Improved Dual to Ratio Cum Dual to Product Estimator in the Stratified Random Sampling

Transcription

1 Amrican Journal of Oprational Rsarc 05 5(3): DOI: 0593/jajor Improvd Dual to Ratio Cum Dual to Product Eimator in t Stratifid Random Sampling Subas Kumar adav S S Misra Cm Kadilar Alok Kumar Sukla 3 Dpartmnt of Matmatics and Statiics (A Cntr of Excllnc) Dr RM Avad Univrsit Faiabad UP India Dpartmnt of Statiics Hacttp Univrsit Btp Ankara Turk 3 Dpartmnt of Statiics DA-V Collg Kanpur UP India Abract In tis articl w propos an improvd dual to ratio cum dual to product imator of t population man undr t ratifid random sampling scm T xprssions for t bias and man squard rror (MSE) of t proposd imator ar found b t fir dgr of approximation T optimum valu of t conant wic minimis t MSE of t proposd imator is also obtaind Efficinc comparisons ar prformd btwn t proposd imator and man imators in itratur undr t ratifid random sampling and t fficinc conditions of t proposd imator ar dtrmind Finall an mpirical ud is carrid out wic sows t prformanc of t proposd imator along wit t xiing imators undr t ratifid random sampling Kwords Ratio and Product imators Stratifid Random Sampling Bias MSE Efficinc Introduction In practic it is vr common tat t us of t auxiliar variabl (x or ) improvs t fficinc of t imators for t population paramtrs of t ud variabl () T auxiliar information supplid b t auxiliar variabls is usd bot at t dsign and imation ags of t surv W av usd it in t imation ag in tis articl T auxiliar variabl is igl corrlatd (positivl or ngativl) wit t ud variabl T ratio mtod of imation providing ratio tp imators is usd for t imation of population paramtrs wn t ud variabl and t auxiliar variabls ar igl positivl corrlatd to ac otr; wras t product mtod of imation giving product tp imators is usd wn and x ar igl ngativl corrlatd to ac otr As in practic w find tat t ud variabl as bot positiv and ngativ corrlations wit two diffrnt variabls at a tim Tis ncourags us to us bot positivl and ngativl corrlatd variabls in our ud and w propos t dual to ratio and dual to product imator of t population man in t ratifid random sampling T improvmnt is a continuous procss of rsarc and t form of t imator using t scalar α as alwas improvd t imators of population paramtrs in t simpl random sampling Bing inspird from tis trut w av proposd t dual to ratio Corrsponding autor: sant_x003@aoocoin (S S Misra) Publisd onlin at ttp://journalsapuborg/ajor Coprigt 05 Scintific & Acadmic Publising All Rigts Rsrvd and dual to product imator of t population man in t ratifid random sampling As w dal wit t mtods in t ratifid random sampling assum tat t finit population u ( u u u ) consis of diinct and idntifiabl units wic ar trognous from ac otr t t wol population b dividd into rata of sis ( ) in wic units ar rlativl omognous to ac otr In addition t ud variabl and two auxiliar variabls x and tak t valus i x i and i ( ; i ) rspctivl for t i t t unit of t ratum It is clar tat t sub-sampls of sis n ( ) ar drawn from ac ratum using t proportional allocation mtod conituting of t rquird sampl of si as n n Following common notations of t ratifid random sampling it can b givn b i : T t ratum population man for i t ud variabl X xi i t auxiliar variabl x : T t ratum population man for

2 58 Subas Kumar adav t al: Improvd Dual to Ratio Cum Dual to Product Eimator in t Stratifid Random Sampling Z i i t auxiliar variabl : T t ratum population man for W : T i i population man of t ud variabl X x X WX : T i i population man of t auxiliar variabl x Z Z WZ : T i i population man of t auxiliar variabl n i n i ud variabl : T t ratum sampl man of t n t x xi : T ratum sampl man of t n i auxiliar variabl x n t i : T ratum sampl man of t n i auxiliar variabl W : Wigt of t t ratum Eimators in itratur Hansn t al [] proposd t classical combind ratio imator for t population man undr t ratifid random sampling as RC X x () wr W and x Wx T MSE of t combind ratio imator in () up to t fir ordr of approximation is RC λ x x + x x () MSE W S R S R S wr λ Rx n X X is t population ratio S is t population varianc of t ud variabl auxiliar variabl and S x is t population varianc of t S x is t population covarianc btwn t ud and auxiliar variabls in t t ratum T combind product imator of t population man in t ratifid random sampling is dfind as PC Z (3) wr W T MSE of t combind product imator in (3) up to t fir ordr of approximation is PC λ + + (4) MSE W S R S R S wr R Z Z Man autors suc as Kadilar and Cingi ([] [3]) Sabbir and Gupta [4] Sing and Viswakarma [5] Kouncu and Kadilar ([6] [7] [8] [9]) Sanaulla t al [0] av improvd t ratio and product imators givn in () and (3) for t population man of t ud variabl in t ratifid random sampling Howvr in tis articl w xamin onl dual imators for t population man in itratur tat can b summarid as follows: Using t combind ratio and product imators Kuswaa t al [] proposd t following dual to ratio and dual to product imators b t Srivnkataramana [] transformation as (5) X x RC PC Z (6) rspctivl wr x Wx and W Hr t Srivnkataramana [] transformations ar X xn Z n x and n n T MSE of dual to ratio and dual to product imators in (5) and (6) rspctivl up to t fir ordr of approximation ar rspctivl givn b RC + x x x x MSE W λ S R g S R g S (7)

