# An Analysis of Generalized Exponential Uniform Distribution

Authors: Muhammad Ahsan-ul-Haq; Ahmed Zogo Memon; Sohail Chand
Journal: Mathematical Problems in Engineering
 (2022)

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## 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.


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## Body

## 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 Rt as the cdf of a random variable T∈a,b−∞<a<b<∞ and Fx as the cdf of another random variable X such that the link function W.:0,1⟶a,b satisfies the following conditions:(i)
WGx∈a,b(ii)
W is differentiable and monotonically nondecreasing(iii)
WGx⟶a as x⟶−∞ and WGx⟶b as x⟶∞Then,(1)Fx=∫aWGxdRtdt=RWGx.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 Section2, 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 Gx=xfor0<x<1 and Rt as the pdf of the exponential distribution. With the generator Rt and the baseline distribution WGx=Gxθ/1−Gxθ, we get the cdf of the GEU random variable X as(2)Fx=∫0Gxθ/1−Gxθλexp−λtdt=1−exp−λxθ1−xθ.over the range 0<x<1, where λ and θ are its two parameters.It can be easily proved that(i)
Fλ1x>Fλ2xforλ1<λ2(ii)
Fθ1x<Fθ2x for θ1<θ2 in the range 0<x<1The corresponding pdf of GEUD is(3)fx=λθxθ−11−xθ2exp−λxθ1−xθ,0<x<1.


### 2.1. GEUD Subfamilies

The distribution under investigation involves the scaleλ and shape θ parameters. The behavior of the pdf in equation (2) when x⟶0 and x⟶1 is as(4)limx⟶0fx=∞forθ<1λθ=10θ>1andlimx⟶1fx=0.The parameterθ 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:(5)SFI=x:λ>0,0<θ<1,SFII=x:λ>0,θ=1,SFIII=x:λ>0,θ>1.Figure1 shows the pdf curves for some selected values of θ.Figure 1
pdf curves of the GEUD.The graphs of corresponding distributions of threeFx subfamilies are given in Figure 2.Figure 2
cdf curves of the GEUD.The density curves of the second subfamily unlike the other subfamilies begin at (0,λ) and are likely to assume upward or downward trends at x=0 depending on the value of λ according to the following theorem. This subfamily further degenerates into three classes.Theorem 1.
For the second subfamily withθ=1,(6)limx⟶0f′x=>0,λ<2=0,λ=2<0,λ>2..The proof of this result immediately follows from the expression(7)limx⟶0f′xfx=limx⟶021−x−λ=2−λ.Figure1 presents the illustration of this theorem with reference to the density curves of three classes of SFII.


## 2.1. GEUD Subfamilies

The distribution under investigation involves the scaleλ and shape θ parameters. The behavior of the pdf in equation (2) when x⟶0 and x⟶1 is as(4)limx⟶0fx=∞forθ<1λθ=10θ>1andlimx⟶1fx=0.The parameterθ 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:(5)SFI=x:λ>0,0<θ<1,SFII=x:λ>0,θ=1,SFIII=x:λ>0,θ>1.Figure1 shows the pdf curves for some selected values of θ.Figure 1
pdf curves of the GEUD.The graphs of corresponding distributions of threeFx subfamilies are given in Figure 2.Figure 2
cdf curves of the GEUD.The density curves of the second subfamily unlike the other subfamilies begin at (0,λ) and are likely to assume upward or downward trends at x=0 depending on the value of λ according to the following theorem. This subfamily further degenerates into three classes.Theorem 1.
For the second subfamily withθ=1,(6)limx⟶0f′x=>0,λ<2=0,λ=2<0,λ>2..The proof of this result immediately follows from the expression(7)limx⟶0f′xfx=limx⟶021−x−λ=2−λ.Figure1 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(8)Qp=−log1−pλ−log1−p1/θ.It is easy to see thatQp decreases for larger values of λ under fixed θ and increases with θ for each fixed λ.


### 3.2. Mode

The general expression for the mode of GEUD is(9)x=2−λθ±2−λθ2−41+θ1−θ21+θ1/θ.provided it exists. The subfamily SFI has no mode. For SFII with λ<2, it is 2−λ/2, while it remains zero for λ ≥ 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(10)μr′=eλ∑j=0∞−1jrθjλjГ1−j,λ.For its proof, we have(11)μr′=∫01xrfxdx=λ∫01θxr+θ−11−xθ2exp−λxθ1−xθdx=λ∫0∞y1+yr/θexp−λydy,where y=xθ/1−xθ.By Lemma1 in Appendix A, it follows that(12)μr′=eλ∑j=0∞−1jrθjλjГ1−j,λ.From this, we can find the variance, coefficients of skewness, and kurtosis of GEUD.


