MODIFIED CLASS OF RATIO AND REGRESSION TYPE ESTIMATORS FOR IMPUTING SCRAMBLING RESPONSE

Transcription

1 Pak. J. tatist. 07 Vol. 33(4), MODIFIED CLA OF RATIO AND REGREION TYPE ETIMATOR FOR IMPUTING CRAMBLING REPONE Muhammad Umai ohail, Javid habbi ad hakeel Ahmed Depatmet of tatistis, Quaid-i-Azam Uivesity, Islamabad, Pakista Coespodig autho ABTRACT I this study, we modify the lass of atio ad egessio type estimatos fo imputig samblig espose o the lies of Ahmed et al. (006) ad Mohamed et al. (06). We popose a ew lass of estimatos by usig the highe ode momets of auxiliay ifomatio. Ou poposed lass of estimatos ae moe effiiet as ompaed to existig estimatos. Two umeial studies by usig simulated ad eal life data set at vaious espose ate ae also aied out fo evaluatig the pefomae of poposed lass of estimatos. KEYWORD Mea squae eo, imputatio, vaiae, samblig espose, sesitive vaiable.. INTRODUCTION I most of suveys, the ommo poblem whih eoute was the missig values. Hase ad Huwitz (946) fist time deal with the missig data ad late may studies have bee oued to hadle this poblem. Rubi (976) suggested the vaious imputatio methods, whih make the data stutually omplete. Late igh ad Deo (003) ad Ahmed et al. (006) suggested may imputatio methods to deal with missig data. Usually the o-espose is oued i the soio-eoomi ad politial suveys, whee eseahes ae ofte iteested to ivestigate the sesitive issues of the soiety. Iitially, Wae (965) povide the idea of adomized espose tehique (RRT) to hadle the edued espose ate. The mai oept behid this idea was to seue the twi objetives: (i) to eate the feelig amog the espodets that thei idetity is seued beside thei tuthful espose, (ii) to podue the eliable data to daw the fuitful ifeee about the populatio. Late o may studies have bee oued to impove the effiiey of adomized devie. I this eseah, we oside the samblig model poposed by Gjestvag ad igh th (009), whee the j espodet has two hoies: he a epot Yj j o Yj j with pobability p = ad ( p) = espetively, whee ( ) ( ) Y j ad j be the sesitive ad samblig vaiables espetively, ad ad ae two positive eal umbes. A simple adom sample of size is seleted fom populatio of N uits. 07 Pakista Joual of tatistis 77

2 78 Modified Class of Ratio ad Regessio Type Estimatos.. Coside a dek of ads, whih has the p popotio of ads that beaig statemet: multiplied the samblig vaiable j with ad add the tue value of Y j, ad ( p) be the popotio of ads that beaig the statemet: multiplied the samblig vaiable j with ad subtated it fom the tue value of his espose as: Y j. The th j espodet a epot z = p Y ( p) Y. (.) j j j j j Let E 3 ad V 3 deoted the expeted value ad vaiae due to adom devie. We assume that E3 ( ) = s, 3 ( ) = V s ae kow. o V z j s s 3 ( ) =. Let be the umbe of espodets i subset A i a sample s who povide the espose by usig the above metioed adomize espose model ad ( ) ae those i subset A, who efuse to povide the espose, so, s = A A. Let the sample mea of the samblig espose is give as: z = j=zj. Now we have the Lemma as: Lemma : The vaiae of sample mea is give as: V ( z ) = y ( s s ) N (.) Poof: Let E ad E deote the expeted values fo the give ad, ad V ad V deoted the vaiae fo the give values of ad espetively. By the defiitio of vaiae, we have: V( z ) = E E V ( z ) E V E ( z ) V E E ( z ) = EE V3 z j EV E3 z j V EE3 z j j= j= j= = E E V 3( z j) E V E 3( z j) V E E 3( z j) j= j= j= = E E { ( s s )} E V y j V E y j j= j= j= = E ( s s ) E sy( ) V y j j= j=

