Improving the efficiency of the ratio/product estimators of the population mean in stratified random samples

Transcription

1 STATISTICS RESEARCH ARTICLE Impovng the effcency of the ato/poduct estmatos of the populaton mean statfed andom samples Receved: 10 Decembe 2017 Accepted: 08 July 2018 Fst Publshed 16 July 2018 *Coespondng autho: Isaac Dalsngh, Depatment of Mathematcs and Statstcs, The Unvesty of the West Indes, Tndad and Tobago E-mal: Revewng edto: Guohua Zou, Chnese Academy of Scences, Chna Addtonal nfomatos avalable at the end of the atcle Bendon Bhagwandeen 1, Andew Vea 1 and Isaac Dalsngh 1 * Abstact: The effcency of a statstc detemnes ts effcacy. In statfed andom samplng, many estmatos fo the populaton mean has been poposed. In ths pape, we popose two new estmatos both of whch ae combned ato/poduct estmatos. We efe to ou estmatos as mxtue estmatos. We deve the mean squae eos (MSEs) up to the fst ode. A compehensve smulaton study was caed out to show the effectveness of ou estmatos as compaed to the conventonal estmatos that utlze auxlay nfomaton. We also compaed the pefomance of ou estmatos and some of the moe popula competng estmatos usng eal data. Both the smulatons and eal data analyss showed ou estmatos wee moe effcent than almost all exstng estmatos consdeed. Subjects: Appled Mathematcs; Mathematcs Educaton; Statstcs & Pobablty Keywods: ato/poduct; effcent; auxlay nfomaton; statfed samplng; smulaton 1. Intoducton and notaton Sample suveys ae usually deployed as the most cost-effectve devce fo estmaton of a populaton paamete. Thee ae many competng estmatos fo a populaton paamete. The best estmato s usually the one wth the smallest mean squaed eo (MSE). The MSE of an estmato s the sum of the vaance and the squae of the bas. It theefoe accounts fo both pecson and accuacy. The smalle the MSE, the bette the estmato. Many estmatos have been poposed fo ABOUT THE AUTHORS Andew Vea and Bendon Bhagwandeen ae gaduate students n the Depatment of Mathematcs and Statstcs, The Unvesty of the West Indes, St Augustne Campus, Tndad and Tobago. D. Isaac Dalsngh s a lectue n the Depatment of Mathematcs and Statstcs at the Unvesty of the West Indes, St Augustne Campus, Tndad and Tobago. PUBLIC INTEREST STATEMENT Have you eve thought about how the estmates fo a smple populaton paamete lke the mean ncome s calculated unde dffeent samplng scenaos? Well t s mpotant to note that the samplng scheme whethe t be smple andom samplng, systematc samplng, statfed samplng o even cluste samplng mpacts on the coect computaton of the mean. In ths pape, we look at some estmatos of the populaton mean unde statfed samplng. In ths type of samplng usually, thee s an auxlay vaable that we eadly obseve that may be coelated wth the man vaable of nteest. Ths addtonal nfomaton can assst us n gettng bette estmates of the mean. In ths pape, we popose two new estmatos and show va smulatons and eal data analyss that ou estmatos ae effcent unde a vaety of statfcaton confguatons The Autho(s). Ths open access atcle s dstbuted unde a Ceatve Commons Attbuton (CC-BY) 4.0 lcense. Page 1 of 17