### 3.4. Remarks on Measures of GEUD

Table1 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 ofθ, 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 whenθ assumes higher values, but it increases with increasing values of λ. For some values of both parameters, distribution in SFIII is likely to be symmetrical. For example, for the parameters λ = 2 and θ = 1.5, values nearly generate almost symmetrical GEU distributions.Table 1
The values of mean, mode, variance, skewness, and kurtosis of GEUD.
λθMeanMedianModeVarianceSkewnessKurtosis0.50.10.04140.0044—0.00592.685211.13510.50.40.28670.2572—0.04950.41292.03740.50.80.47730.5072—0.0589-0.27701.98330.51.00.53850.58090.75000.0563-0.48142.21370.51.30.60720.65850.80870.0505-0.71472.62250.51.70.67200.72650.85410.0425-0.94593.18830.52.00.70800.76220.87610.0373-1.08203.60010.52.50.75300.80470.90100.0302-1.26284.24280.55.00.86030.89710.95050.0129-1.76426.64961.00.10.00950.0001—0.00075.050836.97061.00.40.15990.1072—0.02571.04623.29521.00.80.33800.3275—0.04560.20842.00621.01.00.40370.40940.50000.0480-0.01711.98151.01.30.48160.50310.62610.0473-0.26512.11761.01.70.55890.59130.71680.0432-0.50252.41171.02.00.60350.63980.75980.0395-0.63902.66261.02.50.66060.69960.80810.0336-0.81743.07631.05.00.80300.83640.90410.0159-1.29834.74132.00.10.00130.0000—0.000010.636174.972.00.40.07340.0336—0.00891.82166.39602.00.80.21510.1833—0.02710.68702.66242.01.00.27730.25740.00000.03250.41682.29812.01.30.35680.35200.34800.03630.13322.13382.01.70.44100.45010.50410.0369-0.12742.18472.02.00.49170.50730.57740.0356-0.27312.30082.02.50.55890.58110.66000.0322-0.45972.54372.05.00.73580.76230.82730.0175-0.94573.7218As the parameterθ increases, the coefficient of kurtosis decreases in SFIII 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:Ω⟶0,1 be distributed as GEUD. Let(13)q1x=1q2x=q1xExp−λxθ1−xθ,0<x<1.Then, ther.v., X follows GEUD iff ηx=1/2exθλ/−1+xθ.Proof.
We have(14)1−FxEq1X|X≥x=∫x1θλxθ−11−xθ2Exp−λxθ1−xθdx,which simplifies to(15)Exp−λxθ1−xθ,0<x<1.
Similarly, it can be shown that(16)1−FxEq2X|X≥x=12e2xθλ/−1+xθ,0<x<1.
Hence,(17)ηx=Eq2X|X≥xEq1X|X≥x=12exθλ/−1+xθ.
As(18)ηxq1x−q2x=q1xηx−Exp−λxθ1−xθ,=−Exp−λxθ1−xθ+12Exp−λxθ1−xθ≠0.
Conversely,(19)s′x=η′xq1xηxq1x−q2x=exθλ/−1+xθ−x−1+θ+θθλ/−1+xθ2+x−1+θθλ/−1+xθ2−exθλ/−1−xθ+1/2exθλ/−1+xθ,and hence,(20)sx=−lnExp−λxθ1−xθ,0<x<1.


## 3.1. Quantile Function

The pth quantile of GEUD is(8)Qp=−log1−pλ−log1−p1/θ.It is easy to see thatQp decreases for larger values of λ under fixed θ and increases with θ for each fixed λ.


## 3.2. Mode

The general expression for the mode of GEUD is(9)x=2−λθ±2−λθ2−41+θ1−θ21+θ1/θ.provided it exists. The subfamily SFI has no mode. For SFII with λ<2, it is 2−λ/2, while it remains zero for λ ≥ 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(10)μr′=eλ∑j=0∞−1jrθjλjГ1−j,λ.For its proof, we have(11)μr′=∫01xrfxdx=λ∫01θxr+θ−11−xθ2exp−λxθ1−xθdx=λ∫0∞y1+yr/θexp−λydy,where y=xθ/1−xθ.By Lemma1 in Appendix A, it follows that(12)μr′=eλ∑j=0∞−1jrθjλjГ1−j,λ.From this, we can find the variance, coefficients of skewness, and kurtosis of GEUD.


