Research Article  Open Access
Awad A. Bakery, Wael Zakaria, OM Kalthum S. K. Mohamed, "A New Double Truncated Generalized Gamma Model with Some Applications", Journal of Mathematics, vol. 2021, Article ID 5500631, 27 pages, 2021. https://doi.org/10.1155/2021/5500631
A New Double Truncated Generalized Gamma Model with Some Applications
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 fiveparameter 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, order statistic, and median distributions. The delivery has hazard frequencies that are monotonically increasing or declining, bathtubshaped, or upsidedown bathtubshaped. We use the Newton Raphson approach to approximate model parameters that increase the loglikelihood 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.
1. Introduction
The gamma (M) 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 M 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 loglikelihood 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 threeparameter exponentiated gamma distribution. Agarwal and AlSaleh [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 highresolution 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 (real numbers) or any bounded subset of . 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 (BGM) for analyzing data. The BM 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 bathtubshaped depending on the parameter values and bounded support regions. The remainder of this paper is organized as follows: The BM with its sub models and some shapes describe the hazard rate function are defined in Section 2. Some properties of the BGM 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
The incomplete gamma function is defined by
The probability density function (pdf) of the generalized gamma distribution is given byfor all , where , and . The cumulative distribution function (cdf) of generalized gamma distribution defined as follows:
Let and we denote the indicator function by
We define the pdf of the bounded generalized gamma distribution (BGM) as
In another form, we can write the pdf of the bounded generalized gamma distribution (BGM) aswhere
It is clear to see that
Hence, the cdf of the bounded generalized gamma distribution (BGM) is given by
The parameters and are corresponding to the location, scale, and shape parameters, respectively. Note that can be any kind of distribution, for example, in exponential distribution (ED) [14, 15] be , Weibull distribution (WD) [16â€“18] be , Rayleigh distribution (RD) [19, 20] be , generalized Gaussian distribution (GGD) [21] be , Gaussian distribution (GD) [15] be , Laplacian distribution (LD) [22] be and Gamma distribution (D) [1] be . These distributions are all unbounded with support range . We extend all the past models with range also to the bounded case. The BGM has several models as special cases, which makes it distinguishable scientific importance from other models. We investigate the various special models of the BGM as listed in Table 1. The survival function and hazard rate function for BGM are, respectively, given by

In Figures 1 and 2, we display the plots of the pdf of BM for various parameters. Figure 3 displays the BGM failure rate function which can be increasing, decreasing, bathtub, and upside down bathtubshaped depending on the parameter values.
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3. Properties of BM
In this section, we provide some general properties of the BM including quantile function, mode, moments, mean deviation, mean residual life and mean waiting time, RÃ©nyi entropy, and order statistics.
3.1. Mode and Quantile
The quantile function of the BM is the solution of
The median, denoted by , can be obtained by substituting in 10 and solving the equation
The mode, denoted by of the B distribution, is given by
Remark 1. (1)If, then the Bdistribution is unimodal distribution(2)If, then the Bdistribution is multimodal distribution
3.2. Moments, Generating Function, and Mean Deviation
The moment about zero of B distribution is
The mean of the B distribution is given byThe variance of the B distribution is given by
The central moments of B distribution can be obtained as follows
The moment generating function of B distribution is
The mean deviation of B distribution can be derived as
In Table 2, the Median, Mode, Mean, Variance, Skewness, and Kurtosis of BGM have given for , , , , and and various values of and . From Table 2, we note that for fixed values of , and , the Kurtosis is decreasing function of . Also, for fixed values of , and , the Mode 1, Variance, and Skewness are increasing function and the Mode 2 and Mean are decreasing function of . In Table 3, Median, Mode, Mean, Variance, Skewness, and Kurtosis of BGM have given for , , , , and and various values of and . From Table 3, we note that for fixed values of , and , Mode 1 is decreasing, Median, Mode 2, and Mean are increasing functions of . Also, for fixed values of , and , Mode 1 and Skewness are increasing and Mode 2 and Mean are decreasing functions of .


3.3. Mean Residual Life and Mean Waiting Time
The mean residual life function, say , is given by
The mean waiting time of B distribution, say , can be derived as
3.4. Entropy
The entropy of a random variable measures the variation of the uncertainty. The RÃ©nyi entropy of B distribution, say for and , is derived as
3.5. Order Statistics
Let denote the order statistics obtained from a random sample of size from B distribution. The probability density function of order statistics is given by
The pdf of the minimum and the maximum order statistics of B distribution can be obtained, respectively, as follows:
If is odd. The pdf of B distribution of the median is obtained by substituting in equation (24) as follows:
The joint pdf of the and the order statistics for can be written asSo the joint pdf of the and the order statistics of B distribution is
4. Maximizing the LogLikelihood Function
Here, we consider the estimation of the unknown parameters of the BD by the method of maximum likelihood. Let be a random sample from the BD. The total loglikelihood is given by
4.1. Location Parameter Estimation
To maximize the likelihood function in (28), we consider the derivation of with the location at the iteration step. We have
At that point as [23], we havewhere indicates the random variable that is drawn from the probability distribution , with and is the number of random variables . We use , for our experiments. In the same manner, we can write
By using (31) and (32), we can rewrite (30) aswhere
According to the theory of robust statistics [24], any estimate is defined by an implicit equation:This gives a numerical solution of the location of as a weighted mean:
Now, we can apply (35) to in (33), and the solution of gives the solutions of at the step:
4.2. Scale Parameters Estimation
Putting the derivative of the loglikelihood function with respect to the scale parameter at the iteration step, we have
Similarly as (31) and (32), we can rewrite aswhere
The solution of yields the solutions of at the step:
The next step is to update the estimate of the scale parameter . This includes fixing the other parameters and improving the estimate of by using the Newton Raphson method [25]. Every cycle requires the first and second derivatives of with respect to the parameter .where is a scaling element. The derivative of the function regarding is given bywhere
The term can be approximated as
The term is given bywhere
Also the term can be approximated as
4.3. Shape Parameters Estimation
For shape parameter estimation by using the Newton Raphson method, we have
The derivative of the function with respect to is given bywhere
The term can be approximated as