2 the estmaton of the populaton mean unde smple andom and statfed andom samplng. Some of the notable contbutons to estmatos n samplng lteatue nclude Cochan (1940), Muthy (1967), Kadla and Cng (2005), Sngh and Vshwakama (2007). Ths lst s by no means exhaustve. Fo statfed samples, the use of an auxlay vaable (X) has been shown to mpove on the effcency of the estmatos of the populaton paametes fo the vaable of nteest (Y). Auxlay nfomatos used n the desgn and estmaton stages of a suvey. Ths pape focuses on mpovng effcency when estmatng the populaton mean fom statfed samples wth the help of auxlay nfomaton at the estmaton stage. We assume a statfed andom sample of sze s selected fom a lage multvaate populaton of sze N: A smple andom sample of sze s taken wthout eplacement fom each statum of sze N whee ¼ 1; 2;...; k and k s the numbe of stata n the populaton. Data n each statum ae assumed to have come fom a multvaate nomal supe populaton wth fnte populaton coecton pe statum f ¼ N N, N ; so that f ff 1. Suveys often make use of an auxlay vaable ðx Þ whch s assumed to povde useful nfomaton the estmaton of the mean of the vaable of nteest (Y). The auxlay vaable n each statum X s moe eadly avalable than the vaable of nteest Y. In estmatng the populaton mean Y fo the vaable of nteest, whle assumng that the populaton mean of the auxlay nfomaton X s known, the followng classcal estmatos n statfed samplng have been poposed (Hansen & Huwtz, 1943). 2. Classcal estmatos fo populaton mean The combned ato estmato: by ¼ y st X x st (2:1) and the combned poduct estmato: ð by ¼ y st :x stþ X (2:2) ae by fa the most popula estmatos. We defne y st and x st as the statfed sample means of the man and auxlay vaables, espectvely. In these estmatos, y st x st s the estmate of ato of the populaton means whle y st :x st s the estmate of the poduct of the populaton means. In addton, y st ¼ k w y x st ¼ k w x (2:3) (2:4) whee w ¼ N N s the statum weght, y and x ae the statum sample means fo the man and auxlay vaables, espectvely. The ato estmato s used fo the estmaton of the populaton mean when Y and X ae postvely coelated to each othe whle the poduct estmato s used when Y and X ae negatvely coelated to each othe. Fst-ode appoxmatons to the bas and MSE of the combned ato/poduct estmatos ae deved. To do ths, we let y st ¼ Y ð1 þ ε 0 Þ and x st ¼ X ð1 þ ε 1 Þ, whee ε 0 and ε 1 ae eos whch can be postve o negatve such that Eðε 0 Þ ¼ Eðε 1 Þ ¼ 0. Fo statfed andom samplng, Vaðε 0 Þ ¼ 1 Y 2 k f w 2 S2 Y, Vaðε 1 Þ ¼ 1 2 k f w 2 S2 X X and Covðε 0 ; ε 1 Þ ¼ 1 X Y k f w 2 ρ S X S Y, whee ρ s the coelaton between the two (the man and auxlay) vaables n the th statum. In addton, Page 2 of 17

3 S 2 Y ¼ 1 N N 1 Y j Y 2 (2:5) j¼1 S 2 X ¼ 1 N N 1 X j X 2 (2:6) j¼1 and t s assumed that when the sample s suffcently lage such that jε 0 j and jε 1 j ae small enough so that tems nvolvng ε 0 and/o ε 1 to degees hghe than two ae consdeed neglgble. By substtutng the expessons fo y st and x st n tems of ε 0 and ε 1 nto Eqs. (2.1) and (2.2), the followng ae obtaned: by ¼ ð1 þ ε 0 Þð1 þ ε 1 Þ 1 Y (2:7) by ¼ ð1 þ ε 0 Þð1 þ ε 1 ÞY (2:8) Assume jε 1 j < 1 and expandng ð1 þ ε 1 Þ 1 we obtan: Bas b " Y ¼ R k f w 2 S 2 X X ρ # S X S Y Y (2:9) Hee, R ¼ Y X, the ato of the populaton means. Smlaly, Bas b Y ¼ 1 X k f w 2 S XY (2:10) s obtaned up to the fst ode of appoxmaton (Sngh & Mangat, 2013). In addton, S XY ¼ 1 N N 1 X j X Yj Y j¼1 (2:11) MSE ^ Y ¼ E ^ Y Y 2 h ff E Y 2 ε 2 0 þ ε2 1 2ε 0ε 1 ¼ Vaðy st Þþ k f w 2 R 2 S 2 X n 2RS XY (2:12) Smlaly, MSE ^ Y ¼ E ^ Y Y 2 h ff E Y 2 ε 2 0 þ ε2 1 þ 2ε 0ε 1 ¼ Vaðy st Þþ k f w 2 R 2 S 2 X n þ 2RS XY Theefoe, up to the fst-ode appoxmaton, MSE b Y 2RS XY Þ < 0,.e., f and only f C > 1 2, whee C ¼ k Smlaly, MSE b Y f w 2 S XY R k f w 2 S2 X < Vaðy st Þ f and only f k f w 2 (2:13) R 2 S 2 X (wth expemental data, C s guessable). < Vaðy st Þ f and only f C < 1 2. Ths ndcates that the combned ato/poduct estmatos ae elatvely moe effcent than y st, the unbased statfed sample mean, when C > 1 2 and C < 1 2, espectvely. Thus, b Y = b Y wll not mpove y st when 1 2 C 1 2. Page 3 of 17

