Abstract
A two-parameter distribution under “Generalized Exponential Uniform Distribution (GEUD)” is proposed in this paper by using the idea of Alzaatreh et al. (2013). We investigate its various properties including its characterization, hazard rate, mean residual life, and Shannon entropy. The constraints on its parameters generate three disjoint subfamilies with regard to their distribution characteristics that exhibit more adaptability when compared with other existing models in the literature for fitting real-life datasets with observations limited in the range (0,1). For the estimation of its parameters, we consider and compare the maximum likelihood, distance minimizing Anderson–Darling, and Cramer–von Mises approaches through simulation studies. To display the importance of this model, we provide its important applications. The GEUD model relative to other models is more suitable in describing certain real datasets.
1. Introduction
A number of models have been proposed in the statistical literature for describing datasets related to industrial, medical, engineering, and other fields. Some of the models proposed are based on the extension, generalization, and other approaches of popular models. Marshall–Olkin [2], for instance, introduces the G family of distributions adding a parameter to parent distribution. The other models including their applications are Beta-G [3], Gamma-G [4, 6], Kumaraswamy-G, Gauss-M [5], McDonald-G [7], transformer (T-X) [1], Weibull-G [8], odd Fréchet-G [9], odd Burr III-G [10], power Lindley-G [11], Marshall–Olkin odd Burr III-G [12], flexible Burr X-G [13], and Fréchet Topp Leone-G [14]. These models have been found adequate in their description of certain datasets.
Alzaatreh et al. [1] use the following methodology for the generalization of a distribution. They consider as the cdf of a random variable and as the cdf of another random variable X such that the link function satisfies the following conditions:(i) (ii) is differentiable and monotonically nondecreasing(iii) as and as
Then,
This method is useful in extending the scope of a distribution for its applications through its generalization. It was used in generalizing odd Fréchet-G family in [15] and odd log-logistic family in [16].
In this paper, we follow the above approach in proposing a two-parameter Generalized Exponential Uniform Distribution (GEUD). For its study, we present the material in this paper as follows: In Section 2, we derive GEUD and identify its subfamilies with distinct shapes of their density curves. Section 3 discusses the theoretical properties of GEUD along with the importance of their parameters. A theorem on the characterization of GEUD involving truncated moments is given as well. In Section 4, some results on the reliability analysis of distribution are developed. A theorem on the characterization of GEUD through its hazard function is also introduced. Section 5 includes a critical study of the distribution’s entropy. Estimates of GEUD parameters are given in Section 6 employing the methods of maximum likelihood and minimization of Anderson–Darling and Cramer–von Mises distances. A comparison of these estimation approaches is made by means of simulation studies. Section 7 provides the application of GEUD to two important real datasets and assesses its performance relative to other existing competitive models in the literature. Finally, we present the broad conclusions about our investigation of the proposed distribution.
2. Generalized Exponential Uniform Distribution
We obtain this generalization by using the idea of Alzaatreh et al. [1] mentioned in Section 1. For this, we consider cdf as and as the pdf of the exponential distribution. With the generator and the baseline distribution , we get the cdf of the GEU random variable X asover the range , where and are its two parameters.
It can be easily proved that(i)(ii) for in the range
The corresponding pdf of GEUD is
2.1. GEUD Subfamilies
The distribution under investigation involves the scale λ and shape θ parameters. The behavior of the pdf in equation (2) when and is as
The parameter θ has a pivotal role in producing heterogeneity in the shapes of pdf curves of the GEUD family. We decompose this family into the following three distinct subfamilies of distributions in view of the similarity of pdf curves within each subfamily:
Figure 1 shows the pdf curves for some selected values of θ.
The graphs of corresponding distributions of three subfamilies are given in Figure 2.
The density curves of the second subfamily unlike the other subfamilies begin at () and are likely to assume upward or downward trends at depending on the value of according to the following theorem. This subfamily further degenerates into three classes.
Theorem 1. For the second subfamily with ,
The proof of this result immediately follows from the expression
Figure 1 presents the illustration of this theorem with reference to the density curves of three classes of SFII.
3. An Analytic Study of GEUD
In this section, we study the quantiles, mode, moments, skewness, and kurtosis of GEUD. A theorem is also added on the characterization of GEUD through the use of truncated moments.
3.1. Quantile Function
The pth quantile of GEUD is
It is easy to see that decreases for larger values of under fixed and increases with θ for each fixed λ.
3.2. Mode
The general expression for the mode of GEUD isprovided it exists. The subfamily has no mode. For with , it is , while it remains zero for ≥ 2. The mode of the third subfamily can be determined from the previous expression.
