The XLindley Distribution: Properties and Application

This paper proposes a new distribution called XLindley distribution (XLD), this distribution is generated as a special mixture of two distributions:exponential and Lindley andhencethe nameproposed.Also, thestatistical properties like stochastic ordering, quantile function, the maximum likelihood method and method of moments. An application of the model to a real data set presented finally and compared with the fit and shows that XLD has more flexibility than others one-parameter distributions. © 2021 The Authors . Published by Atlantis Press B.V. This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).


INTRODUCTION
The real-life applications of contemporary numerical techniques in different fields such as medicine, finance, biological engineering sciences and statistics.To this end, statistics plays a critical role in our real-life applications.Often by using the statistical analysis which strongly depends on the assumed probability model or distributions.However, several problems in statistics do not follow any of the classical or standard probability models.
Recently, Zeghdoudi and Nedjar [9,10] introduced a new distribution named Gamma LD distribution based on mixture of gamma distribution with scale parameter , mixture parameter  and shape parameter 2 and LD distribution with parameter .This idea about mixture of two known distributions is not new, there are a lot of mathematicians has use it before like Shanker and Sharma [11] to create a two parameter LD distribution.
The idea of this work is based on special mixture of exponential and LD distributions in order to create the XLindley distribution (XLD).This work is motivated by the following: XLD it is simple and easy to apply; the formulas of the mean, variance, coefficient of variation, skewness, kurtosis and index of dispersion are simple in form and may be used as quick approximations in many cases.However, in general it is applicable to try out simpler distributions than more complicated ones; the XLD can be used quite effectively in analyzing many real lifetime data set: application to Ebola, Corona and Nipah virus and gives adequate fits too many data sets.
The paper is organized as follows: Section 2 is devoted to introduce the methodology and gives survivals properties of XLD.Section 3 discusses the estimation of its parameter using method of moment and maximum likelihood.Finally, we present illustrative example of XLD with other distributions to show the superiority and flexibility of this model that found.

METHODOLOGY AND SURVIVAL PROPERTIES
In this section, a mixture of two known distributions used to give new distribution called XLD.Let X be a random variable following mixture distribution, it's density function (pdf) f (x) given in this form: With: • f i (x) probability density function for each i • p i i = 1……k denote mixing proportions that are no-negative and We consider f 1 (x) ∼ Exp () and f 2 (x) ∼ LD () two independents random variables with p 1 =  1+ and p 2 = 1 −  1+ respectively.Now the density function of X is given by: The first derivative of f XL is: gives: And the second derivative is: Therefore, the mode of XL is given by: mode We can find easily the cumulative distribution function (CDF) of the XLD: The shapes of the PDF, CDF and hazard function of the XLD distribution are given in Figures 1-3 for different values of the parameter theta.

SURVIVAL AND HAZARD RATE FUNCTION
The survival function and failure rate (hazard rate) function for a continuous distribution are defined as: Let: and: ) be the survival and hazard rate function, respectively.

Proposition 1. Let H XL (x) be the hazard rate function of X. Then H XL (x) is increasing.
Proof.According to Glaser [12] and from the density function (2): It follows that: Imply that h XL (x) is increasing.

MOMENTS AND RELATED MEASURES
The rth moment about the origin of the XLindey distribution can be obtained as: Using gamma integral and little algebraic simplification, we get finally a general expression for the rth factoriel moment of XLD as: Substituting r =1,2,3 and 4 in ( 9), the first four moments can be obtained and then using the relationship between moments about origin and moment about mean, the first four moment about origin of XLD were obtained as: Proposition 2. Let X ∼ XL(x), the mean, variance, coefficients of variation, skewness and kurtosis for X are: Skewness, Kurtosis and Coefficient of variation of XLD: The coefficients are increasing functions in  (see Figure 4 for the graphe of C.V () and Skewness √  1 for varying ).

STOCHASTIC ORDERING
Definition 1.Consider two random variables X and Y. Then X is said to be smaller than Y in the: , if for all convex functions  and provided expectation exist, Convex order⇔ Stochastic order.
Proof.We have: For simplification, we use ln ) . Now, we can find ) ≤ 0. This means that X 1 ≺ lr X 2 .Also, according to Remark 1 the theorem is proved.

Method of Moments Estimation
Let X be the sample mean, equating sample mean and population mean E(x): Putting the expression of E(x) from equation (10) in the equation and solving the equation for , We get: We obtain equation of 3rd degree: We take the real part for the solution:

Maximum Likelihood Estimation
Let X i ∼ XL () , i = 1.......n be n random variables.The ln-likelihood function ln l(x i ; ) is: Logarithm of likelihood function is: The derivative of ln l(x i ; ) with respect to  is: To obtain the maximum likelihood estimation (MLE) of  ∶ θMLE can maximize equation (11) directly with respect to , or we can solve the non-linear equation dln l(x i ;) d = 0. Note that θMLE cannot solved analytically; numerical iteration techniques, such as the Newton-Raphson algorithm, are thus adopted to solve the Logarithm of likelihood equation for which (11) is maximized.
The following theorem shows that the estimator of  is positively biased.

Theorem.2. the estimator
and it is easy to find dt 2 > 0, since g (t) is strictly convex.Thus, by Jensen's inequality, we haveE Theorem 3. The estimator θ of  is consistent and asymptotically normal: ) The large-sample 100 (1 − ) % confidence interval for  is given by: The proof is omitted because it is very similar to the proof of Theorem 4 [7]

THE QUANTILE FUNCTION OF XLD
It may be noted that F X (x) in equation ( 5) is continues and strictly increasing, so we for the quantile function of X is defined: For u = F XL (x), we give an explicit expression for Q X (u) in terms of the Lambert W function in the following theorem and results.
Theorem 4. For any  ≻ 0, the Q X (u) of the XLD X is: Where W −1 is the negative branch.
Proof.For any  ≻ 0 let 0 ≺ u ≺ 1.From equation (5) we will solve the equation u = F XL (x) with respect to x, by following the steps bellow: We multiplying the both sides by This in turn means the result that given before in Theorem 4 is complete.

Figure 1 Figure 2 Figure 3
Figure 1 Plots of the density function for some parameter values