Journal of Statistical Theory and Applications

Volume 20, Issue 2, June 2021, Pages 318 - 327

The XLindley Distribution: Properties and Application

Authors
Sarra Chouia, Halim Zeghdoudi*
LaPS Laboratory, Badji Mokhtar University, Annaba, Algeria
*Corresponding author. Email: halimzeghdoudi77@gmail.com
Corresponding Author
Halim Zeghdoudi
Received 19 March 2019, Accepted 31 January 2021, Available Online 10 July 2021.
DOI
https://doi.org/10.2991/jsta.d.210607.001How to use a DOI?
Keywords
Exponential distribution; Lindley distribution; quantile function; method of moment; maximum likelihood method; simulation
Abstract

This paper proposes a new distribution called XLindley distribution (XLD), this distribution is generated as a special mixture of two distributions: exponential and Lindley and hence the name proposed. Also, the statistical 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.

Copyright
© 2021 The Authors. Published by Atlantis Press B.V.
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. 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.

Let X is a random variable following the one-parameter distribution with density function called Lindley (LD) distribution:

fx;θ=θ2(1+x)eθx1+θx,θ00otherwise(1)

It has introduced by Lindley [1]. Sankaran [2] used (1) as mixing distribution of poisson parameter which it named poisson-LD distribution. Recently, Asgharzadeh et al. [3], Zeghdoudi and Nedjar [4], Zeghdoudi and Bouchahed [5], Beghriche and Zeghdoudi [6], Ghitany et al. [7,8] rediscovered and studies the distribution bounded to (1), what that derived is known as Zero-truncated poisson-LD and pareto poisson-LD distributions.

Recently, Zeghdoudi and Nedjar [9,10] introduced a new distribution named Gamma LD distribution based on mixture of gamma distribution with scale parameter θ, mixture parameter β and shape parameter 2 and LD distribution with parameter θ. 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.

2. 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) fx given in this form:

fx=i=1kpi.fix

With:

  • fix probability density function for each i

  • pi i = 1……k denote mixing proportions that are no-negative and i=1kpi=1

We consider

f1x~Expθ and f2x~LDθ two independents random variables with p1=θ1+θ and p2=1θ1+θ respectively. Now the density function of X is given by:

fXLx;θ=θ2(2+θ+x)eθx1+θ2x,θ00otherwise(2)

The first derivative of fXL is:

dfXLxdx=θ2θ2+θ2+x11+θ2eθx=0(3)

gives:

x=θ2+2θ1θ

For:

  1. 0θ21:x=θ2+2θ1θ is critical point which fXLx;θ is maximum

  2. θ21:ddxfXLx;θ0, then the density function fXLx;θ is decreasing in x

And the second derivative is:

d2fXLxdx2=θ3θ2+θ2+x21+θ2eθx(4)

Therefore, the mode of XL is given by:

modeX=θ2+2θ1θfor 0θ210otherwise

We can find easily the cumulative distribution function (CDF) of the XLD:

FXLx;θ=11+θx1+θ2eθx   x0,θ0(5)

The shapes of the PDF, CDF and hazard function of the XLD distribution are given in Figures 13 for different values of the parameter theta.

Figure 1

Plots of the density function for some parameter values

Figure 2

Plots of cumulative distribution function (CDF) for some parameter values

Figure 3

Plots of hazard function for some parameter values: blue (0.25); pink (0.5); red (3); black (4)

3. SURVIVAL AND HAZARD RATE FUNCTION

The survival function and failure rate (hazard rate) function for a continuous distribution are defined as:

Let:

SXLx=1FXLx
SXLx=111+θx1+θ2eθx
SXLx=1+θx1+θ2eθx(6)

and:

HXLx=fXLx1FXLx
HXL(x)=θ2(x+θ+2)(1+θ)2xθ(1+θ)2+1
HXLx=θ2x+θ+21+θ2+xθ(7)
be the survival and hazard rate function, respectively.

Proposition 1.

Let HXL (x) be the hazard rate function of X. Then HXL (x) is increasing.

Proof.

According to Glaser [12] and from the density function (2):

ρx=fXLxfXLx=xθ+θ2+2θ1x+θ+2=1x+θ+2xθ+θ2+2θ1

It follows that:

ρx=1x+θ+22(8)

Imply that hXL(x) is increasing.

