# A New Double Truncated Generalized Gamma Model with Some Applications

Authors: Awad A. Bakery; Wael Zakaria; OM Kalthum S. K. Mohamed
Journal: Journal of Mathematics
 (2021)

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## Abstract

The generalized Gamma model has been applied in a variety of research fields, including reliability engineering and lifetime analysis. Indeed, we know that, from the above, it is unbounded. Data have a bounded service area in a variety of applications. A new five-parameter bounded generalized Gamma model, the bounded Weibull model with four parameters, the bounded Gamma model with four parameters, the bounded generalized Gaussian model with three parameters, the bounded exponential model with three parameters, and the bounded Rayleigh model with two parameters, is presented in this paper as a special case. This approach to the problem, which utilizes a bounded support area, allows for a great deal of versatility in fitting various shapes of observed data. Numerous properties of the proposed distribution have been deduced, including explicit expressions for the moments, quantiles, mode, moment generating function, mean variance, mean residual lifespan, and entropies, skewness, kurtosis, hazard function, survival function,rth order statistic, and median distributions. The delivery has hazard frequencies that are monotonically increasing or declining, bathtub-shaped, or upside-down bathtub-shaped. We use the Newton Raphson approach to approximate model parameters that increase the log-likelihood function and some of the parameters have a closed iterative structure. Six actual data sets and six simulated data sets were tested to demonstrate how the proposed model works in reality. We illustrate why the Model is more stable and less affected by sample size. Additionally, the suggested model for wavelet histogram fitting of images and sounds is very accurate.


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## Body

## 1. Introduction

The gamma (ΓM) model, including Weibull, gamma, exponential, and Rayleigh as special submodels, among others, is a very popular distribution for modeling lifetime data and for modeling phenomenon with monotone failure rates. An advantage of ΓM is that it requires a little measure of parameters for learning. Also, these parameters can be measured by getting the expectation maximization (EM) algorithm [1, 2] to maximize the log-likelihood function. The early generalization of gamma distribution can be traced back to Amoroso [3] who discussed a generalized gamma distribution and applied it to fit income rates. Johnson et al. [4] gave a four parameter generalized gamma distribution which reduces to the generalized gamma distribution defined by Stacy [2] when the location parameter is set to zero. Mudholkar and Srivastava [5] introduced the exponentiated method to derive a distribution. The generalized gamma defined by Stacy [2] is a three-parameter exponentiated gamma distribution. Agarwal and Al-Saleh [6] applied generalized gamma to study hazard rates. Balakrishnan and Peng [7] applied this distribution to develop generalized gamma frailty model. Cordeiro et al. [8] derived another generalization of Stacys generalized gamma distribution using exponentiated method and applied it to life time and survival analysis. Nadarajah and Gupta [9] proposed another type of generalized gamma distribution with application to fit drought data. As of late, Chen et al. [10] used generalized gamma distribution with three parameters for flood frequency analysis, Zhao et al. [11] used generalized gamma distribution with three parameters to give the statistical characterizes of high-resolution SAR images, and Mead et al. [12] defined modified generalized gamma distribution so as to investigate greater flexibility in modeling data from a practical viewpoint and they derived multifarious identities and properties of this distribution, including explicit expressions for the moments, quantiles, mode, moment generating function, mean deviation, mean residual lifetime, and expression of the entropies. We extend all the past models with five parameters to range ℝ (real numbers) or any bounded subset of ℝ. Fulger et al. [13] generate random numbers within any arbitrary interval. We introduce in this paper the high flexibility of a bounded generalized Gamma model with five parameters (BGΓM) for analyzing data. The BGΓM Model is of noticeable significance for image coding, compression applications, sound system, wind speed data, and breast cancer data fitting. This new distribution has a flexibility to fit any kind of observed data whose pdf is monotonically increasing, decreasing, bathtub, and upside down bathtub-shaped depending on the parameter values and bounded support regions. The remainder of this paper is organized as follows: The BGΓM with its sub models and some shapes describe the hazard rate function are defined in Section 2. Some properties of the BGΓM distribution are studied in Section 3 including, quantile, mode, moments, moment generating function, mean deviation, mean residual life and entropy. Section 4 presents the parameter estimation. Section 5 sets out the experimental results. Section 6 presents our conclusions.


## 2. The Bounded Generalized Gamma Model and Its Special Models

The standard form of gamma function is(1)Γη=∫0∞xη−1e−xdx,η>0.The incomplete gamma function is defined by(2)Γη,s=1Γη∫0sxη−1e−xdx,η>0ands≥0.The probability density function (pdf) of the generalized gamma distribution is given by(3)Tx|Θ=λβη2δΓηx−uδηλ−1e−βx−u/δλ,for all x∈ℝ, where Θ=u,δ,β,η,λ′, δ,η,λ,β>0 and u∈ℝ. The cumulative distribution function (cdf) of generalized gamma distribution defined as follows:(4)Dz=∫−∞zTx|Θdx=∫−∞zλβη2δΓηx−uδηλ−1e−βx−u/δλdx=12+12signz−uΓη,βz−uδλ−Γη,βuδλ.LetΩ=a,b⊆ℝ and we denote the indicator function by(5)1Ωx=1,ifx∈Ω,0,if otherwise.We define the pdf of the bounded generalized gamma distribution (BGΓM) as(6)Υ=Υx|Θ=Tx|Θ∫ΩTx|Θdx=λβη/2δΓηx−u/δηλ−1e−βx−u/δλ∫ΩTx|Θdx,for allx∈Ω.In another form, we can write the pdf of the bounded generalized gamma distribution (BGΓM) as(7)Υ=λβη/2δΓηx−u/δηλ−1e−βx−u/δλDb−Da,where(8)Db−Da=12signb−uΓη,βb−uδλ−Γη,βuδλ−12signa−uΓη,βa−uδλ−Γη,βuδλ.It is clear to see that(9)Υ≥0and∫ΩΥx|u,δ,β,η,λdx=1.Hence, the cdf of the bounded generalized gamma distribution (BGΓM) is given by(10)Φx=Dx−DaDb−Da=signx−uΓη,βx−u/δλ−Γη,βu/δλ−signa−uΓη,βa−u/δλ−Γη,βu/δλsignb−uΓη,βb−u/δλ−Γη,βu/δλ−signa−uΓη,βa−u/δλ−Γη,βu/δλ.The parametersuδ,β and η,λ are corresponding to the location, scale, and shape parameters, respectively. Note that Υx|Θ can be any kind of distribution, for example, in exponential distribution (ED) [14, 15] be ϕx|u,δ,β, Weibull distribution (WD) [16–18] be Tx|u,δ,β,λ, Rayleigh distribution (RD) [19, 20] be Tx|u,δ, generalized Gaussian distribution (GGD) [21] be Tx|u,δ,λ, Gaussian distribution (GD) [15] be Tx|u,δ, Laplacian distribution (LD) [22] be Tx|u,δ and Gamma distribution (ΓD) [1] be Tx|u,δ,η,β. These distributions are all unbounded with support range 0,∞. We extend all the past models with range −∞,∞ also to the bounded case. The BGΓM has several models as special cases, which makes it distinguishable scientific importance from other models. We investigate the various special models of the BGΓM as listed in Table 1. The survival function and hazard rate function for BGΓM are, respectively, given by(11)Sx=1−Φx=signb−uΓη,βb−u/δλ−Γη,βu/δλ−signx−uΓη,βx−u/δλ−Γη,βu/δλsignb−uΓη,βb−u/δλ−Γη,βu/δλ−signa−uΓη,βa−u/δλ−Γη,βu/δλ,∧x=Υx|ΘSx=λβη/2δΓηx−u/δηλ−1e−βx−u/δλsignb−uΓη,βb−u/δλ−Γη,βu/δλ−signx−uΓη,βx−u/δλ−Γη,βu/δλ.Table 1
The comparative models are special cases of the BGΓM.
GΓMBGΓM: Ω=ℝBΓMBGΓM: λ=1ΓMBGΓM: Ω=ℝ and λ=1BWMBGΓM: η=1WMBGΓM: Ω=ℝ, η=1BGGMBGΓM: η=1/λ and β=Γ3/λ/Γ1/λλ/2GGMBGΓM: Ω=ℝ, η=1/λ and β=Γ3/λ/Γ1/λλ/2BEMBGΓM: λ=1, and η=1EMBGΓM: Ω=ℝ, λ=1, and η=1BGMBGΓM: λ=2, η=0.5 and β=0.5GMBGΓM: Ω=ℝ, λ=2, η=0.5 and β=0.5BRMBGΓM: λ=2, η=1 and β=0.5RMBGΓM: Ω=ℝ, λ=2, η=1 and β=0.5BLMBGΓM: λ=1, η=1 and β=2LMBGΓM: Ω=ℝ, λ=1, η=1 and β=2In Figures1 and 2, we display the plots of the pdf of BGΓM for various parameters. Figure 3 displays the BGΓM failure rate function which can be increasing, decreasing, bathtub, and upside down bathtub-shaped depending on the parameter values.Figure 1
The pdf of BGΓM for a=−2, b=3, u=1, δ=1, β=1 and (a) η=7 and λ=2,4,7; (b) λ=7 and η=2,4,9; (c) η=0.5,λ=2, η=2,λ=0.6 and η=7,λ=5.
(a)(b)(c)Figure 2
The pdf of BGΓM for a=−2, b=3, u=1, δ=1, λ=2 and (a) β=2 and η=0.5,5,7; (b) η=3 and β=0.5,1,10; (c)η=0.5,β=0.5, η=0.5,β=3, η=3,β=0.5 and η=4,β=4.
(a)(b)(c)Figure 3
The hazard plots of BGΓM for a=−2, b=3, u=1, δ=1 and different values of β, η and λ.