3 ohail, habbi ad Ahmed 79 N = ( ) N j= N s s y y = ( s s ) y. N We pove the lemma, so we have the bette appoximatio fo esults.. OME EXITING METHOD OF IMPUTATION FOR CRAMBLING REPONE i) Mohamed et al. (06) z j if j A zˆ j = z x j if j A x The poit estimato is: x y R = z. x ii) Ahmed et al. (006) z j if j A ˆ z = xi x z j z if j A x ( ) x The poit estimato by this model is give as: zx ya. =, 8 x ( ) x (.3) (.) (.) whee is the ostat. iii) Ahmed et al. (006) z j if j A zˆ j = z X z if j A x ( ) X The poit estimato is defied as: zx ya. =, 9 x ( ) X (.3) whee is the ostat.

4 80 Modified Class of Ratio ad Regessio Type Estimatos.. iv) Ahmed et al. (006) z j if j A zˆ j = z X z if j A 3x ( 3) X The poit estimato is defied as: zx ya =, x ( ) X (.4) whee 3 is the ostat. v) Ahmed et al. (006) z j if j A ˆ j = z w ( X x ) z if j A The poit estimato by this model is give as: y A = z 5 w ( X x ), (.5) whee w is the ostat. vi) Ahmed et al. (006) z j if j A ˆ j = z w ( X x ) z if j A The poit estimato by this model is give as: y A = z 6 w ( X x ), (.6) whee w ae the ukow ostat. vii) Mohamed et al. (06) z j if j A zˆ j = ˆ ( ) z s x j x if j A The poit estimato of y Re. is give as: y = z ˆ ( x x ), (.7) Re. s whee ˆ = s j= j j= ( x x)( y j y) is the simple egessio oeffiiet. ( x x) j

5 ohail, habbi ad Ahmed 8 3. MODIFIED ETIMATOR I this setio, we modify the atio ad egessio type estimatos fo imputig samblig espose by usig the highe ode momets of the auxiliay vaiable. The use of the highe ode momets of a auxiliay vaiable play a impotat ole fo the estimatio of populatio paamete of the study vaiable i suvey samplig. The poposed imputatio methods ae give as: i) The fist poposed method of imputatio is z j if j A ( x j x ) ( x j x ) ja Ij = z x j x ( ) ( )( ) z if j A Ix ( I) x I sx( ) ( I ) sx( ) (3.) whee I ad I ae the ukow ostats whose values ae to be detemied. The poit estimato of populatio mea ude poposed method of imputatio, is give as: y = I j= Ij ( x j x ) ( x j x ) ( ) ja ja ja z x j x ( ) ( )( ) ja = ja z j Ix ( I) x I sx( ) ( I ) sx( ) ( ) z ( x j x) ( xj x ) ja ja z{ x x x} ( ) ( ) ( ) = z z Ix ( I) x I sx( ) ( I ) sx( )

6 8 Modified Class of Ratio ad Regessio Type Estimatos.. ( x j x ) ( xi x ) ja z ja x = z x x s s I ( I) I x( ) ( I ) x( ) zxsx( ) = x x s s I ( I) I x( ) ( I ) x( ) z y z xsx( ) I I I I x I x =. x ( ) x s ( ) ( ) s ( ) (3.) The poit estimato give i Equatio (3.) a be used, whe we utilized the sample ifomatio of the auxiliay vaiable to imputig the missig value. ii) The seod poposed method of imputatio is z j if j A = IIj zxx z if j A II x ( II ) X II sx( ) ( II ) x (3.3) whee II ad II ae the ukow ostats whose values ae to be detemied. The poit estimato y is give as: II = II j= IIj y = zxx jaz j j A II x ( II ) X II sx( ) ( II ) x ja z = zxx z z II x ( II ) XII sx( ) ( II ) x