4 Some othe estmatos nclude the combned estmatos: by eg ¼ y st þ β X x st by eg and sepaate egesson (2:14) whee β ¼ k f w 2 S XY k f w 2 S2 X (2:15) ¼ k w y þ β X x (2:16) whee β ¼ S XY S 2 X (2:17) The MSEs of the combned egesson and sepaate ato estmatos ae gven by: MSE b Yeg ¼ k w 2 f S 2 Y þ β2 c S2 X 2β c S XY (2:18) whee β c ¼ k f w 2 S XY (2:19) k f w 2 S2 X and MSE b Yseg ¼ k w 2 f S 2 Y n 1 ρ2 (2:20) whee ρ ¼ S XY S X S Y (2:21) The mpovement of y st has been an ongong aea of eseach. The dea fo ou poposed estmatos come fom a pape by Shley, Saha, and Dalsngh (2014) whee a desgn paamete θ was used to mpove the populaton mean estmaton unde a smple andom samplng scenao. Ths povdes the motvatng facto fo ou poposed estmatos of the populaton mean the statfed andom samplng scheme. Thus, the am of ths study s to mpove on y st as well as the ato and poduct estmatos usng auxlay nfomaton. 3. Othe estmatos n lteatue Bahl and Tuteja (1991) poposed ato/poduct exponental estmatos fo estmatng the mean of a fnte populaton usng a sngle auxlay vaable. by BT1 ¼ k X x X w y e þx (3:1) by BT2 ¼ k x X x w y e þx (3:2) The MSEs of these estmatos ae gven by MSE b YBT1 ¼ k w 2 ðf = Þ S 2 Y þ R2 4 S2 X R S XY (3:3) whee R ¼ Y X (3:4) Page 4 of 17