4. MOMENTS AND RELATED MEASURES

The rth moment about the origin of the XLindey distribution can be obtained as:

μr=ΕXr=0xrfXLxdx=0xrθ22+θ+x1+θ2eθxdx=θ21+θ20x(r)2+θ+xdx

Using gamma integral and little algebraic simplification, we get finally a general expression for the rth factoriel moment of XLD as:

μr=θ2+2θ+r+1r!1+θ2θr(9)

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:

μ1=θ2+2θ+21+θ2θ=1θ+11+θ2θ
μ2=2θ2+2θ+31+θ2θ2
μ3=6θ2+2θ+41+θ2θ3
μ4=24θ2+2θ+51+θ2θ4

Proposition 2.

Let X ∼ XL(x), the mean, variance, coefficients of variation, skewness and kurtosis for X are:

μ1=ΕX=1+θ2+11+θ2θ(10)
ΕX2=2θ2+2θ+31+θ2θ2
μ2=VarX=1+θ4+4θ2+6θ+11+θ4θ2

Skewness, Kurtosis and Coefficient of variation of XLD:

Skewness=β1=ΕX3VarX32=6θ2+2θ+41+θ41+θ4+4θ2+6θ+132
Kurtosis=β2=ΕX4VarX2=24θ2+2θ+51+θ61+θ4+4θ2+6θ+12
C.V=γ=VarXΕX=1+θ4+4θ2+6θ+11+θ2+1

The coefficients are increasing functions in θ (see Figure 4 for the graphe of C.V (γ) and Skewness β1 for varying θ).

Figure 4

Coefficients for variation (red) and skewness (black)

5. STOCHASTIC ORDERING

Definition 1.

Consider two random variables X and Y. Then X is said to be smaller than Y in the:

  1. Stochastic orderXSY if FXtFYt,t.

  2. Convex order XCXY, if for all convex functions ϕ and provided expectation exist, ΕϕXΕϕY.

  3. Hazard rate order XhrY, if hX(t)hY(t),t.

  4. Likelihood ratio order XlrY, if fXtfYt is decreasing in t.

Remark 1.

Likelihood ratio order Hazard rate order Stochastic order. If ΕX=ΕY, then:

Convex order Stochastic order.

Theorem 1.

Let XiXLθi,i=1,2 be two random variables. If θ1θ2, then X1lrX2, X1hrX2,X1SX2.

Proof.

We have:

fX1tfX2t=θ122+θ1+t1+θ22θ222+θ2+t1+θ12eθ1θ2t

For simplification, we use lnfX1tfX2t. Now, we can find

ddtlnfX1tfX2t=θ1θ2t+θ1+2t+θ2+2θ1θ2

To this end, if θ1θ2, we have ddtlnfX1tfX2t0. This means that X1lrX2. Also, according to Remark 1 the theorem is proved.

6. ESTIMATION OF PARAMETER

6.1. Method of Moments Estimation

Let X¯ be the sample mean, equating sample mean and population mean E(x):

Εx=i=1nxin

Putting the expression of E(x) from equation (10) in the equation and solving the equation for θ, We get:

X¯=1+θ2+11+θ2θ=θ2+2θ+2θ3+2θ2+θ

We obtain equation of 3rd degree: X¯θ3+θ22X¯1+θX¯22=0, We take the real part for the solution:

θ^MoM=13X¯2X¯1+29X¯+19X¯2+193127X¯+1336X¯2+19X¯3+127X¯4+1118X¯+19X¯2+127X¯3+127+3127X¯+1336X¯2+19X¯3+127X¯4+1118X¯+19X¯2+127X¯3+17X¯

6.2. Maximum Likelihood Estimation

Let XiXLθ,i=1.n be n random variables. The ln-likelihood function lnl(xi;θ) is:

Lθ=θ21+θ2ni=1n2+θ+xieθi=1nxi

Logarithm of likelihood function is:

lnlxi;θ=2nlogθ2nlog1+θ+i=1nlog2+θ+xiθi=1nxilnlxi;θ=2nlogθlog1+θ+i=1nlog2+θ+xiθi=1nxi