7 ohail, habbi ad Ahmed 83 y zxx = z x X s II ( II ) II x( ) ( II ) x zxx II II II II x II x =. x ( ) X s ( ) ( ) z (3.4) The poit estimato i Equatio (3.4) a be used i situatio, whe populatio mea ad populatio vaiae of the auxiliay vaiable of the auxiliay vaiable ae kow. iii) The thid poposed method of imputatio z j if ja = IIIj zxx z if ja III 3x ( III 3) X III 3 sx( ) ( III 3) x (3.5) whee III 3 ad III 3 ae the ukow ostats, whose values ae to be detemied. The poit estimato of y is give as: III = III j= IIIj y y = zxx jaz j j A III 3x ( III 3) XIII 3 sx( ) ( III 3) x ja z = zxx z z III 3x ( III 3) XIII 3 sx( ) ( III 3) x = zxx z z III 3x ( III 3) X III 3 sx( ) ( III 3) x zxx III III III III x III x =. 3x ( 3) X 3 s ( ) ( 3) (3.6) The poit estimato i Equatio (3.6) a be used, whe the populatio mea ad populatio vaiae ae kow.

8 84 Modified Class of Ratio ad Regessio Type Estimatos.. iv) Fouth poposed method of imputatio IVj z j if j A N N j ( j ) ( j ) = x x X x x j= j= ja z IV 4 x j IV 4 if j A N( ) ja ( )( N ) ( ) (3.7) whee IV 4 ad IV 4 ae ostats, whose values ae to be detemied. The poit estimato of y is give as: IV = IV j= IVj y = N( ) N ja xj j = jaz j ja z IV 4 ja jax j N ( x X ) ( x x ) ja j= IV 4 ( )( N ) ( ) j j ja ja N ( ) ( ) x X x x j= = z ( ) z IV 4X x IV 4 ( N ) ( ) = z IV 4 X x IV 4 { x sx( ) } IV IV 4 IV 4 x x( ) j j js y = z X x s (3.8) The poit estimato i Equatio (3.8) is the egessio imputatio method, is used whe the populatio paametes ( X, ) ae kow. x

9 ohail, habbi ad Ahmed 85 v) Fifth poposed method of imputatio Vj z j if j A N N j j ( i ) ( i ) = x x x X x x j= ja j= ja z V 5 V 5 if j A N( ) ( ) ( )( N ) ( )( ) (3.9) whee V 5 ad V 5 ae the ostat values whose ae to be detemie. The poit estimato of populatio mea is give as: = V j= Vj y N x ja x j j= j A = ja jazj ja z V 5 N( ) ( ) N ( x X ) ( x x ) ja j= V 5 ( )( N ) ( )( ) j j ja ja N ( ) ( ) x X x x j= = z ( ) z V 5 X x V 5 ( N ) ( ) = z V 5 X x V 5 { x sx( ) } V V 5 V 5 x x( ) j j i ja y = z X x s. (3.0) Expessio give i Equatio (3.0), a be used whe populatio mea ad vaiae ae kow. 4. LARGE AMPLE APPROXIMATION Fo evaluatig the bias ad mea squae eo of the estimatos, we defie some useful esults as:

