Journal of Statistical Theory and Applications

Volume 18, Issue 1, March 2019, Pages 1 - 11

A New Zero-Truncated Version of the Poisson Burr XII Distribution: Characterizations and Properties

Authors
Haitham M. Yousof1, *, Mohammad Ahsanullah2, Mohamed G. Khalil1
1Department of Statistics, Mathematics and Insurance, Benha University, Benha, Egypt
2Department of Management Sciences, Rider University, New Jersey, USA
*Corresponding author. Email: haitham.yousof@fcom.bu.edu.eg
Corresponding Author
Haitham M. Yousof
Received 13 July 2018, Accepted 10 February 2019, Available Online 22 April 2019.
DOI
10.2991/jsta.d.190306.001How to use a DOI?
Keywords
Zero-truncated Poisson; Poisson Topp Leone; Burr XII distribution; simulation; characterizations
Abstract

In this work, a new four-parameter zero-truncated Poisson Topp Leone Burr XII distribution is defined and studied. Various structural mathematical properties of the proposed model including ordinary and incomplete moments, residual and reversed residual life functions, generating functions, order statistics are investigated. Some useful characterizations are also presented.

Copyright
© 2019 The Authors. Published by Atlantis Press SARL.
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 Pearson system of frequency curves was introduced by Pearson [1] who worked out a set of four-parameter probability density functions (PDFs) as solutions to the following differential equation

fx/fx=Px/Qx=xab0+b1x+b2x21
where f is a density function and a,b0,b1, and b2 Pearson family such as Gamma, Gaussian, Beta, and Student's t models. Analogously to the Pearson system, Burr [2] introduced another system of frequency curves that includes 12 types of cumulative distribution functions (CDFs) which yield a variety of density shapes, this system is obtained by considering CDFs satisfying a differential equation which has a solution, given by
Gx=1+expτxdx1,
where τx is chosen such that Gx is a CDF on the real line and has 12 choices which made by Burr, resulted in 12 models which might be useful for modeling data, the principal aim in choosing one of these forms of distributions is to facilitate the mathematical analysis to which it is subjected, while attaining a reasonable approximation. A special attention has been devoted to one of these forms denoted by type XII (for more details see Burr [24], Burr and Cislak [5], Hatke [6], and Rodriguez [7]), whose CDF, Gx, is given as
Gα,βx=11+xαβ,
where both α and β are shape parameters. The location and scale parameters can easily be introduced to make Gα,βx a four-parameter distribution. The corresponding PDF is given by
gα,βx=αβxα11+xαβ1.

The Burr XII (BXII) (see Burr [2]) has many applications in different areas including reliability, acceptance sampling plans, and failure time modeling. Tadikamalla [8] studied the BXII model and its related models. Zimmer et al. [9] proposed a new three-parameter BXII distribution, this distribution, having the Weibull and the logistic as submodels, is a very popular distribution for modeling lifetime data and phenomenon with monotone failure rates. Shao [10] studied the maximum likelihood estimations for the three-parameter BXII model then Soliman [11] studied the estimation of parameters of life from progressively via censored data using Burr-XII model, Wu et al. [12] discussed the estimation problems for BXII model on the basis of progressive type II censoring under random removals where the number of units removed at each failure time has a discrete uniform model. Recently, Silva et al. [13] introduced the log-BXII regression models with censored data, Silva et al. [13] proposed a new location-scale regression model based on BXII model, Silva et al. [14] proposed a residual for the log-BXII regression distribution whose empirical model is close to normality, Afify et al. [15] studied the Weibull BXII distribution, Cordeiro et al. [16] proposed the double BXII model among others. For the other new extensions of the BXII see Altun et al. [17], Altun et al. [18], Paranaíba et al. [19], Yousof et al. [20], and Yousof et al. [21].

The rest of the paper is outlined as follows. In Section 2, we introduce the new model and its physical motivation. Section 3 presents some plots and the justification for introducing the new model. Some useful characterizations are presented in Section 4. In Section 5, we derive some statistical properties for the new model. Finally, we offer some concluding remarks in Section 6.

2. THE NEW MODEL AND ITS PHYSICAL MOTIVATION

The CDF and the PDF of the Topp Leone BXII (TLBXII) distributions (Reyad and Othman [22]) are given by

Hb,α,βx=11+xα2βb
and
hb,α,βx=2bαβxα11+xα2β111+xα2βb1
respectively. Suppose Z1,Z2,,ZN be independent identically random variable (iid rv) with common CDF of the TLBXII model and N be a rv with probability mass function (PMF)
PN=n=an/ea1n!|n=1,2,,a>0
and define
MN=maxZ1,Z2,,ZN
then
Fx=n=0pMNx|N=npN=n

As described in Ramos et al. [23], Eq. (3) can be expressed as

F(x)=n=1  1ea1ann!1(1+xα)2βbn

Using Eqs. (2) and (4), we can write

Fx=1expa11+xα2βb/1ea.

