Egyptian
Mathematical
Society
Journal of the Egyptian Mathematical Society
www.etms-eg.orgwww.elsevier.com/locate/joems
Original
Article
Quartile
ranked
set
sampling
for
estimating
the
distribution
function
Amer Ibrahim Al-Omari
Alal-BaytUniversity,FacultyofScience,DepartmentofMathematics,Mafraq,Jordan Received24January2014;revised5July2014;accepted15April2015
Availableonline21May2015
Keywords
Distributionfunction; Quartilerankedset sampling;
Simplerandomsampling; Rankedsetsampling; Relativeefficiency
Abstract Quartilerankedsetsampling(QRSS)methodissuggestedbyMuttlak(2003)for
estimat-ingthepopulationmean.Inthisarticle,theQRSSprocedureisconsideredtoestimatethe distribu-tionfunctionofarandomvariable.TheproposedQRSSestimatoriscomparedwithitscounterparts basedonsimplerandomsampling(SRS)andrankedsetsampling(RSS)schemes.Itisfoundthatthe suggestedestimatorofthedistributionfunctionofarandomvariableXforagivenxisbiasedand moreefficientthanitscompetitorsusingSRSandRSS.
2010MathematicsSubjectClassification: 62G30;62D05;62G05;62G07
Copyright2015,EgyptianMathematicalSociety.ProductionandhostingbyElsevierB.V. ThisisanopenaccessarticleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction
LetX1,X2,...,Xn bea randomsampleof sizenhaving proba-bilitydensityfunction(pdf) f(x)andcumulativedistribution function(cdf)F(x),withafinitemeanμandvarianceσ2.Let
X11i ,X12i ,...,X1ni ;X21i ,X22i ,...,X2ni ;...;Xn 1i ,Xn 2i ,...,Xnni benindependentsimplerandomsampleseachofsizeninthe ithcycle(i = 1,2,...,m).
Let FSRS(x) denotethe empirical distribution function of
asimplerandomsampleX1,X2,...,Xnm fromF(x).Bahadur [2]showedthatFSRS (x)hasthefollowingproperties:
E-mailaddress:alomari_amer@yahoo.com
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1. FSRS(x)isanunbiasedestimatorofF(x)foragivenx.
2. Var[FSRS (x)] = mn1 F(x)[1−F(x)].
3. FSRS(x)isaconsistentestimatorofF(x).
The RSSwasfirst suggestedbyMcIntyre[3] asamethod forestimatingthemeanofpastureandforageyields.TheRSS is a useful method when the sampling units can be easily rankedthanquantified.McIntyreproposedtherankedset sam-plemeanasanestimatorofthepopulationmeanandshowed thattheRSSmeanestimatorisunbiasedandismoreefficient thantheSRScounterpart.
TheRSScanbedescribedasfollows:randomlyselectnsets eachofsizenfromthetargetpopulation.Then,visuallyrank the units within each sample with respect to the variableof interest.Fromthefirstsetofnunitsthesmallestrankedunit isselected. Fromthesecond setofnunits thesecond small-estrankedunitisselected.The processiscontinueduntilthe largestrankedunitismeasuredfromthenthset.Toincreasethe S1110-256X(15)00027-9Copyright2015,EgyptianMathematicalSociety.ProductionandhostingbyElsevierB.V.Thisisanopenaccessarticle undertheCCBY-NC-NDlicense(http://creativecommons.org/licenses/by-nc-nd/4.0/).
