Estimation of Population Mean Using Exponential Type Imputation Technique for Missing Observations

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1 Joural of Moder Appled Statstcal Methods Volue 5 Issue Artcle Estato of Populato Mea Usg Epoetal Tpe Iputato Techque for Mssg Observatos Rajesh Sgh Baaras Hdu Uverst, rsghstat@gal.co Heat K. Vera Baaras Hdu Uverst, coolheat0089@gal.co Praas Shara Uverst of Petroleu ad Eerg Studes, praasshara0@gal.co Follow ths ad addtoal works at: Recoeded Ctato Sgh, Rajesh; Vera, Heat K.; ad Shara, Praas (06) "Estato of Populato Mea Usg Epoetal Tpe Iputato Techque for Mssg Observatos," Joural of Moder Appled Statstcal Methods: Vol. 5 : Iss., Artcle 9. DOI: 0.37/jas/ Avalable at: Ths Regular Artcle s brought to ou for free ad ope access b the Ope Access Jourals at DgtalCoos@WaeState. It has bee accepted for cluso Joural of Moder Appled Statstcal Methods b a authorzed edtor of DgtalCoos@WaeState.

2 Joural of Moder Appled Statstcal Methods Ma 06, Vol. 5, No., Coprght 06 JMASM, Ic. ISSN Estato of Populato Mea Usg Epoetal Tpe Iputato Techque for Mssg Observatos Rajesh Sgh Baaras Hdu Uverst Varaas, Ida Heat K. Vera Baaras Hdu Uverst Varaas, Ida Praas Shara Uverst of Petroleu ad Eerg Studes Dehradu, Ida Soe putato techques are suggested for estatg the populato ea whe the data values are ssg copletel at rado uder a sple rado saple wthout replaceet schee. Two classes of pot estators are proposed. The bas ad ea squared error epressos of the proposed pot estators are derved up to frst order of approato. It has bee show that the proposed pot estators are ore effcet tha soe estg pot estators due to Lee, Racourt, ad Sardal (994) ad Sgh ad Hor (000). Theoretcal fdgs are supported b a eprcal stud based o fve populatos to show the superort of the costructed estators ad ethods of putato over others. Kewords: Mssg data, putato, bas, ea squared error, sple rado saplg wthout replaceet Itroducto Mssg data s a coo ad serous proble surve saplg. Mssg data aturall occurs saple surves whe a few saplg uts refuse to respod or are uable to partcpate the surve. There are two tpes of o-resposes whch occur surves: ut o-respose ad te o-respose. Ut o-respose occurs whe a elgble saple ut fals to partcpate a surve because of falure to establsh a cotact or eplct refusal to cooperate. Ite o-respose occurs stead whe a respodg ut does ot provde useful aswers to partcular tes Dr. Sgh s a Assstat Professor the Departet of Statstcs. Eal h at: rsghstat@gal.co. Dr. Shara, the correspodg author, s a Professor the Departet of Decso Sceces, College of Maageet ad Ecoocs Studes. Eal at praasshara0@gal.co. 358

3 SINGH ET AL of the questoare. Such stuatos create ssg data proble. The putato s a well-defed ethodolog b vrtue of whch such probles ca be uraveled. I the lterature several putato techques are avalable ad dscussed. Rub (976) addressed three cocepts: observed at rado (OAR), ssg at rado (MAR), ad paraetrc dstrbuto (PD). Rub defed MAR as the probablt of the observed ssgess patter, gve the observed ad uobserved data, does ot deped o the value of the uobserved data. Hetja ad Basu (996) dstgushed the eag of MAR ad ssg copletel at rado (MCAR) a ver ce wa. The putato techque s also applcable whe forato o aular varable s avalable. Lee et al. (994; 995) used the forato o a aular varable for the purpose of putato, Sgh ad Hor (000) suggested a coprosed ethod of putato, Ahed, Al-Tt, Al-Raw, ad Abu-Daeh (006) suggested several ew putato based estators that use the forato o a aular varable ad copared ther perforaces wth the ea ethod of putato, ad Rao ad Stter (995) used the putato techques for varace estato uder two phase saplg. Kadlar ad Cg (008) ad Daa ad Perr (00) also suggested soe putato techques case of ssg data. I the preset stud we plctl assue MCAR. Let Y N N be the populato ea of stud varable Y. A sple rado saple wthout replaceet (SRSWOR), s, of sze s draw fro Ω = {,,, N} to estate the populato ea Y. Let r be the uber of respodg uts out of sapled, the the uber of o-respodg uts s ( r). Let the set of respodg uts be deoted b R ad that of o-respodg uts be deoted b R c. For ever ut R, the value s observed. However for the uts R c, the values are ssg ad puted values are to be derved. We assue that putato s carred out wth the ad of a quattatve aular varable such that, the value of for ut s, kow ad postve for ever s. I other words, the data s = { : s} are kow. Soe Avalable Methods of Iputato ad Estators There are soe classcal ethods of putato whch are cool used ad gve as follows: 359

