Sciresol Sciresol https://indjst.org/author-guidelines Indian Journal of Science and Technology 0974-5645 10.17485/IJST/v13i32.353 A view on characterizations of the J shaped statistical distribution Zaka Azam 1 Akhter Ahmad Saeed 1 Jabeen Riffat drriffatjabeen@cuilahore.edu.pk 2 College of Statistical and Actuarial Sciences, University of the Punjab Lahore Pakistan COMSATS University Islamabad Lahore Campus Lahore, +92-300-4364368 Pakistan 13 32 2020 Abstract

Objectives: In recent years, characterization of any distribution has become important in the field of probability distribution. The objective of the study is to characterize the power function distribution to see its usefulness under different real life situations such as Engineering and medical sciences.Methods: The study proposed the characterization of Power function distribution based on mean inactivity times (MIT), mean residual function (MRF), conditional moments, conditional variance (CV), doubly truncated mean (DTM), incomplete moments and reverse hazard function. Findings: We have characterized the power function distribution using different method, and conclude that the sufficient and necessary conditions of different methods mentioned above meet the results of Power function distribution. Application: Power function distribution has wide applicability in the field of Engineering. The findings of the paper may help the Engineers to know more about the Power function distribution.

Keywords Characterization mean inactivity time mean residual function power function distribution None
Introduction

The characterization of probability distribution is important before its application to any real world phenomena to confirm whether the given probability distribution is suitable for the specific data set or not. There are several characterizing functions associated with a probability distribution that uniquely define it.

We assume that the distribution of random variable called “Z” follows Power function distribution (PFD) with following cumulative density function

Φz=zβω;                    0<z<β

Then W=Z-1has the Pareto distribution with parameters δ=β-1and γ=ω.

Gw=1-δwγ;              0<δw

The application of this distribution was discoursed by Meniconi and Barry 1 to test the reliability of any module. The characterization of PFD has been introduced by Fisz 2, Basu 3, Govindarajulu 4 and Dallas 5. Many authors have studied the characterization of different distributions; see Dallas 6, Deheuvels 7, Gupta 8, Nagaraja 9, Rao et al. 10 and Huang et al. 11.

Su and Huang12 presented the following function for the cdf “ Φ" of “T”

v(t)=E(g(T)T>t),  a<t<b

Where g may be observed as continuous function and (a,b) is the support of Φ. They have also showed that if T has a pdf, then Φ (t), a<t<b, δ(t), a<t<b, or E(T|T>t), a<t<b, are equivalent, in the sense that given on them , the other two can be determined, where:

δt=f(t)1-Φ(t)  ;                 a<t<b

Gupta and Kirmani 13 consider the case a=0 and b=∞ and used the ratio of δtand E(T|T>t) to characterize the distribution of T. Nair and Shudheesh 14 showed that, for a continuous function g and a differential function h, the pdf “f” satisfies the differential equation. Unnikrishnan and Sudheesh15 studied the characterization of continuous distributions by properties of conditional variance. Also, characterizations based on the properties of the failure rate function have been considered by many authors. Elbatal et al. 16, Huang et al. 17, Bhatt 18, Ahsanullah et al. 19 and references therein. Imen et al. 20 discussed the characterization of Exponential q-distribution. 21, 22 and 23 discussed the properties and characterizations of the J shaped distributions. Zaka et al. 24 introduced the exponentiated generalized class of Power function distribution.

This study discusses a novel characterization of PFD established on mean inactivity times (MIT), mean residual function (MRF), conditional moments, conditional variance, doubly truncated mean (DTM), incomplete moments and reverse hazard function (RHF). The paper is organized as follows. We characterized the PFD by MIT, MRF and doubly truncated mean in section-2. Under Section -3, we introduce characterization based on different types of moments. In Section 4, we obtain the PFD from RHF. The concluding remarks are provided in Section 5.

Materials and Methods 2.1. Characterization of PFD under mean inactivity times

We may write the pdf and cdf for the PFD

fz=  ω zω-1   ;                 0<z<1Φz=   zω  ;                        0<z<1

Also let Φ-z as the survival function for the PFD. Then “Z” has PFD with parameter “ ω” if and only if

Mzt=t-ωtω+1

Where Mz(t): Mean Inactivity Times for PFD.

