REVIEW OF RANKED SET SAMPLING: MODIFICATIONS AND APPLICATIONS

Transcription

1 REVISTA INVESTIGACIÓN OPERACIONAL VOL. 35 NO REVIEW OF RANKED SET SAMPLING: MODIFICATIONS AND APPLICATIONS Aer Ibrah Al-Oar * and Carlos N. Bouza** *Departent of Matheatcs-Faculty of Scence Al al-bayt Unversty Jordan. **Facultad de Mateátca y Coputacón Unversdad de La Habana Havana Cuba. ABSTRACT The proble of estatng the populaton ean s consdered by McIntyre (95). A new saplng ethod s suggested naely; ranked set saplng () as effcent ethod copared to the well known sple rando saplng (SRS) ethod. In the last years any authors suggested dfferent odfcatons of the and used t n wde applcatons. In ths paper a lterature revew of the ethod s presented as well as soe ts odfcatons and applcatons are provded. KEYWORDS: Ranked set saplng; Populaton ean; Effcency. MSC: 6G05 94A0 6P RESUMEN El problea de estar la eda poblaconal fue consderado por McIntyre (95). Este sugró un nuevo étodo de uestreo llaado uestreo por conjuntos ordenados (Ranked Set Saplng ) que consderaba coo un étodo efcaz en coparacón con el étodo del usual uestreo aleatoro sple. En los últos años uchos autores sugreron dversas odfcacones del y les han usado en un aplo espectro de aplcacones. En este trabajo se presenta una revsón de la lteratura del étodo así coo algunas de sus odfcacones y aplcacones.. INTRODUCTION In ths paper we wll present soe studes where the well known ranked set saplng () ethod as well as soe of ts odfcatons are appled. The was frst suggested by McIntyre (95) for estatng the populaton ean of pasture and forage yelds. He claed wthout proof that was ore accurate than sple rando saple ts effcency for estatng the hgher populaton oents s better than that of sple rando saplng (SRS) unless f the underlyng dstrbuton s rectangular n shape. The usual saplng desgns are characterzed as follows. Defnton: A randoly selected saple fro a larger saple or populaton gvng all the ndvduals n the saple an equal chance to be chosen. (Cochran 977). ay be consdered as a controlled rando saplng desgn. It can be descrbed as follows: Step : Select randoly unts fro the populaton of nterest. Step : Allocate the unts randoly nto sets each set of sze. Step 3: Rank the unts n each set based on a varable of nterest vsually or by usng any cost free ethod. Step 4: The saple s chosen for actual easureent by selectng fro the frst set the lowest ranked unt fro the second set the second sallest ranked unt and so forth untl fro the last set the axu ranked unt s selected. Step 5: The above steps can be repeated n cycles to get a saple of sze n. For fxng soe deas consder a rando saple fro a dstrbuton F(x) whch adts a densty functon f( x ) wth a ean and a varance. Wth copared to SRS uses one unt naely (:) the sallest unt fro ths set then (:) the second sallest ranked unt fro another ndependent set of unts and fnally (:) the largest ranked unt fro a last set of unts. Ths Correspondng author. Al al-bayt Unversty P.O. Box Mafraq 53 Jordan aeraloar@aabu.edu.jo 5

2 process can be explaned n Fgure. It s portant to ephasze here that although requre dentfcaton of as any as unts but only of the are quantfed. The fnal unts (:) (:) (:) are used for nvestgaton. These unts are ndependent but not dentcally dstrbuted and (:) s the th order statstc n a rando saple of sze fro F(x). Thus akng a coparson of a of sze wth a SRS of the sae sze s eanngful. Obvously would be a good rval to SRS n case where the collectng of the saplng unts s easy and ther relatve rankngs based on the characterstc under study can be done wth trval cost. (: ) (: ) (: ) (: ) (: ) (: ) ( : ) ( : ) ( : ) Fgure : Elucdaton of unts n sets of each The effcency of reles on the saplng allocaton ether balanced or unbalanced. In balanced the rank order statstcs has an equal allocaton. They proved theoretcally and pretended eprcally that the balanced estator has a varance whch s no greater than the SRS estator varance whether errors n rankng or the nature of the parent dstrbuton of the varable of nterest. In SRS the sapler ust ncrease the saple sze to ncrease the chance of coverage the whole range of possble observatons values and there s no other chance. Whle n one can ncrease the representatveness based on a specfc nuber of saple observatons. Hence there s a savng consderably on the easureent costs. Thus based on the easured ranked set saple we can obtan unbased estators of populaton paraeters as the ean and and for ore than one cycle the populaton varance. The relatve precson (RP) of relatve to SRS based on the sae nuber of Var SRS easured unts s defned as RP Var. Note that the ethod cannot be worse than the SRS ethod (Patl 00; Takahas and Wakoto 968).. REVIEW ON SOME PREVIOUS APPLICATIONS OF Fortunately n any felds such as n edcne envronent bology and agrculture the study varable s not easly easured but ts rankng can be done easly wth cheap or free cost. The can be pleented to yeld ore effcent estators of the populaton paraeters as copared to SRS usng the sae nuber of quantfed observatons. Here soe exaples on reported applcatons of n real stuatons wll be gven. Evans (967) appled the to regeneraton surveys n areas drect-seeded to longleaf pne. He noted that the eans based on both of and SRS ethods were not sgnfcantly dfferent but the coputed varances of the eans were very dfferent. Martn et al. (980) appled the procedure for estatng shrub phytoass n Appalachan Oak forests. Cobby et al. (985) conducted four experents at Hurley (UK) durng 983 to nvestgate the perforance of relatve to SRS for estaton of herbage ass n pure grass swards and of herbage ass and clover content n xed grass-clover swards. Johnson et al. (993) appled ethod to estate the ean of forest grassland and other vegetaton resources. Mode et al. (999) nvestgated under whch condtons the becoes a cost-effectve saplng ethod for ecologcal and envronental feld studes where the rough but cheap easureent has a cost. They have ntroduced forula for the total cost for both and SRS and present cost ratos for a real data set consstng of judgent estated and physcally easured strea. Al-Saleh and Al-Shrafat (00) studed the perforance of n estaton lk yeld based on 40 sheep. Al-Saleh and Al-Oar (00) used the ultstage to estate the average of Olves yelds n a feld n West of Jordan. Husby et al. (005) nvestgated on the use of the n estatng of the ean and edan of a populaton usng the crop producton dataset fro the Unted State Departent of Agrculture. They found that the gan n effcency for ean estaton usng s better for syetrc dstrbuton than asyetrc dstrbuton and vce versa n the case of edan estaton. Kowalczyk (005) appled the procedure n arket and consuer surveys. 5