## 3.4. Remarks on Measures of GEUD

Table1 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 ofθ, 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 whenθ assumes higher values, but it increases with increasing values of λ. For some values of both parameters, distribution in SFIII is likely to be symmetrical. For example, for the parameters λ = 2 and θ = 1.5, values nearly generate almost symmetrical GEU distributions.Table 1
The values of mean, mode, variance, skewness, and kurtosis of GEUD.
λθMeanMedianModeVarianceSkewnessKurtosis0.50.10.04140.0044—0.00592.685211.13510.50.40.28670.2572—0.04950.41292.03740.50.80.47730.5072—0.0589-0.27701.98330.51.00.53850.58090.75000.0563-0.48142.21370.51.30.60720.65850.80870.0505-0.71472.62250.51.70.67200.72650.85410.0425-0.94593.18830.52.00.70800.76220.87610.0373-1.08203.60010.52.50.75300.80470.90100.0302-1.26284.24280.55.00.86030.89710.95050.0129-1.76426.64961.00.10.00950.0001—0.00075.050836.97061.00.40.15990.1072—0.02571.04623.29521.00.80.33800.3275—0.04560.20842.00621.01.00.40370.40940.50000.0480-0.01711.98151.01.30.48160.50310.62610.0473-0.26512.11761.01.70.55890.59130.71680.0432-0.50252.41171.02.00.60350.63980.75980.0395-0.63902.66261.02.50.66060.69960.80810.0336-0.81743.07631.05.00.80300.83640.90410.0159-1.29834.74132.00.10.00130.0000—0.000010.636174.972.00.40.07340.0336—0.00891.82166.39602.00.80.21510.1833—0.02710.68702.66242.01.00.27730.25740.00000.03250.41682.29812.01.30.35680.35200.34800.03630.13322.13382.01.70.44100.45010.50410.0369-0.12742.18472.02.00.49170.50730.57740.0356-0.27312.30082.02.50.55890.58110.66000.0322-0.45972.54372.05.00.73580.76230.82730.0175-0.94573.7218As the parameterθ increases, the coefficient of kurtosis decreases in SFIII 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:Ω⟶0,1 be distributed as GEUD. Let(13)q1x=1q2x=q1xExp−λxθ1−xθ,0<x<1.Then, ther.v., X follows GEUD iff ηx=1/2exθλ/−1+xθ.Proof.
We have(14)1−FxEq1X|X≥x=∫x1θλxθ−11−xθ2Exp−λxθ1−xθdx,which simplifies to(15)Exp−λxθ1−xθ,0<x<1.
Similarly, it can be shown that(16)1−FxEq2X|X≥x=12e2xθλ/−1+xθ,0<x<1.
Hence,(17)ηx=Eq2X|X≥xEq1X|X≥x=12exθλ/−1+xθ.
As(18)ηxq1x−q2x=q1xηx−Exp−λxθ1−xθ,=−Exp−λxθ1−xθ+12Exp−λxθ1−xθ≠0.
Conversely,(19)s′x=η′xq1xηxq1x−q2x=exθλ/−1+xθ−x−1+θ+θθλ/−1+xθ2+x−1+θθλ/−1+xθ2−exθλ/−1−xθ+1/2exθλ/−1+xθ,and hence,(20)sx=−lnExp−λxθ1−xθ,0<x<1.


## 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(21)hx=λθxθ−11−xθ2.Its limits atx = 0 and 1 are(22)limx⟶0hx=∞forθ<1λθ=10θ>1andlimx⟶1hx=∞,θ,λ>0.Theorem 4.
The hazard rate function of GEUD is(i)
Increasing forθ≥1(ii)
A bath type ifθ<1
The proof follows from the first derivative ofhx, i.e.,(23)h′x=θ−1x+2θxθ−11−xθ.Forθ≥1, we have h′x>0 for each x.As forθ<1, since the limit of h′x<0 at x=0 and limx⟶1hx=∞, it follows that the hazard curve of GEUD has a minimum value at the critical point xc given by the equation h′x=0.(24)xc=1−θ1+θ1/θ,provided h''xc>0. For this, we can show that(25)h''xc=1−θ1−2/θ1+θ1+2/θ2>0.The first subfamily of GEUD describes bath-type hazard curves with minimum values:(26)hxc=λ4θ1+θ1+1/θ1−θ1/θ−1.Theorem 5.
The hazard rate function of GEUD is an increasing function of the parameterλ for each fixed θ.The proof of the theorem is evident from equation (10).Theorem 6.
Letxc1andxc2 be the critical points of two GEU distributions with parameters λ,θ1 and λ,θ2 where 0<θ1<θ2<1. If their hazard rate functions are h1x and h2x, then(i)
xc1>xc2(ii)
h1xc1<h2xc2Proof (i).
We havexci=1−θi/1+θi1/θi,i=1,2.
Forθ1<θ2, we have(27)1−θ11+θ1>1−θ21+θ2.
Or,(28)1−θ11+θ11/θ11−θ21+θ21/θ1.
Or,(29)>1−θ21+θ21/θ2.
Hence, the result follows.Proof (ii).
We havehxci=λ/4θi1+θi1/θi+1/1−θi1/θi−1,i=1,2.(30)hxc1=λ4θ11+θ11/θ1+11−θ11/θ1−1=λ4θ11+θ11/θ11−θ11/θ11−θ12=λ4θ11+θ11/θ1+11−θ11/θ1−1>λ4θ21+θ21/θ2+11−θ21/θ2−1.
And, asθ1<θ2 and xc1>xc2, 1−θ12>1−θ22; it follows thath1xc1<h2xc2.Theorem 7.
The hazard rate function of GEUD in its third subfamily is a decreasing function ofθ with an upper bound 1/1−x2.Proof.
From equation (10), it can be shown that the derivative of the hazard function with respect to θ is(31)xθ−1−1+xθ−1+xθθlogxxθ−13.
Asxθ1−θlogx>1+θlogx,(32)xθ−13<0,for 0<x<1 and it follows that the above derivative remains negative. It also follows that as θ⟶1, the hazard function approaches the value 1/1−x2.Remark.
We can conclude from the above theorems that the hazard rates increase withx 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 Figure3.Figure 3
Hazard rate curves of the GEUD.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 parameterθ 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 parameterλ has no impact on the critical point xc. Since it is a multiplier in expression (11), the minimum value of hxc 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 withθ = 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 ofθ. 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 variableX follows GEUD if and only if its hazard function hx satisfies the differential equation(33)h′x−θ−1x−1hx=−2λθ2x2θ−2x2θ−13,where h0≥0.Proof.
It is well known that a hazard functionhx satisfies the differential equation(34)f′xfx=h′xhx−hx.
IfX has the hazard function as indicated in equation (10), then the derivative of h(x) w.r.t. x is(35)h′x=λθxθ−2θ−1+xθθ+11−xθ3.
Hence,(36)h′x−θ−1x−1hx=λθxθ−2θ−1+xθθ+11−xθ3−xθ−2λθθ−11−xθ2,which simplifies to equation (14).
Conversely, if differential equation (14) holds, then(37)ddxxθ−1−1hx=2λθ2xθ−11−xθ3.
On integrating the previous equation, we have(38)hx=λθxθ−11−xθ2+C.which 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(39)mt=EX−t|X>t=∫t1xfxdxSt−tfort>0.We have(40)∫t1xfxdx=λ∫t1θxθ−11−xθ2exp−λxθ1−xθdx.On lettingy=xθ/1−xθ, the previous integral simplifies to(41)∫t1xfxdx=λ∫tθ1−tθ∞y1+y1/θexp−λydy.On using Lemma1 in Appendix A with α=tθ/1−tθ, we get(42)∫t1xfxdx=eλ∑j=0∞−1j1θjλjΓ1−j,λ1−tθ.By inserting the previous expression in equation (17), the MRL becomes(43)mt=expλtθ1−tθeλ∑j=0∞−1j1θjλjΓ1−j,λ1−tθ−tfort>0.Figure4 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.Figure 4
MRL curves for GEUD.Table2 given in the Appendix is useful to assess the trends of the critical xc and hazard and MRL functions in SFI for different values of θ and λ at the critical points xc of the hazard function. It shows that the increase in the value of θ in SFI decreases the value of hx at its critical point shifting the bathtub to the left side, but an increase in λ raises hxc. The behavior of mxc is opposite to that of hxc, which is indicated in the table that an increase in λ decreases the MRL.Table 2
hxc and mxc in SFI.
θxcλ=1λ=5λ=10hxcmxchxcmxchxcmxc0.050.13511036.9140.026711184.57240.005414369.1450.0027650.10.13443118.4110.05203492.054910.010833184.1090.0054270.20.1316879.11250.09793245.562500.02173291.12500.0109430.50.1111113.37500.21078616.875000.05634633.75000.0291220.80.0641501.75370.3149388.7685070.10017917.53700.0540330.90.0379461.39090.3547216.9543800.12021713.90880.066260


