Burhan Distribution with Structural Properties and Applications in Distinct Areas of Science

In this work a novel distribution has been explored referred as Burhan distribution. This distribution is obtained through convex combination of exponential and gamma distribution to analyse complex real-life data. The distinct structural properties of the formulated distribution have been derived and discussed. The behaviour of probability density function (pdf) and cumulative distribution function (cdf) are illustrated through different graphs. The estimation of the established distribution parameters are performed by maximum likelihood estimation method. Eventually the versatility of the established distribution is examined through two real life data sets.


Introduction
In biomedicine, engineering, economics, and other fields of science, statistical distribution plays essential role in modelling data. The exponential and gamma distributions are popular distributions for assessing statistical data and are regarded as life time distributions. Among these distributions, the exponential distribution has one parameter and several attractive statistical characteristics, such as memory less and a constant hazard rate. Various extensions of these distributions have been made in the statistical literature to lead greater flexibility. Lindley formulated a one parameter life time distribution with the following probability density function in (1958).

Moments of Burhan distribution
Letbe a random variable follows Burhan distribution. Then . /0 moment denoted by 1 2 3 is given as

Moment generating function of Burhan distribution
Letbe a random variable follows Burhan distribution. Then the moment generating function of the distribution denoted by > ? @ is given .

Renyi Entropy of Burhan Distribution
Ifdenotes a continuous random variable having probability density function , , , . Then Renyi entropy is defined as where F > 0 and F ≠ 1.
Thus, the Renyi entropy of Burhan distribution (1.1) is given as Using generalized binomial theorem, we have After solving the integral, we get

Tsallis Entropy of Burhan Distribution
Tsallis entropy of order F for Burhan distribution (1.1) is given as where F > 0 and F ≠ 1 After solving the integral, we get

Mean Deviation from Mean of Burhan Distribution
The quantity of scattering in a population is evidently measured to some extent by the totality of the deviations. Letbe a random variable from Burhan distribution with mean 1. Then the mean deviation from mean is defined as K.
After solving the integral, we get Using equation (5.2) and (1.2) in equation (5.1), we obtain

Mean Deviation from Median of Burhan Distribution
Letbe a random variable from Burhan distribution with median >. Then the mean deviation from median is defined as

Reliability Measures
Supposebe a continuous random variable with cdf % , ≥ 0. Then its reliability function which is also called survival function is defined as Therefore, the survival function for Burhan distribution is given as The hazard rate function of a random variable is given as    Mean residual function of random variable y can be obtained as c , , , = 1 Q , , , 5 @ @, , , After solving the integral, we get c , , , = + 1 + + 2,

Order Statistics of Burhan Distribution
Let -, -, -: , . . . . ,d denotes the order statistics of a random sample drawn from a continuous distribution with cdf % and pdf , then the pdf ofe is given by ? e -, = f! g − 1 ! f − g ! ? W% ? X e W1 − % ? X d e g = 1,2,3. . . . , f. 8.1 Substitute the equation (1.1) and (2.2) in equation (8.1), we obtain the probability function of g /ℎ order statistics of Burhan distribution is given by Then, the pdf of first order statisticsof Burhan distribution given by And the pdf of fth order statisticsd of Burhan distribution is given by

Maximum Likelihood Estimator of Burhan Distribution
Let -, -, . . ,d be a random sample of size f from Burhan distribution. Then its likelihood function is given by The partial derivatives of equation (9.1), with respect parameters are given as

9.4
From equations (9.2), (9.3) and (9.4), we have obtained a system of non-linear equations which cannot be expressed in compact form and is difficult to solve explicitly for , and . Applying the iterative methods such as Newton-Raphson method, secant method,

Regula-falsi method etc. The MLE of theparameters denoted as
can be obtained by using the above methods.
For interval estimation and hypothesis tests on the model parameters, an information matrix is required. The 3 by 3 observed matrix is where † ‡ denotes the u /ℎ percentile of the standard normal distribution.

Applications
This section exhibits the adaptability of the formulated distribution by applying realworld data sets. The suggested distribution is compared to the Shanker distribution (SHD), the Akash distribution (AD), the Ishita distribution (ID), the Pranav distribution (PD), the Rani distribution (RD), the Prakaamy distribution (PKD), and the Lindley distribution. We were efficient in outranking the specified distribution.
In order to compare the above distribution models, we consider the criteria like AIC (Akaike information criterion), CAIC (corrected Akaike information criterion), BIC (Bayesian information criterion). Among the above distributions, the better distribution is considered having lesser values of AIC, CAIC, HQIC and BIC.

Data set 1:
This data is obtained from Murthy et al. [13], which represents failure times of 84 Aircraft Windshield. The data follows:   Data set 2: The data set represents the survival times (in days) of 72 guinea pigs infected with virulent tubercle bacilli, observed and reported by Bjerkedal [12]. The data are as follows:    It is clear from Table 10.3 and 10.6 that Burhan distribution (BHD) has least values of -GH , AIC, CAIC, BIC and HQIC when compared with competitive distributions. We accomplish that Burhan distribution provides an adequate fit than compared distributions.

Conclusion
In the present paper a novel distribution has been formulated by employing convex combination of exponential and gamma distribution stated as Burhan distribution (BHD). Its several properties including moments, moment generating function, survival function, hazard rate function, mean residual life function, mean deviations, order statistics, Renyi entropy and Tsallis entropy have been discussed. The parameters of the distribution have been estimated by known method of maximum likelihood estimator. Finally the performance of the model has been examined through two data sets and compared which shows that Burhan distribution gives an adequate fit for the data sets.