ON ESTIMATION OF POPULATION MEAN IN THE PRESENCE OF MEASUREMENT ERROR AND NON-RESPONSE

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1 Pak. J. Statst. 015 ol. 31(5), ON ESTIMATION OF POPLATION MEAN IN THE PRESENCE OF MEASREMENT ERROR AND NON-RESPONSE Muhammad Azeem 1 ad Muhammad Haf Natoal College of Busess Admstrato & Ecoomcs, Lahore, Paksta Emal: 1 azeem_stats@hotmal.com drmahaf@gmal.com ABSTRACT I ths paper we study the problem of estmato of populato mea the presece of measuremet error ad o-respose smultaeously usg formato o a sgle auxlary varable. We have developed a ew estmator of populato mea ad compared t wth some exstg estmators uder the stuatos whe measuremet error ad orespose occur smultaeously. The proposed estmators are theoretcally compared wth exstg estmators. Emprcal ad smulato study s also coducted to assess the performace of proposed estmator. KEWORDS Auxlary varable, expoetal estmator, bas, mea square error, measuremet error, o-respose. 1. INTRODCTION I samplg theory, there are two types of o-samplg errors: respose error ad o-respose error. Respose error or measuremet error occurs whe the reported value dffers from the true value. No-respose error occurs whe the researcher fals to collect formato o oe or more tha oe ut of the survey. May estmators are avalable lterature wthout takg to accout the respose ad o-respose errors. Eve f the o-samplg errors have bee take to accout by the researchers, most of the avalable estmators are ether for the case of measuremet error or o-respose separately. I practce, the researcher ofte faces stuatos where some measuremet error ad o-respose occur at the same tme whle collectg formato. Measuremet error may occur due to some over-reportg, uder-reportg, memory falure by the respodets etc. whle collectg data. The problem of o-respose occurs whe the researcher fals to collect formato from some uts the survey due to a umber of reasos lke o-avalablty of the respodets at home, refusal to aswer the questoare, lack of formato etc. The problem of estmato of populato mea the presece of measuremet errors ad o-respose has bee cosdered by may researchers. Hase ad Hurwtz (196) studed the problem of o-respose for the frst tme ad suggested a procedure whch a sub-sample was draw from the o-respodg uts the sample ad a extra effort was made for the collecto of formato from the o-respodets selected the sub- 015 Paksta Joural of Statstcs 657

2 658 O estmato of populato mea the presece of measuremet error sample. Cochra (1968) studed the effect of measuremet errors o ordary least squares estmates of regresso coeffcets ad foud that measuremet errors may lead to cosstet ad based estmates of the regresso coeffcets. Cochra (1977) suggested some ew estmators of populato mea usg formato o a sgle auxlary varable for stuatos where some o-respose occurs the survey. Shalabh (1997) studed the classcal rato estmator of populato mea the presece of measuremet errors. Masha ad Sgh (001) examed the effect of measuremet errors o a ew estmator whch was a lear combato of the rato ad mea per ut estmators. Besdes ths, the problem of measuremet error has also bee studed by Fuller (1995), Masha ad Sgh (00), Srvastava ad Shalabh (001), Alle et al. (003), Sgh ad Karpe (008, 009), Kumar et al. (011) ad Shukla et al. (01) etc. The problem of o-respose has also bee studed by Rao (1986, 1987), Khare ad Srvastava (1993, 1995, 1997, 010), Tabasum ad Kha (00), Sgh ad Kumar (008, 011), Kumar ad Sul (01, 01) ad Shabbr ad Kha (013) etc. May researchers have prevously studed the problem of measuremet error ad orespose separately. I practce, t s possble that the problem of measuremet error ad o-respose may occur at the same tme. The objectve of ths paper s to suggest a ew estmator of populato mea usg formato of auxlary varable the presece of measuremet error ad o-respose. A ew estmator of populato mea has bee preseted ad a comparatve study s made wth the Hase ad Hurwtz (196) estmator, Cochra s (1977) estmator ad Sgh ad Kumar (008) estmator. After troducg the cocept of samplg, auxlary formato, orespose ad measuremet alog wth some mportat refereces, sample desg the case of measuremet error ad orespose s dscussed secto alog wth mportat otatos. I secto three, some exstg estmators are dscussed ad secto four cotas the costructo of ew estmator alog wth the dervato of ts mea square error. The proposed estmator s theoretcally compared wth exstg estmators ad effcecy codtos are deduced secto fve. The emprcal ad smulato study s coducted secto sx, the cocluso s provded last secto.. NOTATIONS A smple radom sample of sze s draw from populato of sze N by smple radom samplg wthout replacemet (SRSWOR). It s assumed that the populato s composed of two mutually exclusve groups, the respodets ad the N orespodets, though ther szes are ukow. Let r 1 respod ad r do ot respod. A sub-sample of sze k (k = r h, h 0 ) s take from the o-respodets the sample. Let (, ) be the observed values ad (, ) be the true values o two characterstcs ( x, y ) respectvely the th ( = 1,,., ) ut the sample. Let the measuremet errors be ad x y u y v x / N1, (.1.1), (.1.)