3 Ganeslnga and Ganesh (006) appled the ethod to estate the populaton ean and the rato usng a real data set on body easureent. The authors used the data of the weght and heght of 507 ndvduals. Halls and Dell (966) coned McIntyre s ethod as and appled t for estatng the weghts of browse and herbage n a pne-hardwood forest of east Texas USA. Wang et al. (009) used the n fsheres research. Twar and Pandey. (03) consdered an applcaton of ranked set saplng desgn n envronental nvestgatons for real data set. For ore about applcatons of see Dong et al. (0). 3. ESTIMATION USING 3.. Estaton of the populaton ean Assue that the populaton under consderaton has a densty functon f( x ) wth ean and varance. Theore : An unbased estator of a populaton ean usng SRS s gven by 5 SRS wth varance Var SRS. McIntyre (95) claed wthout proof that: ) Regardless of any errors n rankng the estator of the populaton ean s unbased. ) Under perfect rankng the effcency of wth respect to SRS s nearly for estatng the ean of typcal unodal dstrbutons based on the sae nuber of quantfed unts. 3) In estatng the hgher populaton oents the effcency of s better than that of SRS unless f the underlyng dstrbuton has a rectangular shape. 4) The rankng errors reduce the effcency of and these errors ncrease as saple sze ncreases. 5) If there are enough nforaton about the underlyng dstrbuton the unequal allocatons ay prove the perforance of. Takahas and Wakoto (968) have gven the atheatcal propertes of and obtan the sae ethod ndependently of McIntyre (95). Theore : Let ( : ) be th order statstcs of a saple of sze fro a probablty densty functon f ( ) ( : ) x wth ean ( : ) and varance s gven by ( : ) An unbased estator of the populaton ean usng. ( : ) Var. wth varance ( : ) f ( x) f ( x) ( : ) ( : ) Also ( : ) ( ( : ) ). Let n unts be randoly chosen fro the populaton of nterest and randoly allocated nto n sets each of sze unts. Fro each set of sze one unt wll be drawn to get n easured unts. The unts to be easured were chosen as n the followng steps. Frst let l l l be postve ntegers such that l l... l n. After rankng the unts wthn each set wth respect to the varable of study the sallest ranked unt s easured fro the frst l sets; the second sallest ranked unt s easured fro the next l sets and so forth untl the largest ranked unt s easured fro the last l sets. Let T be the su of easureents of the th ranked unts for.... So that T are ndependent. T Therefore the unbased estator of s. l Based on both balanced allocatons wth l l... l and Neyan allocatons when each l proportonal to : they copared the perforance of wth respect to SRS usng the relatve

4 Var precson (RP) RP Var SRS and Neyan allocatons we have Var l or the equvalent relatve savngs (RS) ( : ) and ( : ) n RS. For balanced RP Var RS ( : ) where respectvely. Also for balanced allocatons the relatve savng s 0 RS and RP. The lower bound s holds f and only f the underlyng dstrbuton s degenerate and the upper bound s holds f and only f the parent dstrbuton s rectangular. But for Neyan allocatons they showed that 0 RS and RP. See Takahas and Wakoto (968) for ore detals. Al-Saleh and Al-Oar (00) ntroduced a ultstage ranked set saplng (MS) as a generalzaton of the double. The MS procedure can be descrbed as: r Step : Randoly select unts fro the populaton of nterest where r s the nuber of stages and s the saple sze. r Step : Allocate the r selected unts as randoly as possble nto sets each of sze. Step 3: For every set n Step () use the procedure of balanced ranked set saplng as descrbed n r Secton to have a ranked set saple of sze. Ths step yelds ranked set saples each of sze. r Step 4: Repeat Step (3) on the r ranked set saples to obtan second stage saples each of sze. The process contnues untl we end up wth one r th stage of sze. Suppose that the varable of nterest has ean and varance wth a pdf f( x ) and cdf Fx ( ). ( Let... be a MS of sze at stage r wth ean varance r) pdf ( ) ( ) ( ) ( and cdf F r) ( x ).... The MS estator of the ean s ( r) ( r) ( r) MS ( r) ( r) ( r) Var MS. The authors proved the followng denttes ( r) ( r) f ( x) f ( x) ( r) ( r) and defned the RS and RP at stage r respectvely as ( r) ( r) RS and ( ) ( ) r r ( r RP ). ( r f ) ( x ) wth varance The authors defned a steady state effcency of at stage r to be as proved the followng theore. eff l eff and they ( ) ( r) r Theore 3: Suppose that the varable of nterest has an absolutely contnuous dstrbuton functon and ( ) ( ) ( ) let r r... r be a MS of sze then 0 x Q( )/ ( x) ( ) ( x) ( ) Q x Q x Q/ F lf x F ( ) ( r) ( )/ / r where Q s the quantty whch satsfes f ( x) dx (0). Q 5