## 4.1. Hazard Function

The hazard function is(21)hx=λθxθ−11−xθ2.Its limits atx = 0 and 1 are(22)limx⟶0hx=∞forθ<1λθ=10θ>1andlimx⟶1hx=∞,θ,λ>0.Theorem 4.
The hazard rate function of GEUD is(i)
Increasing forθ≥1(ii)
A bath type ifθ<1
The proof follows from the first derivative ofhx, i.e.,(23)h′x=θ−1x+2θxθ−11−xθ.Forθ≥1, we have h′x>0 for each x.As forθ<1, since the limit of h′x<0 at x=0 and limx⟶1hx=∞, it follows that the hazard curve of GEUD has a minimum value at the critical point xc given by the equation h′x=0.(24)xc=1−θ1+θ1/θ,provided h''xc>0. For this, we can show that(25)h''xc=1−θ1−2/θ1+θ1+2/θ2>0.The first subfamily of GEUD describes bath-type hazard curves with minimum values:(26)hxc=λ4θ1+θ1+1/θ1−θ1/θ−1.Theorem 5.
The hazard rate function of GEUD is an increasing function of the parameterλ for each fixed θ.The proof of the theorem is evident from equation (10).Theorem 6.
Letxc1andxc2 be the critical points of two GEU distributions with parameters λ,θ1 and λ,θ2 where 0<θ1<θ2<1. If their hazard rate functions are h1x and h2x, then(i)
xc1>xc2(ii)
h1xc1<h2xc2Proof (i).
We havexci=1−θi/1+θi1/θi,i=1,2.
Forθ1<θ2, we have(27)1−θ11+θ1>1−θ21+θ2.
Or,(28)1−θ11+θ11/θ11−θ21+θ21/θ1.
Or,(29)>1−θ21+θ21/θ2.
Hence, the result follows.Proof (ii).
We havehxci=λ/4θi1+θi1/θi+1/1−θi1/θi−1,i=1,2.(30)hxc1=λ4θ11+θ11/θ1+11−θ11/θ1−1=λ4θ11+θ11/θ11−θ11/θ11−θ12=λ4θ11+θ11/θ1+11−θ11/θ1−1>λ4θ21+θ21/θ2+11−θ21/θ2−1.
And, asθ1<θ2 and xc1>xc2, 1−θ12>1−θ22; it follows thath1xc1<h2xc2.Theorem 7.
The hazard rate function of GEUD in its third subfamily is a decreasing function ofθ with an upper bound 1/1−x2.Proof.
From equation (10), it can be shown that the derivative of the hazard function with respect to θ is(31)xθ−1−1+xθ−1+xθθlogxxθ−13.
Asxθ1−θlogx>1+θlogx,(32)xθ−13<0,for 0<x<1 and it follows that the above derivative remains negative. It also follows that as θ⟶1, the hazard function approaches the value 1/1−x2.Remark.
We can conclude from the above theorems that the hazard rates increase withx 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 Figure3.Figure 3
Hazard rate curves of the GEUD.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 parameterθ 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 parameterλ has no impact on the critical point xc. Since it is a multiplier in expression (11), the minimum value of hxc 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 withθ = 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 ofθ. 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 variableX follows GEUD if and only if its hazard function hx satisfies the differential equation(33)h′x−θ−1x−1hx=−2λθ2x2θ−2x2θ−13,where h0≥0.Proof.
It is well known that a hazard functionhx satisfies the differential equation(34)f′xfx=h′xhx−hx.
IfX has the hazard function as indicated in equation (10), then the derivative of h(x) w.r.t. x is(35)h′x=λθxθ−2θ−1+xθθ+11−xθ3.
Hence,(36)h′x−θ−1x−1hx=λθxθ−2θ−1+xθθ+11−xθ3−xθ−2λθθ−11−xθ2,which simplifies to equation (14).
Conversely, if differential equation (14) holds, then(37)ddxxθ−1−1hx=2λθ2xθ−11−xθ3.
On integrating the previous equation, we have(38)hx=λθxθ−11−xθ2+C.which 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(39)mt=EX−t|X>t=∫t1xfxdxSt−tfort>0.We have(40)∫t1xfxdx=λ∫t1θxθ−11−xθ2exp−λxθ1−xθdx.On lettingy=xθ/1−xθ, the previous integral simplifies to(41)∫t1xfxdx=λ∫tθ1−tθ∞y1+y1/θexp−λydy.On using Lemma1 in Appendix A with α=tθ/1−tθ, we get(42)∫t1xfxdx=eλ∑j=0∞−1j1θjλjΓ1−j,λ1−tθ.By inserting the previous expression in equation (17), the MRL becomes(43)mt=expλtθ1−tθeλ∑j=0∞−1j1θjλjΓ1−j,λ1−tθ−tfort>0.Figure4 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.Figure 4
MRL curves for GEUD.Table2 given in the Appendix is useful to assess the trends of the critical xc and hazard and MRL functions in SFI for different values of θ and λ at the critical points xc of the hazard function. It shows that the increase in the value of θ in SFI decreases the value of hx at its critical point shifting the bathtub to the left side, but an increase in λ raises hxc. The behavior of mxc is opposite to that of hxc, which is indicated in the table that an increase in λ decreases the MRL.Table 2
hxc and mxc in SFI.
θxcλ=1λ=5λ=10hxcmxchxcmxchxcmxc0.050.13511036.9140.026711184.57240.005414369.1450.0027650.10.13443118.4110.05203492.054910.010833184.1090.0054270.20.1316879.11250.09793245.562500.02173291.12500.0109430.50.1111113.37500.21078616.875000.05634633.75000.0291220.80.0641501.75370.3149388.7685070.10017917.53700.0540330.90.0379461.39090.3547216.9543800.12021713.90880.066260