Table1 TherelativeprecisionofFQRSS (x)withrespecttoFSRS (x)andbiasvaluesofFQRSS (x)for4≤n≤11. F(x) n=4 n=6 n=8 n=10 n=5 n=7 n=9 n=11 0.01 RP 0.5270 7.9487 5.3923 9.1690 11.0248 7.0763 4.5962 8.2674 Bias 0.0090 −0.0093 −0.0088 −0.0099 −0.0096 −0.0092 −0.0084 −0.0099 0.10 RP 0.6852 1.4566 1.1587 1.2156 1.7024 1.4095 1.1780 1.1780 Bias 0.0718 −0.0429 −0.0066 −0.0650 −0.0655 −0.0356 0.0006 −0.0592 0.20 RP 0.9939 1.3260 1.0919 1.2896 1.3138 1.4388 1.1553 1.3873 Bias 0.0973 −0.0254 0.0482 −0.0396 −0.0802 −0.0142 0.0527 −0.0257 0.30 RP 1.6994 1.6332 1.5037 1.7397 1.2961 1.9075 1.7570 1.9325 Bias 0.0849 −0.0044 0.0727 0.0098 −0.0673 0.0062 0.0679 0.0201 0.40 RP 2.9902 2.2115 3.3200 2.9599 1.3908 2.5254 3.3480 3.1859 Bias 0.0475 0.0041 0.0505 0.0224 −0.0367 0.0103 0.0447 0.0265 0.50 RP 4.3694 2.5640 7.3573 4.8399 1.4534 2.8914 5.5273 4.8417 Bias 0.0005 0.0008 0.0000 −0.0005 0.0003 0.0005 0.0004 −0.0003 0.60 RP 3.0824 2.2121 3.2736 2.9261 1.3843 2.5467 3.3917 3.2440 Bias −0.0481 −0.0040 −0.0508 −0.0216 0.0354 −0.0105 −0.0443 −0.0256 0.70 RP 1.7065 1.6410 1.5109 1.7521 1.2757 1.8497 1.7220 1.9539 Bias −0.0848 0.0051 −0.0733 −0.0088 0.0662 −0.0067 −0.0686 −0.0194 0.80 RP 1.0080 1.3748 1.1220 1.2898 1.2997 1.4229 1.1603 1.3658 Bias −0.0960 0.0277 −0.0483 0.0391 0.0812 0.0136 −0.0536 0.0256 0.90 RP 0.6628 1.4455 1.1659 1.2054 1.6756 1.4006 1.1633 1.1870 Bias −0.0736 0.0428 0.0063 0.0650 0.0659 0.0352 0.0004 0.0592 0.99 RP 0.5126 7.3353 4.9833 9.4109 11.8336 7.3257 4.4370 8.6399 Bias −0.0098 0.0092 0.0086 0.0099 0.0096 0.0093 0.0084 0.0099
samplesize,thewholeprocesscanberepeatedmtimestoobtain asetofsizenmunits.
Let Xj(1:n)i,Xj(2:n)i,...,Xj(n:n)i be the order statistics
of the jth sample Xj1i ,Xj2i ,...,Xjni (j = 1,2,...,n) in the ith cycle (i = 1,2,...,m). Then, the measured units X1(1:n )i ,X2(2:n )i ,...,Xn (n :n )i aredenotedtotheRSS. Davidand
Nagaraja[4] showed thatthecdfandthepdfofthejthorder statisticX(j:n)aregivenby
F(j:n)(x)= n i=j n i [F(x)]i[1−F(x)]n−i,−∞ <x< ∞ , and f(j:n)(x) = (j −1)n!(n! − j)![F(x)]j−1[1−F(x)]n−jf(x). The mean and the variance of X(j:n) are given by μ(j:n) =
∞
−∞x f(j:n )(x)dx and σ(2j:n)=
∞
−∞(x−μ(j:n ))2f(j:n )(x)dx, re-spectively. Takahasi and Wakimoto [5] independently intro-ducedthesamemethodofRSSwithmathematicaltheoryand showedthat f(x)= 1 n n j=1 f(j:n )(x), μ= 1n n j=1 μ(j:n ), and σ2 = 1 n n j=1 σ2 (j:n )+ 1 n n j=1 μ(j:n )−μ2.
Forafixedx,StokesandSager[6]suggestedanestimatorfor F(x)usingRSSas FRSS (x)= 1 mn m i=1 n j=1 IXj (j :n )i ≤x,
whereI(·)isanindicatorfunction.Theyprovedthefollowing: 1. FRSS (x)isanunbiasedestimatorforF(x).
2. Var[FRSS(x)]= mn 12
n
j=1F(j:n)(x)[1−F(j:n)(x)].
3. FRSS√ (x)−E[FRSS(x)]
Var[FRSS(x)]
converges in distribution tothe standard normalasm → ∞ whenxandnarefixed.
Formore aboutestimationofthedistribution functionin rankedsetsamplingmethodsseeStokesandSager[6],Samawi andAl-Sagheer[7],KimandKim[8],Al-SalehandSamuh[9], andGhoshandTiwari[10].
Therestofthispaperisorganizedasfollows.InSection2, weintroducedthesuggestedestimationofthedistribution func-tionusingQRSSmethod.Theperformanceofthenew estima-toragainstitsSRSandRSScounterpartsisgiveninSection3. Section4,isdevotedforsomeinferencesaboutF(x).InSection 5,someconcludingremarksareprovided.