5 So that f ( x) f ( x) f ( x) for Q( )/ x Q / and zero otherwse. If the varable of nterest ( r) ( ) has a unfor dstrbuton then ( ) f ( x) for x and zero otherwse and eff. For the specal case f we saple fro the standard unfor dstrbuton t ( ) can be shown that an approxate ntegrals for the above functon can be found by Monte Carlo ethods. Ths dea was consdered by Al-Saleh and Saaw (000). It can be noted that the effcency s ncreasng n ters of the saple sze and the nuber of stages. In Table 3. soe values of the effcency of the MS wth respect to the SRS estators are suarzed. See Al-Saleh and Al- Oar (00). Table 3.: The effcency of MS relatve to SRS for =3 Dstrbuton 3 r r r 3 r r U(0 ) N( ) Logstc ( ) Exp( ) LogN ( ) Pareto ( 3) For ean estaton based on and ts odfcatons: Saaw et al. (996) suggested a varety of extree. Muttlak (997) ntroduced a edan ranked set saplng. Saaw (00) suggested double extree ranked set saplng. Yu and Ta (00) proposed the n the presence of concord data. Al-Saleh and Al-Hadhra (003) nvestgated ovng extrees paraetrcally for estatng the locaton paraeter of syetrc dstrbuton. Muttlak (003ab) suggested percentle and quartle ethods. Rahov and Muttlak (003) extend the rando selecton n suggested by L et al. (999) for estatng the populaton ean. Barabs and Psan (00) nvestgated steady state for replcated saplng protocols n order to estate the objectve paraeter usng Horvtz- Thopson estator. For asyetrc dstrbuton Muttlak and Abu-Dayyeh (004) suggested weghted odfed to overcoe the bas of several estators based on odfed ethods such as E M and P ethods. Bouza (008) consdered the ean estaton when soe observatons are ssng usng product type estators. Bouza (009) nvestgated the ean estaton of a senstve quanttatve character based on and randozed response procedures. Bouza (00) consdered the ean estaton usng n non-responses case. Jeelan et al. (04) consdered a role of rank set saplng n provng the estates of populaton ean under stratfcaton. Jozan et al. (0) ntroduced unbased rato estators of the populaton ean usng ranked set saplng. Ban- Mustafa et al (0) suggested folded for estatng the populaton ean of asyetrc dstrbutons. Sngh et al. (04) ntroduced a general procedure for the ean estaton usng. Patl et al. (997) nvestgated the effect of the saple sze upon the perforance of the balanced for estatng the populaton ean. Hossan (00) suggested a nonparaetrc approach for the odfed ethod for the populaton ean estaton naely nonparaetrc selected ranked set saplng. Unlke the usual where we chose only one unt fro each ranked set of sze Wang et al. (004) proposed an estator of the populaton ean usng the general. In whch ore than one unt can be chosen fro each ranked set. Al-Saleh and Al-Kadr (000) proposed the double procedure (D) for the ean estaton. They showed that at the second stage the rankng s easer than rankng at the frst stage and also the D estator s ore effcent than that usng wth respect to SRS based on the sae saple sze. Al-Saleh et al. (000) consdered Bayesan estaton of the paraeter of the underlyng dstrbuton usng. In ters of the Bayes rsk the Bayes rsk of the Bayes estator usng ethod s less 5

6 than the Bayes rsk of the Bayes estator usng SRS. The procedure was used for estatng the average lk yeld of 40 sheep. Also see Jean and Al-Oar (006a 006b 006c 007a 007b). Al-Oar (999 0) Al- Oar and Al-Saleh (0) Jean et al. (007a 007b 008a 008b) Chen et al. (004) Al-Oar and Raqab (03) Konak and Mahdzadeh (04) Takahas (969) Bouza (00 03) Al-Nasser and Al-Oar (04) Haq et al. (04b) Al-Nasser (007) Sya et al. (0) Ohyaa et al. (008) Haq et al. (03) Mehta and Mandowara (03) Alodat and Al-Saleh (00) Wang et al. (008) Al- Nasser and Ban-Mustafa (009) Al-Hadhra and Al-Oar (04) Snha (005) Wolfe (0) and Sya et al. (03a 03b). 3.. Estaton of the varance The SRS estator of the populaton varance s gven by ˆ SRS ( : ) SRS. An earler work for estatng the populaton varance s consdered by Stokes (980a). Based on judgent ordered usng balanced she defned n ˆ [ : ] h n h s the quantfcaton of the th ranked unt n a set of sze n the hth replcate. She where [ : ] h showed that.e. E ˆ n [ : ] ˆ s a based estator of the populaton varance. However the bas approach to zero as the nuber of easureents n becoes large. Based on the rato Var ˆ SRS RP the perforance of MES ˆ s nvestgated and proved that l RP. The author concluded that the gan n effcency of n over SRS s lttle when estatng hgher oents. A nonparaetrc study s consdered by Perron et al. (004) for the estaton of the populaton varance under ranked set saple. Bsws et al. (03) consdered varance estaton usng Jackknfe technque n ranked set saplng based on fnte populaton fraework. Also see MacEachern et al. (00) Al-Hadhra and Al-Oar (006) Al-Hadhra (00a) Chen and L (0) Abu-Dayyeh and Al-Subh (03) Estaton of the populaton rato The populaton rato of two varables and Y s defned as Y R. The SRS estator of the populaton rato s R SRS y y. x x Ths estator s based snce the denonator x as well as the nuerator y are rando varables. Saaw and Muttlak (996) suggested an estator of the populaton rato based on ranked set saplng as () ˆ Y R Y [] 55