## 5. Shannon Entropy of GEUD

The entropy of this distribution is measured by using the following well-known results:(44)∫0∞lnye−λydy=−γ+lnλλ,∫0∞ln1+ye−λydy=−eλE1−λλ,where γ=≅0.577216 and Ei−λ=−∫λ∞e−y/ydy. We find the following theorem.Theorem 9.
The Shannon entropy of GEUD is given by(45)Hx=−logθ−logλθ+1−1θγ+eλE1−λ1θ+1+1.Proof.
We have(46)Hx=E−lnfx,=−∫01lnλθxθ−11−xθ2exp−λxθ1−xθλθxθ−11−xθ2exp−λxθ1−xθdx.
Lety=xθ/1−xθwith dy=θxθ−1/1−xθ2dx. Then,(47)Hx=−λ∫0∞lnλθy1−1θ1+y1θ+1e−λye−λydy,=−λ∫0∞lnλe−λy+lnθe−λy+1−1θlnye−λy+1θ+1ln1+ye−λy−λye−λydy.
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 is(48)1λ+θ+1θeλEi−λ.
It can be used to determine the nature of an entropy’s trend whenλ increases and the parameter θ is kept fixed. The derivative largely depends on θ+1/θ 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 −eλEi−λ 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 λ under fixed θ.
A numerical study of Figure5 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 parameterλ increases. The entropy is very large for distributions for very small as well as large λ values.
For each fixedθ, 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 inθ in SFI enlarges the entropy of a distribution. On the contrary, a larger value of θ causes a decrease in the entropy of distribution in SFIII. For small values of λ, the entropy generally increases in the SFI and decreases in SFIII.Table 3
Shannon entropy of various members of the GEUD family.
λSFISFIISFIIIHλ,0.5H′λ,0.5Hλ,1.0H'λ,1.0Hλ,1.5H'λ,1.50.1–0.32283.9561–0.72675.9707–1.03576.64230.2–0.14520.5199–0.37732.0133–0.62902.51110.3–0.1437–0.3343–0.24110.8883–0.44791.29580.4–0.1949–0.6435–0.17940.4043–0.34860.75360.5–0.2665–0.7687–0.15270.1542–0.28920.46180.6–0.3462–0.8171–0.14500.0108–0.25240.28680.7–0.4288–0.8295–0.1487–0.0768–0.22970.17410.8–0.5115–0.8237–0.1594–0.1325–0.21640.09790.9–0.5932–0.8087-0.1745–0.1688–0.20940.04451.0–0.6731–0.7890–0.1927–0.1927–0.20700.00612.0–1.3544–0.5839–0.4158–0.2227–0.2774–0.10223.0–1.8675–0.4529–0.6228–0.1908–0.3823–0.10354.0–2.2757–0.3690–0.7990–0.1627–0.4812–0.09395.0–2.6142–0.3113–0.9503–0.1408–0.5701–0.0840Figure 5
Shannon entropy curves for GEUD.