3.3. Ordinary Moments of GEUD
The ordinary moments are given by
For its proof, we havewhere .
By Lemma 1 in Appendix A, it follows that
From this, we can find the variance, coefficients of skewness, and kurtosis of GEUD.
3.4. Remarks on Measures of GEUD
Table 1 in the Appendix shows the values 0 on important descriptive measures related to GEUD when its parameters are assumed. Based on this, we state here some important remarks:(i)The median, mean, and mode: the GEU distribution with regard to its mean, median, and mode is an increasing function of θ, while it decreases when its parameter λ decreases.(ii)Variance: a similar behavior of this measure occurs when a parameter varies.(iii)Coefficients of skewness and kurtosis: the coefficient of skewness decreases when θ assumes higher values, but it increases with increasing values of λ. For some values of both parameters, distribution in is likely to be symmetrical. For example, for the parameters λ = 2 and θ = 1.5, values nearly generate almost symmetrical GEU distributions.
As the parameter θ increases, the coefficient of kurtosis decreases in and increases in the other two subfamilies. For some values of both parameters, a distribution in the third subfamily is likely to be with the same coefficient of kurtosis as that of a normal distribution (for example with λ = 1 and θ = 2.4).
3.5. Characterization Based on Moments
We base the characterization of GEU distribution on the approach mentioned in the paper [17].
Theorem 2. Let X: be distributed as GEUD. Let
Then, the r.., X follows GEUD iff
Proof. We havewhich simplifies toSimilarly, it can be shown thatHence,AsConversely,and hence,
4. Reliability Characteristics of GEUD
In this section, we study the hazard rate and mean residual life function of GEUD. We characterize this distribution by using its hazard function as well.
4.1. Hazard Function
The hazard function is
Its limits at x = 0 and 1 are
Theorem 4. The hazard rate function of GEUD is(i)Increasing for (ii)A bath type if The proof follows from the first derivative of , i.e.,
For , we have for each x.
As for , since the limit of at and , it follows that the hazard curve of GEUD has a minimum value at the critical point given by the equation provided . For this, we can show that
The first subfamily of GEUD describes bath-type hazard curves with minimum values:
Theorem 5. The hazard rate function of GEUD is an increasing function of the parameter λ for each fixed θ.
The proof of the theorem is evident from equation (10).
Theorem 6. Let be the critical points of two GEU distributions with parameters and where If their hazard rate functions are and , then(i)(ii)
Proof (i). We have
For we haveOr,Or,Hence, the result follows.
Proof (ii). We have And, as and ,
Theorem 7. The hazard rate function of GEUD in its third subfamily is a decreasing function of θ with an upper bound .
Proof. From equation (10), it can be shown that the derivative of the hazard function with respect to θ isAs ,for and it follows that the above derivative remains negative. It also follows that as , the hazard function approaches the value
Remark. We can conclude from the above theorems that the hazard rates increase with x for each θ. These trends become more increasing for each value of x as θ decreases.
We provide the graphs of hazard curves for each subfamily of GEU distributions in Figure 3.
Remarks. In each subfamily, the hazard curves display similarities in their shapes. The above graphs reveal the following differential features from one subfamily to another subfamily with respect to their hazard rates:(a)Subfamily I: for this subfamily of GEUD, as the parameter θ increases for each fixed λ, the minimum value of hazard rate decreases. The critical point of a hazard curve moves towards the left for a larger value of θ (Figure 3). The parameter λ has no impact on the critical point . Since it is a multiplier in expression (11), the minimum value of increases as this parameter increases. Also, the hazard function decreases (or increases) with a faster rate as λ increases (Figure 3).(b)Subfamily II: the increasing hazard rate for the second subfamily with θ = 1 further increases for the larger value of λ. The J-shaped trend of a distribution moves to the left, that is, the products of an industry that follow the distributions of their lifetimes in this subfamily with lower values of λ are likely to take longer times for their survival.(c)Subfamily III: the increasing J-shaped trend of the hazard rate for the third subfamily moves to the right for the larger value of θ. Also, the effect of the increase in λ on the hazard trend in this subfamily as well as is to increase the hazard rate.
4.2. Characterization Based on Hazard Function
The characterization of GEU distribution can also be done through its hazard rate.
Theorem 8. The random variable X follows GEUD if and only if its hazard function satisfies the differential equationwhere
Proof. It is well known that a hazard function satisfies the differential equationIf X has the hazard function as indicated in equation (10), then the derivative of h(x) w.r.t. x isHence,which simplifies to equation (14).