5 MSE b YBT2 ¼ k w 2 ðf = Þ S 2 Y þ R2 4 S2 X þ R S XY (3:5) (Upadhyaya, Sngh, Chattejee, & Yadav, 2011) poposed an exponental ato-typed estmato: by YE ¼ k X x X w y e þða 1Þx (3:6) The MSE s gven by MSE b YYE ¼ k w 2 f S 2 Y þ R2 S 2 a 2 X 2 R S XY a! 4. Poposed estmatos The bass fo ou estmato comes fom the esults of Shley et al. (2014) and ou dese to extend ths type of estmatos to statfed samplng. Now, snce b Y and b Y ae moe effcent than y st when C > 1 2 and C < 1 2, espectvely, sngle-paamete lnea combnatons of and y st,naddtonto and y st ae used as ou poposed estmatos: ¼ ð1 þ θþb Y θy st (4:1) (3:7) ¼ ð1 þ θþb Y θy st (4:2) In Eqs. (4.1) and (4.2), θ s the desgn paamete fo the poposed estmatos and s to be assgned an optmal value whch mnmse the fst-ode MSEs of the poposed estmatos (Shley et al., 2014). We note that when θ ¼ 0, ¼ and ¼. Based on the guess fo the value of C, a sutable value of θ can be obtaned. 5. Bas and mean squae eo of poposed estmatos The fst-ode appoxmatons of the bas and MSE ae deved usng the notatontoduced n Secton 1 of ths pape by substtutng the expessons fo y st and x st nto Eqs. (4.1) and (4.2). ¼ Y ð1 þ ε 0 Þð1 θε 1 Þð1 þ ε 1 Þ 1 (5:1) Bas Y b ¼ E Y b Y ¼ E Y ð1 þ ε 0 Þð1 θε 1 Þð1 þ ε 1 Þ 1 Y ff E Y ð1 þ ε 0 Þð1 θε 1 Þ 1 ε 1 þ ε 2 1 Y ¼ E Y 1 þ ε 0 θε 1 θε 1 ε 0 ε 1 ε 1 ε 0 þ θε 2 1 þ θε2 1 ε 0 þ ε 2 1 þ ε2 1 ε 0 θε 3 1 θε3 1 ε 0 Y ff E Y þ YE ðε 0 ÞθYE ðε 1 ÞθYE ðε 1 ε 0 ÞYE ðε 1 ÞYE ðε 1 ε 0 ÞþθYE ε 2 1 ¼ YE ε 2 1 þ θye ε 2 1 YE ð ε1 ε 0 ÞθYE ðε 1 ε 0 Þ ¼ Y ð1 þ θþ E ε 2 1 E ð ε1 ε 0 Þ ¼ Y ð1 þ θþ 1 X 2 k f w 2 S2 X 1 X Y k f w 2 ρ S X S Y! þ YE ε 2 1 E Y ¼ ð1 þ θþr k f w 2 S 2 X X S XY Y (5:2) Page 5 of 17

6 MSE Y b ¼ E Y 2 ¼ E Y ð1 þ ε 0 Þð1 θε 1 Þð1 þ ε 1 Þ 1 Y 2 ff E Y ð1 þ ε 0 Þð1 θε 1 Þ 1 ε 1 þ ε 2 1 Y 2 ¼ E Y 1 þ ε 0 θε 1 θε 1 ε 0 ε 1 ε 1 ε 0 þ θε 2 1 þ θε2 1 ε 0 þ ε 2 1 þ ε2 1 ε 0 θε 3 1 θε3 1 ε 0 Y 2 ff EðY ðε 0 ð1 þ θþε 1 Þ 2 ¼ Y 2 E ε þ ð θ Þε 1ε 0 þ ð1 þ θþ 2 ε 2 1 ¼ Y 2 E ε 2 0 þ Y 2 ð1 þ θþ 2 E ε 2 1 2Y 2 ð1 þ θþeðε 1 ε 0 Þ ¼ Y 2 1 f w 2 Y 2 k S2 Y þ Y n 2 ð1 þ θþ 2 1 X 2 k ¼ k f w 2 S2 Y ¼ Vaðy st þ R 2 ð1 þ θþ 2 k Þ þ R 2 ð1 þ θþ 2 k f w 2 S2 X f w 2 S2 X f w 2 S2 X 2Y 2 ð1 þ θþ 1 X Y k f w 2 ρ S X S Y 2Rð1 þ θþ k f w 2 S XY 2Rð1 þ θþ k f w 2 S XY (5:3) Mnmzng Eq. (5.3) wth espect to θ, the optmal value of θ s C 1. Theefoe, n the poposed estmato n Eq. (4.1), θ ¼ C 1 s used, whee C s the guess of C. Smlaly, ¼ Y ð1 þ ε 0 Bas Y b MSE Y b Þð1 þ ð1 þ θþε 1 Þ (5:4) ¼ E Y b Y ¼ E Y ð1 þ ε 0 Þð1 þ ð1 þ θþε 1 ÞY ¼ E Y ð1 þ ε 0 þ ð1 þ θþε 1 þ ð1 þ θþε 1 ε 0 ÞY ¼ E Y þ YE ðε 0 ÞþY ð1 þ θþeðε 1 ÞþY ð1 þ θþeðε 1 ε 0 ÞE Y ¼ Y ð1 þ θþeðε 0 ε 1 Þ ¼ Y ð1 þ θþ 1 X Y k f w 2 ρ S X S Y ¼ ð1 þ θþ 1 X k 2 f w 2 S XY (5:5) ¼ E Y b Y ¼ E Y ð1 þ ε 0 Þð1 þ ð1 þ θþε 1 ÞY 2 ¼ E Y ð1 þ ε 0 þ ð1 þ θþε 1 þ ð1 þ θþε 1 ε 0 ÞY 2 ff EðY ðε 0 þ ð1 þ θþε 1 Þ 2 ¼ Y 2 E ε 2 0 þ ð1 þ θþ2 ε 2 1 þ 21þ ð θþε 1ε 0 ¼ Y 2 E ε 2 0 þ Y 2 ð1 þ θþ 2 E ε 2 1 þ 2Y 2 ð1 þ θþeðε 1 ε 0 Þ Page 6 of 17