2. Estimation of F(x)using QRSS
The quartile ranked setsampling procedure as suggested by Muttlak [1] canbesummarizedasfollows.Randomlyselectn sampleseachofsizenunitsfromthetargetpopulationandrank theunitswithineachsamplewithrespecttothevariableof in-terest.Ifthesamplesizeniseven,selectandmeasurefromthe firstn/2samplestheQ1(n+ 1)thsmallestrankedunitofeach sample,i.e.,thefirstquartile,andfromthesecondn/2samples theQ3(n+ 1)thsmallestrankedunitofeachsample,i.e.,the thirdquartile.Notethat,wealwaystakethenearestintegerof Q1(n+ 1)thandQ3(n+ 1)thwhereQ1 = 25%,andQ3 = 75%. Ifthesamplesizenisodd,selectandmeasure fromthefirst (n−1)/2samplestheQ1(n+1)thsmallestrankedunitofeach sampleandfromtheother(n−1)/2samplestheQ3(n+ 1)th smallestrankedunitofeachsample,andfromonesamplethe median forthatsample.The cyclecanbe repeatedmtimesif neededtogetasampleofsizenmunits.
Ifthesamplesizeniseven,intheithcycle(i = 1,2,...,m), letXj(Q 1(n +1):n )i be the(Q1(n+ 1))thsmallestrankedunitofthe jth sample (j=1,2,...,n 2), and Xj(Q 3(n +1):n )i be the (Q3(n+ 1))th smallest ranked unit of the jth sample
(j = n +2 2 ,
n +4
2 ,...,n). Note that, the measured units
X1(Q 1(n +1):n )i ,X2(Q 1(n +1):n )i ,...,Xn2(Q1(n+1):n)i are independent
and identically distributed (iid) random variables, and also Xn+2
2 (Q 3(n +1):n )i ,Xn+24(Q 3(n +1):n )i ,...,Xn (Q 3(n +1):n )i are iid.
However, all units X1(Q 1(n +1):n )i ,X2(Q 1(n +1):n )i ,...,Xn2(Q1(n+1):n)i, Xn+2
2 (Q3(n+1):n)i,Xn+24(Q3(n+1):n)i,...,Xn(Q3(n+1):n)i are mutually
independent but not identically distributed. These measured unitsaredenotedbyQRSSE.
For odd sample size, let Xj(Q1(n+1):n)i be the Q1(n+ 1)th
smallest ranked unit of the jth sample (j = 1,2,...,n−1 2 ),
Xj(n+1
2 :n )i bethemedianofthejthsampleoftherank j = n +1
2 , andXj(Q3(n+1):n)ibe theQ3(n+ 1)thsmallestrankedunitofthe
jthsample(j= n+23,n+25,...,n).Notethat,theonlymeasured units X1(Q1(m+1):m)i,X2(Q1(n+1):n)i,. . .,Xn−1
2 (Q 1(n +1):n )i are iid,
andalsoXn+3
2 (Q 3(n +1):n )i ,Xn+25(Q 3(n +1):n )i ,...,Xn(q3(n+1):n)i areiid.
However,allunitsX1(Q1(m+1):m)i,X2(Q1(n+1):n)i,. . .,Xn−1
2 (Q 1(n +1):n )i , Xn+1
2 (n+21:n )i ,Xn+23(Q 3(n +1):n )i ,Xn+25(Q 3(n +1):n )i ,. . .,Xn (q 3(n +1):n )i are
mutually independent but not identically distributed. These measuredunitsaredenotedbyQRSSO.
Following Stokes and Sager [6], let V= (V1,V2,...,Vn)
be a multinomial random vector with nm trials and P= (1
n,
1
n,...,
1
n) be a probability vector, and assume that the nm random variables were found by first observing V and thenselectingVj unitsrandomlyfromapopulationwithpdf f(j:n )(x)(j = 1,2,...,n).LetY1,Y2,...,Ynm denotetothe ob-tainedmnunits.
Also,followingStokesandSager[6] wehavethefollowing theorembasedonQRSSmethod.
Theorem 1. BasedonthesameconditionsofTheorem1inStokes andSager[6]wehavethefollowing:
(1) For even sample size, { Y1,Y2,...,Ynm| V = (nm
2 ,0,0,...,0,
nm
2 )} has the same probability
struc-ture as {Xj(Q1(n+1):n)i,Xl(Q3(n+1):n)i; j=1,2,. . ., n 2; l= n +2 2 , n +4
2 ,. . .,n;i=1,2,...,m},aQRSSEfromthesame
population.