7 where () and [] denote that the rankng of s perfect and the rankng of Y has errors. The varance of R ˆ s gven by ˆ R Var R where ( ) ( ) Y[ ] Y [ ] Y. Y Y ( ) Y[ ] Y ( ) Y Y Y Y and Y ( ) ( ) Y[ ] Y Based on Table 3. t s clear that the s ore effcent than SRS for estatng the populaton rato. Table 3.: The effcency and bas values of R ˆ wth respect to R ˆSRS for Effcency Bas Bas SRS Effcency Bas Bas SRS Effcency Bas Bas SRS Effcency Bas Bas SRS Effcency Bas Bas SRS Effcency Bas Bas SRS Saaw and Muttlak (00) used the edan to estate the populaton rato. Saaw and Tawalbeh (00) ntroduced a double edan for estatng the populaton ean and rato. Al- Oar and Bouza (04) consdered rato estators of the populaton ean wth ssng values usng ranked set saplng For ore about rato estaton n see Saaw and Saed (004) Al-Oar et al. (009) Mandowara and Mehta (04) Kadlar et al. (009) and Al-Oar (0) Estaton of the quantle Let be a rando varable wth cuulatve dstrbuton functon Fx ( ). The pth quantle s defned as p nf x : F( x) p for 0 p. The followng authors have done works to estate the pth quantle by dfferent procedures as gven below: Chen (000) studed quantle estaton based on balanced data and concluded that the schee can substantally develop the effcency of quantle estaton. Chen (00b) further generalzed the results n Chen (000) fro balanced to unbalanced schee. Indeed the quantle estator proposed n both Chen (000 00b) s construced n ters of the eprcal dstrbuton of the pooled data. Kaur et al. (00) proposed sgn test for populaton quantles and dentfes the optal allocaton based on the quantle obtaned but not based on the underlyng dstrbuton. Adata and Saleh (004) appled the generalzed ethod n estatng quantles of the unfor dstrbuton. Zhu and Wang (004) consdered quantle estaton usng under perfect rankng. Also see Saaw (00). 55

8 3.5. Estaton of the populaton proporton Lacayo et al. (00) nvestgated a populaton proporton estaton usng n the stuatons where the bnary varable of study s selected fro a contnuous varable. Terpstra (004) used procedure to estate a populaton proporton and consdered two estators the saple proporton and axu lkelhood estator of the data. Chen et al. (005) nvestgated the use of n estatng the populaton proporton. Chen et al. (006b) was the frst to suggest the use of the unbalanced ranked set saplng for estatng a populaton proporton under perfect rankng. Also see Terpstra and Nelson (005) and Bouza (03) Estaton of the dstrbuton functon Ths secton s devoted for estatng the dstrbuton functon under. Indeed t s of nterest to estate the dstrbuton functon to nvestgate the parent dstrbuton propertes such as the skewness and ultodalty. Also the estator of the underlyng dstrbuton s requred for constructng the confdence ntervals and hypothess testng about soe paraeters. Soe results for the dstrbuton functon estaton under are gven as follows. Let FSRS ( x ) denote the eprcal dstrbuton functon of a SRS as FSRS ( x) I x where () I s an ndcator functon. It s clear that E F ( x) F( x) wth SRS fro Fx ( ) defned Var FSRS ( x) F( x) F( x). Also FSRS ( x ) s a consstent estator of Fx ( ) (See Bahadur (996)). Stokes and Sager (998) used to estate Fx ( ) for fxed x and suggested the estator as F ( x) I x ( : ). They proved that F ( x ) s an unbased estator for Fx ( ) wth varance Var F ( x) F( : ) ( x) F( : ) ( x) and F ( x) E F ( x) Var F ( x) converges n dstrbuton to the standard noral dstrbuton as when x and n are fxed. The relatve effcency of wth respect to SRS for estatng the dstrbuton functon s defned as Var FSRS ( x) Eff F ( x) FSRS ( x) Var F ( x). Table 3.3: The relatve precson of F ( x ) wth respect to FSRS ( x ) wth Fx ( ) Fro Table 3.3 we can see that F ( x ) s ore effcent than FSRS ( x ). See Stokes and Sager (998) for ore detals. 555