7 ¼ Y 2 1 f w 2 Y 2 k S2 Y þ Y n 2 ð1 þ θþ 2 1 X 2 k ¼ k f w 2 S2 Y ¼ Vaðy st þ R 2 ð1 þ θþ 2 k ÞþR 2 ð1 þ θþ 2 k f w 2 S2 X f w 2 S2 X f w 2 S2 X þ 2Y 2 ð1 þ θþ 1 X Y k f w 2 ρ S X S Y þ 2Rð1 þ θþ k f w 2 S XY þ 2Rð1 þ θþ k f w 2 S XY (5:6) Mnmzng Eqs. (5.6) wth espect to θ gves the optmal value of θ as ðc þ 1Þ. Theefoe, n the poposed estmato n Eq. (4.2), θ ¼C ð þ 1Þ s used, whee C s the guess of C. 6. Compason of the estmatos Algebac compason of the MSEs of the estmatos s not feasble, theefoe, a smulaton execse s undetaken to facltate ths. We compaed the estmated MSEs of the followng estmatos: by ¼ y st X x st ð by ¼ y st :x stþ X by eg ¼ y st þ β X x st ¼ k w y þ β X x ¼ ð1 þ θþb Y θy st ¼ ð1 þ θþb Y θy st fom 10,000 sets of smulated data wth sample szes n ¼ 30 and 60. These sample szes ae selected snce the combned estmatos ae ecommended when the sample sze wthn each statum s small, 20 (Shley et al., 2014). The elatve effcency of each of these estmatos elatve to y st wee evaluated usng: RE Y b ¼ MSE ð y stþ MSE b 100% (6:1) Y It s assumed the paent populatos lage and data fom each statum come fom a populaton whch s multvaate nomal wth the followng paametes (fo smplcty llustaton): Y 1 ¼ 4:0 Y2 ¼ 5:0 Y3 ¼ 6:0 X 1 ¼ 1:0 X2 ¼ 2:0 X3 ¼ 3:0 S 1Y ¼ 2:0 S 2Y ¼ 2:0 S 3Y ¼ 2:0 S 1X ¼ 1:0 S 2X ¼ 1:0 S 3X ¼ 1:0 Usually the vaablty of the auxlay vaable X s less than the man vaable Y. We vay the coelaton between the man and auxlay vaables n each statum. We defne ρ to be the coelaton between X and Y n the th stata: jρ j ¼ 0:1; 0:4; 0:7. Page 7 of 17