## 3. Properties of BGΓM

In this section, we provide some general properties of the BGΓM including quantile function, mode, moments, mean deviation, mean residual life and mean waiting time, Rényi entropy, and order statistics.


### 3.1. Mode and Quantile

Thepth quantile function of the BGΓM is the solution of(12)Φxp=Dxp−DaDb−Da=p⇒signxp−uΓη,βxp−u/δλ−Γη,βu/δλ−signa−uΓη,βa−u/δλ−Γη,βu/δλsignb−uΓη,βb−u/δλ−Γη,βu/δλ−signa−uΓη,βa−u/δλ−Γη,βu/δλ=p.The median, denoted byμ∗, can be obtained by substituting p=0.5 in 10 and solving the equation(13)signμ∗−uΓη,βμ∗−u/δλ−Γη,βu/δλ−signa−uΓη,βa−u/δλ−Γη,βu/δλsignb−uΓη,βb−u/δλ−Γη,βu/δλ−signa−uΓη,βa−u/δλ−Γη,βu/δλ=0.5.The mode, denoted byxm of the BGΓ distribution, is given by(14)xm=u±δηλ−1βλ1/λ,such thatΥ″xm|Θ<0.Remark 1.
(1)
Ifηλ=1, then the BGΓdistribution is unimodal distribution(2)
Ifηλ>1, then the BGΓdistribution is multimodal distribution


### 3.2. Moments, Generating Function, and Mean Deviation

Therth moment about zero of BGΓ distribution is(15)Exr=∫abxrΥx|Θdx=±∑n=0rrnun±δ/β1/λr−nΓη+r−n/λΓη+r−n/λ,βb−u/δλ−Γη+r−n/λ,βa−u/δλ2ΓηDb−Da.The meanμ of the BGΓ distribution is given by(16)μ=Ex=±12ΓηDb−Da±δβ1/λΓη+1λΓη+1λ,βb−uδλ−Γη+1λ,βa−uδλ+uΓηΓη,βb−uδλ−Γη,βa−uδλ.The variance σ2 of the BGΓ distribution is given by(17)σ2=Ex2−E2x=±12ΓηDb−Daδ2β2/λΓη+2λΓη+2λ,βb−uδλ−Γη+2λ,βa−uδλ±2uδβ1/λΓη+1λΓη+1λ,βb−uδλ−Γη+1λ,βa−uδλ+u2ΓηΓη,βb−uδλ−Γη,βa−uδλ−μ2.The central moments of BGΓ distribution can be obtained as follows(18)Ex−μr=∫abx−μrΥx|Θdx=±12ΓηDb−Da∑n=0rrnun±δβ1/λr−nΓη+r−nλΓη+r−nλ,βb−uδλ−Γη+r−nλ,βa−uδλ∑j=0rrj−μr−j,γ1=Ex−μ3σ3,γ2=Ex−μ4σ4.The moment generating functionMxt of BGΓ distribution is(19)Mxt=∫abetxΥx|Θdx=±∑r=0∞tr/r!∑n=0rrnun±δ/β1/λr−nΓη+r−n/λΓη+r−n/λ,βb−u/δλ−Γη+r−n/λ,βa−u/δλ2ΓηDb−Da.The mean deviationMd of BGΓ distribution can be derived as(20)Md=Ex−μ=∫abx−μΥx|Θdx=2μΦμ−∫aμxΥx|Θdx=2μsignμ−uΓη,βμ−u/δλ−Γη,βu/δλ−signa−uΓη,βa−u/δλ−Γη,βu/δλsignb−uΓη,βb−u/δλ−Γη,βu/δλ−signa−uΓη,βa−u/δλ−Γη,βu/δλ−1ΓηDb−Daδβ1/λΓη+1λΓη+1λ,βμ−uδλ−Γη+1λ,βa−uδλΓηΓη,βμ−uδλ−Γη,βa−uδλ.In Table2, the Median, Mode, Mean, Variance, Skewness, and Kurtosis of BGΓM have given for a=−2, b=3, u=1, δ=1, and β=1 and various values of η and λ. From Table 2, we note that for fixed values of a,b,u,δ,β, and η, the Kurtosis is decreasing function of λ. Also, for fixed values of a,b,u,δ,β, and λ, the Mode 1, Variance, and Skewness are increasing function and the Mode 2 and Mean are decreasing function of η. In Table 3, Median, Mode, Mean, Variance, Skewness, and Kurtosis of BGΓM have given for a=−2, b=3, u=1, δ=1, and λ=2 and various values of η and β. From Table 3, we note that for fixed values of a,b,u,δ,λ, and η, Mode 1 is decreasing, Median, Mode 2, and Mean are increasing functions of β. Also, for fixed values of a,b,u,δ,β, and β, Mode 1 and Skewness are increasing and Mode 2 and Mean are decreasing functions of η.Table 2
Median, Mode, Mean, Variance, Skewness, and Kurtosis of BGΓM.
ηλMedianMode1Mode2MeanVarianceSkewnessKurtosis0.520.9979110.99490.4895−0.05252.88570.541.00371.70710.292910.564201.57080.570.99091.86320.136810.666101.2251220.42072.2248−0.22480.89321.82970.01171.4184240.80712.1502−0.150211.329301.13176489270.92442.0925−0.092411.156801.0458178772−1.38173.5495−1.5495−0.90092.01852.06495.707874−0.14532.6119−0.61190.99592.59580.00351.0358770.97852.3166−0.316611.718101.0121Table 3
Median, Mode, Mean, Variance, Skewness, and Kurtosis of BGΓM.
ηβMedianMode1Mode2MeanVarianceSkewnessKurtosis0.50.50.9732110.94920.8732−0.18242.64720.521110.99990.2499−0.0022.9930.5711110.07140320.5−0.22272.7321−0.73210.45992.70850.29471.4377220.83271.8660.1340.99680.9946−0.00521.4863271.0011.46290.537110.285701.570.5−1.7224.6056−2.6056−1.5470.58864.873728.032672−0.4832.8028−0.80280.58543.03440.32591.2304771.08551.96360.03641101.1429


### 3.3. Mean Residual Life and Mean Waiting Time

The mean residual life function, sayφt, is given by(21)φt=Eτ−t|τ>t=1St∫tbxΥx|Θdx−t=signb−uΓη,βb−u/δλ−Γη,βu/δλ−signa−uΓη,βa−u/δλ−Γη,βu/δλsignb−uΓη,βb−u/δλ−Γη,βu/δλ−signt−uΓη,βt−u/δλ−Γη,βu/δλ×12ΓηDb−Daδβ1/λΓη+1λΓη+1λ,βb−uδλ−Γη+1λ,βt−uδλ±uΓηΓη,βb−uδλ−Γη,βt−uδλ−t.The mean waiting time of BGΓ distribution, say φ¯t, can be derived as(22)φ¯t=t−∫atxΥx|ΘdxΦt=t−12ΓηDb−Daδβ1/λΓη+1λΓη+1λ,βt−uδλ−Γη+1λ,βa−uδλ±uΓηΓη,βt−uδλ−Γη,βa−uδλ×signb−uΓη,βb−u/δλ−Γη,βu/δλ−signa−uΓη,βa−u/δλ−Γη,βu/δλsignt−uΓη,βt−u/δλ−Γη,βu/δλ−signa−uΓη,βa−u/δλ−Γη,βu/δλ.