4 EXPONENTIAL TYPE IMPUTATION TECHNIQUE Mea Method of Iputato I ths ethod of putato, the stud varable after putato takes the for as, R r, R c () ad the pot estator of the populato ea Y s gve b s. () s Thus, uder ths ethod of putato, the pot estator of the populato ea Y s. (3) r r R Lea. The epresso of Bas ad Varace of the pot estator s gve as N S Y Y. N where Rato Method of Iputato Bas 0 (4) r N S V Followg the otatos of Lee et al. (994), the case of sgle value putato, f the th ut requres putato, the value b ˆ s puted. Thus, the stud varable after putato takes the for as (5) 360

5 SINGH ET AL, R b ˆ, R c (6) where ˆ R b. R Uder ths ethod of putato, the pot estator of the populato ea Y s gve b RAT r (7) r where, s r, ad r R r. r R Lea. The epresso of Bas ad Mea Square Error (MSE) of the pot estator RAT s gve as where Bas RAT Y C CC, r (8) MSE RAT S S R S RS, N r (9) S s defed as above ad S X X S Y Y X X N N N N, Y S S S R, C, C, X Y X S S 36

6 EXPONENTIAL TYPE IMPUTATION TECHNIQUE Coprosed Method of Iputato Sgh ad Hor (000) proposed coprosed putato procedure. After putato the stud varable takes for as, R r b ˆ c b ˆ, R (0) where α s a sutabl chose costat such that the varace of the resultat estator s u. Here, we are also usg forato fro puted values for the respodg uts addto to o-respodg uts. Thus, uder coprosed ethod of putato, the pot estator of the populato ea Y s. () COMP r r r Lea 3. The epresso of Bas ad MSE of the pot estator COMP s gve as Bas COMP Y C CC r MSE COMP S S R S RS N r Y C r () (3) C where opt C. Thus MSE S COMP N r (4) 36

7 SINGH ET AL Alog slar les, Ahed et al. (006) proposed several ew putato techques b troducg soe ukow paraeters ad hece proposed the correspodg estators for estatg the fte populato eas Y. Proposed Iputato Methods ad Correspodg Estators The followg two putato ethods are suggested. After putato for the frst proposed putato of techque, the stud varable takes the for as, R h h X r r ep r, R r h h X a r c (5) where a, h, ad α are sutabl chose costats. We optze α such a wa that the MSE of the resultat estator s u. Thus we have the followg theore: Theore. Uder the proposed ethod of putato cosdered (5), the pot estator of the populato ea Y s gve as T P h h X ep r r h h X a r. (6) Proof: TP s R R c (7) where R ad R c are the sets of respodg ad o-respodg uts the saple s of sze. Now puttg the values fro (5) to (7), the pot estator of populato s obtaed as ea Y as defed (6), whch copletes the proof. 363

