Improvement Over General And Wider Class of Estimators Using Ranked Set Sampling
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1 ITERATIOAL JOURAL OF SIETIFI & TEOLOG RESEAR VOLUME ISSUE 7 AUGUST ISS Iprovee Over Geeral Ad ider lass of Esiaors Usi Raked Se Sapli V L Madoara iu Meha Raka Absrac: I his paper Iprovee over eeral ad ider class of esiaors of fiie populaio eas usi raked se sapli is ivesiaed Raked se sapli as firs suesed o icrease he efficiec of esiaor of he populaio ea The firs order approiaio o he bias ad ea square error MSE of he ivesiaed esiaors are obaied Theoreicall i is sho ha hese suesed esiaors are ore efficie ha he eeral ad ider class of esiaors i siple rado sapli Ke ords: Raked se sapli Geeral ad ider class of esiaor Auiliar variable Mea square error Efficiec Iroducio Raked se sapli as firs suesed b McIre 95 o icrease he efficiec of esiaor of populaio ea Kadilar e al 9 used his echique o iprove raio esiaor ive b Prasad 989 ere e shall iprove eeral ad ider class of esiaors ive b Srivasava based o auiliar variable Srivasava 97 proposed a eeral class of esiaors o esiae he populaio ea of he sud variable hich i he case of sile ea of he auiliar variable is ive b here u ad is a paraeric fucio I he Siple rado sapli he bias ad iiu ea square error of he eeral class of esiaor see sih Volue pae 6-65 are ive b f or O iori f u or here ad are he saple ad populaio sizes respecivel ; f ; deoe he coefficie of variaio of ad respecivel ad deoe he correlaio coefficie beee ad ere u ad u u deoe u` he firs ad secod order parial derivaives of ih respec o u ad are he ko cosas The iiu ea square error MSE of he eeral class of esiaor defied a o he firs order of approiaio is Mi MSE f or Mi MSE o iori f or Mi MSE VLMadoara is orki as Professor i he Depare of Maheaics ad Saisics Uiversi ollee of Sciece MLSU Udaipur Eai- adoara_vl@ahoocoi iu Meha Raka is currel pursui PhD i Saisics i he Depare of Maheaics ad Saisics Uiversi ollee of Sciece MLSU Udaipur Eail- iueha8@ailco If e aach a fucio of o he saple ea he aspoic iiu ea square error of he resula esiaor cao be reduced furher ha ha ive i Thus he usual raio esiaor produc esiaor ad poer rasforaio esiaor are he special cases of he class of esiaors defied i hile he reressio esiaor ad differece esiaor are o special cases of he eeral class of esiaors The Srivasava 98 IJSTR ijsror 57
2 ITERATIOAL JOURAL OF SIETIFI & TEOLOG RESEAR VOLUME ISSUE 7 AUGUST ISS defied aoher class of esiaors ad aed a ider class of esiaors as here u u is a fucio of ad u The aspoic bias ad iiu ea square error see sih Volue pae are ive b f o iori f or here u 5 u u ad u u u u The iiu ea squared error of he ider class of esiaor is ive b Mi MSE f or Mi MSE o iori f or Mi MSE 6 If e ake a fucio of ad o esiae he populaio ea he aspoic iiu ea square error of he resula esiaor aai cao be reduced furher ha ha ive i 6 Thus he usual liear reressio esiaor ad differece esiaor are special cases of he ider class of esiaor defied a The Suesed Esiaors I Raked se sapli idepede rado ses each of size are seleced ih equal probabili ad ihou replacee fro he populaio The ebers IJSTR ijsror of each rado se are raked ih respec o he characerisic of he sud variable or auiliar variable The he salles ui is seleced fro he ordered se ad he secod salles ui is seleced fro he secod ordered se his a his procedure is coiued uil h he ui ih he lares rak is chose fro he se uis This ccle a be repeaed r ies so r have bee easured duri his process he e rak o h he auiliar variable le deoe a i jude [ h h orderi i he i se for he sud variable ad i se for he auiliar variable Adapi he esiaor i o he eeral class of esiaor for he populaio ea proposed b Srivasava 97 e sues he folloi esiaor for eeral class of esiaor usi raked se sapli is u here u [ ] i [ i ad is a paraeric fucio such ha i saisfies he folloi codiios: a b The firs ad secod order parial derivaives of ih respec o u eiss ad are ko cosas a a ive poi u u Epadi abou he value i a secod order Talor s series e have u u u u u u oe ha u u u hus he hiher order ers ca be eleced Usi he above o codiios e obai u u here u ad u u u` To obai bias ad MSE of e pu ad E E so ha 58