### 3.4. Entropy

The entropy of a random variableX measures the variation of the uncertainty. The Rényi entropy of BGΓ distribution, say REXν for ν≠1 and ν>0, is derived as(23)REXν=ln∫abxΥνx|Θdx1−ν=−lnλ−νln21−ν+lnδ−νlnΓη1−ν−νlnDb−Da1−ν−lnβλ−1λ+νη1−νlnν+11−νlnΓνη−ν−1λ+11−νln±Γνη−ν−1λ,βνb−uδλ∓Γνη−ν−1λ,βνa−uδλ.


### 3.5. Order Statistics

LetX1:n,X2:n,…,Xn:n denote the order statistics obtained from a random sample of size n from BGΓ distribution. The probability density function of ith order statistics is given by(24)fi:nx=n!i−1!n−i!Φxi−11−Φxn−iΥx|Θ=n!λβηx−u/δηλ−1e−βx−u/δλDx−Dai−1Db−Dxn−i2i−1!n−i!δΓηDb−Dan.The pdf of the minimum and the maximum order statistics of BGΓ distribution can be obtained, respectively, as follows:(25)f1:nx=nλβηx−u/δηλ−1e−βx−u/δλDb−Dxn−12δΓηDb−Dan,fn:nx=nλβηx−u/δηλ−1e−βx−u/δλDx−Dan−12δΓηDb−Dan.Ifn is odd. The pdf of BGΓ distribution of the median is obtained by substituting i=n+1/2 in equation (24) as follows:(26)fn+1/2:nx=n!λβηx−u/δηλ−1e−βx−u/δλDx−Dan−1/2Db−Dxn−1/22n−1/2!2δΓηDb−Dan.The joint pdf of theith and the lth order statistics for x<y can be written as(27)fi,l:nx=n!i−1!l−i−1!n−l!Φxi−11−Φyn−lΦy−Φxl−i−1Υx|ΘΥy|Θ.So the joint pdf of the ith and the lth order statistics of BGΓ distribution is(28)fi,l:nx=n!λ2β2ηDx−Dai−1Db−Dyn−lDy−Dxl−i−14δ2Γ2ηDb−Dani−1!l−i−1!n−l!×x−uy−uδ2ηλ−1e−βx−u/δλ+y−u/δλ.


## 3.1. Mode and Quantile

Thepth quantile function of the BGΓM is the solution of(12)Φxp=Dxp−DaDb−Da=p⇒signxp−uΓη,βxp−u/δλ−Γη,βu/δλ−signa−uΓη,βa−u/δλ−Γη,βu/δλsignb−uΓη,βb−u/δλ−Γη,βu/δλ−signa−uΓη,βa−u/δλ−Γη,βu/δλ=p.The median, denoted byμ∗, can be obtained by substituting p=0.5 in 10 and solving the equation(13)signμ∗−uΓη,βμ∗−u/δλ−Γη,βu/δλ−signa−uΓη,βa−u/δλ−Γη,βu/δλsignb−uΓη,βb−u/δλ−Γη,βu/δλ−signa−uΓη,βa−u/δλ−Γη,βu/δλ=0.5.The mode, denoted byxm of the BGΓ distribution, is given by(14)xm=u±δηλ−1βλ1/λ,such thatΥ″xm|Θ<0.Remark 1.
(1)
Ifηλ=1, then the BGΓdistribution is unimodal distribution(2)
Ifηλ>1, then the BGΓdistribution is multimodal distribution


## 3.2. Moments, Generating Function, and Mean Deviation

Therth moment about zero of BGΓ distribution is(15)Exr=∫abxrΥx|Θdx=±∑n=0rrnun±δ/β1/λr−nΓη+r−n/λΓη+r−n/λ,βb−u/δλ−Γη+r−n/λ,βa−u/δλ2ΓηDb−Da.The meanμ of the BGΓ distribution is given by(16)μ=Ex=±12ΓηDb−Da±δβ1/λΓη+1λΓη+1λ,βb−uδλ−Γη+1λ,βa−uδλ+uΓηΓη,βb−uδλ−Γη,βa−uδλ.The variance σ2 of the BGΓ distribution is given by(17)σ2=Ex2−E2x=±12ΓηDb−Daδ2β2/λΓη+2λΓη+2λ,βb−uδλ−Γη+2λ,βa−uδλ±2uδβ1/λΓη+1λΓη+1λ,βb−uδλ−Γη+1λ,βa−uδλ+u2ΓηΓη,βb−uδλ−Γη,βa−uδλ−μ2.The central moments of BGΓ distribution can be obtained as follows(18)Ex−μr=∫abx−μrΥx|Θdx=±12ΓηDb−Da∑n=0rrnun±δβ1/λr−nΓη+r−nλΓη+r−nλ,βb−uδλ−Γη+r−nλ,βa−uδλ∑j=0rrj−μr−j,γ1=Ex−μ3σ3,γ2=Ex−μ4σ4.The moment generating functionMxt of BGΓ distribution is(19)Mxt=∫abetxΥx|Θdx=±∑r=0∞tr/r!∑n=0rrnun±δ/β1/λr−nΓη+r−n/λΓη+r−n/λ,βb−u/δλ−Γη+r−n/λ,βa−u/δλ2ΓηDb−Da.The mean deviationMd of BGΓ distribution can be derived as(20)Md=Ex−μ=∫abx−μΥx|Θdx=2μΦμ−∫aμxΥx|Θdx=2μsignμ−uΓη,βμ−u/δλ−Γη,βu/δλ−signa−uΓη,βa−u/δλ−Γη,βu/δλsignb−uΓη,βb−u/δλ−Γη,βu/δλ−signa−uΓη,βa−u/δλ−Γη,βu/δλ−1ΓηDb−Daδβ1/λΓη+1λΓη+1λ,βμ−uδλ−Γη+1λ,βa−uδλΓηΓη,βμ−uδλ−Γη,βa−uδλ.In Table2, the Median, Mode, Mean, Variance, Skewness, and Kurtosis of BGΓM have given for a=−2, b=3, u=1, δ=1, and β=1 and various values of η and λ. From Table 2, we note that for fixed values of a,b,u,δ,β, and η, the Kurtosis is decreasing function of λ. Also, for fixed values of a,b,u,δ,β, and λ, the Mode 1, Variance, and Skewness are increasing function and the Mode 2 and Mean are decreasing function of η. In Table 3, Median, Mode, Mean, Variance, Skewness, and Kurtosis of BGΓM have given for a=−2, b=3, u=1, δ=1, and λ=2 and various values of η and β. From Table 3, we note that for fixed values of a,b,u,δ,λ, and η, Mode 1 is decreasing, Median, Mode 2, and Mean are increasing functions of β. Also, for fixed values of a,b,u,δ,β, and β, Mode 1 and Skewness are increasing and Mode 2 and Mean are decreasing functions of η.Table 2
Median, Mode, Mean, Variance, Skewness, and Kurtosis of BGΓM.
ηλMedianMode1Mode2MeanVarianceSkewnessKurtosis0.520.9979110.99490.4895−0.05252.88570.541.00371.70710.292910.564201.57080.570.99091.86320.136810.666101.2251220.42072.2248−0.22480.89321.82970.01171.4184240.80712.1502−0.150211.329301.13176489270.92442.0925−0.092411.156801.0458178772−1.38173.5495−1.5495−0.90092.01852.06495.707874−0.14532.6119−0.61190.99592.59580.00351.0358770.97852.3166−0.316611.718101.0121Table 3
Median, Mode, Mean, Variance, Skewness, and Kurtosis of BGΓM.
ηβMedianMode1Mode2MeanVarianceSkewnessKurtosis0.50.50.9732110.94920.8732−0.18242.64720.521110.99990.2499−0.0022.9930.5711110.07140320.5−0.22272.7321−0.73210.45992.70850.29471.4377220.83271.8660.1340.99680.9946−0.00521.4863271.0011.46290.537110.285701.570.5−1.7224.6056−2.6056−1.5470.58864.873728.032672−0.4832.8028−0.80280.58543.03440.32591.2304771.08551.96360.03641101.1429


## 3.3. Mean Residual Life and Mean Waiting Time

The mean residual life function, sayφt, is given by(21)φt=Eτ−t|τ>t=1St∫tbxΥx|Θdx−t=signb−uΓη,βb−u/δλ−Γη,βu/δλ−signa−uΓη,βa−u/δλ−Γη,βu/δλsignb−uΓη,βb−u/δλ−Γη,βu/δλ−signt−uΓη,βt−u/δλ−Γη,βu/δλ×12ΓηDb−Daδβ1/λΓη+1λΓη+1λ,βb−uδλ−Γη+1λ,βt−uδλ±uΓηΓη,βb−uδλ−Γη,βt−uδλ−t.The mean waiting time of BGΓ distribution, say φ¯t, can be derived as(22)φ¯t=t−∫atxΥx|ΘdxΦt=t−12ΓηDb−Daδβ1/λΓη+1λΓη+1λ,βt−uδλ−Γη+1λ,βa−uδλ±uΓηΓη,βt−uδλ−Γη,βa−uδλ×signb−uΓη,βb−u/δλ−Γη,βu/δλ−signa−uΓη,βa−u/δλ−Γη,βu/δλsignt−uΓη,βt−u/δλ−Γη,βu/δλ−signa−uΓη,βa−u/δλ−Γη,βu/δλ.