10 86 Modified Class of Ratio ad Regessio Type Estimatos.. Let whee z x x sx( ) sx( ) s xz( ) Y X X x x xy e =, e =, e =, e =, e =, e =, E( ei ) = 0 fo i = 0,,,3, 4,5 To fist ode of appoximatio, we have 0 s s, N y, N x, N x E e = ( ) C, E e = C, E e = C, Y E e =, E e =, E e e = C C, 3, N 04 4, N 04 0, N xy x y 0 =, Nxy x y, 0 3 =, N y, 0 4 = C, E e e = C, E e e C C E e e C E e e, N y, N x 3 =, N x03, 4 =, N x03, 3 = C, E e e = C, E e e C E e e C E e e, N x 03 4, N x , N 04 5, Nxy x 5, Nxy x E e e =, E e e = C, E e e = C, =, =, =, =, =, =, N N N Y j= j C R, N X N, N =,, =, = j=( j )( j ), N N ab N a b ab =, = / / ab j=( y j Y ) ( x j X ), a b N 0 0 whee = x, y ad = x, y. 5. PROPERTIE OF IMPUTATION METHOD I this setio, we defie the popeties of existig ad poposed imputatio methods ude samblig espose as: i) Fom Equatio (.), the bias ad mea squae eo of y. R is give by: Bias( y R ), Y Cx xycxc y (5.) ad ME( y ) ( ) ( R R ). (5.) R, N y s s, y x xy ii) Afte simplifyig the Equatio (.), The bias ad mea squae eo is give as:

11 ohail, habbi ad Ahmed 87 ad. 8, Bias( y) A N Y C x xy C x C y (5.3) A 8, N y, x, xy x y s s ME( y) Y C C C C ( ) whee = Cy xy, C x xy A 8( mi.), N y, s s x ME( y) ( ). (5.4) iii) By solvig the Equatio (.3), bias ad mea squae a be appoximated as: ad A 9, N x xy x y Bias( y) Y ( C C C ) (5.5) A 9, N y, N x, N xy x y s s ME( y) C C C C ( ) whee = Cy xy, so C x xy A 9( mi.), N y, N s s x ME( y) ( ). (5.6) iv) By solvig the Equatio (.4), bias ad mea squae a be appoximated as: ad A 0, N 3 x 3 x y Bias( y) Y ( C C C ) (5.7) A 0, N y, N 3 x, N 3 x y s s ME( y) C C C C ( ) whee 3 = Cy xy, so C x xy ME( y) A 0(.), Ny ( ). mi s s (5.8) x v) Fo Equatio (.5), the mea squae eo a be omputed as: A 5, N y, N x, N xy s s V( y) w w ( ) (5.9) whee w = xy, so x A 5( mi.), N y, N xy s s V ( y ) ( ). (5.0)

12 88 Modified Class of Ratio ad Regessio Type Estimatos.. vi) Fo Equatio (.6), the mea squae eo a be omputed as: A 6, N y x xy s s V( y) ( w w ) ( ) (5.) whee w = xy x, so V ( y ) ) ( ). (5.) A 6( mi.), N xy s s vii) The bias ad mea squae eo of y Re. ae give as by solvig Equatio (.7): Bias (5.3) ad ME( y Re. )( mi.), N y ( s s ), y xy. (5.4) 5. Modified Imputatio Methods The Equatio (3.) i tem of eo a be witte as: y Y ( e0 ) X ( e ) x ( e4 ) = I I I I x I x X ( e ) ( ) X ( e ) ( e3 ) ( ) ( e4 ) 0 4 I I I 3 I 4 = Y ( e )( e )( e ){ e ( ) e } ( e ( ) e ) = Y ( e )( e )( e ) ( e ( ) e ) ( e ( ) e ) 0 4 I I I I ( e ( ) e ) ( e ( ) e ) I 3 I 4 I 3 I 4 = Y ( e )( e )( e ) ( e ( ) e ) ( e ( ) e ) 0 4 I I I I ( e e ) ( e ( ) e ) I 3 I 4 I 3 I 4 Afte implifiatio, we have I 3 I 4 I 3 I 4 I I 3 4 I = Y e ( ) e e ( ) e ( ) e e e I I ee 3 I( I) e e4 ( I) e I( I) ee3 I I ee4 Ie ( I) e I( I) ee 4 Ie3e4 ( I) e4 Ie e4 ( I) ee4 e Iee3 I ee4 Ie e ( I) e ee4 e0 Ie0e3 ( I) e0e4 Ie0e ( I) e0e e0e4 ( )( ) e Afte simplifiatio ( )., I 0 Bias( y) N YE I e e e, NYE I Ie e3 Ie0e3 Ie3 Ie e0e

