Computer Systems Science & Engineering DOI:10.32604/csse.2021.014909 | |

Article |

Extended Rama Distribution: Properties and Applications

1Department of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 UKM, Bangi, Selangor, Malaysia

2Department of Mathematics, Faculty of Science, Al al-Bayt University, Mafraq, Jordan

*Corresponding Author: Khaldoon M. Alhyasat. Email: p105764@siswa.ukm.edu.my

Received: 26 October 2020; Accepted: 05 February 2021

Abstract: In this paper, the Rama distribution (RD) is considered, and a new model called extended Rama distribution (ERD) is suggested. The new model involves the sum of two independent Rama distributed random variables. The probability density function (pdf) and cumulative distribution function (cdf) are obtained and analyzed. It is found that the new model is skewed to the right. Several mathematical and statistical properties are derived and proved. The properties studied include moments, coefficient of variation, coefficient of skewness, coefficient of kurtosis and moment generating function. Some simulations are undertaken to illustrate the behavior of these properties. In addition, the reliability analysis of the distribution is investigated through the hazard rate function, reversed hazard rate function and odds function. The parameter of the distribution is estimated based on the maximum likelihood method. The distributions of order statistics for ERD are also presented. The performance of the suggested model is compared with several other lifetime distributions based on some goodness of fit tests on a real dataset. It turns out that the suggested model is more flexible than its competitors considered in this study, for modeling real lifetime data.

Keywords: Coefficient of kurtosis; coefficient of skewness; order statistics; two independent Rama random variables; Maximum likelihood estimation; Rama distribution

Recently, some studies on the distribution of sum of random variables have acquired some significant importance in various branches of science for fitting real data. Many authors have studied the distribution of the sum of random variables and their applications based on various base distributions, to suggest new flexible distributions. As an example, [1] suggested a power length-biased Suja distribution, whereas [2] derived the distribution for the sums of uniformly distributed random variables. Moreover, [3] introduced the distribution of the sum of mixed independent random variables pertaining to special functions. Also, [4] proposed the sum and difference of two squared correlated Nakagami variates which are in connection with the McKay distribution. In contrast, the sum of t and Gaussian random variables is suggested by [5], while [6] proposed the sum of independent gamma random variables. Reference [7] studied the sum of independent gamma random variables, while generalizations of two-sided power distributions and their convolution are introduced by [8]. Besides, [9] derived the distribution of the mixed sum of independent random variables where one of them is associated with the H-function. On the other hand, [10] obtained the distribution of the sum of mixed independent random variables pertaining to certain special functions, while [11] proposed the weighted Suja distribution and its applications in engineering. Moreover, [12] suggested a Darna distribution as a mixture of exponential and gamma distributions. Reference [13] proposed a power size biased two-parameter Akash distribution, and [14] suggested a generalization of the new Weibull-Pareto distribution. In 2019, [15] introduced a transmuted Ishita distribution and [16] proposed a Marshall-Olkin length-biased exponential distribution.

In this paper, we modified the Rama distribution, which was firstly suggested by [17], while considering the sum of two independent variables in order to propose a new lifetime distribution, named extended Rama distribution. The probability density function (pdf) of the Rama distribution is given by

and the corresponding cumulative distribution function (cdf) is given by

The rest of the paper is organized as follows. In Section 2, the new distribution is presented in terms of its functions and shapes. Some statistical properties are given in Section 3, while in Section 4, the parameter of the distribution is estimated using the maximum likelihood method. Distributions of order statistics based on samples selected from ERD are presented in Section 5. Moreover, the reliability analysis based on ERD is given in Section 6. An application using a real data set is provided in Section 7. Finally, the paper is concluded in Section 8.

Let X and Y be two independent random variables defined on the interval

and

Hence, a random variable X is said to follow the ERD if its pdf is given by

It is easy to show and validate Eq. (5) as follows:

It is of interest to note here that

Theorem 1: Let X be a random variable follows the ERD with parameter θ. The cumulative distribution function of X is defined as

Proof: The cdf of the ERD can be derived as follows:

To study the behavior of the ERD, we consider x ∈ (0, 2]. In Fig. 1, the plots of the pdf and cdf of the ERD for various values of the parameter of the distribution are presented.

Referring to Fig. 1, it is clear that ERD is asymmetric and skewed to the right over the interval (0, 2]. The degree of the skewness depends on the parameter value.