## 3.4. Entropy

The entropy of a random variableX measures the variation of the uncertainty. The Rényi entropy of BGΓ distribution, say REXν for ν≠1 and ν>0, is derived as(23)REXν=ln∫abxΥνx|Θdx1−ν=−lnλ−νln21−ν+lnδ−νlnΓη1−ν−νlnDb−Da1−ν−lnβλ−1λ+νη1−νlnν+11−νlnΓνη−ν−1λ+11−νln±Γνη−ν−1λ,βνb−uδλ∓Γνη−ν−1λ,βνa−uδλ.


## 3.5. Order Statistics

LetX1:n,X2:n,…,Xn:n denote the order statistics obtained from a random sample of size n from BGΓ distribution. The probability density function of ith order statistics is given by(24)fi:nx=n!i−1!n−i!Φxi−11−Φxn−iΥx|Θ=n!λβηx−u/δηλ−1e−βx−u/δλDx−Dai−1Db−Dxn−i2i−1!n−i!δΓηDb−Dan.The pdf of the minimum and the maximum order statistics of BGΓ distribution can be obtained, respectively, as follows:(25)f1:nx=nλβηx−u/δηλ−1e−βx−u/δλDb−Dxn−12δΓηDb−Dan,fn:nx=nλβηx−u/δηλ−1e−βx−u/δλDx−Dan−12δΓηDb−Dan.Ifn is odd. The pdf of BGΓ distribution of the median is obtained by substituting i=n+1/2 in equation (24) as follows:(26)fn+1/2:nx=n!λβηx−u/δηλ−1e−βx−u/δλDx−Dan−1/2Db−Dxn−1/22n−1/2!2δΓηDb−Dan.The joint pdf of theith and the lth order statistics for x<y can be written as(27)fi,l:nx=n!i−1!l−i−1!n−l!Φxi−11−Φyn−lΦy−Φxl−i−1Υx|ΘΥy|Θ.So the joint pdf of the ith and the lth order statistics of BGΓ distribution is(28)fi,l:nx=n!λ2β2ηDx−Dai−1Db−Dyn−lDy−Dxl−i−14δ2Γ2ηDb−Dani−1!l−i−1!n−l!×x−uy−uδ2ηλ−1e−βx−u/δλ+y−u/δλ.


## 4. Maximizing the Log-Likelihood Function

Here, we consider the estimation of the unknown parameters of the BGΓD by the method of maximum likelihood. Let x1,x2,…,xN be a random sample from the BGΓD. The total log-likelihood LΘ is given by(29)LΘ=∑i=1NlnTxi|u,δ,β,η,λ−ln∫ΩTx|u,δ,β,η,λdx.


### 4.1. Location Parameter Estimation

To maximize the likelihood function in (28), we consider the derivation of L with the location u at the t+1 iteration step. We have(30)∂L∂u=∑i=1Nu−xiu−xi−2ηλ−1−βλδλu−xiλ−∫Ωsignu−xiu−xi−1ηλ−1−βλ/δλu−xiλTx|Θdx∫ΩTx|Θdx.At that point as [23], we have(31)∫ΩTx|Θdx≈1M∑i=1M1Ωvi,where vi∼Tx|Θt indicates the random variable that is drawn from the probability distribution Tx|Θt, with Θt=ut,δt,βt,ηt,λt′ and M is the number of random variables vi. We use M=106, for our experiments. In the same manner, we can write(32)∫Ωsignu−xiu−xi−1ηλ−1−βλδλu−xiλTx|Θdx≈1M∑i=1M1Ωvisignut−viut−vi−1ηtλt−1−βtλtδtλtut−viλt.By using (31) and (32), we can rewrite (30) as(33)∂L∂u|u=ut≈∑i=1Nut−xi−2ut−xiηtλt−1−βtλtδtλtut−xiλt−Hut−xi2,where(34)H=∑i=1M1Ωvisignut−viut−vi−1ηtλt−1−βtλt/δtλtut−viλt∑i=1M1Ωvi.According to the theory of robust statistics [24], any estimate u is defined by an implicit equation:(35)∑iFxi−u=0.This gives a numerical solution of the location of u as a weighted mean:(36)u=∑iωixi∑iωi;whereωi=Fxi−uxi−u.Now, we can apply (35) to ∂L/∂u in (33), and the solution of ∂L/∂u=0 gives the solutions of u at the t+1 step:(37)ut+1=∑i=1Nut−xi−2xiηtλt−1−βtλt/δtλtut−xiλt+H∑i=1Nut−xi−2ηtλt−1−βtλt/δtλtut−xiλt.


### 4.2. Scale Parameters Estimation

Putting the derivative of the log-likelihood functionL with respect to the scale parameter δ at the t+1 iteration step, we have(38)∂L∂δ=δ−1∑i=1Nβλδ−λu−xiλ−ηλ−∫Ωβλδ−λu−xiλ−ηλTx|Θdx∫ΩTx|Θdx.Similarly as (31) and (32), we can rewrite ∂L/∂δ as(39)∂L∂δ=δ−1∑i=1Nβλδ−λu−xiλ−ηλ−G,where(40)G=∑i=1Mβtλtδt−λtut−viλt−ηtλt1Ωvi∑i=1M1Ωvi.The solution of∂L/∂δ=0 yields the solutions of δ at the t+1 step:(41)δt+1=βtλt∑i=1Nut−xiλt∑i=1Nηtλt+G1/λt.The next step is to update the estimate of the scale parameterβ. This includes fixing the other parameters and improving the estimate of β by using the Newton Raphson method [25]. Every cycle requires the first and second derivatives of LΘ with respect to the parameter β.(42)βt+1=βt−∂L/∂β∂2L/∂β2+ε|β=βt,where ε is a scaling element. The derivative of the function LΘ regarding β is given by(43)∂L∂β=∑i=1Nfxi,Θ−∫Ωfx,ΘTx|Θdx∫ΩTx|Θdx,where(44)fxi,Θ=1Txi|Θ∂Txi|Θ∂β=ηβ−u−xiδλ.The term∂L/∂β can be approximated as(45)∂L∂β|β=βt≈∑i=1Nfxi,Θt−∑i=1M1Ωvifvi,Θt∑i=1M1Ωvi.The term∂2L/∂β2 is given by(46)∂2L∂β2=∑i=1N∂fx,Θ∂β+∫ΩTx|Θfdx2∫ΩTx|Θdx2−∫Ω∂f/∂β+f2Tx|Θdx∫ΩTx|Θdx,where(47)∂f∂β=−ηβ2.Also the term∂2L/∂β2 can be approximated as(48)∂2L∂β2|β=βt≈∑i=1N−ηtβt2+∑i=1M1Ωvifvi,Θt2∑i=1M1Ωvi2−∑i=1M1Ωvi−ηt/βt2+fvi,Θt2∑i=1M1Ωvi.