Conversely, if differential equation (14) holds, thenOn integrating the previous equation, we havewhich under the boundary condition implies C = 0.
4.3. Mean Residual Life
The mean residual life (MRL) is the expected additional life of a unit at the age and is given by
We have
On letting , the previous integral simplifies to
On using Lemma 1 in Appendix A with , we get
By inserting the previous expression in equation (17), the MRL becomes
Figure 4 displays the behavior of MRL of GEUD for different values of θ and λ = 1. The MRL curves of GEUD are inverted bathtubs in SFI but decreasing in SFII and SFIII.
Table 2 given in the Appendix is useful to assess the trends of the critical and hazard and MRL functions in SFI for different values of θ and λ at the critical points of the hazard function. It shows that the increase in the value of θ in SFI decreases the value of at its critical point shifting the bathtub to the left side, but an increase in raises The behavior of is opposite to that of , which is indicated in the table that an increase in λ decreases the MRL.
5. Shannon Entropy of GEUD
The entropy of this distribution is measured by using the following well-known results:where and . We find the following theorem.
Theorem 9. The Shannon entropy of GEUD is given by
Proof. We haveLet . Then,On using the above results and simplifying, we prove the theorem.
Remarks on Shannon Entropy. Since the subfamilies of GEUD depending on θ have heterogeneous shapes of their density curves, it is useful to study their entropies. Given a subfamily with some specified value of θ, the parameter λ plays a significant role in affecting a distribution’s entropy. To look into this aspect, we consider the first derivative of entropy of a GEUD distribution with respect to λ. This derivative isIt can be used to determine the nature of an entropy’s trend when λ increases and the parameter θ is kept fixed. The derivative largely depends on which assumes positive values in a decreasing order as θ is assigned increasing values; that is, for small θ, this term takes up a large value. The other contributing term to the derivative is which assumes negative values and increases with higher values of λ. Thus, derivative (19) can be a positive or negative number or zero depending on the values of GEUD parameters involved. Table 3 in Appendix, for the comprehension of entropic trends, presents Shannon entropies of several GEU distributions as well as the derivatives at selected positions of λ when fixed θ is given different values. We provide Figure 4 of entropy curves showing the relationship of GEUD entropy with under fixed θ.
A numerical study of Figure 5 reveals some following entropic properties about GEUD.
The entropy of GEUD family members assumes negative values, and for all GEUD subfamilies, there appear downward tublike shapes when the parameter λ increases. The entropy is very large for distributions for very small as well as large λ values.
For each fixed θ, there occurs some different values of λ where the entropy is minimum; this value can be found by solving expression (11) when equated to zero. The change in the values of derivatives shifts the positions of these inverted tubs to the right for a larger value of λ as θ increases. Entropies are relatively lower for the subfamilies with a lower θ value when λ is fixed.
An increase in θ in enlarges the entropy of a distribution. On the contrary, a larger value of θ causes a decrease in the entropy of distribution in . For small values of λ, the entropy generally increases in the and decreases in .
6. Estimation of Parameters
This section considers three commonly used techniques to estimate the parameters λ and θ of GEUD. For this purpose, we use the maximum likelihood method and Anderson–Darling as well as Cramer–von Mises minimum distance measures. The comparative performance of these methods is investigated through Monte Carlo simulations.
6.1. Maximum Likelihood Estimates of GEUD Parameters
The log-likelihood function of the sample from GEUD is
The equations used for the estimation of are
Setting these two equations to zero, we get the MLEs of .
6.2. Anderson–Darling Minimum Distance Estimates of GEUD Parameters
The Anderson–Darling estimates (ADEs) of θ and λ can be obtained by minimizing the function
These estimators can be obtained from the following equations by taking the derivatives of the above expression with respect to λ and θ and equating them to zero.andwhereand
We can estimate the parameters from the previous equations and so the estimate of
6.3. Cramèr–von Mises Minimum Distance Estimates of GEUD Parameters
The Cramèr–von Mises minimum distance is given by the expression
For the estimation of Cramèr–von Mises minimum distance of we solve the nonlinear equationsand. .Here, and are defined in equations (21)–(22). The estimate of can now be found.
6.4. Simulation Study
In this section, we use the Monte Carlo simulation procedure to assess the comparative performance of estimates of the GEUD parameters by the above three methods. The simulation procedure is applied by selecting 10000 random samples of size n and estimating the parameters when their values are specified. The estimated and true values of parameter are used for the calculation of mean square errors (MSEs).