Proof:

Necessary part:

From (1), by taking β=1.

Φz= zω

Since Mzt=t-E(z|zt)

Mzt=t-0tz ωzω-1 dzΦ(t) Mzt=t-ω0tzω dzΦ(t)

Since Φ (t) = tω

Mzt=t-ωtω+1

Also

Mz(t)-t=ωtω+1=E(zzt) E(zzt)=ωtω+1

Sufficient Part:

Now E(zzt) can be written as:

E(zzt)=0tzf(z)dzΦ(t)

Using integration by parts

E(zzt)=t-0tΦ(z)dzΦ(t)

Equating (2) and (3), we get

t-0t Φ(z)dzΦ(t)=ωtω+1 0t Φ(z)dz=tω+1Φ(t)

Differentiating “t” on both sides

Φ(t)1-1ω+1=tω+1f(t) f(t)Φ(t)=ωt

Apply Integral on both sides

ln ( Φ (t)) =  ω ln(t)

Therefore Φ (t) = tω

and f(t)= ωtω-1

so the random variable “z” has PFD, “f(z) = ωzω-1” with parameter “ ω” if and only if

Mzt=t-ωtω+1  ;                   where Mzt:Mean Inactivity Times
2.2 Characterization of PFD under mean residual function

We may write the pdf and cdf for the PFD

f(z)=ωzω-1;0<z<1Φ(z)=zω

Also let Φ-z as the survival function for the PFD. Then the random variable “Z” has PFD with parameter “ ω” if and only if

Mt=ωω+1 1-tω+1Φ-t- t  ;

Where M (t): Mean Residual Function.

Proof:

Necessary part:

Φz=zω

Since

M(t)=E(zzt)-t E(zzt)=t1zωzω-1dzΦ¯(t) E(zzt)=ωΦ¯(t)1-t(ω+1)ω+1

Therefore, (4) becomes

Mt=ωω+1 (1-t(ω+1))Φ-(t)- t   Mt+ t=ωω+1 (1-t(ω+1))Φ-(t) E(zzt)=ωω+11-t(ω+1)Φ¯(t))

Sufficient Part:

Now E(zzt) can be written as:

E(zzt)=t1zf(z)dzΦ¯(z)

Using integration by parts

E(zzt)=-t-t1Φ¯(z)dzΦ¯(z)

By equating (5) and (6)

-t ϕ-t-t1 ϕ-zdz= ωω+1 1-tω+1 t1 ϕ-zdz= -t ϕ-t- ωω+1 1-tω+1

Since ϕ (t) = tω

1-t-t1ϕ(z) dz= -t+ tω+1- ωω+1 1-tω+1 t1ϕ(z) dz=1+ωω+1 1-tω+1

Differentiating “t” on both sides

-ϕt=2ω+1ω+1(-ω+1 tω) ϕt=2ω+1 tω

By Differentiating “t” on both sides

ft=2ω+1ω tω-1 12ω+101ft dt=1

Therefore, ft=ω tω-1

Therefore, it is proved that the random variable “z” has PFD, f(z) = ω zω-1” with parameter “ ω” if and only if

Mt=ωω+1 (1-t(ω+1))Φ-(t)- t  ;                   where Mt:Mean Residual Function
2.3 Characterization Of PFD under doubly truncated mean

We may write the pdf and cdf for the PFD

f(z)=ωzω-1;  0<z<1ϕ(z)=zω;  0<z<1

And let Φ-z be the survival function .Then the random variable “Z” has PFD with parameter “ ω” if and only if

EZz<Z<w)=ωω+1w ϕw-z ϕzϕw-ϕz  ;     where EZz<Z<w):Doubly Truncated Mean .

