SciresolSciresolhttps://indjst.org/author-guidelinesIndian Journal of Science and Technology0974-564510.17485/IJST/v13i32.353A view on characterizations of the J shaped statistical distributionZakaAzam1AkhterAhmad Saeed1JabeenRiffatdrriffatjabeen@cuilahore.edu.pk2College of Statistical and Actuarial Sciences, University of the PunjabLahorePakistanCOMSATS University Islamabad Lahore CampusLahore, +92-300-4364368Pakistan13322020Abstract
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 distributionto 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 thesufficient 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.
KeywordsCharacterizationmean inactivity timemean residual functionpower function distributionNoneIntroduction
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<δ≤w
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 Methods2.1. Characterization of PFD under mean inactivity times
We may write the pdf and cdf for the PFD
fz=ωzω-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|z≤t)
Mzt=t-∫0tzωzω-1dzΦ(t)Mzt=t-ω∫0tzωdzΦ(t)
Since Φ (t) = tω
Mzt=t-ωtω+1
Also
Mz(t)-t=ωtω+1=E(z∣z≤t)E(z∣z≤t)=ωtω+1
Sufficient Part:
Now E(z∣z≤t) can be written as:
E(z∣z≤t)=∫0tzf(z)dzΦ(t)
Using integration by parts
E(z∣z≤t)=t-∫0tΦ(z)dzΦ(t)
Equating (2) and (3), we get
t-∫0tΦ(z)dzΦ(t)=ωtω+1∫0tΦ(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)) =ωln(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;whereMzt:MeanInactivityTimes2.2 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
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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