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) [16–18] 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 bathtub-shaped 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, 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 .
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 aswhere
The calculation of the term is obtained aswhere
The term can be approximated aswhere
For shape parameter estimation by using the Newton Raphson method, we haveThe derivative of the function with respect to is given bywhere
The term can be approximated aswhere
The calculation of the term is obtained aswhere
The term can be approximated as
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 find such that , , , and 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 parameters . Step 2: Reestimate the parameters , where the most common value of scaling parameter is for our experiments. +Update the parameter in (37). +Update the parameter in (41). +Update the parameter in (42). +Update the parameter in (49). +Update the parameter in (58). Step 3: Check the convergence, if , for all under the constrains is negative definite, where
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 matrix be an symmetric matrix and let be the submatrix of obtained by taking the upper left-hand corner submatrix of . Furthermore, let = det, the principal minor of . Then is negative definite if and only if for . In comparison with the standard EM algorithm, our methodology can make it simple to evaluate the parameters , , and 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 BM is compared with the WM [16], RM [19], EM [14], LM [22], GM [15], GGM [25], M [1], GM [2], BWMM [26], BRM [27], BEM [28], BLM [22], BGM [29, 30], BGGM [22], and BM [31]. To measure the fitting precision of every model, we use the corresponding −2 Log-likelihood values of models fitted to data. In general, the smaller values of , is the better fit to the data.
5.1. Simulation Study
We generate 40000 random numbers from BGM with different parameters and bounded support regions see Figures 4–6. The corresponding values of models fitted to simulated data are listed in Table 4. We find that BGM is the most powerful and has the least . The pdf of BGM 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].
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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 Figure 7–12. The corresponding 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 .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 and , respectively. Hence, BGM is high flexible than MGG for this data. Furthermore, we compare McDonald log-logistic distribution (McLL) [36] with our model BGM. 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 BGM is more flexible than McLL in this case.
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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 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 BGM 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 BGD in video coding and image denoising as a future work.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Ethical Approval
This article does not contain any studies with human participants or animals performed by any of the authors.
Conflicts of Interest
The authors declare that they have no competing interests.
Authors’ Contributions
All the authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.