8 Fo smplcty, we used thee stata. Theefoe, whee equal allocatos used, the stata weghts wee: w 1 ¼ w 2 ¼ w 3 ¼ 1 3. Fo popotonal allocaton, we used two weght confguatons. Confguaton 1 used the stata weghts, w 1 ¼ 1 5 ; w 2 ¼ 1 5 ; w 3 ¼ 3 5 whle Confguaton 2 used the weghts w 1 ¼ 1 6 ; w 2 ¼ 2 6 ; w 3 ¼ 3 6.UsngaguessofC, wedefnedc ¼ Cð1 þ Þ,whee accommodates fo unde/ove guess (Shley et al., 2014). The followng values of ae used: 0; 0:02; 0:06; 0:08. The statstcal softwae package R Development Coe Team (2008)wasusedfothesmulatons. Numecal compasons of b Y ; b Yeg ; b Yseg ; b Y fo postve values of ρ and b Y ; b Yeg ; b Yseg fo negatve values of ρ ae obseved. The esults of these smulatons ae summazed n Tables 1 and 2 (as well as Tables A1 A12 n Appendx A). 7. Applcaton to eal data To assess the pefomance of ou poposed estmatos aganst classcal (and competng) estmatos, we appled ou methods to a eal dataset. Table 3 gves the summay statstcs fom the dataset fom Muthy (1967). Table 1. Relatve effcences (n %) of the ato estmatos when n ¼ 30; ¼ 0 and w 1 ¼ w 2 ¼ w 3 ¼ 1 3 ρ 1 ρ 2 ρ 3 C by by eg Table 2. Relatve effcences (n %) of the poduct estmatos when n ¼ 30; ¼ 0 and w 1 ¼ w 2 ¼ w 3 ¼ 1 3 ρ 1 ρ 2 ρ 3 C by by eg Table 3. Summay statstcs Statum 1 Statum N 5 5 Y 1, , X S Y S X S XY 39, , Page 8 of 17

9 Table 4. MSE and RE (%) of estmatos Estmato MSE RE (%) y st 21, by 10, by 120, by eg 4, by BT by BT by YE 10, , , The MSE and elatve effcency values ae gven Table Results and dscusson The esults of the othe smulatons ae shown Tables A1 A12 n Appendx A.Fogvenvaluesofn,the elatve effcences of b Y ; b Y ; b Yeg and b Yseg do not depend on. These ae not ncluded n the man body of the tables poduced; but ae nstead stated at the top fo each set of ρ s. Usng n ¼ 30; 60 fo each value of ¼ 0; 0:02; 0:06 and 0:08, the values of RE Y b fo ρ 1 ¼ ρ 2 ¼ ρ 3 ¼ 0:7; ρ 1 ¼ ρ 2 ¼ ρ 3 ¼ 0:4; ρ 1 ¼ ρ 2 ¼ ρ 3 ¼ 0:1; ρ 1 ¼ 0:7; ρ 2 ¼ 0:4; ρ 3 ¼ 0:1; ρ 1 ¼ 0:7; ρ 2 ¼ 0:7; ρ 3 ¼ 0:1; ρ 1 ¼ 0:1; ρ 2 ¼ 0:1; ρ 3 ¼ 0:7; and RE Y b fo ρ 1 ¼ ρ 2 ¼ ρ 3 ¼0:7; ρ 1 ¼ ρ 2 ¼ ρ 3 ¼0:4; ρ 1 ¼ ρ 2 ¼ ρ 3 ¼0:1; ρ 1 ¼0:7; ρ 2 ¼0:4; ρ 3 ¼0:1; ρ 1 ¼0:7; ρ 2 ¼0:7; ρ 3 ¼0:1; ρ 1 ¼0:1; ρ 2 ¼0:1; ρ 3 ¼0:7ae gven. The elatve effcences n Tables 1 and 2 show the desed mpovement the poposed estmatos acheve vesus b Y and b Y (and how well these poposed estmatos ae than bette b Yeg and ). When 1 2 C 1 2, the poposed estmatos ae consdeably moe effcent than y st despte the combned ato/poduct estmatos b Y and b Y beng wose than y st (whch uses no auxlay nfomaton). Howeve, when jcj s sgnfcantly less than 1 2, unlke eg and seg, the poposed estmatos do make pope use of auxlay nfomaton and poduces mpoved esults. Obsevng futhe smulatons, fom the elatve effcences n Tables A1-A12, the poposed estmatos consstently pefomed bette the combned ato/poduct estmatos. It s also obseved, egadless of sample sze o statum weght used, fo smla coelaton values wthn each statum, the poposed estmatos pefomed bette than the combned and sepaate egesson estmatos. Howeve, when the coelaton each statum vaes, coupled wth the senstvty to the unde/ove guess of C, the pefomance of the poposed estmatos seem to fluctuate when compaed wth the egesson estmatos. Applyng ou poposed estmatos to a eal data, we obseve that they pefom emakably bette than the classcal estmatos y st, b Y and b Y, even when b Y s less effcent than y st. The poposed estmatos even matched the combned egesson estmato eg fo ths dataset. In addton, ou poposed estmatos outpefomed othe exstng exponental-type estmatos. Page 9 of 17

