A New Generalized Two-Sided Class of the Distributions Via New Transmuted Two-Sided Bounded Distribution
- 10.2991/jsta.d.190306.003How to use a DOI?
- Hazard rate function; order statistics; maximum likelihood estimator; transmutation map; transmuted two-sided distribution
In the present paper, we first consider a generalization of the standard two-sided power distribution so-called the transmuted two-sided distribution, and then extend proposed idea to generalized two-sided class of distributions, introduced by Korkmaz and Genç . Some statistical and reliability properties including explicit expressions for quantiles, hazard rate function, order statistics, and maximum likelihood estimation are obtained in general setting. Generalized transmuted two-sided exponential distribution is considered as a especial case and denoted with the name . A simulation study is presented to investigate the bias and mean square error of the maximum likelihood estimators. We use a real data set and obtain the maximum likelihood and parametric bootstrap estimator of the parameters of distribution. Finally, the superiority of the new model to some common statistical distributions is shown through the different criteria of selection model including log-likelihood values, Akaike information criterion, and Kolmogorov–Smirnov test statistic values.
- © 2019 The Authors. Published by Atlantis Press SARL.
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- This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).
The statistical distribution theory has been widely explored by researchers in recent years. Given the fact that the data from our surrounding environment follow various statistical models, it is necessary to extract and develop appropriate high-quality models. One of the most important models in the statistical theory is the change point models. In the distribution theory, the change point distributions are used in the different branch of sciences such as economic, engineering, agriculture, and so on. Van Dorp and Kotz  introduced a family of the change point distributions so-called two-sided power (TSP) distribution with the probability density function (pdf),and with the cumulative distribution function , where and . The parameter is the location parameter called “turning point” and is the shape parameter that control the shape of distribution on the left and right of .
The distribution for can be used for modeling unimodal phenomena on a bounded domain when a peak in data is observed. Van Dorp and Kotz  introduced an extension of the three-parameter triangular distribution utilized in risk analysis. Their model includes the distribution as a special case. Van Dorp and Kotz  considered a family of continuous distributions on a bounded interval generated by convolutions of the distributions. In recent years, a number of researchers have studied some generalization of the distribution such as Nadarajah , Oruç and Bairamov , Vicari et al. , Herrerías-Velasco et al. , and Soltani and Homei . Korkmaz and Genç  proposed a new generalization of Weibull distribution by making use of a transformation of the standard TSP distribution. Also, Korkmaz and Genç  considered the log transformation of the distribution instead of uniform distribution and introduced a generalization of the exponential distribution. Korkmaz and Genç  extended the idea of two-sidedness to other ordinary distributions like normal and introduced the two-sided generalized normal distribution.
One of the interesting methods for constructing new distributions is the transmutation map approach. Recently, some new distributions have been generalized based on the transmutation method. The transmuted distribution based on the is defined aswhere is the of the parent distribution.
Aryal and Tsokos  generated a flexible family of probability distributions taking extreme value distribution as the base value distribution by using the quadratic rank transmutation map (QRTM). Aryal and Tsokos  generalized the two-parameter Weibull distribution using the QRTM. Aryal  introduced a generalization of the log-logistic distribution so-called the transmuted log-logistic distribution. Abd El Hady  introduced a new generalization of the two-parameter Weibull distribution by using the QRTM. This new distribution is named exponentiated transmuted Weibull (ETW) distribution. Elgarhy and Shawki  introduced a new generalized version of the quasi Lindley distribution which is called the transmuted generalized quasi Lindley (TGQL) distribution.
The transmutation method, as an important method for developing statistical distributions, hasn't been used for the change point distributions, yet. The main motivation of the present paper is to apply the transmutation technique for increasing the flexibility and usefulness of the distribution and generalized class of distributions.