## 6. Estimation of Parameters

This section considers three commonly used techniques to estimate the parametersλ 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 samplex1,x2,x3,…,xn from GEUD is(49)logL=nlogλθ+θ−1∑i=1nlogxi−2∑i=1nlog1−xiθ+∑i=1nlogexp−λxiθ1−xiθ.The equations used for the estimation ofθ,λ are(50)∂logL∂λ=nλ−∑i=1nxiθ1−xiθ.∂logL∂θ=nθ−∑i=1nlogxi+2∑i=1nlogxixiθ1−xiθ−λ∑i=1nlogxixiθxiθ−12.Setting these two equations to zero, we get the MLEs ofθ,λ.


### 6.2. Anderson–Darling Minimum Distance Estimates of GEUD Parameters

The Anderson–Darling estimates (ADEs) ofθ and λ can be obtained by minimizing the function(51)ADEθ,λ=−n−1n∑i=1n2i−1logFxi:n;θ,λ+log1−Fxn+1−i:n;θ,λ.These estimators can be obtained from the following equations by taking the derivatives of the above expression with respect toλ and θ and equating them to zero.(52)∑i=1n2i−1η1xi:n;θ,λFxi:n;θ,λ−η1xn+1−i:n;θ,λF¯xn+1−i:n;θ,λ=0and(53)∑i=1n2i−1η2xi:n;θ,λFxi:n;θ,λ−η2xn+1−i:n;θ,λF¯xn+1−i:n;θ,λ=0,where(54)η1xi:n;θ,λ=λxθlogx1−xθ2exp−λxθ1−xθand (55)η2xi:n;θ,λ=xθ1−xθexp−λxθ1−xθ.We can estimate the parametersθ,λ from the previous equations and so the estimate of ADEθ,λ.


### 6.3. Cramèr–von Mises Minimum Distance Estimates of GEUD Parameters

The Cramèr–von Mises minimum distance is given by the expression(56)CVMEθ,λ=112n+∑i=1nFxi:n;θ,λ−2i−12n2.For the estimation of Cramèr–von Mises minimum distance ofθ,λ, we solve the nonlinear equations(57)∑i=1nFxi:n;θ,λ−2i−12nη1xi:n;θ,λ=0and. ∑i=1nFxi:n;θ,λ−2i−1/2nη2xi:n;θ,λ=0.Here, η1xi:n;θ,λ and η2xi:n;θ,λ are defined in equations (21)–(22). The estimate of CVMEθλ 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 sizen 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 Table4, we provide the estimates of GEUD parameters along with their MSEs by each method. We have provided the efficiencies MSEADE/MSEMLE and MSECVME/MSEMLE of MLEs relative to estimates by other methods.Table 4
Estimates of GEUD parameters, MSEs, and relative efficiencies.
Para.nEst.MLEAdeCvmeMSEADE/MSEMLEMSECVME/MSEMLEEst.MSEEst.MSEEst.MSEλ=0.5θ=0.520λ0.78950.48110.65230.38890.91391.72880.80843.5934θ0.65520.14410.55670.13160.66360.26400.91331.832150λ0.60100.08680.54730.08190.62860.18050.94352.0795θ0.55940.04030.51940.04280.56510.07601.06201.8859100λ0.54630.03150.52730.03590.56090.06311.13972.0032θ0.52840.01720.51260.02120.53410.03301.23261.9186200λ0.52180.01360.51160.01670.52890.02631.22791.9338θ0.51340.00800.50440.01020.51580.01551.27501.9375λ=0.5θ=1.020λ0.79680.51110.63990.36710.94751.85720.71833.6337θ1.30760.56971.10600.51001.35781.13190.89521.986850λ0.59870.08140.55160.08220.62810.17591.00982.1609θ1.11670.15461.04660.17401.13030.29581.12551.9133100λ0.54620.03060.52760.03610.55940.06171.17972.0163θ1.05600.06701.02660.08381.06360.13081.25071.9522200λ0.52390.01360.51020.01660.52930.02581.22061.8971θ1.03030.03221.00800.04151.03200.06061.28881.8820λ=0.5θ=1.520λ0.80500.57900.66810.44590.92151.67180.77012.8874θ1.98131.34991.69511.24522.01042.48260.92241.839150λ0.59870.08370.55110.08580.62400.17681.02512.1123θ1.67610.35771.56380.39921.68620.66821.11601.8680100λ0.54860.03100.52680.03620.56040.06241.16772.0129θ1.59260.15181.53480.18641.59760.29791.22791.9625200λ0.52320.01340.51190.01640.52630.02511.22391.8731θ1.54220.07091.51620.09171.54350.13551.29341.9111λ=1.5θ=0.520λ2.14992.75811.88822.75422.56185.54400.99862.0101θ0.57670.04840.52940.04660.58430.08380.96281.731450λ1.73250.51011.63110.58761.78500.94691.15191.8563θ0.53240.01560.51150.01650.53190.02441.05771.5641100λ1.60430.18731.55910.19871.61990.30221.06091.6135θ0.51450.00700.50470.00780.51320.01071.11431.5286200λ1.54900.07811.53300.09061.55140.12811.16011.6402θ0.50700.00330.50340.00380.50550.00511.15151.5455λ=1.5θ=1.020λ2.18423.20191.86612.18212.50915.41800.68151.6921θ1.15870.20481.04930.18061.17010.31320.88181.529350λ1.72250.46581.62170.47301.77790.88541.01551.9008θ1.05980.05861.01910.06481.06280.09721.10581.6587100λ1.60140.18281.55860.19611.12490.31511.07281.7237θ1.02880.02791.00990.03101.02780.04351.11111.5591200λ1.54580.08051.52880.08951.55780.12571.11181.5615θ1.01280.01341.00500.01521.01380.02011.13431.5000λ=1.5θ=1.520λ2.15272.90651.89112.44092.564329.6570.839810.204θ1.72170.44021.59420.43001.75230.71730.97681.629550λ1.71510.47791.62010.48431.75150.81451.01341.7043θ1.58680.13751.52890.14361.58300.20541.04441.4938100λ1.59830.17881.56240.19971.63340.31131.11691.7411θ1.53950.06121.51750.07071.54860.09621.15521.5719200λ1.55310.08261.52650.09031.55950.12891.09321.5605θ1.52350.03081.50660.03511.52030.04611.13961.4968Table4 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.