The derivative of lnl(xi;θ) with respect to θ is:

dlnlxi;θdθ=0dlnlxi;θdθ=2nθ2n1+θ+i=1n12+θ+xii=1nxidlnlxi;θdθ=2θ21+θ+1ni=1n12+θ+xiX¯dlnlxi;θdθ=2θ1+θ+1ni=1n12+θ+xiX¯(11)

To obtain the maximum likelihood estimation (MLE) of θ:θ^MLE can maximize equation (11) directly with respect to θ, or we can solve the non-linear equation dlnlxi;θdθ=0. Note that θ^MLE 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 θ is positively biased.

Theorem 2.

the estimator θ^ of θ is positively biased, i.e: Εθ^θ>0

Proof.

Let g(x¯)=θ^

gt=13t2t1+29t+19t2+193127t+1336t2+19t3+127t4+1118t+19t2+127t3+127+3127t+1336t2+19t3+127t4+1118t+19t2+127t3+17t
and it is easy to find d2gtdt2>0, since gt is strictly convex. Thus, by Jensen's inequality, we haveΕgx¯>gΕx¯.

Finally, since Εgx¯=gμ=g1+θ2+11+θ2θ=θ, we obtain Εθ^MoMθ

Theorem 3.

The estimator θ^ of θ is consistent and asymptotically normal:

nθ^θPN0,1σ2

The large-sample 1001α% confidence interval for θ is given by:

θ^±zα21nσ2

The proof is omitted because it is very similar to the proof of Theorem 4 [7]

7. THE QUANTILE FUNCTION OF XLD

It may be noted that FX(x) in equation (5) is continues and strictly increasing, so we for the quantile function of X is defined:

QXu=xu=FX1u(12)

For u=FXLx, we give an explicit expression for QXu in terms of the Lambert W function in the following theorem and results.

Theorem 4.

For any θ0, the QXu of the XLD X is:

QXu=xu=1+θ2θ1θW11+θ2exp1+θ2u1    ,u0,1(13)

Where W1 is the negative branch.

Proof.

For any θ0 let 0u1. From equation (5) we will solve the equation u=FXL(x) with respect to x, by following the steps bellow:

1+θx1+θ2eθx=1u
1+θ2+θxeθx=1u1+θ2(14)

We multiplying the both sides by exp1θ2 of the equation (14), we get:

1+θ2+θxe1+θ2+θx=u11+θ2e1+θ2(15)

By using the definition of Lambert W function WzexpWz=z [see Jodrá [13] for more details], we observe that 1+θ2+θx is the Lambert W function of the real argument u11+θ2e1+θ2.

So, we have

Wu11+θ2e1+θ2=1+θ2+θxW1+θ2e1+θ2u1e1+θ2=1+θ2+θx(16)

In addition to that, for any θ0 and x0 0 it's obviously that 1+θ2+θx0 and it also checked that u11+θ2e1+θ21e,0 since 0u1 Thus, by taking into account the properties of the negative branch W1 of the Lambert W function, so the equation above (16) become:

W11+θ2e1+θ2u1e1+θ2=1+θ2+θx(17)

This in turn means the result that given before in Theorem 4 is complete.

8. SIMULATION

We can see that the equation Fx=u, where u is an observation from the uniform distribution on (0; 1), can be solved explicitly in x (we're going to use Lambert W function, because in this case k = 1).

In this subsection, we investigate the behavior of the ML estimators for a finite sample size (n). A simulation study consisting of the following steps is being carried out N = 10000 times for selected values of θ;n, where θ = 0.1; 0.5; 1; 3; 5 and n = 20; 40; 100

  • Generate Ui Uniform(0;1), i=1.n

  • Generate Yi exponential (θ), i=1.n

  • Generate Zi LD (θ), i=1.n

  • If Uipθ, then set Xi=Yi, otherwise, set Xi=Zi, i=1.n

average bais=1Ni=1Nθ^θ
and the average square error:
MSEθ=1Ni=1Nθ^θ2

The result of the simulation is presented in Tables 1 and 2. The following observations are made from the simulation study:

  • For some given value of θ, the average of: bias of θ^ and mean square error of θ^ are decreases as sample size n increases

  • The mean square error (MSE) gets higher and following a similar ways for larger value of θ as we mentioned before.