This paper organized as follows. In Section 2, we introduce a new distribution so-called transmuted two-sided distribution. In Section 3, we propose a generalization of the transmuted two-sided distribution and consider the hazard function, quantiles, and order statistics of this distribution. We consider the exponential distribution as a parent distribution and introduce transmuted two-sided generalized exponential (TSGE) distribution, in Section 4. In this section, we plot the shape of density function and hazard function. In Section 5, the estimation of parameters of the generalized transmuted two-sided distribution are obtained by using two methods maximum likelihood estimation (MLE) and bootstrap estimation. Also, we study the performance of MLEs of parameters of the transmuted TSGE distribution via a simulation study. In Section 6, the superiority of new model to some competitor statistical models is shown through the different criteria of selection model. Finally, the paper is concluded in Section 7.
2. TRANSMUTED TWO-SIDED DISTRIBUTION
In this section, we introduce the transmuted two-sided distribution and then we consider its shape for different values of parameters. The main motivation for introducing this new family is to provide the more flexibility for the distribution by compounding two-sided distributions family and transmutation map approach.
Let the random variable have a beta distribution with ,
Assume that be a random variable associated with the truncated on the right at , aswhere denotes identically distributed. The of the random variable is given by
Now, suppose that be a random variable with the density function . By definingthe of is given by
So, based on Eq. (7), we have the following definition.
A random variable is said to be transmuted two-sided distribution if its is given byand its is given by where, is a shape parameter, is a scale parameter, and is a transmuted parameter. We denote the transmuted two-sided distribution by .
2.1. Density Shape of the Distribution
Here, we consider a discussion about the shape of the proposed density function. In the end points of the support, the behaviour of the Eq. (8) is given as follows:
The derivative is
The right- and left-hand limits of at are given by
These limits are different. So, does not exist and has a corner at .
When , and , if then and when , , and , if then . So, when and , the maximum of occurs at points and . It implies when and , is a bimodal .
When , , and , we have and when , , and , we have . So, for , , is a unimodal . The density shapes of the distribution for different choices of the parameters are plotted in Fig. 1.
3. TRANSMUTED TWO-SIDED GENERALIZED-G FAMILY OF THE DISTRIBUTIONS
Consider a continuous random variable with and . Using Definition 2.1, we define a generalization of the transmuted two-sided distribution as follows:
A random variable is said to be transmuted two-sided generalized-G (TSG-G) distribution if its is given byand its is given by where, , is a parameter vector in the and is its inverse.
We denote transmuted TSG-G family of distributions by .
If , we have the and of the TSG-G family introduced by Korkmaz and Genç . If , , and , we have the of the base distribution.
In the next subsections, we study hazard function, random variate generation, order statistics and relative entropy of the distribution.
3.1. Hazard Function
The hazard rate is a fundamental tools in reliability modelling for evaluation the ageing process. Knowing the shape of the hazard rate is important in reliability theory, risk analysis, and other disciplines. The concepts of increasing, decreasing, bathtub-shaped (first decreasing and then increasing) and upside-down bathtub-shaped (first increasing and then decreasing) hazard rate functions are very useful in reliability analysis. The lifetime distributions with these ageing properties are designated as the increasing failure rate (), decreasing failure rate (), bathtub-shape (), and upside-down bathtub-shaped () distributions, respectively. The hazard function of the distribution is given by
3.2. Random Variate Generation
For generating random variables from the distribution, we use the inverse transformation method. The quantile of order of the distribution is
Let be a random variable generated from a uniform distribution on , thenis a random variable generated from the distribution by the probability integral transform.
3.3. Order Statistics
Order statistics play a vital role in the theory of probability and statistics. Let be a random sample from the distribution. Let denote the th order statistics. Then the of is given bywhere .
3.4. Kullback–-Leibler Divergence
The Kullback–Leibler divergence (or relative entropy) is an informational measure for comparing the similarity between two pdfs. The Kullback–Leibler divergence between the proposed distribution and the distribution, introduced by Korkmaz and Genç , is obtained as
On the other hand, if be a uniform random variable on then the corresponding transmuted distribution is given as
The Shannon entropy of transmuted uniform distribution is computed by
4. TRANSMUTED TSGE DISTRIBUTION
The distribution is specialized by taking as the well-known distribution. We suppose that the base distribution has an exponential distribution with and inverse function and , respectively. The of the parent distribution is . By considering this distribution, the of the distribution can be given asand its is given by
We call this distribution the transmuted TSGE distribution and denote by .