13 ohail, habbi ad Ahmed 89, NY ICx xycxcy ( ), NY I ICx 03 ICy I( 04 ) ICx xycxcy (5.5) Fo mea squae eo, it a be desibe as 0 I I I 4 3 ME( y) Y E e ( e e ) ( e e ) 0 I( ) I( 4 3) Y E e e e e e I Afte simplifiatio, we have e ( e e ) e ( e e ) ( e e )( e e ) 0 I I I 4 3 I, N y s s, y I x I 04 I xycxcy ICy I ICx 03 ME( y) ( ) Y C C ( ) (5.6) The optimum values of the ukow ostats ae detemied by miimizig the mea squaed eo. The optimum value fo I ad I is give as ( ) = ad =, ( ) ( ) xy 04 y x 03 y x xy 03 I I Rx xy (5.7) Afte substitutig the optimum value of II ad II i Equatio (5.5), we have ( mi.), N y I s s ME( y) ( ). xy ( 04 ) xy 03, y The Equatio (3.4) i tem of eo a be desibe as y Y ( e0 ) Xx = II II II II x II x X ( e ) ( ) X ( e4 ) ( ) 0 II II 4 = Y ( e )( e ) ( e ) ad to fist ode appoximatio ad = ( 0)( II II ) ( II 4 II 4 ) II ( ), N II x II ( 04 ) II II II Cx 03 II xycxcy II Cy y Y e e e e e Bias y Y C (5.8) (5.9)

14 90 Modified Class of Ratio ad Regessio Type Estimatos.. 0 II II II 4 ME( y) Y E( e e e ) 0 II II 4 II 0 II 0 4 II II 4 Y E e e e e e e e e e Afte simplifiatio, we have II s s, N y, N II x II 04 ME( y) ( ) Y C ( ) (5.0) II xycxcy II Cy II II Cx 03 ad the optimum values of II ad II ae give as ( ) = ad =, xy 04 y x 03 y x xy 03 II II Rx( 04 03) Yx( 04 03) (5.) Puttig he optimum values of II ad II fom Equatio (5.) put i (5.0), so we have ( ). ( mi.), N y II s s ME y xy ( 04 ) xy 03, N y The Equatio (3.6) i tem of eo a be defied as y Y ( e0 ) Xx = III III III III x III x 3X ( e ) ( 3) X 3 ( e3 ) ( 3). = ( 0)( 3 ) III ( III 3 3) III y Y e e e ad fo fist ode appoximatio ad y = Y ( e ) e e e e III 0 III 3 III 3 III 3 3 III 3 3 ( ), N III 3 x III 3( 04 ) III III 3III 3 x 03 III 3 xycxcy III 3Cy Bias y Y C C (5.) (5.3) 0 III III 3 III 3 3 ME( y) Y E( e e e ) 0 III 3 III 3 3 III 3 0 III III 3 III 3 3 Y E( e e e e e e e e e ) Afte simplifiatio, we have

15 ohail, habbi ad Ahmed 9 III s s, N y III 3 x III 3 04 III 3 xycxcy III 3Cy III 3 III Cx 03 ME( y) ( ) Y C C ( ) (5.4) ad the optimum values of II ad II ae give as ( ) = ad =, xy 04 y x 03 y x xy 03 III 3 III 3 Rx( 04 03) Yx( 04 03) substitutig the Equatio (5.5) i (5.4), so (5.5) xy ( 04 ) xy 03 ME( y) ( mi.) ( s s ), N y. III (5.6) The vaiae of ( y ). IV i tems of eo is defied as. 0 IV E Y e0 IV 4X e IV 4xe4 IV 4YXe 0e V ( y) E Y e IV Xe IV x e IV 4Y x e0e4 IV 4 IV 4Xx ee 3 Afte simplifiatio, we have 4 IV s s, N y, N IV 4 x IV 4 x 04 V ( y) ( ) ( ) 3 IV 4xy IV 4x y IV 4 IV 4x 03 (5.7) optimum values of IV 4 ad IV 4 ae give as ( ) = ad =, xy 04 y x 03 y x xy 03 IV 4 IV 4 3 x( 04 03) x( 04 03) Afte simplifiatio, we have ( xy 03) V y (.) s s, N y, N IV mi y xy (5.8) ( ) ( ). (5.9) The vaiae of y i tem of eo is defied as V V ( y) E Y e0 V 5 X X ( e ) V 5 x x ( e3 ) V E Y e0 V 5Xe V 5xe 3