## 6.1. Maximum Likelihood Estimates of GEUD Parameters

The log-likelihood function of the samplex1,x2,x3,…,xn from GEUD is(49)logL=nlogλθ+θ−1∑i=1nlogxi−2∑i=1nlog1−xiθ+∑i=1nlogexp−λxiθ1−xiθ.The equations used for the estimation ofθ,λ are(50)∂logL∂λ=nλ−∑i=1nxiθ1−xiθ.∂logL∂θ=nθ−∑i=1nlogxi+2∑i=1nlogxixiθ1−xiθ−λ∑i=1nlogxixiθxiθ−12.Setting these two equations to zero, we get the MLEs ofθ,λ.


## 6.2. Anderson–Darling Minimum Distance Estimates of GEUD Parameters

The Anderson–Darling estimates (ADEs) ofθ and λ can be obtained by minimizing the function(51)ADEθ,λ=−n−1n∑i=1n2i−1logFxi:n;θ,λ+log1−Fxn+1−i:n;θ,λ.These estimators can be obtained from the following equations by taking the derivatives of the above expression with respect toλ and θ and equating them to zero.(52)∑i=1n2i−1η1xi:n;θ,λFxi:n;θ,λ−η1xn+1−i:n;θ,λF¯xn+1−i:n;θ,λ=0and(53)∑i=1n2i−1η2xi:n;θ,λFxi:n;θ,λ−η2xn+1−i:n;θ,λF¯xn+1−i:n;θ,λ=0,where(54)η1xi:n;θ,λ=λxθlogx1−xθ2exp−λxθ1−xθand (55)η2xi:n;θ,λ=xθ1−xθexp−λxθ1−xθ.We can estimate the parametersθ,λ from the previous equations and so the estimate of ADEθ,λ.


