Journal of Mathematics

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Research Article | Open Access

Volume 2021 |Article ID 5500631 | https://doi.org/10.1155/2021/5500631

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

Academic Editor: Ghulam Mustafa
Received07 May 2021
Accepted06 Aug 2021
Published17 Aug 2021

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, 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.

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 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 (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 bathtub-shaped 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) [1618] 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


MBM:
BMBM:
MBM: and
BWMBM:
WMBM: ,
BGGMBM: and
GGMBM: , and
BEMBM: , and
EMBM: , , and
BGMBM: , and
GMBM: , , and
BRMBM: , and
RMBM: , , and
BLMBM: , and
LMBM: , , and

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 bathtub-shaped depending on the parameter values.

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, Rényi 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 .


MedianMode1Mode2MeanVarianceSkewnessKurtosis

0.520.9979110.99490.4895−0.05252.8857
0.541.00371.70710.292910.564201.5708
0.570.99091.86320.136810.666101.2251
220.42072.2248−0.22480.89321.82970.01171.4184
240.80712.1502−0.150211.329301.13176489
270.92442.0925−0.092411.156801.04581787
72−1.38173.5495−1.5495−0.90092.01852.06495.7078
74−0.14532.6119−0.61190.99592.59580.00351.0358
770.97852.3166−0.316611.718101.0121


MedianMode1Mode2MeanVarianceSkewnessKurtosis

0.50.50.9732110.94920.8732−0.18242.6472
0.521110.99990.2499−0.0022.993
0.5711110.071403
20.5−0.22272.7321−0.73210.45992.70850.29471.4377
220.83271.8660.1340.99680.9946−0.00521.4863
271.0011.46290.537110.285701.5
70.5−1.7224.6056−2.6056−1.5470.58864.873728.0326
72−0.4832.8028−0.80280.58543.03440.32591.2304
771.08551.96360.03641101.1429

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 Rényi 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 Log-Likelihood 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 log-likelihood 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 log-likelihood 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