(2) For odd sample size, {Y1,Y2,...,Ynm |V=v1 =
(n−1)m
2 ,0,0,...,0;v(n+1
2 )= m;0,0,...,0,vn = (n−1)m
2 } ,
hasthesameprobabilitystructureas
Xj(Q 1(n +1):n )i ,Xt (t:n )i Xl(Q 3(n +1):n )i ; j=1,2,. . ., n−1 2 ; t n+ 1 2 : l = n+ 3 2 , n+ 5 2 ,...,n; i = 1,2,...,m , aQRSSOfromthesamepopulation.
Proof. TheproofisdirectlyandsimilartotheproofofTheorem 1inStokesandSager[6].
ThesuggestedQRSSEestimatorofF(x)isgivenby
FQRSSE(x)= 1 nm m i=1 n 2 j=1 IXj(Q1(n+1):n) i ≤x 1 nm m i=1 n j=n+2 2 IXj(Q3(n+1):n) i≤x , (1) withvariance VarFQRSSE(x)= 1 2nm FQ1(x) 1−FQ1(x) +FQ3(x) 1−FQ3(x) . (2) AndthesuggestedQRSSOestimatorofF(x)isgivenby
FQRSSO(x)= 1 nm m i=1 n−1 2 j=1 IXj(Q1(n+1):n) i≤x + 1 nmI Xn+1 2( n+1 2:n)i≤x +nm1 m i=1 n j=n+3 2 IXj(Q3(n+1):n) i≤x , (3) withvariance VarFQRSSO(x)= 1 n2m n−1 2 FQ1(x) 1−FQ1(x) +FQ3(x) 1−FQ3(x) +FQ2(x) 1−FQ2(x) , (4) where FQ1(x)=B F(x);n−1 4 , 3n−3 4 , FQ2(x)=B F(x);n+1 2 , n+1 2 , FQ3(x)=B F(x);3n−3 4 , n−1 4 ,
and B(w;a,b) is the incomplete beta function defined as B(w;a,b)=
w
0 u
a−1(1−u)b−1du.
Assume that n is odd, and n +1
2 is odd, and let Q1 =
X(n+3 4 ),Q2 = X(n+21),andQ3 = X(3n+41),then fQ 1(x)= 1 Bn −1 4 , 3n −3 4 [F(x)]n−41[1−F(x)] 3n−3 4 f(x), (5) fQ2(x) = 1 Bn+21,n+21[F(x)(1−F(x))] n−1 2 f(x), (6) and fQ3(x) = 1 B3n−3 4 , n−1 4 [F(x)]3n4−3[1−F(x)]n−41f(x), (7)
whereB(δ,ϑ )istheBetafunctiondefinedas B(δ,ϑ )= (δ)(ϑ )(δ+ ϑ ) = (δ(δ−+ 1)ϑ!(ϑ − −1)!
1)! . (8)
SomepropertiesoftheQRSSEandQRSSOestimatorsof thedistributionfunctionareprovidedinthefollowing proposi-tions. Proposition 1. (1) E[FQRS S E (x)]= 12[FQ 1(x)+FQ 3(x)]. (2) E[FQRS S O (x)] = n2−n 1{ FQ 1(x)[1−FQ 1(x)]+ FQ 3(x)[1− FQ 3(x)]}+ 1 n FQ 2(x), where FQ 1(x),FQ 2(x),FQ 3(x) , are definedabove.
Proposition 2. FQRS S O (x)andFQRS S E (x)arebiasedestimators ofF(x),withbiasgivenby
BiasFQRS S E (x)= 1 2 FQ 1(x)+ FQ 3(x) −FQRS S E (x), and BiasFQRSSO(x)= n−1 2n FQ1(x) 1−FQ1(x) +FQ 3(x)1−FQ 3(x)+1 nFQ 2(x)−FQRS S O (x).
ItiseasytoprovePropositions1and2 basedonsome alge-bra.Fromthispropositionwecanseethatthesuggested estima-torsarebiased,butbasedonthecalculationsgiveninSection3, thebiasisclosetozero.
Proposition 3. Forfixednandm→∞,wehavethefollowing: 1. F QRS√SE(x )−E[F QRSSE(x )]
Var [F QRSSE(x )] convergesindistributiontoN(0,1).