3 Amrican Journal of Oprational Rsarc 05 5(3): PC + + MSE W λ S R g S R g S (8) wr g n n Sing t al [3] suggd following xponntial ratio and product tp imators in t ratifid random sampling basd on Bal and Tutja [4] imators of t population man undr t simpl random sampling rspctivl as follows: W ( X x ) (9) X x R xp xp + X x W( X + x) W ( Z) (0) Z P xp xp + Z W( + Z) T MSE of t imators in (9) and (0) up to t fir ordr of approximation ar rspctivl ( R) Rx MSE W λ S RxS x + Sx () 4 R MSE( P) W λ S + S + RS 4 () Tailor t al [5] proposd t following dual to ratio and product tp imators using t Sing t al [3] imators as follows: ( ) W x X x X R xp xp + (3) x X W( x + X) ( ) W Z Z xp xp P + (4) Z W( Z + ) T MSE of t imators in (3) and (4) up to t fir ordr of approximation ar rspctivl ( R ) Rx MSE W λ S RxgS x + gsx (5) 4 ( P ) R MSE W λ S + RgS + gs (6) 4 3 Proposd Eimator Motivatd b t imators mntiond in Sction and t fact tat t us of a scalar in t currnt forms (3) and (4) alwas improvs t imator w propos t following dual to ratio cum dual to product imator of t population man in t ratifid random sampling b combining t dual to ratio and dual to product imators as x X Z t α xp + ( α) xp x X Z (3) + + wr α is a suitabl conant to b dtrmind suc tat t MSE of t proposd imator t is minimum It is wort notabl tat it bcoms t dual to product imator givn in (4) for α 0 and tat it rducs to dual to ratio imator givn in (3) for α To ud t larg sampl proprtis of t proposd imator t lt us dfin t following notations: ( + 0 ) x X( + ) and Z( + ) suc tat E( 0) E( ) E( ) 0 and E ( 0 ) λ C E ( ) λ C x ( ) ( 0 ) ( 0 ) E λ C E λρ x C C x E λρ C C E λρ x C x C Exprssing t proposd imator in (3) in trms of i ( i 0 ) w av W x X W Z t α W xp ( α) W xp + W( x + X) W( Z + )

4 60 Subas Kumar adav t al: Improvd Dual to Ratio Cum Dual to Product Eimator in t Stratifid Random Sampling wr 0 α W ( 0 ) xp WX WgX + + ( ) ( + ) xp WgX α W 0 WZ WgZ WgZ ( 0) xp ( 0) xp α + + α + (3) W 0 and E E(0 ) X WgX X and 0 W λs W λgs x 0 E E( ) WgZ suc tat Z W λg Sx X W λgs Z E E E 0 0 E ( ) W λg S Z λ x XZ E( ) W g S On simplifing t xprssions aftr t xpansion on t rigt and sid of (3) up to t fir ordr of approximation w av t α ( α) α α α + ( α) + ( α) + ( α) W can writ (33) as t 0 α α α + ( α) + ( α) + ( α) α α α + α + α + α (33) (34) Taking t xpctation on bot sids of (34) w av t bias of t proposd imator up to t fir ordr of approximation as Bt () gs gs + g S g S α α x 3α α x Wλ (35) Z X 8 Z 8 X From (34) up to t fir ordr of approximation t MSE of t proposd imator is MSE() t E 0 α + α E 0 α + α

5 Amrican Journal of Oprational Rsarc 05 5(3): α α E 0+ + α 0+ α0 αα (36) 4 4 wic is minimum for wr E + E + E ( ) E E E E A α B A E + E + E + E W gs + gs + + gs x gsx 0 0 λ Z X Z XZ B E + E + E W + + gsx gs gsx λ X Z XZ Finall t minimum MSE of t proposd imator is MSE t W S gs gs A min () λ + + 4Z Z 4B A W λ S RgS RgS 4 4B (37) Efficinc Comparisons T varianc of t sampl man in t ratifid random sampling λ is givn b V( ) W S (4) W obtain t following conditions undr wic t proposd dual to ratio cum dual to product tp xponntial imator is bttr tan t imators ˆRC ˆPC ˆ ˆ RC ˆ PC ˆ R ˆ P ˆ R and PC rspctivl T proposd imator t is mor fficint tan if It is mor fficint tan t imator It is mor fficint tan t imator It is mor fficint tan t imator W gs RgS A R λ + > 0 4B 4 (4) RC if R A W λ RS x x RS x x gs RgS + > 0 (43) 4 4B PC if g A W λ ( ) RS + ( g) RS + > 0 (44) 4 4B RC if R A W λ RgS x x RgS x x gs RgS + > 0 (45) 4 4B

