Estimation of Population Mean with. a New Imputation Method

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1 Aled Mathematal Sees, Vol. 9, 015, o. 34, HIKARI Ltd, htt://d.do.og/ /ams Estmato of Poulato Mea wth a New Imutato Method Abdeltawab A. Ga Isttute of Statstal Studes & Reseah Cao Uvest, Gza, Egt Coght 015 Abdeltawab A. Ga. Ths s a oe aess atle dstbuted ude the Ceatve Commos Attbuto Lese, whh emts uestted use, dstbuto, ad eoduto a medum, ovded the ogal wok s oel ted. Abstat I ths ae, we todue a ew method of ato te mutato ad oesodg ot estmato has bee oosed. We obta the bas mea squae eo equatos fo ths estmato. It s omaed wth the samle mea ad ato estmatos the ase of mssg data. I addto, t s omaed wth estmatos Sgh ad Ho (000), Sgh ad Deo (003), ad Sgh (009). The oosed estmato emas bette tha the mea ad ato estmatos. Numeal llustatos ae efomed to vef theoetal esults though thee eal data set eamles. Kewods: Imutato; Effe; Mssg data; estmato of mea; ato-te estmatos 1. Itoduto Mssg data s a ve ommo oblem samle suves. Imutato s oe of the most oedue wheeb mssg values fo oe o moe stud vaables ae flled wth substtutes. These substtutes a be ostuted aodg to some ules, o the a be obseved values but fo elemets othe tha the oesodets. Imutato usg aula data ma odue estmatos that ae moe effet tha the oe ostuted b gog o-esodets ad eweghtg. A atual questo ases, what oe eeds to assume ode to justf gog the omlete mehasm. Rub (1976) addessed two ke oets: mssg omletel at adom (MCAR), whe the esose dato to the suve s deedet of all othe vaables the suve. Mssg at adom (MAR), whe the esose dato deeds ol o some haatests obseved the

2 1664 Abdeltawab A. Ga suve ad avalable also fo o-esodets. Dffeet mutato methods ae used suh as mea, ato, odut ad egesso to move the estmato of oulato mea wth oesose. Sgh ad Ho (000) todued a omomsed method of mutato. Sgh ad Deo (003) suggested a ew owe tasfomato estmato of oulato mea. Sgh (009) eseted a alteatve estmato of oulato mea the esee of o-esose that omes the fom of Walsh s estmato. Followg them, we assume MCAR the eset wok. Let N Y 1 N be the mea of the fte oulato 1,,,,, N. A smle adom samle wthout elaemet (SRSWOR), s, of sze s daw fom to estmate Y. Let be the umbe of esodg uts out of samled uts. Let the set of the esodg uts be deoted b R ad that of o-esodg uts be deoted br. Fo eve R, the value of s obseved. Howeve, fo the uts R, the values ae mssg ad muted values ae deved usg dffeet methods. The mutato s aed out wth the ad of a aula vaable,, suh that s the value of aula vaable fo ut s kow ad ostve fo eve s : s ae kow..e., the data s The mea mutato method s to elae eah mssg datum wth the mea of the obseved value. The data afte mutato beomes. f R f R Ude ths method of mutato, the ot estmato of oulato mea gve b, s 1 () s beomes, (1) 1 1 (3). Followg the otatos of Lee, et al. (1994), the ase of sgle mutato method, f the th ut eques mutato, the value b ˆ s muted, whee bˆ data afte mutato beomes, 1 1 j. f R b ˆ f R (4)

3 Estmato of oulato mea wth a ew mutato method 1665 Ths method of mutato s all the ato method of mutato. Ude ths method of mutato, the ot estmato () beomes (5) whee, 1, s 1 ad R 1. R Sgh ad Ho (000) suggested a omomsed method of mutato. It based o usg fomato fom muted values fo the esodg uts addto to o-esodg uts. I the ase of omomsed mutato oedues, the data take the fom:. / (1 ) ˆ b f R (1 ) b ˆ f R (6) whee s a a sutabl hose ostat, suh that the vaae of the esultat estmato s mmum.. The ot estmato () ude omomsed method of mutato method beomes: Com (1 ) (7) Sgh ad Deo (003) suggested a ew owe tasfomato estmato of oulato mea. I the ase of the mutato oedue, the data take the fom:. f R f R R (8) whee s a sutabl hose ostat, suh that the vaae of the esultat estmato s mmum. The ot estmato () ude omomsed method of mutato method beomes: (9) If 0 the ad f 0 the.