2. FQRS√ SO(x)−E[FQRSSO(x)]
Var[FQRSSO(x)] convergesindistributiontoN(0,1).
Proof. FollowingSamawiandAl-Sagheer[7]andKimandKim [8],theproofofthefirstpartcanbedonebyassuming
Zi = 1 n n 2 j=1 IXj(Q1(n+1):n)i ≤x + 1 n n j=n+2 2 IXj(Q3(n+1):n)i ≤x , i = 1,2,...,m.
AsZi’sareindependentandidenticallywith finitemeanand
variancerandomvariables,thenbasedonthecentrallimit the-oremwecanconclude √m√ [Zi−E(Zi)]
Var(Zi) convergesindistributionto
thestandardnormaldistribution.
Thesecondpartcanbeprovedsimilarlybyassuming
Zi = 1 n n−1 2 j=1 IXj(Q 1(n +1):n )i ≤x + 1 nI Xn+1 2 n+1 2 :n i ≤x +1 n n j=n+3 2 IXj(Q 3(n +1):n )i ≤x . 3. Numerical comparisons
Numericalcomparisonsareconsideredinthissectionto inves-tigatetheperformanceofthesuggestedQRSSestimatorofthe
distributionfunctionwithrespecttoSRSandRSSestimators basedonthesamenumberofmeasuredunits.Weconsidered differentsamplesizes4≤n ≤11.Therelativeprecisions(RP) ofFRSS (x)andFQRSS (x)withrespecttoFSRS (x)aredefinedas: RP[FRSS (x),FSRS (x)]= Var[FSRS (x)]
Var[FRSS (x)], and RPFQRSS (x),FSRS (x)= Var[FSRS (x)]
MSEFQRSS(x)
, (9)
whereMSEisthemeansquarederrorofanestimator.
TheresultsarepresentedinTables1and2 usingQRSSand RSS, respectively.Theresultsinboldfontsinbothtablesare thebestRPvaluesofthedistributionfunctionestimatorsQRSS andRSSwithrespecttoSRSfordifferentvaluesofF(x).Based onQRSSfromTable1,thelargestRPvaluesareobservedas F(x)goestozeroor1.Forexample,withn=5andF(x)= 0.1,0.99,theRPvaluesare11.0248and11.8336,respectively. Otherwise,theRPvaluesincreasewhenF(x)iscloseto0.5.In allcasesthebiasisnegligibleandclosetozero.
FromTable2 weobservethatFRSS (x)ismoreefficientthan FSRS(x). However, FQRSS(x) is better than FRSS(x) for most
ofcasesconsidered inthisstudy.Asan example,whenn = 9 andF(x)=0.4,therelativeprecisionsofRSSandQRSSOare 2.6347and3.3480,respectively.
The optimal values ofthe sample sizeusing QRSSEand QRSSOmethodsforestimatingF(x)aresummarizedinTable 3.
FromTable3 itcanbenotedthatQRSSOismoreefficient thanQRSSEforestimatingF(x).Also,thesamplesizen = 5 has mostoccurrenceamong other samplesizesconsideredin thisstudy.
4. Inferences about the distribution function
In this section, a pointwise estimate of F(x) is consid-ered and some inferences about the population distribution are presented using QRSS. As m gets large, then based on Proposition3,anapproximate100(1−α)%confidenceinterval forE[F(x)], =QRSSE,QRSSO,RSS,SRS,isgivenby F(x) −Zα/2 Var[F(x)]<E[F(x)]<F(x) +Zα/2 Var[F(x)], (10)
whereZα/2istheupper100(α/2)%quantileoftheN(0,1),and
Table2 TherelativeprecisionofFRSS(x)withrespecttoFSRS(x)for4≤n≤11.