8 EXPONENTIAL TYPE IMPUTATION TECHNIQUE Table. Mebers of the class of estators TP Estators Costats α = α = a h X r r X TP r ep X TP5 r ep X X r r X TP r ep TP6 r ep X X X r r X TP3 r ep TP7 r ep X r X r X r r X TP4 r ep TP8 r ep X r X r Because the pot estator proposed (6) after putg the ssg values, belogs to a class of estators. Soe ebers of the proposed class of pot estator defed (6) are show Table for dfferet choce of a, h, ad α. The stud varable after putato for the secod proposed putato of techque becoes, R h h X r ep r, R r h h X a c (8) where a, h, ad α are sutabl chose costats. We optze α such a wa that the MSE of the resultat estator s u. Thus we have the followg theore: Theore. Uder the proposed ethod of putato cosdered (8), the pot estator of the populato ea Y s gve as 364

9 SINGH ET AL T g h h X X ep r h h X a X (9) Proof: Tg s R R c (0) where R ad R c are the sets of respodg ad o-respodg uts the saple, s, of sze. Puttg the values fro (8) to (0), we get the for of the pot estator of populato ea Y as defed (9), whch copletes the proof. Soe ebers of the proposed class of pot estator defed (9) are show Table for dfferet choces of a, h, ad α. Propertes of the Estators TP ad Tg To obta the bas ad MSE epressos of the estators to the frst degree of approato, we defe r Y r X X e0, e, e Y X X such that E(e ) = 0; = 0,,, ad E e0 C, E e C, E e0e CC, r N r N r N E e C, E ee C, Ee0e C C N N N Usg above terolog, the bas ad MSE of the proposed estators are gve below. 365

10 EXPONENTIAL TYPE IMPUTATION TECHNIQUE Table. Mebers of the class of estators Tg Estators Costats α = α = a h T = ep g r T = ep g r X - X X - X X - T = ep g3 r X + T = ep g 4 r X - X + T = ep g5 r T = ep g6 r T = ep g7 r T = ep g8 r - - X + - X X X X - X X + X Theore 3. The Bas of the estator T P s gve b BasT Y a C C C P r N ah ah () ad the MSE of the estator T P s gve b MSE TP Y C C CC r N a h ah, () where the optu value of α s gve b opt C ah C Proof: Epressg the estator T P ters of the e s, we have e h TP Y e0 ep a e h 366

11 SINGH ET AL e e e Y e0 ep ah h ah e a Y e0 ep ep e ah a h e a Y e0 e e ah a h a h a TP Y e0 e e e ah a h a h Takg epectato o both sdes, we get the bas epresso of estator T P as BasT Y a C C C P r N ah ah To fd the MSE of the estator T P, we have MSE TP E TP Y Y E e0 e ah Y E e E e E e e a h ah 0 0 MSETP Y C C CC r N a h ah Partall dfferetatg above equato wth respect to α ad equatg to zero, we have T P MSE Y C 0 CC r N a h ah Splfg the above equato, we get the optu value of α as 367

12 EXPONENTIAL TYPE IMPUTATION TECHNIQUE opt C ah C Theore 4. The Bas of the estator T g s gve b BasT Y a C C C g N ah ah (3) ad the MSE of the estator T g s gve b MSE Tg Y C C CC r N N a h ah, (4) where the optu value of α s gve b opt C ah C Proof: The above theore ca be proved a slar wa to the proof of Theore 3. Effcec Coparso Estator T P s ore effcet tha estator f T V MSE P 0. But Y C Y C 0 r N r N Y C 0 r N 368

13 SINGH ET AL sce. Therefore, T P s ore effcet tha. Slarl, r N RAT r r N MSE COMP MSE TP 0 f r r N MSE Tg MSE TP 0 f r MSE MSE TP 0 f C CC C 0 Thus, fro the above results, we ca sa that the estator T P s ore effcet tha other estators. Eprcal Stud Fve populatos, A, B, C, D, ad E, are cosdered. Populato A s the artfcal populato of sze N = 00 fro Shukla, Thakur, Pathak, ad Rajput (009), populato B s fro Ahed et al. (006), populato C s fro Dass (988), populato D s fro Murth (967, p. 8), ad populato E s fro Sgh, Sgh, ad Kuar (976, p. 6) wth paraeters as gve Table 3. Let = 40, r = 35 for populato A, = 00, r = 80 for populato B, = 80, r = 7 for populato C, = 3, r = 0 for populato D, ad = 6, r = 5 for populato E respectvel. The the bas ad MSE of the proposed pot estators are gve Table 4 ad Table 5 for populatos A, B, C, D ad E respectvel. Table 3. Paraeters for stud populatos Paraeters Populato N Y X S S ρ C C A B C D E