9 Saaw and Al-Sageer (00) studed the dstrbuton functon estaton usng extree and edan ethods. Barabes and Fattorn (00) suggested kernel estators of probablty densty functons based on. K et al. (005) xed extree and edan to produce extree edan for estatng the dstrbuton functon. La et al. (00) suggested nonparaetrc estators for the dstrbuton functon and the ean usng the auxlary nforaton and concotant varable n process. Ozturk (00a) suggested a new estator of the dstrbuton functon and the center of syetry. Gulat (004) have consdered sooth non-paraetrc estates of the dstrbuton functon based on. Frey. (04) consdered bootstrap confdence bands for the CDF usng rankedset saplng. L et al. (04) nvestgated a kernel densty estaton usng. Abu-Dayyeh et al. (00) consdered the dstrbuton functon estaton usng double ranked set saplng ethod. Also see Wolfe (004) Baraneso and Fattorn (00) and Huang (97) Regresson n ranked set saplng Phlp and La (997) assued that the rankng s done on the bass of a concotant varable assocated wth the dependent varable Y whch s expensve to easure. In the case of the ean of s known and both varables are postvely correlated the authors follow the sae dea of Stokes (977) where the rankng s done by eans of the concotant varable whch can be easured easly copared to the an varable Y. Accordng to Stokes (997) they assued that the regresson of Y on s lnear as Y Y Y where and are ndependent has zero ean and varance Y consdered the dfference estator wth varance and YD Y B Var Y Var Y D B B Var Var Y Bˆ n ( : ) h Y[ : ] h Y h n h ( : ) h. For known they where ( : ) h s the th sallest and Y [ : ] h s the correspondng value of Y get fro the th saple n the hth cycle. The regresson estator for Y s gven by Y Y Bˆ whch s an unbased of Y wth varance where Z ( : ) h reg Y z Var Yreg E n sz ( : ) h n n Z Z( : ) h S Z( : ) Z n n h and Z h. h If s unknown the double saplng schee can be used to estate. If s pleented n the second-phase saplng the double saplng regresson estator of Y s gven by Y Y Bˆ x wth varance Var Y ds ds Z Z Y n ns n Z 55

10 x where Z. The authors showed that the regresson estator s ore effcent than and SRS nave estators f and Y are jontly fro a bvarate noral dstrbuton unless 0.4. For ore about regresson estaton n see Patl et al. (993) Muttlak (995) Muttlak (996) Barreto and Barnett (999) Barnett and Moore (997) Chen (00a) Ozturk (00b) Badus et al. (0) Alodat et al. (00) Murff and Sager (006) Al-Odat et al. (009) Al-Odat and Jetschke (0) Saaw and Ababneh (00) Alodat et al. (009) Saaw and Abu-Dayyeh (00) Der and Çng (000) and Saaw and Al-Saleh (00) Estaton of the dstrbuton paraeters Ths secton suarzes soe of the works where the s used to estate the unknown paraeters of the dstrbuton functon. Stokes (976) used for estatng the scale and locaton paraeters varance nterval and for estatng a correlaton coeffcent and test of correlaton. Stokes (980b) nvestgated the perforance of for correlaton coeffcent estaton n a bvarate noral dstrbuton usng the axu lkelhood estator. Fe et al. (994) nvestgated the perforance of n the estaton of the paraeters of Webull dstrbuton. Kva and Saanego (993994) proposed estaton of the populaton ean and populaton dstrbuton functon under unbalanced ranked-set saples. La et al. (994) used to estate the locaton scale and quantle of the exponental dstrbuton. L et al. (999) ntroduced the concept of rando selecton n wth applcaton to estate the ean and the varance of the noral dstrbuton and also appled the ethod when the parent dstrbuton s exponental and logstc. Shen (994) nvestgated the perforance of ranked set saplng for estatng the ean of a lognoral dstrbuton wth a known coeffcent of varaton. Stokes (995) has consdered a best lnear unbased and the axu lkelhood estaton of and for the locaton-scale faly of rando varables wth dstrbuton functon F( y ) / wth known F. Adata (000) generalzed the ranked set saplng and used t for estatng the locaton and scale paraeters of the half-logstc dstrbuton. Raqap et al. (00) extend the works of Snha et al. (996) and Stokes (995) by proposng best lnear nvarant estators n ( BLIEs) of the scale and locaton paraeters separately provded thata the nusance paraeter s unknown for syetrc and syetrc dstrbutons. Al-Saleh (004a) proposed balanced ranked set saplng as a non-paraetrc ethod for estatng the populaton paraeters other than the ean. Al-Saleh and Saaw (005) nvestgated the correlaton coeffcent estaton usng bvarate wth an applcaton to the bvarate dstrbuton. Al-Saleh (004b) ntroduced steady state ranked set saplng and nvestgated ts relaton to stratfed saplng. Abu-Dayyeh et al. (004) studed the estaton of the logstc dstrbuton paraeters usng SRS and wth soe of ts odfcatons as extree and edan. Modarres and Zheng (004) used to nvestgate axu lkelhood estaton of the dependence paraeter of a general bvarate dstrbuton. Sengupta and Mukhut (004) studed the unbased estaton of an exponental dstrbuton varance usng ranked set saplng. Ozturk (005) has consdered jont estaton of the locaton and scale paraeters of a locaton-scale faly usng ranked set saplng ethod. Hanandeh and Al-Saleh (03) consdered the nference on Downton s bvarate exponental dstrbuton based on ovng extree ranked set saplng Hussan. (04) nvestgated Bayesan and axu lkelhood estaton for Kuarasway dstrbuton based on ranked set saplng. Chen et al. (03) used a paraetrc estaton for the scale paraeter for scale dstrbutons usng ovng extrees. 55