If , we have the and of the TSGE distribution introduced by Korkmaz and Genç . If , and , we have the of the base distribution.
4.1. Density Shape of the TTSG — E Distribution
In the end points of the support, the behaviour of the of the distribution is given as follows:
The derivative is
The right- and left-hand limits of at are given by
These limits are not equal. So, does not exit and the distribution has a corner at . The shape of the distribution for different values of the parameters are plotted in Figs. 2 and 3. Figures 2 and 3 indicate that for and , the distribution is a bimodal distribution and for and , the distribution is a unimodal distribution.
In the next section, we consider the hazard shape of the distribution.
4.2. Hazard Function of the TTSG — E Distribution
The hazard function of the distribution is
Because of complicated form of the hazard function, we couldn't explore this function analytically. We only consider the end points of the support. The behaviour of the hazard function in the end points is given as follows:
5. ESTIMATION OF THE PARAMETERS OF THE TTSG — G DISTRIBUTION
In this section, we obtain the estimation of parameters the distribution by using two methods: MLE and bootstrap estimation. Also, a simulation study is conducted for MLEs of parameters of the distribution.
5.1. Maximum Likelihood Estimation
Let be a random sample of size from the distribution and denote the corresponding order statistics. The log-likelihood function is given bywhere for and , .
For estimating the parameters, we obtain the partial derivatives of the log-likelihood function with respect to the parameters. At the corner point , the log-likelihood function for the distribution is not differentiable and we can not find the estimate of in a regular way. According to Van Dorp and Kotz , we can find the of parameters. We first consider the 's of and when the parameters and are known. Without loss of generality, we assume that . So, the log-likelihood function will be
By taking the derivative of the log-likelihood function with respect to parameter vector and parameter , the s of parameters and are obtained by equating it to zero. These derivatives are given aswhere and .
However, these equations are nonlinear and there are no explicit solutions. Thus, they have to be solved numerically. So, the package is used for estimating the parameters in software.
5.2. Bootstrap Estimation
The parameters of the fitted distribution can be estimated by parametric (resampling from the fitted distribution) bootstrap resampling (see Efron and Tibshirani ). The parametric bootstrap procedure is described as follows:
Parametric bootstrap procedure:
Estimate (vector of unknown parameters), say , by using the procedure based on a random sample.
Generate a bootstrap sample using and obtain the bootstrap estimate of , say , from the bootstrap sample based on the procedure.
Repeat Step 2 times.
Order as . Then obtain -quantiles and confidence intervals (CIs) of parameters.
In case of the distribution, the parametric bootstrap estimators (PBs) of , and , say , and , respectively.
Here, we assess the performance of the s of the parameters with respect to sample size for the distribution. The assessment of performance is based on a simulation study by using the Monte Carlo method. Let , and be the s of the parameters , and , respectively. We calculate the mean square error () and bias of the s of the parameters, and based on the simulation results of 3000 independence replications. results are summarised in Table 1 for different values of, and . From Table 1 the results verify that of the s of the parameters decrease with respect to sample size for all parameters. So, we can see the s of , and are consistent estimators.
