A New Double Truncated Generalized Gamma Model with Some Applications

,e 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 Gammamodel, the boundedWeibull 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. ,is 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, r th order statistic, and median distributions.,e 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.


Introduction
e 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. e 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 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. e BGΓM Model is of noticeable significance for image coding, compression applications, sound system, wind speed data, and breast cancer data fitting. is 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. e remainder of this paper is organized as follows: e 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.

The Bounded Generalized Gamma Model and
Its Special Models e standard form of gamma function is e incomplete gamma function is defined by e probability density function (pdf ) of the generalized gamma distribution is given by for all x ∈ R, where Θ � (u, δ, β, η, λ) ′ , δ, η, λ, β > 0 and u ∈ R. e cumulative distribution function (cdf ) of generalized gamma distribution defined as follows: Let Ω � [a, b] ⊆ R and we denote the indicator function by We define the pdf of the bounded generalized gamma distribution (BGΓM) as In another form, we can write the pdf of the bounded generalized gamma distribution (BGΓM) as where It is clear to see that Hence, the cdf of the bounded generalized gamma distribution (BGΓM) is given by e parameters u(δ, β) and (η, λ) are corresponding to the location, scale, and shape parameters, respectively. Note that Υ(x|Θ) can be any kind of distribution, for example, in exponential distribution (ED) [14,15] be ϕ(x|u, δ, β), Weibull distribution (WD) [16][17][18] be T(x|u, δ, β, λ), Rayleigh distribution (RD) [19,20] be T(x|u, δ), generalized Gaussian distribution (GGD) [21] be T(x|u, δ, λ), Gaussian distribution (GD) [15] be T(x|u, δ), Laplacian distribution (LD) [22] be T(x|u, δ) and Gamma distribution (ΓD) [1] be T(x|u, δ, η, β). ese distributions are all unbounded with support range (0, ∞). We extend all the past models with range (−∞, ∞) also to the bounded case. e 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. e survival function and hazard rate function for BGΓM are, respectively, given by In Figures 1 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.

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.

Mode and Quantile.
e p th quantile function of the BGΓM is the solution of e median, denoted by μ * , can be obtained by substituting p � 0.5 in 10 and solving the equation   Journal of Mathematics 5 e mode, denoted by x m of the BGΓ distribution, is given by   Journal of Mathematics

Moments, Generating Function, and Mean Deviation.
e r th moment about zero of BGΓ distribution is ) . (15) e mean μ of the BGΓ distribution is given by e variance σ 2 of the BGΓ distribution is given by  16 18 e central moments of BGΓ distribution can be obtained as follows e mean deviation Md of BGΓ distribution can be derived as 8 Journal of Mathematics In Table 2, 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.

Mean Residual Life and Mean Waiting Time.
e mean residual life function, say φ(t), is given by e mean waiting time of BGΓ distribution, say φ(t), can be derived as Journal of Mathematics

Entropy.
e entropy of a random variable X measures the variation of the uncertainty. e Rényi entropy of BGΓ distribution, say RE X (]) for ] ≠ 1 and ] > 0, is derived as 3.5. Order Statistics. Let X 1:n , X 2:n , . . . , X n:n denote the order statistics obtained from a random sample of size n from BGΓ distribution. e probability density function of i th order statistics is given by If n is odd. e pdf of BGΓ distribution of the median is obtained by substituting i � (n + 1)/2 in equation (24) as follows: e joint pdf of the i th and the l th order statistics for x < y can be written as So the joint pdf of the i th and the l th order statistics of BGΓ distribution is

Maximizing the Log-Likelihood Function
Here, we consider the estimation of the unknown parameters of the BGΓD by the method of maximum likelihood. Let x 1 , x 2 , . . . , x N be a random sample from the BGΓD. e total log-likelihood (L(Θ)) is given by