Equation (5) is the CDF of the zero-truncated Poisson Topp Leone BXII (ZTPTLBXII) model. Henceforward fx=fa,b,α,βx and Fx=Fa,b,α,βx. The corresponding PDF of Eq. (5) reduces to

fx=2abαβ1eaxα11+xα2β111+xα2βb1expa11+xα2βbA.

Now we can provide a useful linear representation for the ZTPTLBXII density function in Eq. (6). Expanding the quantity A in power series, we can write

fx=i=021iai+1bαβi!1eaxα11+xα2β111+xα2βbi+11.

Consider the power series

1ζτ1=r=01rΓτζr/j!Γτj=r=01rτ1  rζr,
which holds for |ζ|<1 and τ>0 real non-integer. Using the power series in Eq. (8) and after some algebra the PDF of the ZTPTLBXII model in Eq. (7) can be expressed as
fx=i=021iai+1bαβi!1eaxα11+xα2β111+xα2βbi+11=i,r=021i+rai+1bi!1eaαβxα11+xα2β11+xα2rβbi+11r=i,r=021i+rai+1bi!1eabi+11rαβxα11+xα2β1+r1=i,r=021i+rai+1bi!1ea21+rbi+11rα21+rβxα11+xα2β1+r1=r=0i=01i+rai+1bi!1ea1+rbi+11rvrα21+rβxα11+xα21+rβ1gα,21+rβx=r=0vrgα,21+rβx,
where
gα,21+rβx=2α1+rβxα11+xα21+rβ1,
is the BXII density with parameters α and 21+rβ.

vr=i=01i+rai+1bi!1ea1+rbi+11     r

Via integrating Eq. (9), we obtain the same mixture representation

Fx=r=0vrGα,21+rβx,
where Gα,21+rβx is the CDF of the BXII density with parameters α and 21+rβ. The new model has a wide application in many types of data such as Guinea pigs (Bjerkedal [24]) and many other data types.

3. PLOTS AND JUSTIFICATION

In this section, we provide some graphical plots of the PDF and hazard rate function (HRF) of the ZTPTLBXII model to show its flexibility. Figure 1(a) displays some plots of the ZTPTLBXII density for some parameter values a,b,α, and β. Plots of the HRF of the ZTPTLBXII model for selected parameter values are given in Fig. 1(b), where the HRF can be upside down bathtub (unimodal) and decreasing.

Figure 1

Plots of the zero-truncated Poisson Topp Leone BXII (ZTPTLBXII PDF) (right panel) and HRF (left panel).

The justification for introducing the ZTPTLBXII lifetime model is based on the wider use of the BXII model. As well as we are motivated to introduce the ZTPTLBXII lifetime model because it exhibits the unimodal hazard rate as illustrated in Fig. 1(b). It is shown above that the ZTPTLBXII lifetime model can be viewed as a linear mixture of the BXII densities as illustrated in Eqs. (9) and (10).

4. CHARACTERIZATIONS

We will need the following two Lemmas for the characterization of the distribution:

Assumption A.

X is an absolutely continuous rv with CDF Fx and PDF fx. We assume EX exists and fx is differentiable. We assume further

α=supx  |  fx>0 and βx|fx<1.

Lemma 1.

If

EX  |  Xx=gxfx/Fx,
where gx is a continuous differentiable function in α,β, then fx=cexpxgxgxdx, c is determined by the condition αβfxdx=1.

Proof.

gx=αxufudufx, thus αxufudu=fxgx

Differentiating both sides of the above equation, we obtain xfx=fxgx+fxgx on simplification, we get

fx/fx=xgx/gx.

On integrating both sides of the above equation, we obtain

fx=cexpxgxgxdx,
where c is determined by the condition
αβfxdx=1.

Lemma 2.

Under the assumption A, if

EX|Xx=hxfx/1Fx,
where hx is a continuous differentiable function in α,β, then
fx=cexp=x+hxhxdx,
where c is determined by the condition αβfxdx=1.

Proof.

hx=xufudufx, thus xufudu=fxhx

Differentiating both sides of the above equation, we obtain xfx=fxhx+fxhx on simplification, we obtain

fx/fx=x+hx/hx.

On integrating both sides of the above equation, we obtain

fx=cexpx+hxhxdx,
where c is determined by the condition
αβfxdx=1.

Theorem 1.

Suppose that the random variable X satisfies the conditions of the assumption A with α=0 and β=. Then, EX|Xx=gxτx, where τx=fxFx and

gx=r=0νr21+rβBxα,1+1α,21+rβ1αr=0νrgα,21+rβx,
if and only if
fx=r=0νrgα,21+rβx.