|30||0.0274 (0.0316)||0.0165 (0.0099)||0.0855 (0.0481)||0.0272 (0.0229)|
|50||0.0153 (0.0269)||0.0092 (0.0034)||0.0721 (0.0502)||0.0141 (0.0147)|
|100||0.0081 (0.0174)||0.0045 (0.0030)||0.0514 (0.0451)||0.0063 (0.0071)|
|200||0.0035 (0.0098)||0.0020 (0.0018)||0.0243 (0.0245)||0.0033 (0.0046)|
|30||0.0286 (0.0357)||0.0181 (0.0093)||0.0870 (0.0552)||0.2462 (0.0742)|
|50||0.0156 (0.0294)||0.0097 (0.0057)||0.0720 (0.0552)||0.1245 (0.0412)|
|100||0.0079 (0.0202)||0.0042 (0.0011)||0.0484 (0.0470)||0.0632 (0.0352)|
|200||0.0035 (0.0097)||0.0019 (−0.0007)||0.0264 (0.0299)||0.0285 (0.0138)|
|30||32.5268 (0.6036)||0.0291 (0.0479)||0.0914 (−0.0422)||2.9212 (0.1673)|
|50||13.0018 (0.3243)||0.0167 (0.0174)||0.0449 (0.0192)||1.1566 (0.0768)|
|100||0.1474 (0.0976)||0.0063 (0.0019)||0.0236 (0.0454)||0.0132 (0.0142)|
|200||0.0412 (0.0524)||0.0022 (−0.0016)||0.0138 (0.0327)||0.0025 (0.0072)|
|30||13.8834 (0.5349)||0.0291 (0.0320)||0.0946 (−0.0477)||16.7689 (0.4470)|
|50||26.9195 (0.4725)||0.0174 (0.0109)||0.0460 (0.0255)||12.5857 (0.3298)|
|100||2.0210 (0.1251)||0.0067 (−0.0021)||0.0249 (0.0396)||1.5966 (0.0752)|
|200||0.0467 (0.0624)||0.0023 (−0.0025)||0.0150 (0.0388)||0.0256 (0.0241)|
|30||16.0098 (0.6477)||0.0589 (0.0013)||0.8003 (0.8346)||3.5771 (0.2223)|
|50||0.5018 (0.2976)||0.0619 (0.0477)||0.7082 (0.7889)||0.2463 (0.0554)|
|100||0.2836 (0.2171)||0.0600 (0.0730)||0.4518 (0.6418)||0.1114 (0.0304)|
|200||0.0460 (0.1789)||0.0595 (0.0878)||0.3221 (0.5549)||0.0172 (0.0254)|
|30||5.7339 (0.5136)||0.0664 (0.0481)||0.8871 (0.8801)||13.4951 (0.4215)|
|50||2.1495 (0.3433)||0.0707 (0.0714)||0.6986 (0.7841)||8.0247 (0.2459)|
|100||0.0741 (0.2105)||0.0666 (0.0826)||0.4685 (0.6529)||0.2054 (0.0803)|
|200||0.0449 (0.1787)||0.0615 (0.0871)||0.3225 (0.5558)||0.1457 (0.0802)|
|30||83.3071 (3.0259)||0.0518 (0.0820)||0.8762 (0.8905)||1.5715 (0.1644)|
|50||24.7184 (1.9937)||0.0413 (0.0725)||0.9042 (0.9040)||0.4906 (0.0413)|
|100||4.6772 (1.5489)||0.0344 (0.0633)||0.9048 (0.9032)||0.0876 (0.0049)|
|200||2.0712 (1.3450)||0.0255 (0.0435)||0.7545 (0.8364)||0.0120 (−0.0076)|
MLE, maximum likelihood estimation; MSE, mean square error.
MSE and bias (values in parentheses) of the MLEs of the parameters α, β, λ, and θ.
6. APPLICATION OF THE TTSG — E DISTRIBUTION
To investigate the advantage of the proposed distribution, we consider a real data set provided by Bjerkedal . This real data set consists of survival times of 72 guinea pigs injected with different amount of tubercle. This species of guinea pigs are known to have high susceptibility of human tuberculosis, which is one of the reasons for choosing. We consider only the study in which animals in a single cage are under the same regimen. The data represents the survival times of guinea pigs in days. The data are given below:
12 15 22 24 24 32 32 33 34 38 38 43 44 48 52 53 54 54 55 56 57 58 58 59 60 60 60 60 61 62 63 65 65 67 68 70 70 72 73 75 76 76 81 83 84 85 87 91 95 96 98 99 109 110 121 127 129 131 143 146 146 175 175 211 233 258 258 263 297 341 341 376.