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

Journal of Mathematics
At that point as [23], we have where v i ∼ T(x|Θ (t) ) indicates the random variable that is drawn from the probability distribution T(x|Θ (t) ), with , λ (t) ) ′ and M is the number of random variables v i . We use M � 10 6 , for our experiments.
In the same manner, we can write By using (31) and (32), we can rewrite (30) as where According to the theory of robust statistics [24], any estimate u is defined by an implicit equation: is gives a numerical solution of the location of u as a weighted mean: Now, we can apply (35) to zL/zu in (33), and the solution of zL/zu � 0 gives the solutions of u at the (t + 1) step:

Scale Parameters Estimation.
Putting the derivative of the log-likelihood function L with respect to the scale parameter δ at the (t + 1) iteration step, we have Similarly as (31) and (32), we can rewrite zL/zδ as where e solution of zL/zδ � 0 yields the solutions of δ at the (t + 1) step: .
(41) e next step is to update the estimate of the scale parameter β. is 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 L(Θ) with respect to the parameter β.
where ε is a scaling element. e derivative of the function L(Θ) regarding β is given by where e term zL/zβ can be approximated as e term z 2 L/zβ 2 is given by where Also the term z 2 L/zβ 2 can be approximated as

Shape Parameters Estimation.
For shape parameter estimation η by using the Newton Raphson method, we have e derivative of the function L(Θ) with respect to η is given by where e term zL/zη can be approximated as where e calculation of the term z 2 L/zη 2 is obtained as Journal of Mathematics where e term z 2 L/zη 2 can be approximated as where For shape parameter estimation λ by using the Newton Raphson method, we have zL/zλ z 2 L/zλ 2 + ε λ�λ (t) .
(58) e derivative of the function L(Θ) with respect to λ is given by where (60) e term zL/zλ can be approximated as where (62) e calculation of the term z 2 L/zλ 2 is obtained as where e term z 2 L/zλ 2 can be approximated as

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 find u (0) , δ (0) , β (0) , η (0) , λ (0) such that lim t⟶∞ u (t) , lim t⟶∞ δ (t) , lim t⟶∞ β (t) , lim t⟶∞ η (t) , and lim t⟶∞ λ (t) 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). e various steps of the proposed model can be summarized as follows: Step 1: Initialize the parameters Θ(u, δ, β, η, λ).
+Update the parameter u in (37). +Update the parameter δ in (41). +Update the parameter β in (42). +Update the parameter η in (49). +Update the parameter λ in (58). Step 3: Check the convergence, en 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 matrix AL(Θ) be an 5 × 5 symmetric matrix and let A k L(Θ) be the submatrix of AL(Θ) obtained by taking the upper left-hand corner 5 × 5 submatrix of AL(Θ). Furthermore, let Δ k � det(A k L(Θ)), the k th principal minor of AL(Θ). en AL(Θ) is negative definite if and only if (−1) k Δ k > 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.

Simulation Study.
We generate 40000 random numbers from BGΓM with different parameters and bounded support regions see Figures 4-6. e 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. e 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].

Real Data Study.
We give here six real data as follows: (      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). e Daubechies channel bank (db1) is used. e 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) e wavelet approximation coefficient is an essential issue in computer vision as it assumes an important part in an extensive range of applications. e image of (lena) is decomposed into three high-pass subbands (CH, CV, CD) and one low-pass subband (CA). e Daubechies filter bank (db4) is used. e sixth data set is the wavelet coefficients of the highpass subband (CD), level 1 in the interval (−0.5, 0.5).
e histogram for all real sets and their estimated pdfs for the fitted models are displayed in Figure 7-12. e corresponding −2L values of models fitted to real data are listed in Table 5. erefore, 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. e model selection is carried out using the following statistics: AIC (Akaike information criterion), CAIC (consistent Akaike information criterion), and BIC (Bayesian information criterion). e 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.

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. e predictive ability of our model is found to be comparable or superior to widely accepted distributions. e 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. e power of the new model is illustrated by means of application to six real data sets. e BGΓM performs significantly better than the others distributions when sample sizes are  small. us, 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.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Ethical Approval
is article does not contain any studies with human participants or animals performed by any of the authors.