This section presents the

Theorem 2: Let

Proof: The

Based on Eq.7, we can obtain the first four moments of the ERD as

respectively. Therefore, the variance of the ERD is

2.2 The Coefficient of Skewness

The coefficient of skewness determines the degree of skewness of a distribution and for the ERD it is given by

2.3 The Coefficient of Kurtosis

The coefficient of variation of the ERD is defined as

2.4 The Coefficient of Variation

The coefficient of variation of the ERD is defined as

To investigate the behavior of these measures, we calculate the values of

Based on Tab. 1, it can be deduced that:

1. As the values of

2. As the values of

3. The coefficient of skewness and coefficient of kurtosis are decreasing as the values of theta are increasing up to

Theorem 3: The moment generating function of the ERD is given by

Proof: The moment generating function can be derived as follows

3 Maximum Likelihood Estimation

Let

Then, the log-likelihood function is

By taking the first derivative of Eq.14 with respect to

Since there is no closed form solution for Eq. 15, i.e. the MLE of

Assume that

By substituting the pdf and cdf of the ERD in Eq. (16), the pdf of

From Eq. (17), the pdfs of the smallest order statistic

and

This section defines the reliability (survival) function, hazard rate function, reversed hazard rate function and odd function for the suggested model. The reliability function of the ERD is given by

On the other hand, the hazard rate function of the ERD is given by

Meanwhile, the reversed hazard rate function for the ERD distribution is defined as

The odds function for ERD is defined as

Figs. 2–4 reveal that the reliability and reversed hazard rate functions decrease as the values of x increase. In contrast, the hazard and odd functions are increasing with x for various values of the parameter θ.

This section compares the performance of ERD with Rama distribution, exponential distribution, Rani distribution and Maxwell length-biased distribution for fitting a real data set. The probability density functions for these distributions are as follows:

a) Rama distribution (RD) [17]:

b) Exponential distribution (ED):

c) Rani distribution (RND) [18]:

d) Maxwell length-biased distribution (MLBD) [19]:

The dataset explains the strength of the aircraft window glass as recorded by [20], which is given as follows:

Data: 18.83, 20.80, 21.657, 23.03, 23.23, 24.05, 24.321, 25.50, 25.52, 25.80, 26.69, 26.77, 26.78,27.05, 27.67, 29.90, 31.11, 33.20, 33.73, 33.76, 33.89, 34.76, 35.75, 35.91, 36.98, 37.08,37.09, 39.58, 44.045, 45.29, 45.381.

For comparison, we consider the Akaike Information Criterion (AIC), consistent Akaike information criterion (CAIC), Bayesian information criterion (BIC), Hannan-Quinn information criterion (HQIC) and Kolmogorov-Smirnov (KS) test statistics for measuring the goodness of fit (GOF) of the different models to the data. Parameters of the models are estimated using maximum likelihood. These (GOF) measures are defined as follows:

where

Based on Fig. 5, we notice that ERD provides the best fit for modelling the dataset. The p-value of the KS test is 0.39 which is greater than the 0.05 level of significance, indicating that ERD model is adequate for the data. In addition, the other measures of goodness of fit are found the smallest for the ERD, with the respective larger p-values, which further support that ERD as the best model.

In the present paper, an extended Rama distribution is proposed. Many mathematical and statistical properties of the new distribution are provided. The reliability, hazard rate, reversed hazard rate and odd functions of the ERD are also presented. The parameter of this distribution is obtained using MLE. In addition to a simulation study, the performance of ERD in fitting a real dataset is compared to several other models which are often used to describe the lifetime data based on several goodness of fit tests. It turns out that the ERD can be considered as a viable alternative model when modelling lifetime data. As for future work, one can estimate the parameter of the distribution using the ranked set sampling method [21–24].

Acknowledgement: The authors are sincerely grateful to the anonymous referees and the editor for their time and effort in providing constructive, and valuable comments and suggestions that have led to a substantial improvement in the paper.

Funding Statement: The authors extend their appreciation to Universiti Kebangsaan Malaysia for providing a partial funding for the work under the grant number GGPM-2017-124 and TAP-K017073 which were obtained by Mohd Aftar Abu Bakar.

Conflicts of Interest: The authors declare that they have no conflicts of interest to report the findings of the present study.

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