NEW COMPOUND PROBABILITY DISTRIBUTION USING BIWEIGHT KERNEL FUNCTION AND EXPONENTIAL DISTRIBUTION

In this paper, a new continuous probability distribution is proposed for fitting real data using Biweight kernel function and the exponential distribution. The suggested distribution is named the Biweight-exponential distribution (BiEd). Some statistical properties of this distribution are derived and illustrated mathematically. The probability density function and the cumulative distribution function are derived. Some reliability analysis functions are defined. The moments and moment generating function are derived. Re’nyi entropy is derived. The maximum likelihood method of estimation is used to derive the parameter estimates. The Bonferroni and Lorenz curves and Gini index equations are derived. The distribution of the order statistic and the quantile function are derived as well. The mean and median absolute deviations of the new distribution are derived. A numerical study was conducted to the quantile equation. An application to real data set is conducted to investigate the usefulness of the suggested distribution. In the real data application, the values of Cramer-von misses (W), Anderson Darling statistic (A), KolmogorovSmirnov (D) statistic, the p-value, the maximum likelihood estimates (MLE), Akaike information criterion (AIC), consistent Akaike information criterion (CAIC), Bayesian information criterion (BIC), Hannan Quinn information 5879 BIWEIGHT KERNEL FUNCTION AND EXPONENTIAL DISTRIBUTION criterion (HQIC) and minimum value of the log-likelihood function are obtained. The distribution is compared with the exponential (base) distribution based on these criteria. The results showed that the BiED fits better than the exponential distribution. This means that the suggested distribution can replace the exponential distribution for analysis of some real data sets.


INTRODUCTION
The most significant one parameter of life distributions is the exponential distribution. There are many important problems where the real data does not follow any of the classical known probability distributions [1]. There are many generalizations of new continuous distribution based on exponential distribution such that: generalized exponential [2], beta exponential [3], beta generalized exponential [4], Kumaraswamy exponential [5], gamma exponentiated exponential [6], Transmuted exponentiated exponential distribution [7]. Numerous researchers proposed new distributions using different families. For examples; [8] suggested the transmuted Janardan distribution. [9] generated the transmuted Burr type XII distribution: a generalization of the Burr type XII distribution. [10] suggested a generalization of the new Weibull-Pareto distribution. The transmuted two-parameter Lindley distribution is proposed by [11]. [12] proposed the transmuted Shanker distribution. [13] used the same map to develop Mukherjee-Islam distribution. [14] worked out the transmuted Ishita distribution using the quadratic transmutation map. In this article, we present a new generalization of the exponential distribution via Biweight kernel function [15] and call it the Biweight exponential distribution (BED).
The rest of this article is organized as follows: in Section 2, we defined the materials and methods of this article. Section 3 defines the new distribution and its pdf and cdf. The reliability analysis is defined in Section 4. The moments and moment generating function are derived in Section 5. We have used the maximum likelihood method of estimation to estimate the parameters in Section 6.

MATERIALS AND METHODS
The kernel is any function having the following properties . Many kinds of kernel function can be initiated in the related literature [16] - [18].
One of the symmetric kernels is the Biweight kernel function [24]. It is defined as Definition: A random variable X is said to have a Biweight probability Distribution with its cumulative distribution function (CDF) () Gx and probability density function (PDF) () gx are obtained respectively as follows

BIWEIGHT-EXPONENTIAL DISTRIBUTIONS
A random variable X is said to have an exponential distribution with parameter 0   if its pdf and cdf are given by respectively Now using (2.2) and (3.2), the cdf of BiED is defined as Hence, the pdf of BiED is given by

RELIABILITY ANALYSIS
The reliability is concerned with the calculation and prediction of the probability of the limit state The reliability function or the survival function is defined as ( ) The reliability of the BiED is defined as: The hazard rate function of the BiED is defined as

MOMENTS
The r th moment of the BiED is given by the following theorem: The integrals can be solved by the u-substitution as Then, we substitute these assumptions in equation (5.2), we have Simplify and rearrange, we have ( ) We can find the first moment (mean) The following theorem defines the moment generating function of BiED.
Theorem 5.2: Let X be a random variable that follows a BiED, therefore the moment generating function (MGF) is defined as: By using the u-substitution as follows

MAXIMUM LIKELIHOOD ESTIMATES
Maximum likelihood method of estimation is a well-known method of estimating the distribution parameters. The likelihood function used the parameter as a variable conditional to the observations. For the BiED, the maximum likelihood estimator of the distribution parameter is given as the following. The joint pdf of 1 ,..., n XX is given by By taking the log of () L  as By taking the partial derivative with respect to  , we have This solution of equation gives the maximum likelihood estimates of parameter  . The solution can be solved numerically with use the appropriate software like R when data set are available.

QUANTILE FUNCTION
The quantile q of the random variable, say X that follows BiED is the solution of the equation x q q = . Hence, 5 15  Table 1.
Therefore, the pdf of BiED of ( ) j X is given by Moreover, the pdf of the largest order statistic ( ) n X of the BiED is given by 1 And the pdf of the smallest order statistic ( ) 1 X of the BiED is given by

RÉNYI ENTROPY
The Rényi entropy for the BiED is defined by the following theorem:

BONFERRONI AND LORENZ CURVES AND GINI INDEX OF THE BIED
Assume that the random variable X is non-negative with continuous and twice differentiable cumulative distribution function G(x The Gini index of the BiED is given by So, the Gini index of the BiED is defined as

MEAN AND MEDIAN DEVIATIONS OF THE BIED
The mean deviation about the mean 1 8.8, 8.8, 8.9, 8.9, 9.5, 9.6, 9.7, 9.8, 10.7, 10.9, 11.0, 11.0, 11.1, 11.2, 11.2, 11.5, 11.9, 12.4, 12.5, 12.9, 13.0, 13 The forth real data set is given by [22] and it represents the number of million revolutions before failure for each of 23  The aim of generalizing any distribution is to make it more flexible. As presented in Tables   2-6 for all datasets, the BiED was recognized to be more flexible than the exponential distribution.

CONCLUSION
In this paper, we use the Biweight Kernel function (BKF) and exponential distributions to propose a new distribution called the Biweight exponential distribution. The new distribution is introduced without adding any new parameters to the original distributions. The statistical properties are considered including the estimation of model parameters and its application demonstrated using real datasets. Applications show that the new distribution fits better than the exponential distribution.