11 Tahaseb and Jafar (0) studed the scale paraeter of Morgenstern type bvarate unfor dstrbuton usng ranked set saplng. Oar and Ibrah (03) estated the shape and scale paraeters of the Pareto dstrbuton based on extree. Sarkavanj et al. (04) studed the locaton and scale estators of a two-paraeter exponental dstrbuton usng sple rando saple and ranked set saple n ters of generalzed varance. Also see K and Arnold (999) Zhao and Chen (00) Bhoj (000) Badus et al. (0) Shabu and Muttlak (00) Al-Rawwash et al. (04) Abu-Dayyeh et al. (03) El-Neweh and Snha (000) Al-Odat and Oar (0) Chen et al. (04) Chacko and Thoas ( ) Al- Saleh and Dab (009) Al-Saleh and Ananbeh ( ) Bhoj and Ahsanullah (996) Al- Hadhra (00b) Sngh and Mehta (04) Sadek and Alharb (04) and Modarres et al. (006). 4. THE ISSUE OF RANKING AND CONCOMITANT VARIABLE The effcency of s affected by any factors. One portant aspect of the ethod s the rankng steps. The rankng can be based by judgent or on concotant (auxlary) varable that s correlated to the varable of nterest. The effcency depends on the success n rankng. Dell and Clutter (97) was the frst one who nvestgated the perforance of when the rankng s done wth errors. They used the fundaental dentty f ( x) f[ : ]( x) and showed that [ : ] and [ : ] of the th realzed order statstcs can be estated fro the recorded data where ( : ) represent the realzed order statstcs obtaned by the rankng process and [ : ] denote that the orderng s based on the perceved ranks. Also the fundaental dentty s stll true the relatve savng can be wrtten as RS [ : ] and reans unbased estator of the populaton ean wth errors n rankng. Moreover f the rankng process s copletely done wth errors then Var SRS Var Stokes (977) studed the ranked set saplng wth concotant varables. She supposed that the study varable has a lnear relaton wth other varable Y that s easy to rank and showed that RS RS where s the correlaton between Y and and [ : Y ] Y RS [ Y : ] where RS [ Y : ] s the RS n estatng x whle d RS Y s the RS n estatng ( : ) x and EY( : ) y RSY x y f the rankng s pleented by the concotant varable Y Y under perfect rankng on Y. She showed that RSY RS f Y up to a lnear transforaton snce the RSs' are unaffected by lnear transforatons of the varable of nterest. Ths ples that RS RS. [ : Y ] Kaur et al. (996) ade a coparson between and stratfed SRS when usng a concotant varable based on equal and optu allocatons of unts for estatng the populaton ean. Muttlak (998a) conducted a study of the perforance of edan (M) for estatng the populaton ean of nterest when the rankng s perfored usng a concotant varable. Also based on an auxlary varable the regresson estator s proposed to estate the populaton ean. Accordng to ths study Muttlak showed that the M estator s ore effcent than and regresson estators. Rdout and Cobby (987) nvestgated the perforance of wth non-rando selecton of sets based on the concotants odel of Davd and Lavne (97) and Stokes (977) and also they nvestgated the perforance of wth errors n rankng based on cluster saplng. Muttlak (988c) studed the wth sze based probablty of chosen to estate populaton paraeters as populaton total based on a concotant varable populaton ean populaton sze usng perfect and perfect rankng process and the populaton total usng a two-stage saplng ethod n whch the s used at the second stage whle at the frst stage the saplng s done wth probablty proportonal to the sze. 55

12 Muttlak and McDonald (990a 999b) have suggested wth concotant varable and sze based probablty of chosen proportonal to sze usng both perfect rankng and n the presence of errors n rankng. Bohn and Wolfe (99) generalzed the Mann-Whtney-Wlcoxon two saple test to the case when a balanced ranked set saple s selected fro each of the two populatons under perfect rankng. Also the authors extended the test to the case of perfect rankng and study the effects of judgent errors on the features of the test. Bohn and Wolfe (994) extended the work of Bohn and Wolfe (99) to the stuatons of perfect rankng and nvestgate the effects of judgent error on features of Mann-Whtney-Wlcoxon test. The proposed Mann-Whtney-Wlcoxon statstc U s the U-statstc whch s coputed n as the sae way as for under perfect judgent rankng by Bohn and Wolfe (99) based on the sae expresson of the ean and varance. Norrs et al. (995) dscussed two alternatve ethods. The frst s a odfcaton of an dea of Takahas (970). The second s to use an unbalanced data allocaton usng Neyan allocaton for the characterstc of prary nterest treated ths as a concotant varable for the other varables under consderaton. Kaur and Talle (000) apponted the optal allocatons for two classes of syetrc dstrbutons. A general for for the Fsher nforaton atrx s ntroduced by Barabes and El-Sharaaw (00). If s a contnuous rando varable wth dstrbuton functon F( ; ) they showed that based on a saple of sze the Fsher nforaton atrx can be seperated nto the su of the SRS Fsher nforaton atrx and a se-postve defnte atrx as I ( ) I ( ) I ( ) SRS R where I ( ) SRS s the Fsher nforaton atrx based on a SRS of sze k whch s gven by ln f( ; ) I SRS ( ) ke T and ln F( ; ) ln F( ; ) I R ( ) k( k ) E T. In the case of error n rankng the authors showed that I ( ) I SRS ( ). Barabes and Marchesell (004) proposed protocol n saplng surveys when an auxlary varable s avalable n addton to the target varable. Based on the newly suggested saplng ethod the estators proposed n surveys wth auxlary nforaton such as the regresson or the rato estators. They showed that the suggested ethod provdes ore precse estaton than the sple rando saplng counterpart. Ba and Chen (003) studed the and ts rafcatons nvolvng wth perfect rankng wth ultvarate saples and by a concotant varable rankng. Chen and Shen (003) suggested two layers wth concotant varable. In the frst layer of the procedure saplng unts are ranked wth respect to one concotant varable and n the second layer the saplng unts are ranked wth respect to another concotant varable. The authors showed that the two-layers ethod satsfes the fundaental equalty F( x) F( : ) ( x) whch s portant for the usual and then ore effcent than SRS ethod. Snce the two-layers falls nto the fraework of the general they claed that the all features of the general can be appled for the two-layers wthout any detals of the proofs. The results of the sulatons llustrated the superorty of the two-layers over the argnal. Nahhas et al. (004) consdered a vsual judgent error odel whch s based on ratos of szes of pars of observatons; that s Y... where... are d lognoral wth paraeters 0 and where the s and s are utually ndependent. They derved the effcency wth balanced ranked set saplng under ths ultplcatve odel and consdered ethods for estatng the varance of the error. 55