Proof:

Necessary Condition:

EZz<Z<w)=zwz fz dzϕw-ϕ(z) EZz<Z<w)=ωω+1wω+1-zω+1ϕw-ϕ(z)

Since ϕ (z) = zω

EZz<Z<w)=ωω+1w ϕw-z ϕ(z)ϕw-ϕ(z)

Sufficient Condition:

Also EZz<Z<w) can be written as

EZz<Z<w)=zwz fz dzϕw-ϕ(z) EZz<Z<w)=w ϕw-z ϕ(z)-zwϕz dzϕw-ϕ(z)

By equating (7) and (8), we get

w ϕw-z ϕ(z)-zwϕz dzϕw-ϕ(z)= ωω+1w ϕw-z ϕ(z)ϕw-ϕ(z)

Differentiating with respect to ‘w’ on both sides

w fw+ϕw-ϕ(w)= ωω+1w ϕw+ϕ(w) 1ω+1w fw= ωω+1 ϕw fwϕ(w)= ωw

By integrating on both sides

ln ( ϕ (w)) = ln ( wω)

Therefore

f(w)= ω wω-1

Characterization Of PFD by conditional variances, conditional moments and incomplete moments 3.1. Characterization of PFD by conditional variances

We may write the pdf and cdf for the PFD

f(z)=ωzω-1;  0<z<1ϕ(z)=zω

And let ϕ-z be the survival function respectively. Then the random variable “z” has PFD with parameter “ ω” if and only if

V(zzt)=t2ωω+2-ω2(ω+1)2 where EZz<Zt):Conditional Variance(with respect to F(z)) .

Proof:

Necessary part:

ϕ (z) = zω

Since

V(zzt)=Ez2zt-{E(zzt)}2 E(z2|zt)=ω0tzω+1 dzϕ(t)

Since ϕ (t) = tω

E(z2|zt)=ω t2ω+2

Also

Mz(t)-t=ωtω+1=E(zzt)

Therefore (9) becomes

V(zzt)=ωt2ω+2-ωtω+12 V(zzt)=t2ωω+2-ω2(ω+1)2

Sufficient Part:

Now V(zzt) can be written as:

V(zzt)=0tz2f(z)dzϕ(t)-0tzf(z)dzϕ(t)2

Using integration by parts

V(zzt)=t2F(t)-0t2zϕ(z)dzϕ(t)-tϕ(t)-0tϕ(z)dzϕ(t)2 V(zzt)=t2-2otzϕ(z)dzϕ(t)-t-0tϕ(z)dzϕ(t)2

Equating (10) and (11), we get

t2-20tzϕ(z)dzϕ(t)-t-0tϕ(z)dzϕ(t)2=t2ωω+2-ω2(ω+1)2

Since t-0t ϕz dzϕt=ωtω+1

t2-20tzϕ(z)dzϕ(t)-ωtω+12=t2ωω+2-ω2(ω+1)2 t2-20tzϕ(z)dzϕ(t)=t2ωω+2 0tz ϕz dz= t2ω+2ϕ(t)

Differentiating “t” on both sides

tF(t)=1ω+2t2f(t)+ϕ(t)2tωϕ(t)=tf(t)f(t)ϕ(t)=ωt

ln ( ϕ (t)) =  ω ln(t)

Therefore ϕ (t) = tω

and f(t)= ω tω-1

Therefore, it is proved that the “z” has PFD f(z) = ω zω-1 with parameter “ ω” if and only if

V(zzt)=t2ωω+2-ω2(ω+1)2 where E(zr | zt):Conditional rth Moment (with respect to Φ(z))
3.2 Characterization of PFD By Conditional Moments

We may write the pdf and cdf for the PFD

f(z)=ωzω-1,0<z<1ϕ(z)=zω

And let ϕ-z be the survival function respectively. Then the random variable “z” has PFD with parameter “ ω” if and only if

Ezrzt=ωtrω+r, where Ezrzt: Conditional rth Moment (with respect to Φ(z)

Proof:

Necessary part:

ϕ (z) = zω

E(zr|zt)=0tzr ω zω-1dzϕ(t)

Since ϕ (t) = tω

E(zr|zt)=ωtrω+r

Sufficient Part:

Now E(zr|zt) can be written as:

Ezrzt=0tzrf(z)dzϕ(t)

Using integration by parts

Ezrzt=trϕ(t)-r0tzr-1ϕ(z)dzϕ(t) Ezrzt=tr-r0tzr-1ϕ(z)dzϕ(t)

Equating (12) and (13), we get

tr-r0tzr-1ϕ(z)dzϕ(t)=ωtrω+r0tzr-1ϕ(z)dz=ϕ(t)trω+r

Differentiating “t” on both sides

tr-1ϕ(t)=1ω+rtrf(t)+ϕ(t)rtr-1ωϕ(t)=tf(t)f(t)ϕ(t)=ωt

Apply Integral on both sides

ln ( ϕ (t)) =  ω ln(t)