4 1666 Abdeltawab A. Ga Sgh (009) suggested a alteatve estmato of oulato mea the esee of o-esose. I the ase of oosed mutato, the data afte mutato take the fom:. f R ( ) ( ) f R (1 ) R (10) whee s a a ostat hose sutabl, suh that the vaae of the esultat estmato s mmum. The ot estmato () ude Sgh method of mutato method beomes: Sgh (1 ) (11) I the et seto, we defe otato ad eetatos that ae useful to fd the odtoal bas ad vaae of the bas ad mea squaed eos of the estmatos (3), (5), (7), (9) ad (11) ad the estmato esultat fom the oosed methods of mutato.. Poetes of Imuted Estmatos Defe 1, = 1 ad = 1 Y X X Usg the oet of two-hase samlg (Rao ad Stte (1995) ad Aab ad Sgh (006)) ad the mehasm of MCAR, fo gve ad, we have E ( ) E ( ) E ( ) 0 ad E C E C E C C N N N ( ), ( ), ( ) E C E C E C C N N N ( ), ( ), ad ( ) S S whee C, C ae the oulato oeffets of vaato of X Y aula ad stud vaables, esetvel, =S / SS s oeffet of

5 Estmato of oulato mea wth a ew mutato method 1667 oelato betwee the aula ad the stud vaable ad S, S ad S have the usual meag. The odtoal bas of estmatos at (3), (5), (7), (9) ad (11) ae the followg: 1 1 B ( ) Y C C C 1 1 B ( ) (1 )Y C C C Com (1) (13) 1 1 ( 1) B ( ) Y C C C (14) C C C (1 ) C N N N B ( ) Y C C C (1 ) C N N N N (1 ) C C C N N Sgh (15) The mea squae eo (MSE) of estmatos at (3), (5), (7), (9) ad (11) ae the followg: 1 1 V ( ) S N MSE S R S RS N ( ) 1 1 C M. MSE ( ) MSE ( ) Y 1 C 1 1 M. MSE ( ) ( ) MSE at S B R Com at C 1 1 S Y XY M. MSE ( Sgh ) MSE ( at ) S X S X X (16) (17) (18) (19) (0) whee R Y X ad B S XY S X.

6 1668 Abdeltawab A. Ga I the et seto, we suggest a ew method of mutato. The esultat estmato has show to ema bette the tha estmatos fom mea ad ato methods of mutato ad s effet as othe estmatos. 3- A New Poosed Method of Imutato I the oosed method of mutato, the data afte mutato take the fom:. f R f R R (1) whee s a a sutabl hose ostat, suh that the vaae of the esultat estmato s mmum. Note that f 0 the. Theoem (3.1): The ot estmato () of oulato mea Y ude ths oosed method of mutato s: = Poof: We have () 1 1 s R R (3) Usg (1) ad elae t (3) we get (). Note that f 0 the. The estmato at () s a aalogue of the kow estmato of oulato mea oosed b Svekataamaa ad Ta (1980) = ST X The oetes of () ae the followg two theoems: Theoem (3.): The bas of the oosed estmato s gve b: 1 1 B ( ) Y C C X whee X Poof : the estmato tems of, ad a be wtte as follows: (4)

7 Estmato of oulato mea wth a ew mutato method 1669 Y (1 )(1 )(1 ) 1 = Y (1 )(1 )(1...) = Y 1 ( ) (5) Negletg the hghe ode of aomato, the bas B ( ) E ( Y) =YE 1 Y (6) Takg the eetatos of (6), we get the (4) that ove theoem (3.). Theoem (3.3): The mmum mea squae eo (MSE) of the oosed estmato s gve b: m. MSE ( ) S B S N fo the otmum value gve b (7) X( R B) (8) B Poof: The MSE of a be foud u to the fst ode of aomato b ewtg (5) as follows: Y Y ( ) (9) the MSE ( ) E Y Y Y E ( ) Y E ( ) ( ) C C C C N N N N CC N N CC S R S RS N (30)

