Extended Rama Distribution: Properties and Applications

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.


Introduction
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.

The New Model
Let X and Y be two independent random variables defined on the interval ½0; 1Þ follows the Rama density with parameter h. Suppose that the density of the random variable Z ¼ X þ Y indicates the distribution of interest, and f X ; f Y and f Z denote the densities for random variables X, Y and Z, respectively. Therefore, 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.

Some Properties of ERD
This section presents the r th moment, mean, variance, coefficient of variation, coefficient of skewness and coefficient of kurtosis for the ERD. Some numerical calculations for these properties are also provided.

Moments
Theorem 2: Let X $ f ðx; hÞ. Then, the r th moment of X is given by Proof: The r th moment of the ERD can be obtained by Based on Eq.7, we can obtain the first four moments of the ERD as respectively. Therefore, the variance of the ERD is

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

The Coefficient of Kurtosis
The coefficient of variation of the ERD is defined as

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 l; r; Cv; Sk and Ku for the ERD for various values ofh. The results are presented in Tab. 1. Theorem 3: The moment generating function of the ERD is given by Proof: The moment generating function can be derived as follows …; X n be a random sample of size n chosen from the ERD with parameter h. The maximum likelihood estimator for the ERD parameter can be derived as follows. The likelihood function of h is given by Then, the log-likelihood function is By taking the first derivative of Eq.14 with respect to h and setting the results to 0, we obtain Since there is no closed form solution for Eq. 15, i.e. the MLE of h, denoted asĥ, is the numerical solution for this equation.

Order Statistics
Assume that X 1 ; X 2 ; …X n denote a random sample of size n from the ERD distribution. Also, suppose that X ð1:nÞ ; X ð2:nÞ ; …; X ðn:nÞ denote the corresponding order statistics of the sample. The density function of the i th order statistic X ði:nÞ for 1 i n is given by By substituting the pdf and cdf of the ERD in Eq. (16), the pdf of X ði:nÞ is f ði:nÞ ðx; hÞ From Eq. (17), the pdfs of the smallest order statistic X ð1:nÞ and the largest order statistic X ðn:nÞ are, respectively be given by and

Reliability Analysis
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

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The odds function for ERD is defined as

Practical Illustration
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:   CAIC ¼ À2LL þ 2'n n À ' À 1 ; where ' is the number of parameters and n is the sample size. The Kolmogorov-Smirnov (KS) test is defined as KS ¼ Sup n F n ðxÞ À FðxÞ , where F n ðxÞ ¼ 1 n 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.

Conclusions
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][22][23][24].