13 Chen et al. (006a) nvestgated the eprcal evaluaton of the accuracy n rankngs. They showed that the rankng accuracy can be assessed through an atrx P pj where p j s the probablty that the unt wth real rank n a gven set wth sze s consdered to be the jth judgent p P. order statstcs (Bohn & Wolfe 994; Stark & Wolfe 00). In partcular j ( : ) [ j: ] Now the value of p depends on the process of rankng.e. under perfect rankng p for... and pj 0 for j... whle p j for any and j f the rankng process s copletely rando. Snce the jth judgent order statstcs wll equal to the th true order statstcs for soe then p j. They llustrated that under any rankng ssue the rankng errors ncrease progressvely when the set sze ncreases whch ndeed has a negatve effect of the effcency of a estator. On the other hand the effcency of estator ncreases wth larger saple sze under perfect rankng. Hatef and Jozan (03) nvestgated a Fsher nforaton n dfferent types of perfect and perfect ranked set saples selected fro fnte xture odels. Chen (00) consdered ultple concotant varables n adaptve ranked set saplng. Also see Mode et al. (00) Park and L (0) Badus and Ikegwu (03) Scara and Nar (999) Presnell and Bohn (999) Nahhas (004) Wang et al. (006) Vock and Balakrshnan (0) L and Balakrshnan (008) Frey et al. (007) Bouza (00) Ozturk (000c 007b ) Zheng (004) and Patl et al. (994). 5. STATISTICAL QUALITY CONTROL CHARTS BASED ON In ths secton dscussons are gven on the usual or any of ts odfcatons ay consdered to prove control charts for the ean. The charts are copared to the coonly qualty control charts for varables usng the usual SRS. Let... j... n be n ndependent saples each of sze fro the noral dstrbuton N saple ean s N / j j. It s well known that the dstrbuton of the. Shewhart's control charts have been consdered wdely n ndustres to detect the shft n process ean. Salazar and Snha (997) are fst who used the n qualty control charts for varables. If both and are known then the upper control lt (UCL) center lt (CL) and lower control lt (LCL) for the charts as suggested by Salazar and Snha (997) are gven by UCL 3 CL LCL 3 where ( : ). The control charts are copared wth the SRS control charts n ters of ARL whch s the average nuber of ponts that ust be plotted before a pont ndcates an out-of-control condton. The ARL s defned as ARL / for any Shewhart control chart f the process s n control and the process observatons are uncorrelated and ARL / ( ) f the process s get out of control where and denote the probabltes of Type-I and Type-II errors respectvely. Based on the ARL the process s n control wth ean o and standard devaton o otherwse the process has a shft about o /.e. a shft n ean fro o to o o / where s nonnegatve and selected to donate on the shft n the ean. When the process s n control and follow a noral dstrbuton wth paraeters o / o o and o the shft n the process ean s known as see Montgoery (005). Table 5. shows soe ARL values usng and SRS n qualty control charts. As shown n Table 5. The ARL values based on wll decrease uch faster as copared to the values under SRS. As the saple sze ncreases generally the ARL values decreases. 55