## 6.3. Cramèr–von Mises Minimum Distance Estimates of GEUD Parameters

The Cramèr–von Mises minimum distance is given by the expression(56)CVMEθ,λ=112n+∑i=1nFxi:n;θ,λ−2i−12n2.For the estimation of Cramèr–von Mises minimum distance ofθ,λ, we solve the nonlinear equations(57)∑i=1nFxi:n;θ,λ−2i−12nη1xi:n;θ,λ=0and. ∑i=1nFxi:n;θ,λ−2i−1/2nη2xi:n;θ,λ=0.Here, η1xi:n;θ,λ and η2xi:n;θ,λ are defined in equations (21)–(22). The estimate of CVMEθλ 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 sizen 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 Table4, we provide the estimates of GEUD parameters along with their MSEs by each method. We have provided the efficiencies MSEADE/MSEMLE and MSECVME/MSEMLE of MLEs relative to estimates by other methods.Table 4
Estimates of GEUD parameters, MSEs, and relative efficiencies.
Para.nEst.MLEAdeCvmeMSEADE/MSEMLEMSECVME/MSEMLEEst.MSEEst.MSEEst.MSEλ=0.5θ=0.520λ0.78950.48110.65230.38890.91391.72880.80843.5934θ0.65520.14410.55670.13160.66360.26400.91331.832150λ0.60100.08680.54730.08190.62860.18050.94352.0795θ0.55940.04030.51940.04280.56510.07601.06201.8859100λ0.54630.03150.52730.03590.56090.06311.13972.0032θ0.52840.01720.51260.02120.53410.03301.23261.9186200λ0.52180.01360.51160.01670.52890.02631.22791.9338θ0.51340.00800.50440.01020.51580.01551.27501.9375λ=0.5θ=1.020λ0.79680.51110.63990.36710.94751.85720.71833.6337θ1.30760.56971.10600.51001.35781.13190.89521.986850λ0.59870.08140.55160.08220.62810.17591.00982.1609θ1.11670.15461.04660.17401.13030.29581.12551.9133100λ0.54620.03060.52760.03610.55940.06171.17972.0163θ1.05600.06701.02660.08381.06360.13081.25071.9522200λ0.52390.01360.51020.01660.52930.02581.22061.8971θ1.03030.03221.00800.04151.03200.06061.28881.8820λ=0.5θ=1.520λ0.80500.57900.66810.44590.92151.67180.77012.8874θ1.98131.34991.69511.24522.01042.48260.92241.839150λ0.59870.08370.55110.08580.62400.17681.02512.1123θ1.67610.35771.56380.39921.68620.66821.11601.8680100λ0.54860.03100.52680.03620.56040.06241.16772.0129θ1.59260.15181.53480.18641.59760.29791.22791.9625200λ0.52320.01340.51190.01640.52630.02511.22391.8731θ1.54220.07091.51620.09171.54350.13551.29341.9111λ=1.5θ=0.520λ2.14992.75811.88822.75422.56185.54400.99862.0101θ0.57670.04840.52940.04660.58430.08380.96281.731450λ1.73250.51011.63110.58761.78500.94691.15191.8563θ0.53240.01560.51150.01650.53190.02441.05771.5641100λ1.60430.18731.55910.19871.61990.30221.06091.6135θ0.51450.00700.50470.00780.51320.01071.11431.5286200λ1.54900.07811.53300.09061.55140.12811.16011.6402θ0.50700.00330.50340.00380.50550.00511.15151.5455λ=1.5θ=1.020λ2.18423.20191.86612.18212.50915.41800.68151.6921θ1.15870.20481.04930.18061.17010.31320.88181.529350λ1.72250.46581.62170.47301.77790.88541.01551.9008θ1.05980.05861.01910.06481.06280.09721.10581.6587100λ1.60140.18281.55860.19611.12490.31511.07281.7237θ1.02880.02791.00990.03101.02780.04351.11111.5591200λ1.54580.08051.52880.08951.55780.12571.11181.5615θ1.01280.01341.00500.01521.01380.02011.13431.5000λ=1.5θ=1.520λ2.15272.90651.89112.44092.564329.6570.839810.204θ1.72170.44021.59420.43001.75230.71730.97681.629550λ1.71510.47791.62010.48431.75150.81451.01341.7043θ1.58680.13751.52890.14361.58300.20541.04441.4938100λ1.59830.17881.56240.19971.63340.31131.11691.7411θ1.53950.06121.51750.07071.54860.09621.15521.5719200λ1.55310.08261.52650.09031.55950.12891.09321.5605θ1.52350.03081.50660.03511.52030.04611.13961.4968Table4 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 Section6. 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=−2L+2k and Bayesian Information Criterion BIC =  −2L+klogn.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 Section5. Also, the Kolmogorov–Smirnov (KS) test,(58)KS=maxin−Fxi;θ^,λ^,Fxi;θ^,λ^−i−1n,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.Figure 6
The empirical pdf and cdf plots for the first dataset.Figure 7
The empirical pdf and cdf plots for the second dataset.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 Table5.Table 5
Summary of descriptive measures of first and second datasets.
DatasetMin.Max.MeanSdCSCKI (n = 153)0.01800.22400.110750.059960.229111.98291II (n = 50)0.02000.32000.163200.081050.072332.21664The maximum likelihood estimates (standard errors) and measures of model selection criteria for the both datasets are presented in Table6 and 7, respectively.Table 6
MLEs of the parameters, their standard errors, and goodness-of-fit measures (first dataset).
ModelEstimatesLogLAICBICAWKSGEUDλ,θ8.1254 (1.9302)2.0103 (0.1951)74.670−145.34−139.730.7943 [0.4849]0.1118 [0.5307]0.0637 [0.7050]BDα,β2.6434 (0.32066.5294 (0.8377)70.742−137.48−131.881.4367 [0.1924]0.2154 [0.2396]0.0891 [0.2874]UWDα,β0.3627 (0.0481)2.2359 (0.1417)57.768−111.54−105.933.9269 [0.0095]0.6242 [0.0194]0.1269 [0.0392]UBIIIDα,β0.1466 (0.0331)6.4949 (1.3677)67.384−130.77−125.163.6546 [0.0129]0.7871 [0.0077]0.1598 [0.0039]UGomDα,β0.3748 (0.09990.8018 (0.0951)36.802−69.604−63.9968.4247 [0.0000]1.4241 [0.0002]0.1915 [0.0003]Table 7
MLEs of the parameters, their standard errors, and goodness-of-fit measures (second dataset).
ModelEstimatesLogLAICBICAWKSGEUDλ,θ29.976 (13.371)2.0313 (0.2652)56.251−108.50−104.680.6461 [0.6047]0.1016 [0.5788]0.1104 [0.5758]UJSUμ,σ3.7875 (0.4046)2.9152 (0.2917)56.394−108.78−104.960.5914 [∗]0.0970 [∗]0.1121 [0.5555]BDα,β2.6827 (0.5072)13.867 (2.8282)54.606−105.21−101.390.9124 [0.4061]0.1539 [0.3787]0.1415 [0.2695]UWDα,β0.0876 (0.0287)3.0518 (0.3079)48.663−93.325−89.5011.8124 [0.1170]0.3063 [0.1296]0.1819 [0.0731]UBIIIDα,β0.0050 (0.0024)109.84 (54.716)28.982−53.965−50.1418.5722 [0.0000]1.7080 [0.0000]0.3600 [0.0000]UGomDα,β0.0925 (0.0405)1.0747 (0.14194)40.672−77.344−73.5193.0812 [0.0251]0.5213 [0.0347]0.2046 [0.0304]∗Not mentioned by the authors.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 JohnsonSU 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 is(59)fx,μ,σ=σx1−x1+logx/1−x2Φμ+σsinh−1logx1−x,0<x<1,is 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.



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