### 4.3. Shape Parameters Estimation

For shape parameter estimationη by using the Newton Raphson method, we have(49)ηt+1=ηt−∂L/∂η∂2L/∂η2+ε|η=ηt.The derivative of the functionLΘ with respect to η is given by(50)∂L∂η=∑i=1Ngxi|Θ−∫Ωgx|ΘTx|Θdx∫ΩTx|Θdx,where(51)gxi|Θ=1Txi|Θ∂Txi|Θ∂η=lnβ−Ψη+λlnu−xi−lnδ.The term∂L/∂η can be approximated as(52)∂L∂η|η=ηt≈∑i=1Ngtxi|Θ−∑m=1M1Ωvmgtvm|Θ∑i=1M1Ωvi,where(53)gtxi|Θ=lnβt−Ψηt+λtlnut−xi−lnδt.The calculation of the term∂2L/∂η2 is obtained as(54)∂2L∂η2=∑i=1N∂gxi|Θ∂η+∫ΩTx|Θgx|Θdx2∫ΩTx|Θdx2−∫Ω∂gx|Θ/∂η+g2x|θTx|Θdx∫ΩTx|Θdx,where(55)∂g∂η=−Ψ′η.The term∂2L/∂η2 can be approximated as(56)∂2L∂η2|η=ηt≈∑i=1N∂gxi|Θ∂ηt+∑m=1M1Ωvmgtvm|Θ2∑i=1M1Ωvi2−∑m=1M1Ωvm∂gvm|Θ/∂ηt+gtvm|Θ2∑i=1M1Ωvi,where(57)∂gxi|θ∂ηt=−Ψ′ηt.For shape parameter estimationλ by using the Newton Raphson method, we have(58)λt+1=λt−∂L/∂λ∂2L/∂λ2+ε|λ=λt.The derivative of the function LΘ with respect to λ is given by(59)∂L∂λ=∑i=1Nhxi|Θ−∫Ωhx|ΘTx|Θdx∫ΩTx|Θdx,where(60)hxi|Θ=1Txi|Θ∂Txi|Θ∂λ=1λ+η−βu−xiδλlnu−xiδ.The term∂L/∂λ can be approximated as(61)∂L∂λ|λ=λt≈∑i=1Nhtxi|Θ−∑m=1M1Ωvmhtvm|Θ∑m=1M1Ωvm,where(62)htxi|Θ=1λt+ηt−βtut−xiδtλtlnut−xiδt.The calculation of the term∂2L/∂λ2 is obtained as(63)∂2L∂λ2=∑i=1N∂hxi|Θ∂λ+∫ΩTx|Θhx|Θdx2∫ΩTx|Θdx2N−∫Ω∂hx|Θ/∂λ+h2x|ΘTx|Θdx∫ΩTx|Θdx,where(64)∂h∂λ=−1λ2−βu−xiδλln2u−xiδ.The term∂2L/∂λ2 can be approximated as(65)∂2L∂λ2|λ=λt≈∑i=1N∂hxi|Θ∂λt+∑m=1M1Ωvmhtvm|Θ2∑m=1M1Ωvm2−∑m=1M1Ωvm∂hvm|Θ/∂λt+htvm|Θ2∑m=1M1Ωvm.


### 4.4. Algorithm

To study the stability of our model, we have to find the set of initial points that generate a convergent sequence which called stable points of the dynamical system, i.e., we have to findu0,δ0,β0,η0,λ0 such that limt⟶∞ut,limt⟶∞δt, limt⟶∞βt, limt⟶∞ηt, and limt⟶∞λt exist. Indeed for fixed initial, it is difficult to predict how the approximation sequence behaves; hence, for this purpose, we take a random numbers of initial points until the convergence is verified (two successive approximations of each parameter correct to 4 decimal places). The various steps of the proposed model can be summarized as follows:Step 1: Initialize the parametersΘu,δ,β,η,λ.Step 2: Reestimate the parametersΘu,δ,β,η,λ, where the most common value of scaling parameter ε is 10−30 for our experiments.+Update the parameteru in (37).+Update the parameterδ in (41).+Update the parameterβ in (42).+Update the parameterη in (49).+Update the parameterλ in (58).Step 3: Check the convergence, if∂LΘ/∂ΘiΘ=Θt|<10−4, for all 1≤i≤5 under the constrains ALΘt is negative definite, where(66)ALΘ=∂2L∂u2∂2L∂u∂δ∂2L∂u∂β∂2L∂u∂η∂2L∂u∂λ∂2L∂u∂δ∂2L∂δ2∂2L∂δ∂β∂2L∂δ∂η∂2L∂δ∂λ∂2L∂u∂β∂2L∂δ∂β∂2L∂β2∂2L∂β∂η∂2L∂β∂λ∂2L∂u∂η∂2L∂δ∂η∂2L∂β∂η∂2L∂η2∂2L∂η∂λ∂2L∂u∂λ∂2L∂δ∂λ∂2L∂β∂λ∂2L∂η∂λ∂2L∂λ2.Then evaluate the function in (29). When the convergence is not verified, then go to step 1 to update the initial point.Recall that since the matrixALΘ be an 5×5 symmetric matrix and let AkLΘ be the submatrix of ALΘ obtained by taking the upper left-hand corner 5×5 submatrix of ALΘ. Furthermore, let Δk = detAkLΘ, the kth principal minor of ALΘ. Then ALΘ is negative definite if and only if −1kΔk>0 for k=1,2,3,4,5. In comparison with the standard EM algorithm, our methodology can make it simple to evaluate the parameters β, η, and λ by maximizing the higher bound on the data log-likelihood function as appeared in (42), (49), and (58) separately. In the following section, we will explain the robustness, accuracy, and effectiveness of the proposed model, as compared with other models.


## 4.1. Location Parameter Estimation

To maximize the likelihood function in (28), we consider the derivation of L with the location u at the t+1 iteration step. We have(30)∂L∂u=∑i=1Nu−xiu−xi−2ηλ−1−βλδλu−xiλ−∫Ωsignu−xiu−xi−1ηλ−1−βλ/δλu−xiλTx|Θdx∫ΩTx|Θdx.At that point as [23], we have(31)∫ΩTx|Θdx≈1M∑i=1M1Ωvi,where vi∼Tx|Θt indicates the random variable that is drawn from the probability distribution Tx|Θt, with Θt=ut,δt,βt,ηt,λt′ and M is the number of random variables vi. We use M=106, for our experiments. In the same manner, we can write(32)∫Ωsignu−xiu−xi−1ηλ−1−βλδλu−xiλTx|Θdx≈1M∑i=1M1Ωvisignut−viut−vi−1ηtλt−1−βtλtδtλtut−viλt.By using (31) and (32), we can rewrite (30) as(33)∂L∂u|u=ut≈∑i=1Nut−xi−2ut−xiηtλt−1−βtλtδtλtut−xiλt−Hut−xi2,where(34)H=∑i=1M1Ωvisignut−viut−vi−1ηtλt−1−βtλt/δtλtut−viλt∑i=1M1Ωvi.According to the theory of robust statistics [24], any estimate u is defined by an implicit equation:(35)∑iFxi−u=0.This gives a numerical solution of the location of u as a weighted mean:(36)u=∑iωixi∑iωi;whereωi=Fxi−uxi−u.Now, we can apply (35) to ∂L/∂u in (33), and the solution of ∂L/∂u=0 gives the solutions of u at the t+1 step:(37)ut+1=∑i=1Nut−xi−2xiηtλt−1−βtλt/δtλtut−xiλt+H∑i=1Nut−xi−2ηtλt−1−βtλt/δtλtut−xiλt.


## 4.2. Scale Parameters Estimation

Putting the derivative of the log-likelihood functionL with respect to the scale parameter δ at the t+1 iteration step, we have(38)∂L∂δ=δ−1∑i=1Nβλδ−λu−xiλ−ηλ−∫Ωβλδ−λu−xiλ−ηλTx|Θdx∫ΩTx|Θdx.Similarly as (31) and (32), we can rewrite ∂L/∂δ as(39)∂L∂δ=δ−1∑i=1Nβλδ−λu−xiλ−ηλ−G,where(40)G=∑i=1Mβtλtδt−λtut−viλt−ηtλt1Ωvi∑i=1M1Ωvi.The solution of∂L/∂δ=0 yields the solutions of δ at the t+1 step:(41)δt+1=βtλt∑i=1Nut−xiλt∑i=1Nηtλt+G1/λt.The next step is to update the estimate of the scale parameterβ. This includes fixing the other parameters and improving the estimate of β by using the Newton Raphson method [25]. Every cycle requires the first and second derivatives of LΘ with respect to the parameter β.(42)βt+1=βt−∂L/∂β∂2L/∂β2+ε|β=βt,where ε is a scaling element. The derivative of the function LΘ regarding β is given by(43)∂L∂β=∑i=1Nfxi,Θ−∫Ωfx,ΘTx|Θdx∫ΩTx|Θdx,where(44)fxi,Θ=1Txi|Θ∂Txi|Θ∂β=ηβ−u−xiδλ.The term∂L/∂β can be approximated as(45)∂L∂β|β=βt≈∑i=1Nfxi,Θt−∑i=1M1Ωvifvi,Θt∑i=1M1Ωvi.The term∂2L/∂β2 is given by(46)∂2L∂β2=∑i=1N∂fx,Θ∂β+∫ΩTx|Θfdx2∫ΩTx|Θdx2−∫Ω∂f/∂β+f2Tx|Θdx∫ΩTx|Θdx,where(47)∂f∂β=−ηβ2.Also the term∂2L/∂β2 can be approximated as(48)∂2L∂β2|β=βt≈∑i=1N−ηtβt2+∑i=1M1Ωvifvi,Θt2∑i=1M1Ωvi2−∑i=1M1Ωvi−ηt/βt2+fvi,Θt2∑i=1M1Ωvi.