F(x) n=4 n=6 n=8 n=10 n=5 n=7 n=9 n=11 0.01 0.9886 1.0388 1.1355 1.1018 1.0851 1.0644 1.0712 1.0749 0.10 1.3071 1.4832 1.6539 1.7929 1.3798 1.5550 1.7035 1.9029 0.20 1.5471 1.8193 2.0421 2.2984 1.6973 1.9610 2.2084 2.3711 0.30 1.6880 2.0402 2.3306 2.6122 1.8933 2.2060 2.4760 2.7136 0.40 1.7891 2.2099 2.4696 2.7623 1.9969 2.2982 2.6347 2.8966 0.50 1.8487 2.2449 2.5601 2.8271 2.0644 2.3500 2.6801 2.9600 0.60 1.7841 2.2061 2.4787 2.7775 1.9545 2.3337 2.6349 2.9724 0.70 1.6816 2.0456 2.3528 2.6052 1.9025 2.1952 2.4917 2.7661 0.80 1.5507 1.8057 2.0610 2.3004 1.6731 1.9757 2.1988 2.4008 0.90 1.2972 1.4716 1.6241 1.8179 1.3743 1.5503 1.7450 1.8920 0.99 1.1206 1.0458 1.0523 1.0764 1.0146 1.0781 1.1562 1.1301
Table3 ThebestvaluesofthesamplesizeusingQRSSforestimatingF(x). F(x) 0.01 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.99 n 5 5 7 11 9 8 9 11 7 5 5 VarFQRS S E (x)= 1 (m−1)n2 ⎧ ⎨ ⎩ n 2 j=1 ˆ Fj(Q 1(n +1):n )(x) 1− ˆFj(Q 1(n +1):n )(x) + n j=n+2 2 ˆ Fj(Q 3(n +1):n )(x) 1− ˆFj(Q 3(n +1):n )(x) ⎫⎪⎬ ⎪ ⎭ (11) VarFQRSSO(x)= 1 (m−1)n2 ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ n−1 2 j=1 ˆ Fj(Q1(n+1):n) (x) 1− ˆFj(Q1(n+1):n) (x) +FˆQ2(x) 1− ˆFQ2(x) +n j=n+3 2 ˆ Fj(Q3(n+1):n) (x) 1− ˆFj(Q3(n+1):n) (x) ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ , (12) Var[FRSS(x)]= 1 mn2 n j=1 F(j:n) (x)1−F(j:n) (x), and Var[FSRS(x)]= 1 nm−1 ˆ F(x)1− ˆF(x),
wherebasedonQRSS,RSS,andSRS,respectively,wehave ˆ Fi(t:n) (x)= 1 m m i=1 IXj(t:n) i≤x ,t=Q1(n+1),Q3(n+1), n+1 2 ;j=1,2,...,n, (13) ˆ Fi(j:n) (x)= 1 m m i=1 IXj(j:n) i≤x ,j=1,2,...,n, (14) and ˆ FSRS(x)= 1 nm nm i=1 I(Yi≤x). (15)
Based on the 100(1−α)% confidence interval of E[F(x)],=
QRSSE,QRSSO,RSS,SRS, a 100(1−α)% confidence interval for
F(x)canbefoundas P Zα/ 2≤ F(x)−E[F(x)] Var[F(x)] ≤ Z1−α/ 2 =1−α. (16) Solving(16)forE[F(x)], =QRSSE,QRSSO,RSS,SRStoget
P F(x)−Z1−α/ 2 Var[F(x)]≤E[F(x)]≤F(x) +Zα/ 2 Var[F(x)] =1−α,
andthelimitswillbe
LowerBound(LB)=F(x)−Z1−α/ 2 Var[F(x)], and UpperBound(UB)=F(x)+Zα/ 2 Var[F(x)].
Finally,anapproximate100(1−α)%confidenceforF(x)canbeobtained bysolvingtheequations2LB=h(F)and2UB=h(F),numerically,orby anysuitablemethod,wherebasedonQRSSEwehave
h(F)=FQ1(x)+FQ3(x), andusingQRSSOitwillbe
h(F)= 1 n (n−1)FQ1(x) 1−FQ1(x) +FQ3(x) 1−FQ3(x) +2FQ2(x) .
Itisofinteresttonoteherethatanypossiblesolutionforlasttwoequations shouldbesingularsinceh(F)isincreasingfunctioninF(x).
5. Conclusion
Inthisstudy,QRSSprocedureisconsideredtoestimatethe dis-tributionfunctionofarandomvariable.TheQRSSiscompared withRSSandSRSinestimatingthedistributionfunction.Itis foundthatQRSSismoreefficientthanSRSandalsoitismore efficientthanRSSinmostcasesconsideredinthisstudy.The RPvaluesarepreferredasF(x)goestozeroor1andincrease whenF(x)iscloseto0.5.ThebiasoftheQRSSestimatorof thedistributionfunctionisnegligible.
Acknowledgments
The authoris thankfulto theeditor andtheanonymous re-viewersfortheirvaluablecommentsandsuggestionsthat sig-nificantlyimprovedtheoriginalversionofthearticle.
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