14 EXPONENTIAL TYPE IMPUTATION TECHNIQUE Table 4. Bases of estators Populatos Estators A B C D E RAT COMP TP() Tg() Table 5. MSEs of estators Populatos Estators A B C D E RAT COMP TP() Tg() Tables 4 ad 5 ehbts the bas ad MSE of dfferet pot estators ad t has bee observed fro the tables that the estators based o aular forato are ore effcet tha the oe whch does ot use the aular forato such as to overcoe the putato probles. Both the proposed classes of estators T P ad T g are ore effcet tha the estators,, RAT ad COMP, scrupulousl, T P has u MSE aog all the estators cosdered here. Cocluso Two putato techques are suggested usg aular forato followed b two class of estators for estatg the populato ea case of data values are MCAR uder a SRSWR schee. I addto, soe ew ebers are also geerated fro two proposed class of estators usg the sutable values of costats. The u bases ad ea square errors of the proposed class of estators were detered up to the frst order of approato. It was establshed theoretcall ad eprcall that the proposed class of estator perfors best aog the other estators cosdered, ad cosequetl the 370

15 SINGH ET AL correspodg (frst proposed) ethod of putato s better tha the other estg ethods ad a be recoeded for further use. Refereces Ahed, M. S., Al-Tt, O., Al-Raw, Z., & Abu-Daeh, W. (006). Estato of a populato ea usg dfferet putato ethods. Statstcs Trasto, 7(6), Dass, A. K. (988). Cotrbutos to the theor of saplg strateges based o aular forato (Upublshed doctoral dssertato). Bdha Chadra Agrcultural Uverst, West Begal, Ida. Daa, G., & Perr, P. F. (00). Iproved estators of the populato ea for ssg data. Coucatos Statstcs Theor ad Methods, 39(8), do: 0.080/ Hetja, D. F., & Basu, S. (996). Dstgushg ssg at rado ad ssg copletel at rado. The Aerca Statstca, 50(3), do: 0.080/ Kadlar, C., & Cg, H. (008). Estators for the populato ea the case of ssg data. Coucatos Statstcs Theor ad Methods, 37(4), do: 0.080/ Lee, H., Racourt, E., & Sardal, C. E. (994). Eperets wth varace estato fro surve data wth puted values. Joural of Offcal Statstcs, 0(3), Retreved fro: Lee, H., Racourt, E., & Sardal, C. E. (995). Varace estato the presece of puted data for the geeralzed estato sste. I Proceedgs of the Surve Research Methods Secto, Aerca Statstcal Assocato, Retreved fro: Murth, M. N. (967). Saplg Theor ad Methods. Calcutta, Ida: Statstcal Publshg Socet. Rao, J. N. K., & Stter, R. R. (995). Varace estato uder two-phase saplg wth applcato to putato for ssg data. Boetrca, 8(), p do: 0.093/boet/ Rub, D. B. (976). Iferece ad ssg data. Boetrka, 63(3), do: 0.093/boet/

16 EXPONENTIAL TYPE IMPUTATION TECHNIQUE Shukla, D., Thakur, N. S., Pathak, S., & Rajput, D. S. (009). Estato of ea uder putato of ssg data usg factor-tpe estator two-phase saplg, Statstcs Trasto, 0(3), Sgh, D., Sgh, P., & Kuar, P. (976). Hadbook o Saplg Methods. New Delh, Ida: Ida Agrcultural Statstcs Research Isttute (ICAR). Sgh, S., & Hor, S. (000). Coprosed putato surve saplg. Metrka, 5(3), do: 0.007/s