## 4.3. Shape Parameters Estimation

For shape parameter estimationη by using the Newton Raphson method, we have(49)ηt+1=ηt−∂L/∂η∂2L/∂η2+ε|η=ηt.The derivative of the functionLΘ with respect to η is given by(50)∂L∂η=∑i=1Ngxi|Θ−∫Ωgx|ΘTx|Θdx∫ΩTx|Θdx,where(51)gxi|Θ=1Txi|Θ∂Txi|Θ∂η=lnβ−Ψη+λlnu−xi−lnδ.The term∂L/∂η can be approximated as(52)∂L∂η|η=ηt≈∑i=1Ngtxi|Θ−∑m=1M1Ωvmgtvm|Θ∑i=1M1Ωvi,where(53)gtxi|Θ=lnβt−Ψηt+λtlnut−xi−lnδt.The calculation of the term∂2L/∂η2 is obtained as(54)∂2L∂η2=∑i=1N∂gxi|Θ∂η+∫ΩTx|Θgx|Θdx2∫ΩTx|Θdx2−∫Ω∂gx|Θ/∂η+g2x|θTx|Θdx∫ΩTx|Θdx,where(55)∂g∂η=−Ψ′η.The term∂2L/∂η2 can be approximated as(56)∂2L∂η2|η=ηt≈∑i=1N∂gxi|Θ∂ηt+∑m=1M1Ωvmgtvm|Θ2∑i=1M1Ωvi2−∑m=1M1Ωvm∂gvm|Θ/∂ηt+gtvm|Θ2∑i=1M1Ωvi,where(57)∂gxi|θ∂ηt=−Ψ′ηt.For shape parameter estimationλ by using the Newton Raphson method, we have(58)λt+1=λt−∂L/∂λ∂2L/∂λ2+ε|λ=λt.The derivative of the function LΘ with respect to λ is given by(59)∂L∂λ=∑i=1Nhxi|Θ−∫Ωhx|ΘTx|Θdx∫ΩTx|Θdx,where(60)hxi|Θ=1Txi|Θ∂Txi|Θ∂λ=1λ+η−βu−xiδλlnu−xiδ.The term∂L/∂λ can be approximated as(61)∂L∂λ|λ=λt≈∑i=1Nhtxi|Θ−∑m=1M1Ωvmhtvm|Θ∑m=1M1Ωvm,where(62)htxi|Θ=1λt+ηt−βtut−xiδtλtlnut−xiδt.The calculation of the term∂2L/∂λ2 is obtained as(63)∂2L∂λ2=∑i=1N∂hxi|Θ∂λ+∫ΩTx|Θhx|Θdx2∫ΩTx|Θdx2N−∫Ω∂hx|Θ/∂λ+h2x|ΘTx|Θdx∫ΩTx|Θdx,where(64)∂h∂λ=−1λ2−βu−xiδλln2u−xiδ.The term∂2L/∂λ2 can be approximated as(65)∂2L∂λ2|λ=λt≈∑i=1N∂hxi|Θ∂λt+∑m=1M1Ωvmhtvm|Θ2∑m=1M1Ωvm2−∑m=1M1Ωvm∂hvm|Θ/∂λt+htvm|Θ2∑m=1M1Ωvm.


## 4.4. Algorithm

To study the stability of our model, we have to find the set of initial points that generate a convergent sequence which called stable points of the dynamical system, i.e., we have to findu0,δ0,β0,η0,λ0 such that limt⟶∞ut,limt⟶∞δt, limt⟶∞βt, limt⟶∞ηt, and limt⟶∞λt exist. Indeed for fixed initial, it is difficult to predict how the approximation sequence behaves; hence, for this purpose, we take a random numbers of initial points until the convergence is verified (two successive approximations of each parameter correct to 4 decimal places). The various steps of the proposed model can be summarized as follows:Step 1: Initialize the parametersΘu,δ,β,η,λ.Step 2: Reestimate the parametersΘu,δ,β,η,λ, where the most common value of scaling parameter ε is 10−30 for our experiments.+Update the parameteru in (37).+Update the parameterδ in (41).+Update the parameterβ in (42).+Update the parameterη in (49).+Update the parameterλ in (58).Step 3: Check the convergence, if∂LΘ/∂ΘiΘ=Θt|<10−4, for all 1≤i≤5 under the constrains ALΘt is negative definite, where(66)ALΘ=∂2L∂u2∂2L∂u∂δ∂2L∂u∂β∂2L∂u∂η∂2L∂u∂λ∂2L∂u∂δ∂2L∂δ2∂2L∂δ∂β∂2L∂δ∂η∂2L∂δ∂λ∂2L∂u∂β∂2L∂δ∂β∂2L∂β2∂2L∂β∂η∂2L∂β∂λ∂2L∂u∂η∂2L∂δ∂η∂2L∂β∂η∂2L∂η2∂2L∂η∂λ∂2L∂u∂λ∂2L∂δ∂λ∂2L∂β∂λ∂2L∂η∂λ∂2L∂λ2.Then evaluate the function in (29). When the convergence is not verified, then go to step 1 to update the initial point.Recall that since the matrixALΘ be an 5×5 symmetric matrix and let AkLΘ be the submatrix of ALΘ obtained by taking the upper left-hand corner 5×5 submatrix of ALΘ. Furthermore, let Δk = detAkLΘ, the kth principal minor of ALΘ. Then ALΘ is negative definite if and only if −1kΔk>0 for k=1,2,3,4,5. In comparison with the standard EM algorithm, our methodology can make it simple to evaluate the parameters β, η, and λ by maximizing the higher bound on the data log-likelihood function as appeared in (42), (49), and (58) separately. In the following section, we will explain the robustness, accuracy, and effectiveness of the proposed model, as compared with other models.


## 5. Experiments

We explain the proposed technique in different examinations. The execution of BGΓM is compared with the WM [16], RM [19], EM [14], LM [22], GM [15], GGM [25], ΓM [1], GΓM [2], BWMM [26], BRM [27], BEM [28], BLM [22], BGM [29, 30], BGGM [22], and BΓM [31]. To measure the fitting precision of every model, we use the corresponding −2 Log-likelihood −2L values of models fitted to data. In general, the smaller values of −2L, is the better fit to the data.


### 5.1. Simulation Study

We generate 40000 random numbers from BGΓM with different parameters and bounded support regions see Figures 4–6. The corresponding −2L values of models fitted to simulated data are listed in Table 4. We find that BGΓM is the most powerful and has the least −2L. The pdf of BGΓM is monotonically increasing, decreasing, bathtub, and upside down bathtub-shaped depending on the parameter values and bounded support regions. So this model is of noticeable importance for image coding and compression applications [32, 33].Figure 4
The estimated histogram (a) the histogram of the observed data withu=0.3; δ=1.5; η=2, λ=3 and β=0.1 in the interval 1.5,4; (b) the estimated histogram of RM, LM, WM, GM, EM, GGM, ΓM, GΓM, and GΓM; (c) the estimated histogram of BRM, BLM, BWM, BGM, BEM, BGGM, BΓM, and BGΓM.
(a)(b)(c)Figure 5
The estimated histogram. (a) The histogram of the observed data withu=0; δ=0.5; η=2, λ=3 and β=1 in the interval −1.5,1.5; (b) the estimated histogram of RM, LM, WM, GM, EM, GGM, ΓM, GΓM, and GΓM; (c) the estimated histogram of BRM, BLM, BWM, BGM, BEM, BGGM, BΓM, and BGΓM.
(a)(b)(c)Figure 6
The estimated histogram. (a) The histogram of the observed data withu=−0.3; δ=1.5; η=2, λ=3 and β=0.1 in the interval −4,−1.5; (b) the estimated histogram of RM, LM, WM, GM, EM, GGM, ΓM, GΓM, and GΓM; (c) the estimated histogram of BRM, BLM, BWM, BGM, BEM, BGGM, BΓM, and BGΓM.
(a)(b)(c)Table 4
The corresponding−2L values of models fitted to simulated data in Figures 4–6.
ModelFigure4Figure5Figure6LM340369.6127185.1339937.1RM236056.858161.5235563.1GM321759.593056.9321116.7EM342290.4113226.3342259.8GGM328885.379528327945.2WM172977.959251.4172863.8ΓM496904.999792497023.3GΓM199109.318948.2199442.2BLM157338.1123729.1156760.5BRM132502.252067.8132867BGM321759.593056.9321116.7BEM89868.5103830.290370.9BGGM189787.759251.4188722.3BWM76569.878813.476857.7BΓM71019.363381.970210BGΓM55227.918944.455581


