset given that the p-value and the K-S statistic is highest and very close to 1. Besides, the p-value of the K-S statistic for all the fitted distributions is greater than the threshold of 0.05 level of significance showing that all the distributions fitted the data sufficiently and that the proposed Gu-EW{L} distribution greatly presented the best fit.

The dataset two represents the height of 100 Australian athletes collected at the Australian Sport Institute. The dataset was reported by [17]. The dataset is skewed to the left. The data is unimodal and leptokurtic (Skewness = -0.5514, excess kurtosis = 1.1131).

Four distributions were used to fit the dataset namely: the proposed Gu-EW{L}D, the WD, the EWD, and the BND. The results of the maximum likelihood fit of all the distributions are contained in Table 3.

Table 3. Maximum likelihood fit of the Australian athletes’ height data

Figure 8. Fitted densities for the Australian athletes’ height data

Results in Table 3 clearly show that the proposed Gu-EW{L}D not only provided a good fit to the data but also performed better the other distribution in fitting the dataset given that its p-value of the K-S statistic is highest. Again, the p-value of the K-S statistic for all the fitted distributions is greater than the nominal 0.05 level of significance indicating that all the distributions fitted the data considerably well and that the proposed Gu-EW{L}D presented the best fit.

Dataset three represents the Kevlar 49/epoxy strands failure times data (pressure at 90%) collected from [18] and obtained from [19]. The data set is unimodal, highly skewed to the right and leptokurtic (Skewness = 2.9573, excess kurtosis = 13.3798). Gumbel-exponentiated Weibull {Logistic}(Gu-EW{L}) distribution, Weibull Distribution (WD), exponentiated Weibull distribution (EWD) and beta-Normal distribution (BND) were fitted to the data set and presented below.

Table 4. Model Parameter estimates for the Kevlar 49/epoxy strands failure times data (pressure at 90%)

Figure 9. Histogram and fitted density functions of Gu-EW{L}D, WD, EWD and BND

From the result of dataset three shown in Table 4, it can be observed that the proposed Gu-EW{L}distribution provided a good fit to the data and also performed better than other competing distributions by providing the highest p-value of the K-S statistic. Again, the p-value and the K-S statistic for all the fitted distributions is greater than the threshold of 0.05 level of significance indicating that all the distributions fitted the data substantially well and that the proposed Gu-EW{L}distribution presented the best fit which can be observed in Figure 9.

5. Conclusion

This work has introduced a new generalization of the 3-parmater exponentiated Weibull univariate continuous distribution called the T–exponentiated Weibull {Y} distribution. Five new generalized exponentiated Weibull families using the quantile function of Logistic, Log-logistic, Rayleigh, Exponential and Lomax distributions are provided. Several mathematical properties of the new families such as the quantiles, Shannon entropies, non-central moments, absolute mean deviation from the mean and median were obtained.

Five new distributions, Gumbel-Exponentiated Weibull {Logistic} (Gu-EW{L}) (Anyiam and Onyeagu,2021), Frechet - Exponentiated Weibull {Log - Logistic} (F - EW{LL}), Chen - exponentiated Weibull{Rayleigh} (Chen - EW{R}), Burr(III) - exponentiated Weibull{Exponential} (Burr (III) - EW{E}) and Exponential-Exponentiated Weibull {Lomax} (E-EW{Lx}) distributions are presented.

Three real-life datasets are fitted using the Gumbel-Exponentiated Weibull{Logistic} (Gu-EW{L} distribution to exhibit the flexibility and applicability of the proposed generalized exponentiated Weibull family of distributions. The results of the comparison of the T-exponentiated Weibull with other existing distributions showed that the new proposed family performed better when the AICs, K-S statistics and P-values were compared.

Acknowledgements

The reviewers are highly appreciated for making out time to review this article.

Funding

There is no funding for this work.

Conflict of Interest

We declare that there is no conflict of interest.

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