In Table 4, we provide the estimates of GEUD parameters along with their MSEs by each method. We have provided the efficiencies and of MLEs relative to estimates by other methods.
Table 4 shows that the estimates by each method improve with the increase in sample size but each parameter with its larger value yields a higher MSE, implying that larger samples are required for the efficient estimation of such parameters. For small sample sizes, an estimate by AD has lower MSE relative to other methods. Generally, a maximum likelihood estimator is a more desirable procedure, whereas the CVM estimates perform poorly.
7. Applications
We now apply the GEUD model for describing two real-life datasets by using the ML estimates as we find the method of estimation of GEUD parameters to be more efficient than the other methods considered in Section 6. Its adequacy is also compared with the following four competitive probability models:(i)Beta distribution (BD)(ii)Unit-Weibull distribution (UWD) [18](iii)Unit-Burr III distribution (UBIIID) [19](iv)Unit-Gompertz distribution (UGomD) [20]
The parameters of each model are estimated by the MLE method, and for the assessment of their adequacy, we use Akaike Information Criterion AIC and Bayesian Information Criterion BIC =
Assuming that the sample is from GEUD against an alternative hypothesis that it arises from one of the above-mentioned four models, we apply Anderson–Darling (A) and Cramèr–von Mises (W) tests as defined in Section 5. Also, the Kolmogorov–Smirnov (KS) test,is used for this comparison. We also present the graphs of real dataset, density and cumulative distribution function relating to fitted GEUD, and the two other compatible models in Figure 6 and 7.
Dataset 1. We have data on the rate with regard to daily COVID-19-affected patients over 153 days among randomly selected people in Pakistan from April 1, 2020, to August 31, 2020 [21].
Dataset 2. The second dataset in [22] refers to 50 observations on burr (in millimeters), with a hole diameter and sheet thickness of 12 mm and 3.15 mm, respectively. The dataset is as given as follows: 0.04, 0.02, 0.06, 0.12, 0.14, 0.08, 0.22, 0.12, 0.08, 0.26, 0.24, 0.04, 0.14, 0.16, 0.08, 0.26, 0.32, 0.28, 0.14, 0.16, 0.24, 0.22, 0.12, 0.18, 0.24, 0.32, 0.16, 0.14, 0.08, 0.16, 0.24, 0.16, 0.32, 0.18, 0.24, 0.22, 0.16, 0.12, 0.24, 0.06, 0.02, 0.18, 0.22, 0.14, 0.06, 0.04, 0.14, 0.26, 0.18, and 0.16. The data were recently modelled in [23] using distribution under ‘Unit Johnson SU distribution”.
The descriptive characteristics of both datasets and other relevant statistics are given in Table 5.
The maximum likelihood estimates (standard errors) and measures of model selection criteria for the both datasets are presented in Table 6 and 7, respectively.
Based on the above analysis, we find that for the first dataset, the GEUD model among the selected competing models is highly preferable when compared in view of logL, AIC, and BIC. The goodness-of-fit tests also outrightly support this model for the description of the first dataset.
As for the second dataset, both GEUD and “Unit Johnson SU distribution” appear preferable when compared with other models. These two models do not show much difference on all measures. But, Unit Johnson SU distribution, for which its pdf isis a complicated model for its fitting to a dataset, so the GEUD model, being more viable, can be recommended for the second dataset as well.
8. Conclusion
Datasets with measurements ranging from zero to one are frequently encountered in the industry and other applied sciences where it becomes imperative to seek their description by statistical models. For such purposes, a number of models exist in the literature but the need for a more efficient model cannot be ignored. In this paper, we have developed a new probability distribution entitled “Generalized Exponential Uniform Distribution (GEUD).” Important theoretical properties of the family of this distribution are provided that include its characterization, hazard rates, mean residual life, and Shannon entropy. The two shape parameters of this distribution cause a large variation in their characteristics, and in view of this consideration, the family of members of this distribution is classified into subfamilies, each possessing density curves with similar shapes. The members of GEUD are assumed to be increasing as well as bathtub hazard curves under specified parametric constraints. These distributions have negative entropies following an increasing and decreasing pattern like an inverted bathtub as one of the parameters increases. For the selection of its parametric estimation, we compare the MLE estimates with AD and CVM minimum distance estimators on the basis of a simulation study and find that the MLE estimates are more efficient. To validate the importance of the GEUD model, we fit it to two datasets provided in the literature and make its comparison with other relevant competing models.
Appendix
Lemma 1.
Proof. On letting previous integral simplifies toand with ,The result follows as .
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.