Therefore ϕ (t) = tω

and f(t)= ω tω-1

Therefore it is proved that “z” has PFD, “f(z) = ωzω-1” with parameter “ ω” if and only if

Ezrzt=ωtrω+r where V(zr|zt):Conditional Variance(with respect to (z)) .
3.3 Characterization of PFD by incomplete moments

We may write the pdf and cdf for the PFD

f(z)=ωzω-1;0<z<1ϕ(z)=zω

And let ϕ-z be the survival function respectively. Then the random variable “z” has PFD with parameter “ ω” if and only if

μz| ω,β=1;rt=ωtr+ωω+r; where μz| ω,β=1;rt:Incomplete rth Moment

Proof:

Necessary part:

ϕz = zω

μz| ω,β=1;rt=0tzr ωzω-1dz μz| ω,β=1;rt=ω tr+ωω+r

Sufficient Part:

Now μz| ω,β=1;rt can be written as:

μz| ω,β=1;rt=0tzr f(z)dz

Using integration by parts

μz| ω,β=1;rt=trϕ(t)-r0t zr-1ϕz dz

Equating (14) and (15), we get

trF(t)-r0tzr-1ϕ(z)dz=ωtr+ωω+r

Since ϕt=tω

r0tzr-1ϕ(z)dz=trtω-ωtr+ωω+r

Differentiating “t” on both sides

tr-1ϕ(t)=(ω+r)tr+ω-1ω+r

Therefore ϕ (t) = tω

and f(t)= ω tω-1

Therefore it is proved that the “z” has PFD, f(z) = ωzω-1 with parameter “ ω” if and only if

μz| ω,β=1;rt=ω tr+ωω+r; where μz| ω,β=1;rt:Incomplete rth Moment
Characterization Of PFD, using reverse hazard function

We may assume any real number “t” such that Φ z(t)>0 , rw(t)=a rz(t) with a>0 iff

aWnt= -lntγn+na ϕW(t)0t-lnwγn-1fWw dw

Where U(t)=  -lnϕZ(t) , Ut=ddtUt=- rZt,  aWnt=E(UnW|W<t ) ,

“rz(t)” and “rw(t)” are the reverse hazard rates. Also follows the PFD.

Proof:

Necessary Part:

aW(n)(t)=EUn(W)W<t aWnt=1 ϕW(t)0t-lnϕW(w)n fWw dw aWnt=-lnϕW(t)n ϕW(t)ϕW(t)- nϕW(t)0t-lnϕWwn-1(-rz(w)) ϕWw dw

Since rw(t)=a rz(t)

aWnt=-lnϕW(t)n+ nϕW(t)0t-lnϕWwn-1(rW(w)a) ϕWw dw aWnt=-lnϕW(t)n+ na ϕW(t)0t-lnϕWwn-1 fWw dw

Since ϕWt=tγ

aWnt=-lntγn+ na ϕW(t)0t-lnwγn-1 fWw dw

Sufficient Part:

1 ϕW(t)0t-lnwγn fWw dw=-lntγn+ na ϕW(t)0t-lnwγn-1 fWw dw 0t-lnwγn fWw dw=ϕW(t)-lntγn+ na 0t-lnwγn-1 fWw dw

Taking derivative on both sides

-lntγnfW(t)=-lntγnfW(t)+ϕW(t)n-lntγn-1-1tγγtγ-1+na-lntγn-1fW(t) ϕWt-lntγn-1γt=1a -lntγn-1 fWt aγt=fWtϕWt aγln(t)=lnϕw(t)ϕW(t)=taγfW(t)=aγtaγ-1

Also (16) becomes

ϕWt n -lntγn-1(-rZ(t))+na -lntγn-1 fWt=0 ϕWt  -lntγn-1rZ(t)=1a -lntγn-1 fWt rZ(t)= 1a rW(t)rWt=a rZ(t)
Conclusion

The objective of the study is to acquire the characterization of PFD based on mean inactivity times (MIT), mean residual function (MRF), conditional moments, conditional variance, doubly truncated mean (DTM), incomplete moments and reverse hazard function (RHF). We hope that this study will be useful for the statisticians in various fields of studies.

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