3 ITERATIOAL JOURAL OF SIETIFI & TEOLOG RESEAR VOLUME ISSUE 7 AUGUST ISS V V E = S [ r i siilarl E ad = [ V = ov [ ] ov E = S r r here i S S MSE E o aai eleci hiher order ers e have E MSE MSE [ The opiu value of o iiize he MSE of ca be easil foud as follos MSE S [ & r r i [ r i i ere e ould also like o reid ha ad [ [ [ Furher o validae firs deree of approiaio e assue ha he saple size is lare eouh o e ad as sall so ha he ers ivolvi ad or i a deree reaer ha o ill be eliible The proposed esiaor ive i ca easil be rie i ers of ad as o o ias ad MSE of he esiaor deree of approiaio are respecivel ive b ere = E - o he firs E E IJSTR ijsror ov[ [ ] ] = V[ ] [becauseov V ] he e replace b i e obai iiu MSE of he proposed esiaor as follos A Mi MSE A here [ 5 his a e ca rie 5 as MSE MSE A I is easil sho ha iiu ea square error of he proposed esiaor ive i 5 usi raked se sapli is alas saller ha he MSE of esiaor suesed b Srivasava i because A is a o eaive value Thus he raio esiaor produc esiaor ad poer rasforaio esiaor usi raked se sapli are he special cases of he proposed esiaors defied i hile he reressio esiaor ad differece esiaor are o special cases of he esiaor The adapi he esiaor i o he ider class of esiaor for he populaio ea proposed b Srivasava 98 e develop he folloi esiaor for ider class of esiaor usi raked se sapli is 59
4 ITERATIOAL JOURAL OF SIETIFI & TEOLOG RESEAR VOLUME ISSUE 7 AUGUST ISS u 6 here u is a fucio of [] folloi reulari codiios: The poi u ad u saisfies he assues he value i a closed cove subse R of o coaii he poi The fucio u bouded i R diesioal real space is coiuous ad = ad here deoes he firs order parial derivaive of ih respec o [] The firs ad secod order parial derivaives of u i R Epadi u Talor series e have eis ad are coiuous ad bouded abou he poi i a secod order u= u [ ] Usi he above reulari codiios ad u [ ] = e have u u 7 here [ ] u [ ] u [ ] u [ ] u & o ias ad MSE of he esiaor u deree of approiaio are respecivel ive b = E - [ ] u [ ] u o he firs o E E E [ ad MSE E = MSE E ` 8 9 [ O differeiai 9 ih respec o ad equai o zero e obai he opiu value of deoed b is MSE Replaci b he esiaor as follos Mi MSE i 9 e obai iiu MSE of [ A Mi MSE Aai e ca rie as MSE MSE A The esiaor ad as ca easil be rie i ers of IJSTR ijsror So i is proved ha he Mi MSE of he esiaor usi raked se sapli is alas saller ha he MSE 6
5 ITERATIOAL JOURAL OF SIETIFI & TEOLOG RESEAR VOLUME ISSUE 7 AUGUST ISS of esiaor suesed b Srivasava ive i 6 because A is a o eaive value As a resul sho ha he suesed esiaor is ore efficie ha he esiaor Refereces Kadilar Uazici ad ii 9 Raio esiaor for he populaio ea usi raked se sapli Sa papers5-9 McIre GA95 A ehod of ubiased selecive sapli usi raked ses Ausralia Joural of Ariculural Research 85-9 Prasad 989 Soe iproved raio pe esiaors of populaio ea ad raio i fiie populaio saple surves ou Sa Theor Mehods 8:79 9 Sih Sarjider Advaced Sapli Theor ih Applicaio Vol Kluer Acadeic Publishers eherlads 5 Srivasava SK97 A eeralized esiaor for he ea of a fiie populaio usi uli auiliar iforaio J Aer Sais Assoc Srivasava SK98 A class of esiaor usi auiliar iforaio i saple surves aad J Sais IJSTR ijsror 6
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