6 6 Subas Kumar adav t al: Improvd Dual to Ratio Cum Dual to Product Eimator in t Stratifid Random Sampling It is mor fficint tan t imator t is mor fficint tan t is mor fficint tan R if PC if 3g A W λ RS 3RS + > 0 (46) 4 4B Rx R A W λ Sx RS x x gs RgS + > 0 (47) 4 4 4B P if T proposd imator t is mor fficint tan R A W λ ( g ) S + ( g) RS + > 0 4 4B (48) R if Rx R A W λ gs x RgS x x gs RgS + > 0 (49) 4 4 4B T proposd imator t is mor fficint tan 5 Empirical Stud PC if A 0 4B > (40) Tabl Data Statiics Stratum I Stratum II n X Z S S x S S x S x S To xamin t fficinc of t proposd imator ovr otr imators w considr t data st in Murt [6] wos atiics ar givn in Tabl T MSE and prcnt rlativ fficinc (PRE) valus ar givn in Tabl Tabl MSE and PRE Valus of Eimators Eimator MSE PRE RC PC RC PC R P R P t W wis to laborat t tabulatd valus in various columns of t Tabl In t fir column of t tabl t various paramtrs of main and auxiliar variabls suc as population si sampl si population mans variancs and covariancs of bot rata rspctivl av bn givn Scond and tird columns prsnt rspctiv paramtr valus for fir and scond rata Tabl prsnts t man squar rror and t Prcntag Rlativ Efficinc (PRE) for t xiing and t proposd imators T PRE of an imator wit rspct to t sampl man is dfind b MSE PRE( ) 00 MSE()

7 Amrican Journal of Oprational Rsarc 05 5(3): Conclusions Sampl survs ar lgitimatl considrd as co ffctiv apparatus for imation of t population paramtr T Statiician wiss to minimi t man squar rror of t imator to idall infr about t paramtr of t givn population In t prsnt problm w av proposd a dual to ratio cum dual to product imator of population man in ratifid random sampling as in man situations w nd bot positivl and ngativl corrlatd information wit t main variabl undr ud Furtr larg sampl proprtis of t proposd imator av bn udid up to t fir ordr of approximation W av also mad t comparisons of dsird rsults wit prvious rsarcrs Finall w av judgd t prformancs of diffrnt imators along wit t proposd imator troug an mpirical ud undr ratifid random sampling From t tortical discussion and t numrical rsults from Tabl w conclud tat t proposd imator is bttr tan t mntiond imators in Sction undr t ratifid random sampling scm as t proposd imator as smallr man squard rror B tis numrical xampl w also sow tat t fficinc conditions in tor for t proposd imator obtaind in (4)-(40) ar satisfid in practic as wll Tus t proposd imator sould b prfrrd for t imation of t population man undr t ratifid random sampling REFERECES [] Hansn MH Hurwit W and Gurn M (946) Problm and Mtods of t Sampl Surv of Businss JASA [] Kadilar C and Cingi H (003) Ratio Eimators in Stratifid Random Sampling Biomtrical Journal [3] Kadilar C and Cingi H (005) A w Ratio Eimator in Stratifid Random Sampling Communications in Statiics: Tor and Mtods [4] Sabbir J and Gupta S (006) A w Eimator of Population Man in Stratifid Sampling Communications in Statiics: Tor and Mtods [5] Sing HP and Viswakarma GK (008) A Famil of Eimators of Population Man Using Auxiliar Information in Stratifid Sampling Communications in Statiics: Tor and Mtods [6] Kouncu and Kadilar C (009a) Ratio and Product Eimators in Stratifid Random Sampling Journal of Statiical Planning and Infrnc [7] Kouncu and Kadilar C (009b) Famil of Eimators of Population Man Using Two Auxiliar Variabls in Stratifid Random Sampling Communications in Statiics: Tor and Mtods [8] Kouncu and Kadilar C (00a) On Improvmnt in Eimating Population Man in Stratifid Random Sampling Journal of Applid Statiics [9] Kouncu and Kadilar C (00b) On t Famil of Eimators of Population Man in Stratifid Random Sampling Pakian Journal of Statiics [0] Sanaulla A Ali H A oor ul Amin M and Hanif M (04) Gnralid xponntial cain ratio imators undr ratifid two-pas random sampling Applid Matmatics and Computation [] Kuswaa KS Upadaa and Dub SP (990) A Dual to Ratio Eimator in Stratifid Random Sampling Proc Mat Soc 6-5 [] Srivnkataramana T (980) A Dual to Ratio Eimator in Sampl Survs Biomtrika [3] Sing R Cauan P and Sawan (008) On inar Combination of Ratio-Product Tp Exponntial Eimator for Eimating Finit Population Man Statiics in Transition [4] Bal S and Tutja RK (99) Ratio and Product Tp Exponntial Eimator Information and Optimiation Scincs XII I [5] Tailor R Jatwa K Tailor R and Garg (03) Dual to Ratio and Product Tp Exponntial Eimators in Stratifid Random Sampling Using Two Auxiliar Variats Journal of Rliabilit and Statiical Studis [6] Murt M (967) Sampling Tor and Mtods Statiical Publising Socit Calcutta