16 9 Modified Class of Ratio ad Regessio Type Estimatos.. 4 Y e0 V 5X e V 5xe3 V 5YXe0e E V 5Yx e0e3 V 5V 5 Xx e e3 Afte simplifiatio, we have V s s, N y V 5 x x 04 V ( y) ( ) ( ) 3 V 5xy V 5x y V 5 V 5x 03 (5.30) optimum values of V 5 ad V 5 ae give as ( ) = ad =, xy 04 y x 03 y x xy 03 V5 V5 3 x( 04 03) x( 04 03) Afte simplifiatio, we have ( xy 03) V y (.) s s, N V mi y xy ( ) ( ). 5. Theoetial Compaiso (5.3) (5.3) I this setio, we osideed the theoetial ompaiso of the poposed method with the mea imputatio method is give as ( ) ( ) o ( ) > 0 i) V y ME y ME y I ( mi.) III ( mi.) xy 04 xy 03 ( ) > 0 (5.33) ii) V( y) ME( y). > 0 N N II ( mi.) xy ( 04 ) xy >0 (5.34) iv) V ( y) ME( y) o ME( y) (.) > 0 IV mi V ( mi.) ( xy03 ) xy >0 (5.35) Ou study pla is pefom bette as ompae to the mea imputatio method, if the above metiooditio ae satisfied. Futhe thei pefomae a be illustated umeially i the ext setio.

17 ohail, habbi ad Ahmed APPLICATION The umeially ompaiso of the poposed imputatio method is osideed by used the simulated ad eal life data set as: 6. Empiial tudy Fo the empiial study, we oside the followig steps as follow: tep : We geeate 000 adom umbes usig bivaiate omal distibutio. A simple adom sample of size 00 is seleted fom the populatio of 000 uits ad suppose that 70 out of 00 samples a povide the espose ad we wat to impute the values fo the emaiig uits who does ot povide the espose. Y =.6968, X = 9.939, = , y x = , xy =.93743, xy = , = 0.0,, IV 4 = V5 = 0.03 = 0.04, = = 3 = , = = 0.04, I = = = 0.890, = = 0.046, II III 3 IV 4 V 5 = = = I II III 3 tep : We epeat the poess of seletio of the sample times. Thus we obtai the values of the y k. tep 3: Fo bias ad mea squae eo, we used the follow expessio: g= ykg Bias( y ) = k Y (6.) ad fo mea squae eo, we have ME( y ) = k g = ( ykg Y ) (6.) Fo aessig the pefomae of existig ad poposed estimatos with espet to mea, we oside the followig expessio as: ME( y) M. P. R. E( i) = 00 (6.3) M. E( yk ) whee i =,,3,,3 ad k = R, Re, A,, A,, A,, A,, A,. 6 V 8 I 9 II 5 IV 0 III