8 1670 Abdeltawab A. Ga Dffeetatg (30) wth eset to ad set t equal to zeo, we get X( R B) B the substtute the value of, we get whh oves B R theoem (3.3). It s lea that MSE of the oosed estmato s moe effet tha the mea estmato (3) egadless the value of B. I addto, the oosed estmato s alwas moe effet tha the usual ato estmato (5). We ote that oosed method s fee fom the assumtos of a model fo the ato method of mutato. I addto, MSE s smla to the othe metoed estmato MSE ( ) MSE ( ) MSE ( ) MSE ( ). SH S Choosg of s a ma dffult eseall whe t deeds o ukow aametes. We suggest a estmato of that s gve b: s (31) s whee ( )( ) / 1, ( ) / 1 ad ( ) / s s s 3. Numeal Illustatos A smulato stud s aed out to evaluate the efomae of dffeet estmato dsussed evous setos. We osdeed thee dffeet eal oulatos ad omaed the effe of estmatos (5), (7), (9), (11) ad () wth the samle mea that does ot emlo a aula fomato. We omuted aomate values of MSE fo eah estmato usg (17), (18), (19), (0) ad (7). The efomae of the estmatos s dsussed tems of eet elatve effe (PRE) of estmato. wth eset to : V ( ) PRE 100, Com ad, Sgh, (3) MSE ( ) Table.1 esets the aametes of thee oulato. We hoose these oulatos wth dffeet samle szes to be ea fom eal suves. The esose ate s 80%, 85% ad 90%. Also we selet the dffeet oelato oeffet. Table. esets the PRE fo the estmatos volved the emal stud. We ote that fo all the osdeed oulatos, the oosed estmatos efom bette tha the mea ad the ato estmatos. The oosed estmato () s effet as othe estmatos (5), (7), (9) ad (11) all dffeet oulatos.

9 Estmato of oulato mea wth a ew mutato method 1671 Thus, oluso, we a sa that the oosed method of mutato emas bette tha the mea ad ato estmatos. Table 1: Poulato Paametes of Thee Dffeet Poulato. aametes Poulato A (Kadla ad Cg (008)) Poulato B (Sgh (009)) Poulato C ( Daa ad Pe (010)) N Y X S S S R B Table.: PRE of the Cosdeed Estmatos ude thee Dffeet Poulatos Poulato A Poulato B Poulato C Estmato V( ) V( ) V( ) Com Sgh Coluso We todue a ew method of mutato ad the esultat estmato emas bette tha tadtoal estmatos the eset of o-esose. The efomae of the oosed estmato s justfed theoetall ad umeall. The elatve effe dated that the oosed estmato s equvalet to othe metoed estmatos.

10 167 Abdeltawab A. Ga Refeees [1] C. Kadla ad H. Cg, Estmatos fo the Poulato Mea the Case of Mssg Data, Commuatos Statsts Theo ad Methods, 37 (008), htt://d.do.og/ / [] G. Daa, ad P. F. Pe, Imoved Estmatos of the Poulato Mea fo Mssg Data, Commuatos Statsts Theo ad Methods, 39 (010), htt://d.do.og/ / [3] H. Lee, E. Raout ad C.E. Sädal, Eemets wth Vaae Estmato fom Suve Data wth Imuted Values, Joual of Offal Statsts, 10 (1994), [4] J. N. K Rao ad R.R Stte, Vaae Estmato ude Two-Phase Samlg wth Alato to Imutato fo Mssg Data, Bometka, 8 (1995), htt://d.do.og/ /bomet/ [5] R. Aab, ad S. Sgh, A New Method fo Estmatg Vaae fom Data Imuted wth Rato Method of Imutato, Statsts & obablt lettes, 76 (006), htt://d.do.og/ /j.sl [6] R. B. Rub, Ifeee ad mssg data, Bometka, 63, (1976), htt://d.do.og/ /bomet/ [7] S. Sgh, A ew method of mutato suve samlg. Statsts: A Joual of Theoetal ad Aled Statsts, 43 (009), htt://d.do.og/ / [8] S. Sgh ad B. Deo, Imutato b Powe Tasfomato, Statstal Paes, 44 (003), htt://d.do.og/ /bf [9] S. Sgh ad S. Ho, Comomsed Imutato Suve Samlg, Metka, 51 (000), htt://d.do.og/ /s [10] T. Svekataamaa ad D. S Ta, A Alteatve to Rato Method Samle Suves, Aals of the Isttute of Statstal Mathemats, 3 (1980), htt://d.do.og/ /bf Reeved: Febua 14, 015; Publshed: Mah 5, 015