10 9. Concluson The esults of the smulatons do acheve the man objectve of ths pape whch was to obtan a moe effcent estmato of y st. Ou estmatos mpove the effcency of the tadtonal estmatos even when the coelaton between stata- s elatvely smla. In the futue, the challenge would be to poduce moe effcent egesson and poduct estmatos fo othe samplng desgns. Fundng The authos eceved no dect fundng fo ths eseach. Autho detals Bendon Bhagwandeen 1 E-mal: bendon.bhagwandeen@gmal.com ORCID ID: Andew Vea 1 E-mal: andew.vea03@gmal.com ORCID ID: Isaac Dalsngh 1 E-mal: saac.dalsngh@sta.uw.edu ORCID ID: 1 The Depatment of Mathematcs and Statstcs, The Unvesty of the West Indes, St Augustne Campus, Tndad and Tobago. Ctatonfomaton Cte ths atcle as: Impovng the effcency of the ato/ poduct estmatos of the populaton mean statfed andom samples, Bendon Bhagwandeen, Andew Vea & Isaac Dalsngh, Cogent Mathematcs & Statstcs (2018), 5: Refeences Bahl, S., & Tuteja, R. K. (1991, Januay). Rato and poduct type exponental estmatos. Jounal of Infomaton and Optmzaton Scences, 12(1), do: / Cochan, W. G. (1940, Apl). The estmaton of the yelds of ceeal expements by samplng fo the ato of gan to total poduce. The Jounal of Agcultual Scence, 30(2), do: / S Hansen, M. H., & Huwtz, W. N. (1943, Decembe). On the theoy of samplng fom fnte populatons. The Annals of Mathematcal Statstcs, 14(4), do: /aoms/ Kadla, C., & Cng, H. (2005, Ma). A new ato estmato n statfed andom samplng. Communcatons n Statstcs Theoy and Methods, 34(3), do: /sta Muthy, M. N. (1967). Samplng theoy and methods. Calcutta, Inda: Statstcal Publshng Socety. R Development Coe Team. (2008). R: A language and envonment fo statstcal computng. Venna: R Foundaton fo Statstcal Computng. Shley, A., Saha, A., & Dalsngh, I. (2014). Ompovng ato/poduct estmato by ato/poduct-cum-meanpe-unt estmato tagetng moe effcent use of auxlay nfomaton. Jounal of Pobablty and Statstcs, Sngh, H. P., & Vshwakama, G. K. (2007). Modfed exponental ato and poduct estmatos fo fnte populaton mean double samplng. Austan Jounal of Statstcs, 36(3), do: /ajs.v Sngh, R., & Mangat, N. S. (2013, Mach 9). Elements of suvey samplng. Nethelands: Spnge Scence & Busness Meda. Upadhyaya, L. N., Sngh, H. P., Chattejee, S., & Yadav, R. (2011, June). Impoved ato and poduct exponental type estmatos. Jounal of Statstcal Theoy and Pactce, 5(2), do: / Page 10 of 17

11 Appendx A Table A1. Relatve effcences (n %) of, eg, seg and when n ¼ 30 and w 1 ¼ w 2 ¼ w 3 ¼ 1 3 n=30 ρ 1 = 0.4 ρ 3 = 0.4 by by eg Table A2. Relatve effcences (n %) of, eg, seg and when n ¼ 60 and w 1 ¼ w 2 ¼ w 3 ¼ 1 3 n=60 ρ 1 = 0.4 ρ 3 = 0.4 by by eg value Page 11 of 17