6.1. Bootstrap Inference for Parameters of the TTSG — E Distribution
In this section, we obtain point and CI estimation of parameters of the distribution by parametric bootstrap method for the real data set. We provide results of bootstrap estimation based on 10,000 bootstrap replicates in Table 2. It is interesting to look at the joint distribution of the bootstrapped values in a scatter plot in order to understand the potential structural correlation between parameters (see Fig. 6).
CI, confidence interval.
Parametric bootstrap point and interval estimation of the parameters α, β, λ, and θ.
6.2. Inference and Comparing with Other Models
We fit the proposed distribution to the real data set by method and compare the results with the gamma, Weibull, TSGE, generalized exponential , and weighted exponential distributions with respective densities
For each model, Table 3 includes the 's of parameters, Kolmogorov–Smirnov distance between the empirical distribution and the fitted model, its corresponding -value, log-likelihood, and Akaike information criterion for the real data set. We fit the distribution to the real data set and compare it with the mentioned distributions. The selection criterion is that the lowest and statistic corresponding to the best fitted model. The distribution provides the best fit for the data set as it has lower and statistic than the other competitor models. The histogram of data set, fitted of the distribution and fitted s of other competitor distributions for the real data set are plotted in Fig. 7. Also, the plots of empirical and fitted s functions, plots and plots for the and other fitted distributions are displayed in Fig. 7. These plots also support the results in Table 3. The asymptotic covariance matrix of s for model parameters which is the inverse of the Fisher information matrix, is given byand the 95% two-sided asymptotic CIs for , , , and are given by , , , and , respectively.
AIC, Akaike information criterion; GE, generalized exponential; K -- S, Kolmogorov -- Smirnov; MLE, maximum likelihood estimation; TSGE, two-sided generalized exponential, WE, weighted exponential.
The MLEs of parameters for real data set.
6.3. Likelihood Ratio Test
We use the likelihood ratio test for testing the null hypothesis that the distribution, proposed by Korkmaz and Genç , is equally close to the pig data against the alternative hypothesis that the distribution is closer. That is, we wish to testand equivalently, by considering the estimated value of the parameter in Table 3 we should test a one-tailed test as
According to the LRT, the test statistic is given bywhere is the log-likelihood function of distribution. Based on Table 3, . Since is distributed asymptotically chi-squared distribution with 1 degrees of freedom, we can conclude that the null hypothesis is rejected in significance level . Also, the -value is 0.085.
7. CONCLUDING REMARKS
In this paper, we propose a new family of distributions that is a compounding of two-sided distributions family and transmuted technique. The proposed model generalizes TSP distribution and generalized two-sided family of distributions and contains these distributions as its submodels. Some reliability and statistical properties of the proposed family of distribution are discussed through the paper. Estimation and inference procedure for distribution parameters are investigated by two well-known maximum likelihood and bootstrap methods in general setting. The distribution considered as a special case of this family. One of the advantage of this new distribution is that it can be fitted to the data sets with one or two modes. Data analysis shows that the distribution provides the best fit and the best performance. The proposed distribution may be a better alternative than the other well-known distributions commonly used in literature for fitting statistical data.
The authors would like to thank the associate editor and referees for their constructive comments that improved presentation of the paper.
Cite this article
TY - JOUR AU - O. Kharazmi AU - M. Zargar PY - 2019 DA - 2019/06/18 TI - A New Generalized Two-Sided Class of the Distributions Via New Transmuted Two-Sided Bounded Distribution JO - Journal of Statistical Theory and Applications SP - 87 EP - 102 VL - 18 IS - 2 SN - 2214-1766 UR - https://doi.org/10.2991/jsta.d.190306.003 DO - 10.2991/jsta.d.190306.003 ID - Kharazmi2019 ER -