### 5.2. Real Data Study

We give here six real data as follows:(1)
The first data set arose in tests on endurance of deep groove ball bearings which is from Lawless (1982, p. 288). The data set is 17.88, 28.92, 33, 41.52, 42.12, 45.6, 48.48, 51.84, 51.96, 54.12, 55.56, 67.80, 68.64, 68.64, 68.88, 84.12, 93.12, 98.64, 105.12, 105.84, 127.92, 128.04, 173.40.(2)
The second data set of the yearly maximum wind speed data in miles/hour, used in this study has been quoted from Castillo (1988) [34].(3)
The third data set of the tensile strength of 100 observations of carbon fibers, the data was obtained from Ref [35]. The data are 3.7, 2.74, 2.73, 2.5, 3.6, 3.11, 3.27, 2.87, 1.47, 3.11, 4.42, 2.41, 3.19, 3.22, 1.69, 3.28, 3.09, 1.87, 3.15, 4.9, 3.75, 2.43, 2.95, 2.97, 3.39, 2.96, 2.53, 2.67, 2.93, 3.22, 3.39, 2.81, 4.2, 3.33, 2.55, 3.31, 3.31, 2.85, 2.56, 3.56, 3.15, 2.35, 2.55, 2.59, 2.38, 2.81, 2.77, 2.17, 2.83, 1.92, 1.41, 3.68, 2.97, 1.36, 0.98, 2.76, 4.91, 3.68, 1.84, 1.59, 3.19, 1.57, 0.81, 5.56, 1.73, 1.59, 2, 1.22, 1.12, 1.71, 2.17, 1.17, 5.08, 2.48, 1.18, 3.51, 2.17, 1.69, 1.25, 4.38, 1.84, 0.39, 3.68, 2.48, 0.85, 1.61, 2.79, 4.7, 2.03, 1.8, 1.57, 1.08, 2.03, 1.61, 2.12, 1.89, 2.88, 2.82, 2.05, 3.65.(4)
The fourth data set of the survival times of 121 patients with breast cancer, the data was obtained from Ref [36].(5)
In this part “Leleccum.wav”(leleccum (1 : 3920)) is disintegrated into three high-pass subbands (CH, CV, CD) and one low-pass subband (CA). The Daubechies channel bank (db1) is used. The fifth data set is the approximation of the wavelet coefficient (db1, CD, level 1) of “leleccum.wav” in the interval (−20.32, 20.32).(6)
The wavelet approximation coefficient is an essential issue in computer vision as it assumes an important part in an extensive range of applications. The image of (lena) is decomposed into three high-pass subbands (CH, CV, CD) and one low-pass subband (CA). The Daubechies filter bank (db4) is used. The sixth data set is the wavelet coefficients of the high-pass subband (CD), level 1 in the interval (−0.5, 0.5).The histogram for all real sets and their estimated pdfs for the fitted models are displayed in Figure7–12. The corresponding −2L values of models fitted to real data are listed in Table 5. Therefore, the proposed model provides a better fit to these data and has the least −2L.Secondly, if we compare the power of our model with modified generalized gamma distribution (MGG) having 6-parameters defined and studied in [12] on real data 3, we have −2L=280.608 and −2L=282.692, respectively. Hence, BGΓM is high flexible than MGG for this data. Furthermore, we compare McDonald log-logistic distribution (McLL) [36] with our model BGΓM. The model selection is carried out using the following statistics: AIC (Akaike information criterion), CAIC (consistent Akaike information criterion), and BIC (Bayesian information criterion). The corresponding values of models fitted to real data 4 are listed in Table 6. We find that BGΓM is more flexible than McLL in this case.Figure 7
The estimated histogram. (a) The histogram of the tests on endurance of deep groove ball bearings; (b) the estimated histogram of RM, LM, WM, GM, EM, GGM,ΓM, GΓM, and GΓM; (c) the estimated histogram of BRM, BLM, BWM, BGM, BEM, BGGM, BΓM, and BGΓM.
(a)(b)(c)Figure 8
The estimated histogram. (a) The histogram of the yearly maximum wind speed data; (b) the estimated histogram of RM, LM, WM, GM, EM, GGM,ΓM, GΓM, and GΓM; (c) the estimated histogram of BRM, BLM, BWM, BGM, BEM, BGGM, BΓM, and BGΓM.
(a)(b)(c)Figure 9
The estimated histogram. (a) The histogram of the tensile strength of 100 observations of carbon fibers; (b) the estimated histogram of RM, LM, WM, GM, EM, GGM,ΓM, GΓM, and GΓM; (c) the estimated histogram of BRM, BLM, BWM, BGM, BEM, BGGM, BΓM, and BGΓM.
(a)(b)(c)Figure 10
The estimated histogram. (a) The histogram of the survival times of 121 patients with breast cancer; (b) the estimated histogram of RM, LM, WM, GM, EM, GGM,ΓM, GΓM, and GΓM; (c) the estimated histogram of BRM, BLM, BWM, BGM, BEM, BGGM, BΓM, and BGΓM.
(a)(b)(c)Figure 11
Approximation of the wavelet coefficients. (a) “Leleccum.wav”(leleccum (1 : 3920)); (b) approximation of the wavelet coefficient (db1, CD, level 1) of “leleccum.wav;” (c) the estimated histogram of RM, LM, WM, GM, EM, GGM,ΓM, GΓM, and GΓM; (d) the estimated histogram of BRM, BLM, BWM, BGM, BEM, BGGM, BΓM, and BGΓM.Figure 12
Approximation of the wavelet coefficients. (a) The original image (lena); (b) the wavelet coefficient (db4, CD, level 1) of lena image in (−0.5, 0.5); (c) the estimated histogram of RM, LM, WM, GM, EM, GGM,ΓM, GΓM, and GΓM; (d) the estimated histogram of BRM, BLM, BWM, BGM, BEM, BGGM, BΓM, and BGΓM.Table 5
The corresponding−2L values of models fitted to real data in Figures 7–12.
ModelFigure7Figure8Figure9Figure10Figure11Figure12LM231.689372.636294.2011202.92054.79691622.7872RM258.488445.485311.8051343.812980.335420236.904GM230.955396.412285.5411204.76672.022110248.145EM231.691372.636294.2011202.91597.17471.4026GGM231.743370.125285.7271201.74469.57351.6748WM230.41415.379287.881202.27595.6214534.9312ΓM305.696373.761314.1421218.73552.8825739.3194GΓM257.824441.071421.3141198.61557.8015440.4421BLM225.714343.052282.12611522020.37451294.9581BRM224.418343.064285.8741154.820206.642720527.3021BGM224.632343.072281.67411556672.022110248.145BEM225.714343.052282.0981156.61597.168881.9213BGGM232.148342.794328.5751154.92104.64411361.9667BWM224.164432.44279.4861155.73551.7368515.5414BΓM245.613346.724296.56311741.71557.7918626.7688BGΓM223.82342.289280.6081151.31493.2769366.1614Table 6
The corresponding AIC, BIC, and CAIC on real data 4.
ModelAICBICCAICBGΓM1161.3081175.2881161.831McLL1164.6611178.641165.183


## 5.1. Simulation Study

We generate 40000 random numbers from BGΓM with different parameters and bounded support regions see Figures 4–6. The corresponding −2L values of models fitted to simulated data are listed in Table 4. We find that BGΓM is the most powerful and has the least −2L. The pdf of BGΓM is monotonically increasing, decreasing, bathtub, and upside down bathtub-shaped depending on the parameter values and bounded support regions. So this model is of noticeable importance for image coding and compression applications [32, 33].Figure 4
The estimated histogram (a) the histogram of the observed data withu=0.3; δ=1.5; η=2, λ=3 and β=0.1 in the interval 1.5,4; (b) the estimated histogram of RM, LM, WM, GM, EM, GGM, ΓM, GΓM, and GΓM; (c) the estimated histogram of BRM, BLM, BWM, BGM, BEM, BGGM, BΓM, and BGΓM.
(a)(b)(c)Figure 5
The estimated histogram. (a) The histogram of the observed data withu=0; δ=0.5; η=2, λ=3 and β=1 in the interval −1.5,1.5; (b) the estimated histogram of RM, LM, WM, GM, EM, GGM, ΓM, GΓM, and GΓM; (c) the estimated histogram of BRM, BLM, BWM, BGM, BEM, BGGM, BΓM, and BGΓM.
(a)(b)(c)Figure 6
The estimated histogram. (a) The histogram of the observed data withu=−0.3; δ=1.5; η=2, λ=3 and β=0.1 in the interval −4,−1.5; (b) the estimated histogram of RM, LM, WM, GM, EM, GGM, ΓM, GΓM, and GΓM; (c) the estimated histogram of BRM, BLM, BWM, BGM, BEM, BGGM, BΓM, and BGΓM.
(a)(b)(c)Table 4
The corresponding−2L values of models fitted to simulated data in Figures 4–6.
ModelFigure4Figure5Figure6LM340369.6127185.1339937.1RM236056.858161.5235563.1GM321759.593056.9321116.7EM342290.4113226.3342259.8GGM328885.379528327945.2WM172977.959251.4172863.8ΓM496904.999792497023.3GΓM199109.318948.2199442.2BLM157338.1123729.1156760.5BRM132502.252067.8132867BGM321759.593056.9321116.7BEM89868.5103830.290370.9BGGM189787.759251.4188722.3BWM76569.878813.476857.7BΓM71019.363381.970210BGΓM55227.918944.455581