12 Table A3. Relatve effcences (n %) of, eg, seg and when n ¼ 60 and w 1 ¼ 1 5 ; w 2 ¼ 1 5 ; w 3 ¼ 3 5 n=30 ρ 1 = 0.4 ρ 3 = 0.4 by by eg Table A4. Relatve effcences (n %) of, eg, seg and when n ¼ 60 and w 1 ¼ 1 5 ; w 2 ¼ 1 5 ; w 3 ¼ 3 5 n=60 ρ 1 = 0.4 ρ 3 = 0.4 by by eg Page 12 of 17

13 Table A5. Relatve effcences (n %) of, eg, seg and when n ¼ 30 and w 1 ¼ 1 6 ; w 2 ¼ 2 6 ; w 3 ¼ 3 6 n=30 ρ 1 = 0.4 ρ 3 = 0.4 by by eg Table A6. Relatve effcences (n %) of, eg, seg and when n ¼ 60 and w 1 ¼ 1 6 ; w 2 ¼ 2 6 ; w 3 ¼ 3 6 n=60 ρ 1 = 0.4 ρ 3 = 0.4 by by eg Page 13 of 17

14 Table A7. Relatve effcences (n %) of, eg, seg and when n ¼ 30 and w 1 ¼ w 2 ¼ w 3 ¼ 1 3 n=30 ρ 1 = -0.4 ρ 3 = -0.4 by by eg Table A8. Relatve effcences (n %) of, eg, seg and when n ¼ 60 and w 1 ¼ w 2 ¼ w 3 ¼ 1 3 n=60 ρ 1 = -0.4 ρ 3 = -0.4 by by eg Page 14 of 17

15 Table A9. Relatve effcences (n %) of, eg, seg and when n ¼ 30 and w 1 ¼ 1 5 ; w 2 ¼ 1 5 ; w 3 ¼ 1 3 n=30 ρ 1 = -0.4 ρ 3 = -0.4 by by eg Table A10. Relatve effcences (n %) of, eg, seg and when n ¼ 60 and w 1 ¼ 1 5 ; w 2 ¼ 1 5 ; w 3 ¼ 1 3 n=60 ρ 1 = -0.4 ρ 3 = -0.4 by by eg Page 15 of 17

16 Table A11. Relatve effcences (n %) of, eg, seg and when n ¼ 30 and w 1 ¼ 1 6 ; w 2 ¼ 2 6 ; w 3 ¼ 3 6 n=30 ρ 1 = -0.4 ρ 3 = -0.4 by by eg Table A12. Relatve effcences (n %) of, eg, seg and when n ¼ 60 and w 1 ¼ 1 6 ; w 2 ¼ 1 6 ; w 3 ¼ 3 6 n=60 ρ 1 = -0.4 ρ 3 = -0.4 by by eg Page 16 of 17

17 2018 The Autho(s). Ths open access atcle s dstbuted unde a Ceatve Commons Attbuton (CC-BY) 4.0 lcense. You ae fee to: Shae copy and edstbute the mateal n any medum o fomat. Adapt emx, tansfom, and buld upon the mateal fo any pupose, even commecally. The lcenso cannot evoke these feedoms as long as you follow the lcense tems. Unde the followng tems: Attbuton You must gve appopate edt, povde a lnk to the lcense, and ndcate f changes wee made. You may do so n any easonable manne, but not n any way that suggests the lcenso endoses you o you use. No addtonal estctons You may not apply legal tems o technologcal measues that legally estct othes fom dong anythng the lcense pemts. Cogent Mathematcs & Statstcs (ISSN: ) s publshed by Cogent OA, pat of Taylo & Fancs Goup. Publshng wth Cogent OA ensues: Immedate, unvesal access to you atcle on publcaton Hgh vsblty and dscoveablty va the Cogent OA webste as well as Taylo & Fancs Onlne Download and ctaton statstcs fo you atcle Rapd onlne publcaton Input fom, and dalog wth, expet edtos and edtoal boads Retenton of full copyght of you atcle Guaanteed legacy pesevaton of you atcle Dscounts and waves fo authos n developng egons Submt you manuspt to a Cogent OA jounal at Page 17 of 17