18 94 Modified Class of Ratio ad Regessio Type Estimatos.. Table Bias, ME ad Peetage Relative Effiiey ( i ). Estimatos Bias ME Effiiey % Existig Imputatio Methods Mea Imputatio Method y M Ratio Imputatio Methods y R. y y y Regessio Imputatio Methods y y y Re Poposed Imputatio Methods Ratio Imputatio Methods y I y II y III Regessio Imputatio Methods y IV y V I Table, we easily udestad that, the bias, mea squae eo ad peetage elative effiieies of existig ad poposed estimatos. I olum, the value of bias of all the estimatos is expessed i fot of thei espetive estimato. The value of bias of the y M., is miimum as ompae to all othe estimatos. I olum ad 3, the mea squae eo ad peetage elative effiieies ae expessed. The mea squae eo of the atio estimato is 0.00 ( % ), fo y is (50.98 % ), fo y is ( % ), fo 9 0 fo the y. 6 A 5 8 y the value is ( % ) ad, the value of mea squae eo is ( % ), fo the estimato y (3.690 % ) ad fo egessio estimato the value is ( % ). The value of mea squae eo fo the poposed imputatio methods is, y is ( % ), fo y. is ( % ), fo y. fo. I II III

19 ohail, habbi ad Ahmed 95 the value is ( % ), fo y is (3.883 % ) ad fo y IV V is ( % ). The quatity iside the baes shows the amout of peetage elative effiiey elative to mea imputatio method. If oelatio betwee the both the study ad the auxiliay vaiable is high tha ou poposed lass of imputatio methods a pefom bette as ompae to othe existig estimato. The fifth method imputatio a pefom outstadig as ompae to othe methods, but othe methods of imputatio ae also pefom elatively bette as ompae to mea imputatio method. Fo the othe values of the paametes, aothe simulatio study is osideed ude the poedue whih was desibed above at the vayig espose. The values of the paamete ae give as Y =.6368, X = 9.93, = , y x xy xy = , =.9375, = , = 0.0, = 0.04, = = 3 = 0.0, = = 0.0, = = = 0.836, = = , I II III 3 IV 4 V 5 = = = 0.045, = = I II III 3 IV 4 V 5 Table Bias, ME ad Peetage Relative Effiiey( i ) fo Vaious Respose Rate Bias ME Effiiey% Existig Imputatio Method Mea Imputatio Method y M Ratio Imputatio Method y R y y y Regessio Imputatio Method y y y Re

20 96 Modified Class of Ratio ad Regessio Type Estimatos.. Bias ME Effiiey% Poposed Imputatio Method Ratio Imputatio Method y I y II y III Regessio Imputatio Method y IV y V. Existig Imputatio Method Mea Imputatio Method y M Ratio Imputatio Method y R y y y Regessio Imputatio Method y y y Re Poposed Imputatio Method Ratio Imputatio Method y I y II y III Regessio Imputatio Method y IV y V.

21 ohail, habbi ad Ahmed 97 Bias ME Effiiey% Existig Imputatio Method Mea Imputatio Method y M Ratio Imputatio Method y R y y y Regessio Imputatio Method y y y Re Poposed Imputatio Method Ratio Imputatio Method y I y II y III Regessio Imputatio Method y IV y V. I Table, we aied out the simulatio study ove the vaious espose ate to aess out the pefomae of the existee estimatos. It is otied that ou poposed lass of estimatos is pefom bette as ompae tha thei outepat. 6. Real Life Appliatio Fo the eal life appliatio, we use the data set of FEV.DAT whih was attah with the text Rose (05). Fo this eseah wok we oside the FEVstatus ad age of the hilde as the study ad auxiliay vaiable espetively. Fom the populatio of 654 hilde, we selet the sample of size 65 uits fo this study. The data desiptio is give i Table. Table 3 Disiptive tatitsis of Vaiables (FEV ad Age). Vaiable Co. Mii. st Qu. Media Mea st 3 Qu. Max. t.dev. FEV Age