14 Table 5.: ARL values based on 3-sga lts usng SRS and SRS SRS SRS Muttlak and Al-Sabah (003a) developed any control charts for the saple ean by usng par and selected ethods. Muttlak and Al-Sabah (003b) used the and soe of ts odfcatons naely edan and extree ethods to develop dfferent qualty control charts of the ean. Abujya and Muttlak (004) nvestgated qualty control charts of the saple ean usng the edan and double ethods. For ore about n qualty control charts see Al-Oar and Haq (0) Al-Nasser and Al-Rawwash (007) Al-Sabah (00) Abbas and Mller (0) Al-Oar and Al-Nasser (0) Haq (0) Pongpullponsak and Sontsaran (03) Mehood et al. (03 04) Raz et al. (0) Haq et al. (04b) Lee and Raz (04) and Abujya et al. (0). 6. WITH FINITE POPULATIONS Several researchers have consdered dfferent probles belongng to the ranked set saplng schee when the saplng s proceed on an nfnte populaton. Recently soe authors proposed procedure when the populaton s fnte. Kowalczyk (004) studed the effcency of n estatng the fnte populaton ean and total. The author showed that the new aspect of ncludes unt rankng or se-rankng done based on grouped data. If Y and y denote the th populaton eleent and the correspondng value of M unts (... M) so that each eleent n the populaton has probablty M. The fnte populaton ean s y y ( : ) wth varance M where M D y S E y Y M ( : ) S Yj Y. Whle the unbased estator of the populaton total s y M y M j wth varance M M M D y S E y( : ) Y. The author concluded that y and y M y are unbased estators of the populaton ean and total respectvely and they are ore effcent than ther counterparts usng SRS wth replaceent. Under SRS wthout replaceent the author showed that when the saple sze s coparatvely sall D y D y and the should also be superor as copared to the populaton sze then SRS 55 SRSWOR to SRS. Deshpande et al. (006) suggested three protocols for drawng a fro a fnte populaton. For sall populaton sze the saplng ethods lead to hghly dvergent saplng dstrbuton whle when the populaton sze ncreases the dfferences decreases. Nonparaetrc confdence ntervals are found to be shorter than those produced by SRS based on the three saplng protocols. Gokpnar and Ozdr (04) consdered ncluson probabltes n ranked set sat saplng n fnte populatons.

15 7. CONFIDENCE INTERVALS AND HYPOTHESIS TESTING IN Hartlaub and Wolfe (999) extended the concept of ranked set saples to the -saple locaton settng when the treatent effect paraeters follow a restrcted ubrella pattern. They developed the dstrbuton-free testng for both the case where the peak of the ubrella s known and for the case where t s unknown. Bohn (994) consdered technque based on ranked-set saples analogues of the standard one-saple sgn statstcs. Hettansperger (995) proposed procedures based on analogues of the standard one-saple sgned-rank test. Kaur et al. (996) exaned the perforance of the sgn test under unequal allocatons and found that the optal allocaton for testng the edan assgns all observatons to the ddle rank order regardless of the parent dstrbuton. Kaur et al. (997) consdered the unequal allocatons usng for skew dstrbutons. Ozturk and Wolfe (000) suggested alternatve ranked set saplng ethods for the sgn test statstc. Al-Saleh and Zheng (003) suggested a ethod based on ranked set saplng to get an approxate preferable saple fro a populaton of nterest. Ths ethod can be consdered to get saple ponts that goes to be fro a specfc porton of a dstrbuton such as the upper or lower quartles. K and K (003) suggested ranked orderng-set saplng (ROSS) and copared t to the usual. They proposed the test statstc usng sgn test on ROSS and found that the ROSS has ore nforaton than where the asyptotc effcency of ROSS relatve to s always greater than except when saple sze s. Ozturk et al. (004) developed a nonparaetrc ult-saple nference for sple-tree saples alternatves s dstrbuton-free. They have constructed the ult-saple nference based on copare the sultaneous one-saple sgn confdence ntervals for the edans. Hu et al. (005) nvestgated the resaplng ethods to obtan confdence ntervals for the regresson estator of the populaton ean. Ozturk and Deshpande (006) studed the propertes of the quantle ntervals usng and proved dstrbutonal propertes of the order statstcs. The nterpolaton of adjacent order statstcs suggested by Hettansperger and Sheather (996) was consdered by the authors to have confdence nterval by extendng the nterpolated confdence ntervals to the data for sall saple szes. The authors concluded that the quantle ntervals usng have shorter expected lengths or hgher coverage probabltes than ther sple rando saple copettors. Terpstra and Mller (006) studed the hypothess testng and confdence nterval for a populaton proporton based on a. For a data let ( : ) h where... and h... n conssts of a n atrx of ndependent bnary order statstcs. The dstrbuton of ( : ) h s assued to be B ( p) where p denotes the success probablty and ( p) P B( p).... The estator of the populaton proporton s gven by ( ). pˆ ( : ). n n where () B n ( p) rando varable. They showed that the nference procedures are generally ore effcent than ther copettors usng SRS referrng to the shorter confdence ntervals and ore powerful test. Ozturk (007) proved an exact nonparaetrc test for the equalty of edans n a two-saple or desgn. Based on general rankng ethod he showed that the asyptotc null dstrbuton s noral wth perfect rankng. Albatneh et al. (04) studed a confdence nterval of the sgnal-to-nose rato usng. Zhang et al. (04) consdered sgn tests usng ranked set saplng wth unequal set szes. Gaur et al. (03) nvestgated a nonparaetrc test for a ult-saple scale proble usng data. Frey and Wang (03) consdered ost powerful rank tests for perfect rankngs. Oar et al. (0) studed cconfdence nterval estaton of the shape paraeter of Pareto dstrbuton usng extree order statstcs. Also see Saaw et al. (006) Vock and Balakrshnan (03) Ozturk and Wolfe (00 000a 000b 000d) K et al. (999) Saaw and Abu-Dayyeh (003) Langyong and aofang (00) Kot and Babu (996) Al-Rawwash et al. (00) Ozturk (999) Barabes (998) Ozturk and Balakrshnan (009) s the nuber of success whch s a 55

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