## 5.2. Real Data Study

We give here six real data as follows:(1)
The first data set arose in tests on endurance of deep groove ball bearings which is from Lawless (1982, p. 288). The data set is 17.88, 28.92, 33, 41.52, 42.12, 45.6, 48.48, 51.84, 51.96, 54.12, 55.56, 67.80, 68.64, 68.64, 68.88, 84.12, 93.12, 98.64, 105.12, 105.84, 127.92, 128.04, 173.40.(2)
The second data set of the yearly maximum wind speed data in miles/hour, used in this study has been quoted from Castillo (1988) [34].(3)
The third data set of the tensile strength of 100 observations of carbon fibers, the data was obtained from Ref [35]. The data are 3.7, 2.74, 2.73, 2.5, 3.6, 3.11, 3.27, 2.87, 1.47, 3.11, 4.42, 2.41, 3.19, 3.22, 1.69, 3.28, 3.09, 1.87, 3.15, 4.9, 3.75, 2.43, 2.95, 2.97, 3.39, 2.96, 2.53, 2.67, 2.93, 3.22, 3.39, 2.81, 4.2, 3.33, 2.55, 3.31, 3.31, 2.85, 2.56, 3.56, 3.15, 2.35, 2.55, 2.59, 2.38, 2.81, 2.77, 2.17, 2.83, 1.92, 1.41, 3.68, 2.97, 1.36, 0.98, 2.76, 4.91, 3.68, 1.84, 1.59, 3.19, 1.57, 0.81, 5.56, 1.73, 1.59, 2, 1.22, 1.12, 1.71, 2.17, 1.17, 5.08, 2.48, 1.18, 3.51, 2.17, 1.69, 1.25, 4.38, 1.84, 0.39, 3.68, 2.48, 0.85, 1.61, 2.79, 4.7, 2.03, 1.8, 1.57, 1.08, 2.03, 1.61, 2.12, 1.89, 2.88, 2.82, 2.05, 3.65.(4)
The fourth data set of the survival times of 121 patients with breast cancer, the data was obtained from Ref [36].(5)
In this part “Leleccum.wav”(leleccum (1 : 3920)) is disintegrated into three high-pass subbands (CH, CV, CD) and one low-pass subband (CA). The Daubechies channel bank (db1) is used. The fifth data set is the approximation of the wavelet coefficient (db1, CD, level 1) of “leleccum.wav” in the interval (−20.32, 20.32).(6)
The wavelet approximation coefficient is an essential issue in computer vision as it assumes an important part in an extensive range of applications. The image of (lena) is decomposed into three high-pass subbands (CH, CV, CD) and one low-pass subband (CA). The Daubechies filter bank (db4) is used. The sixth data set is the wavelet coefficients of the high-pass subband (CD), level 1 in the interval (−0.5, 0.5).The histogram for all real sets and their estimated pdfs for the fitted models are displayed in Figure7–12. The corresponding −2L values of models fitted to real data are listed in Table 5. Therefore, the proposed model provides a better fit to these data and has the least −2L.Secondly, if we compare the power of our model with modified generalized gamma distribution (MGG) having 6-parameters defined and studied in [12] on real data 3, we have −2L=280.608 and −2L=282.692, respectively. Hence, BGΓM is high flexible than MGG for this data. Furthermore, we compare McDonald log-logistic distribution (McLL) [36] with our model BGΓM. The model selection is carried out using the following statistics: AIC (Akaike information criterion), CAIC (consistent Akaike information criterion), and BIC (Bayesian information criterion). The corresponding values of models fitted to real data 4 are listed in Table 6. We find that BGΓM is more flexible than McLL in this case.Figure 7
The estimated histogram. (a) The histogram of the tests on endurance of deep groove ball bearings; (b) the estimated histogram of RM, LM, WM, GM, EM, GGM,ΓM, GΓM, and GΓM; (c) the estimated histogram of BRM, BLM, BWM, BGM, BEM, BGGM, BΓM, and BGΓM.
(a)(b)(c)Figure 8
The estimated histogram. (a) The histogram of the yearly maximum wind speed data; (b) the estimated histogram of RM, LM, WM, GM, EM, GGM,ΓM, GΓM, and GΓM; (c) the estimated histogram of BRM, BLM, BWM, BGM, BEM, BGGM, BΓM, and BGΓM.
(a)(b)(c)Figure 9
The estimated histogram. (a) The histogram of the tensile strength of 100 observations of carbon fibers; (b) the estimated histogram of RM, LM, WM, GM, EM, GGM,ΓM, GΓM, and GΓM; (c) the estimated histogram of BRM, BLM, BWM, BGM, BEM, BGGM, BΓM, and BGΓM.
(a)(b)(c)Figure 10
The estimated histogram. (a) The histogram of the survival times of 121 patients with breast cancer; (b) the estimated histogram of RM, LM, WM, GM, EM, GGM,ΓM, GΓM, and GΓM; (c) the estimated histogram of BRM, BLM, BWM, BGM, BEM, BGGM, BΓM, and BGΓM.
(a)(b)(c)Figure 11
Approximation of the wavelet coefficients. (a) “Leleccum.wav”(leleccum (1 : 3920)); (b) approximation of the wavelet coefficient (db1, CD, level 1) of “leleccum.wav;” (c) the estimated histogram of RM, LM, WM, GM, EM, GGM,ΓM, GΓM, and GΓM; (d) the estimated histogram of BRM, BLM, BWM, BGM, BEM, BGGM, BΓM, and BGΓM.Figure 12
Approximation of the wavelet coefficients. (a) The original image (lena); (b) the wavelet coefficient (db4, CD, level 1) of lena image in (−0.5, 0.5); (c) the estimated histogram of RM, LM, WM, GM, EM, GGM,ΓM, GΓM, and GΓM; (d) the estimated histogram of BRM, BLM, BWM, BGM, BEM, BGGM, BΓM, and BGΓM.Table 5
The corresponding−2L values of models fitted to real data in Figures 7–12.
ModelFigure7Figure8Figure9Figure10Figure11Figure12LM231.689372.636294.2011202.92054.79691622.7872RM258.488445.485311.8051343.812980.335420236.904GM230.955396.412285.5411204.76672.022110248.145EM231.691372.636294.2011202.91597.17471.4026GGM231.743370.125285.7271201.74469.57351.6748WM230.41415.379287.881202.27595.6214534.9312ΓM305.696373.761314.1421218.73552.8825739.3194GΓM257.824441.071421.3141198.61557.8015440.4421BLM225.714343.052282.12611522020.37451294.9581BRM224.418343.064285.8741154.820206.642720527.3021BGM224.632343.072281.67411556672.022110248.145BEM225.714343.052282.0981156.61597.168881.9213BGGM232.148342.794328.5751154.92104.64411361.9667BWM224.164432.44279.4861155.73551.7368515.5414BΓM245.613346.724296.56311741.71557.7918626.7688BGΓM223.82342.289280.6081151.31493.2769366.1614Table 6
The corresponding AIC, BIC, and CAIC on real data 4.
ModelAICBICCAICBGΓM1161.3081175.2881161.831McLL1164.6611178.641165.183


## 6. Conclusions

A bounded generalized Gamma model with five parameters, whose hazard function can be monotonically increasing, decreasing, bathtub, and upside down bathtub-shaped depending on the parameter values, has been introduced and studied. Some mathematical and statistical properties of the new model are investigated. We estimate the model parameters using maximum log-likelihood function and find a closed form of some parameters by the Newton Raphson method. The predictive ability of our model is found to be comparable or superior to widely accepted distributions. The performance of the model has the smallest−2L values. A simulation study was carried out to evaluate the predictive ability of our model to fit any kind of data with bounded support regions and compare it with other distributions. The power of the new model is illustrated by means of application to six real data sets. The BGΓM performs significantly better than the others distributions when sample sizes are small. Thus, it is less affected by sample size and is more robust. Also the accuracy of the proposed model for wavelet histogram fitting of image and sound is high. We hope that this model may attract wider applications on the modeling of the probability density function of the data via BGΓD in video coding and image denoising as a future work.



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*Source: 5500631-2021-08-17.xml*