Proof.

Suppose that

fx=r=0νrgα,21+rβx,
then
fxgx=r=0νr0xugα,21+rβudu=r=0νr0x2α1+rβuα1+uα21+rβ1du=r=0νr0xα21+rβ1+tβ21+rβ1t1αdt=r=0νr21+rβB1+1α,21+rβ1α.

Thus

gx=r=0νr21+rβBxα,1+1α,21+rβ1αr=0νrgα,21+rβx.

Suppose

gx=r=0νr21+rβBxα,1+1α,21+rβ1αr=0νrgα,21+rβx,
then
gx=xr=0νr21+rβBxα,1+1α,21+rβ1αr=0νrgα,21+rβx×r=0νr2α1+rβxα2α12αβ1+r+1xα1+xα21+rβ+2r=0νrgα,21+rβx=xgxr=0νrα21+rβxα2α12αβ1+r+1xα1+xα21+rβ+2r=0νrgα,21+rβx

Thus

xgxgx=r=0νr2α1+rβxα2α12αβ1+r+1xα1+xα21+rβ+2r=0νrgα,21+rβx.

By Lemma 1, we have

fxfx=r=0νrα1+rβxα2α12αβ1+r+1xα1+xα21+rβ+2r=0νrgα,21+rβx.

Integrating both sides of the above equation we obtain

fx=cr=0νrgα,21+rβx,
where c is constant. Using the condition 0fxdx=1, we obtain
fx=r=0νrgα,21+rβx.

Theorem 2.

Suppose that the random variable X satisfies the conditions of the assumption A with α=0 and β=.

Then, EX|Xx=hxrx, where rx=fxFx and

gx=r=0νr21+rβBcxα,1+1α,β1αr=0νrgα,21+rβx,
if and only if
fx=r=0νrgα,1+rβx,
where
Ba1,a2=0ta111+ta1+a2dt
and
Bz,a1,a2=0zta111+ta1+a2dt
are the beta and the incomplete beta functions of the second type, respectively.

Proof.

Suppose that fx=r=0νrgα,21+rβx, then

fxgx=r=0νrxugα,21+rβudu=r=0νrx2α1+rβuα1+uα21+rβ1du=r=0νrx21+rβ1+t21+rβ1t1αdt=r=0νr21+rβBc1+1α,21+rβ1α.

Thus

g(x)=r=0vr2(1+r)βBc1+1α,2(1+r)β1αr=0vrgα,2(1+r)β(x).

Suppose

g(x)=r=0νr2(1+r)βBc(1+1α,2(1+r)β1α)r=0νrgα,2(1+r)β(x),
then
gx=xr=0νr21+rβBxα,1+1α,21+rβ1αr=0νrgα,21+rβx×r=0νr2α1+rβxα2α12αβ1+r+1xα1+xα21+rβ+2r=0νrgα,21+rβx=xgxr=0νrα21+rβxα2α12αβ1+r+1xα1+xα21+rβ+2r=0νrgα,21+rβx

Thus

xg(x)g(x)=r=0νr2α(1+r)βxα2α12αβ(1+r)+1xα(1+xα)2(1+r)β+2r=0νrgα,(1+r)β(x).

By Lemma 2, we have

f(x)f(x)=r=0νrα(1+r)βxα2α12αβ(1+r)+1xα(1+xα)2(1+r)β+2r=0νrgα,2(1+r)β(x).

Integrating both sides of the above equation we obtain

fx=cr=0νrgα,2(1+r)βx,
where c is constant. Using the condition
0fxdx=1,
we obtain fx=r=0νrgα,2(1+r)βx.

5. MATHEMATICAL PROPERTIES

5.1. Moments and Incomplete Moments

The rth ordinary moment of X is given by

μn=EXn=xnfxdx.

Then we obtain

μr=r=0vr0xngα,21+rβx=r=0vr21+rβB1+nα,21+rβnα.

By setting n=1 in Eq. (11), we have the mean of X. The sth incomplete moment, say Ist, of X can be expressed from (9) as Ist=txsfxdx. Then

Ist=r=0vrtxs gα,21+rβ(x)dx=r=0vr21+rβBtα;1+sα,21+rβsα,

The general equation for the first incomplete, I1t, can be derived from Eq. (12) as

I1t=r=0vrtx gα,21+rβxdx=r=0vr21+rβBtα;1+1α,21+rβ1α.

5.2. Moment Generating Function

The moment generating function (MGF) of X, say MXt=EexptX, can be obtained from Eq. (9) as MXt=r=0vrM21+rβt, where M21+rβt is the mgf of the BXII distribution with parameters α, 21+rβ, then we have

MXt=r=0vrMr+1t=2αβr=01+rvr0exptxxα11+xα21+rβ1dx=2αβr=01+rvrk=0tkk!0xα+k11+xα21+rβ1dx.