22 98 Modified Class of Ratio ad Regessio Type Estimatos.. I Table 3, the oelatio value betwee betwee the study ad auxiliay vaiable is The miimum ad maximum values of the study ad the auxiliay vaiable is 0.79 ad 5.793, 3 ad 9 espetively. The mea ad stadad deviatio of the both vaiables is.637 ad 0.867, ad.964 espetively. Fo aessig the pefomae of existig ad poposed imputatio methods, we use the expessio whih is give i Equatio (6.3) with diffeet values of,,, ad. s s s s = 0.00, 0.00, 0.900, 0.090, = 0.00, 0.00, 0.400, 0.040, 0.5. = 0.090, 0.0, 0.50, 0.80, 0.0. = 0.00, 0.080, 0.060, 0.040ad0.00. I Table 4 (see Appedix): we shows the PRE() i ove the diffeet values of s s,, ad at vayig espose (3.08% 9.3%) by usig the eal life data set. The maximum P. R. E( i ) of the k estimato is 76.7%, 30.47%, %, 5.865%, 8.797%, 90.88%,3.839%, %,3.839%, 4.757%, % ad 5.865% espetively. I etie ombiatio of the s, s, ad with ou poposed estimato a pefom bette as ompae to the existig elated estimatos. 7. CONCLUION Based o the esults of Table, ad 4, we obseved that, ou poposed imputatio methods a pefom bette as ompae to all of the existig imputatio methods. Futhemoe, we otie that the estimatos, (amely y, y III ) have the highe V level of effiiey as ompae to othe poposed imputatio methods. Oveall by the use of highe ode momets of the auxiliay vaiable would lead to impove the effiiey of imputatio methods. I Table 4, we see that by the use of highe ode momets of the auxiliay vaiable ove the diffeet values of s, s, ad povide moe peise esults as ompae to the existig estimatos. Fom ou small sale study, we oluded that the poposed estimatos a pefom bette as ompae to the existig estimato. We see by ompaiso of Table ad Table 4, the behavio of poposed lass of estimatos i empiial study ad i eal life data set is quite simila but we see some of the flutuatio i esults of empiial study ad eal life data beause i simulatio study ou esults ae based o some kid of imagiatios but i eal life these imagiatios may o may ot be aeptable, so thei is slightly vaiability exists. As a whole, ou poposed lass of estimatos ude simulated ad empiial study a pefomed bette as ompae to existig estimatos. ACKNOWLEDGEMENT The authos ae thakful to the efeees fo thei suggestios, whih led to the impovemet of this atile.

23 ohail, habbi ad Ahmed 99 REFERENCE. Ahmed, M., Al-Titi, O., Al-Rawi, Z. ad Abu-Dayyeh, W. (006). Estimatio of a populatio mea usig diffeet imputatio methods. tatistis i Tasitio, 7, 6: 47, 64.. Gjestvag, C.R. ad igh, (009). A impoved adomized espose model: estimatio of mea. Joual of Applied tatistis, 36(), Hase, M.H. ad Huwitz, W.N. (946). The poblem of o-espose i sample suveys. Joual of the Ameia tatistial Assoiatio, 4(36), Mohamed, C., edoy, ad igh, (06). Compaiso of Diffeet Imputig Methods fo ambled Resposes. Hadbook of tatistis, 34, Rose, B. (05). Fudametals of Biostatistis, Nelso Eduatio. 6. Rubi, D.B. (976). Ifeee ad missig data. Biometika, 63(3), igh, ad Deo, B. (003). Imputatio by powe tasfomatio. tatistial Papes, 44(4), Wae, L. (965). Radomized espose: A suvey tehique fo elimiatig evasive aswe bias. Joual of the Ameia tatistial Assoiatio, 60(309),

24 300 Modified Class of Ratio ad Regessio Type Estimatos.. RR μ s APPENDIX Table 4: P.R.E(i) Values ude Diffeet ituatios η γ s P.R.E () P.R.E () P.R.E (3) P.R.E (4) P.R.E (5) P.R.E (6) P.R.E (7) P.R.E (8) P.R.E (9) P.R.E(0) P.R.E() P.R.E()