Bias θ=0.1 θ=0.5 θ=1.5 θ=3 θ=6
n=20 0.0032 0.01067 0.0485 0.276 0.785
n=40 0.00183 0.0150 0.0135 0.126 0.1770
n=100 0.000321 0.00404 0.0147 0.0452 0.0598

MSE, mean square error.

Table 1

Average bias of the estimator θ^.

MSE θ=0.1 θ=0.5 θ=1 θ=3 θ=5
n=20 1,0854.10-5 0.000113 0.00236 0.0765 0.6177
n=40 3,357.10-6 0.000225 0.000183 0.01599 0.03135
n=100 1,0334.10-7 1,640.10-5 0.000217 0.00204 0.00358
Table 2

The average square error of the estimator θ^.

9. APPLICATION AND GOODNESS OF FIT

Now we have used data of survival times (in months) of 94 sierra leone individus infected with Ebola virus showing in Table 3, which we compare LD, Zeghdoudi (ZD), see (Messaadia and Zeghdoudi [14]), Exponential, Xgamma see (Sen et al. [15]) and XLD.

Survival Time m = 3.17 s = 2.095 Obs freq LD θ^=0.522 Xgamma θ^=0.689 ZD θ^=0.852 Exp θ^=0.315 XLD θ^=0.467
[0,2] 45 38. 262 37. 652 30. 339 43. 937 41. 028
[2,4] 22 28. 164 27. 197 37. 27 23. 4 25. 855
[4,6] 17 15. 075 16. 342 17. 743 12. 463 13. 984
[6,8] 7 7. 1187 7. 7769 6. 1658 6. 6375 6. 9986
[8,10] 3 3. 1423 3. 2015 1. 828 3. 5351 3. 3409
Total 94 94 94 94 94 94
χ2 2. 7899 3. 2040 14. 236 1. 8619 1. 6446

LD, Lindley; XLD, XLindley distribution; ZD, Zeghdoudi.

Table 3

Comparison between LD, Xgamma, ZD, Exp and XLD.

10. CONCLUSION

In this work, we present a one-parameter distribution called XLD which is mixture of two known distribution Exponential θ and LDθ distributions. Its survival properties have been discussed: moments, skewness, kurtosis, hazard and rate function, stochastic ordering, quantile function, the maximum likelihood method and method of moments. The XLD is flexible and likely model in describing real life time-to-event data. Lot of properties and simulation are given which confirm the goodness of fit and it's better than exponential, LD, Xgamma and ZD distributions.

CONFLICTS OF INTEREST

The authors declare they have no conflicts of interest.

AUTHORS' CONTRIBUTIONS

Study conception and design: H. Zeghdoudi; analysis and interpretation of results: S. Chouia; draft manuscript preparation: S. Chouia and H. Zeghdoudi. All authors reviewed the results and approved the final version of the manuscript.

Funding Statement

This research was funded by PRFU Program of Ministry of Higher Education and Scientific Research No. C00L03UN230120180014.

ACKNOWLEDGMENTS

The authors acknowledge editor in chief, Prof. Dr. Mohammad Ahsanullah and the referee, of this journal for the constant encouragement to finalize the paper.

Journal
Journal of Statistical Theory and Applications
Volume-Issue
20 - 2
Pages
318 - 327
Publication Date
2021/07/10
ISSN (Online)
2214-1766
ISSN (Print)
1538-7887
DOI
https://doi.org/10.2991/jsta.d.210607.001How to use a DOI?
Copyright
© 2021 The Authors. Published by Atlantis Press B.V.
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

Cite this article

TY  - JOUR
AU  - Sarra Chouia
AU  - Halim Zeghdoudi
PY  - 2021
DA  - 2021/07/10
TI  - The XLindley Distribution: Properties and Application
JO  - Journal of Statistical Theory and Applications
SP  - 318
EP  - 327
VL  - 20
IS  - 2
SN  - 2214-1766
UR  - https://doi.org/10.2991/jsta.d.210607.001
DO  - https://doi.org/10.2991/jsta.d.210607.001
ID  - Chouia2021
ER  -