Let u=xα, then

MXt=2βr=01+rvrk=0tkk!0ukα1+u21+rβ1du=2βr=01+rvrk=0tkk!B1+nα,21+rβnα,
which also means that the rth ordinary moment of X is
μr=2βr=0vr1+rB1+nα,21+rβnα.

5.3. Order Statistics

Let X1,,Xn be a random sample from the ZTPTLBXII model of distributions and let X1:n,,Xn:n be the corresponding order statistics. The PDF of ith order statistic, say Xi:n, can be written as

fi:nx=Bi,ni+11j=0ni1jnijfxFj+i1x,
where B, is the beta function. Substituting Eqs. (5) and (6) in Eq. (13) and using a power series expansion, we get that
fxFxj+i1=h=0vhgα,1+hβx,
where
vh=w,m,k=02bw+1mbaw+11w+m+k+hk+1ww!h!1+h1eaj+ij+i1kbw+11  m×1+βw+1+m  h2+βw+1+m  h

The PDF of Xi:n can be expressed as

fi:nx=j=0ni1jnijBi,ni+11h=0vhgα,1+hβx.

For example, the moments of Xi:n can be expressed as

EXi:nq=j=0ni1jnijBi,ni+1h=0vh1+hβB1+qα,1+hβqα.

5.4. Quantile Spread Ordering

The quantile spread (QS) of the rv T ZTPTLBXII a,b,α,β having CDF (5) is given by

QSTν|ν0.5,1=F1νF11ν,
which implies
QSTν=S11νS1ν,
where
F1ν=S11ν and St=1Ft
is the survival function. The QS of a distribution describes how the probability mass is placed symmetrically about its median and hence can be used to formalize concepts such as peakedness and tail weight traditionally associated with kurtosis. So, it allows us to separate concepts of kurtosis and peakedness for asymmetric models. Let T1 and T2 be two rvs following the ZTPTLBXII a,b,α,β model with QST1 and QST2. Then T1 is called smaller than T2 in QS order, denoted as T1QST2, if
QST1ν|ν(0.5,1)QST2ν.

Following properties of the QS order can be obtained:

  • The order QS is a location-free

    T1QST2 if T1+CQST2|CR.

  • The order QS is dilative

    T1QSCT1 whenever C1 and T2QSCT2|C1.

  • Let FT1 and FT2 be symmetric, then

    T1QST2 if, and only if FT11τFT21ν|ν0.5,1.

  • The order QS implies ordering of the mean absolute deviation around the median, say γMedian(Ti)|i=1,2,

    γMedianT1=E|T1MedianT1|
    and
    γMedianT2=E|T2MedianT2|,
    where
    T1QST2 implies γMedianT1QSγMedianT2.

    Finally

    T1QST2 if, and only if T1QST2.

6. CONCLUSIONS

In this paper, a new four-parameter ZTPTLBXII distribution is defined and studied. The new model has a strong physical motivation. Various structural mathematical properties of the proposed model including ordinary and incomplete moments, residual and reversed residual life functions, generating functions, order statistics are investigated also the QS ordering is defined for formalizing concepts such as peakedness and tail weight traditionally associated with kurtosis on the new model. Some useful characterizations are also presented.

REFERENCES

18.E. Altun, H.M. Yousof, S. Chakraborty, and L. Handique, Int. J. Math. Stat., Vol. 19, 2018, pp. 46-70. http://www.ceser.in/ceserp/index.php/ijms/article/view/5664
24.T. Bjerkedal, Am. J. Hyg., Vol. 72, 1960, pp. 130-148.
Journal
Journal of Statistical Theory and Applications
Volume-Issue
18 - 1
Pages
1 - 11
Publication Date
2019/04/22
ISSN (Online)
2214-1766
ISSN (Print)
1538-7887
DOI
10.2991/jsta.d.190306.001How to use a DOI?
Copyright
© 2019 The Authors. Published by Atlantis Press SARL.
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  - Haitham M. Yousof
AU  - Mohammad Ahsanullah
AU  - Mohamed G. Khalil
PY  - 2019
DA  - 2019/04/22
TI  - A New Zero-Truncated Version of the Poisson Burr XII Distribution: Characterizations and Properties
JO  - Journal of Statistical Theory and Applications
SP  - 1
EP  - 11
VL  - 18
IS  - 1
SN  - 2214-1766
UR  - https://doi.org/10.2991/jsta.d.190306.001
DO  - 10.2991/jsta.d.190306.001
ID  - Yousof2019
ER  -