3 Azeem ad Haf 659 whch are radom ature ad are ucorrelated wth mea zero ad varaces S respectvely for the respodg part of populato. Let S ad be the S varaces assocated wth the measuremet errors study varable ad auxlary varable respectvely for the o-respodg part of populato. Let ad be the populato varaces of ad respectvely for the respodg part of populato. Let S ad be the populato varaces of ad respectvely for the S o-respodg part of populato. Let ad be the populato meas of ad respectvely ad let y ad x be the sample meas of ad respectvely. Let be the populato correlato co-effcet betwee varable ad for the respodg part of populato ad be the populato correlato co-effcet betwee varable ad for the o-respodg part of populato. Let C ad be the populato C co-effcet of varato for varable ad respectvely for the respodg part of populato ad C ad be the populato co-effcet of varato for varable C ad respectvely for the o-respodg part of populato. We further assume that the mea of the study varable s ukow ad the mea of auxlary varable s kow. 3. SOME EISTING ESTIMATORS 3.1 Hase ad Hurwtz (196) Estmator Hase ad Hurwtz (196) suggested the followg estmator whe orespose occur wth varace r r y y y 1 r1 k ar y S S ad, (3.1.1), (3.1.) W h1 N 1 r 1 where,, W, ad. yr1 y N N k yk y r1 1 If measuremet error s take to accout, the varace of the Hase ad Hurwtz (196) estmator ca be wrtte as: ar y S S S S. (3.1.3) 3. Cochra s (1977) Estmator Cochra (1977) proposed the followg rato-type estmator of populato mea S k 1 S S

4 660 O estmato of populato mea the presece of measuremet error t C y x. (3..1) The mea square error of measuremet error s gve as: the presece of o-respose ad wthout If measuremet error s take to accout, the mea square error of. (3..) s gve as: S S S S. (3..3) 3.3 Sgh ad Kumar (008) Estmator Sgh ad Kumar (008) suggested the followg cha-rato-type estmator of populato mea y. (3.3.1) x x The mea square error of measuremet error s gve as: t C C MSE t C C C C C C C C C MSE t C C C C C C C C t SK t SK the presece of o-respose ad wthout MSE t. SK C C CC C C C C (3.3.) If measuremet errors are take to accout, the mea square error of s gve as: MSE t SK C C CC C C C C S S S S. (3.3.3). PROPOSED ESTIMATOR As real lfe stuato, at a tme we ca have such populatos for those the correlato betwee study ad auxlary varables ca be egatve or postve. We requre rato estmator the case of postve correlato ad product estmator the case of egatve correlato. So there s eed of such estmator that ca hadle both stuatos. I ths coecto we proposed the followg estmator that combes the expoetal rato ad expoetal product estmator usg probablty weghtg approach. The suggested estmator s t C t SK

5 Azeem ad Haf 661 x x y exp 1 exp, (.1.1) x x N x where s a costat to be sutably chose, ad x. N I order to derve the mea square error of the proposed estmator, we troduce some further otatos. Let ad t Addg (.1.3) ad (.1.), we have 1 1 O smplfcato, we get y, (.1.), (.1.3), (.1.). (.1.5) =. 1 Smlarly from (.1.) ad (.1.5), we get x (.1.6). (.1.7) Further E S S S S E S S S S E SS SS (.1.8)

6 66 O estmato of populato mea the presece of measuremet error sg (.1.) (.1.1), we get N x x. (.1.9) t y exp 1 exp N x N x N sg (.1.6) ad (.1.7) (.1.9), we have 1 1 t exp N N. 1 exp 1 1 N N Smplfyg ad gorg terms of order greater tha two, we have 1 1 t 1 N N N N 1 1. (.1.10) N N Squarg both sdes of (.1.10) ad takg expectato, we have MSE t E t E E 1 1 N N 1 E. (.1.11) N N sg (.1.8) (.1.11) ad smplfyg, we have

7 Azeem ad Haf MSE t C C C C 1 C C C C S 1 S S 1 S, (.1.1) where 1. N N Dfferetatg (.1.1) wth respect to ad equatg to zero, the optmum value of s gve by opt CC C C (say). (.1.13) 0 S S S S sg (.1.13) (.1.1), the optmum mea square error of 1 MSEopt t C 0C 0CC 1 C 0C 0C C S 1 S S 1 S. (.1.1) EFFICIENC COMPARISON 5.1 Proposed Estmator vs. Hase ad Hurwtz (196) Estmator The proposed estmator s more effcet tha the Hase ad Hurwtz (196) estmator f 0 ar y MSE t. (5.1.1) sg (3.1.5) ad (.1.1) (5.1.1), we have S S 1 S S C C C C 1 C C C C S 1 S S 1 S 0, t s:

8 66 O estmato of populato mea the presece of measuremet error or f 1 S S 1 S S C C C C 0. (5.1.) (5.1.) s true f ad ad From (5.1.3a), we have 0, (5.1.3a), (5.1.a) 1 C S. (5.1.b) 1 C S 5. Proposed Estmator vs. Cochra s (1977) Estmator The proposed estmator s more effcet tha Cochra s (1980) estmator f or f 1 S 1 S S 1 C C S CC sg (3..3) ad (.1.13) (5..1), we have. (5.1.3b) C C 0 S 1 S 0 MSE t MSE t C. (5..1) C C C C C C C C S S S S 1 C C C C 1 C C C C S 1 S S 1 S 0,

9 Azeem ad Haf 665 (5..) s true f ad 1 S S 1 1. (5..) C C S S 1 1 CC 1 0, (5..3a) 1 S S 1 1. (5..3b) C C 1 0 From (5..3a), we have 1 1 C S 1 1, (5..a) C S ad from (5..3b), we have 1 1 C S. (5..b) 1 1 C S 5.3 Proposed Estmator vs. Sgh ad Kumar (008) Estmator The proposed estmator s more effcet tha Sgh ad Kumar (008) estmator f or f 1 S S 1 1 C 1 C 0 MSE t MSE t SK sg (3.3.3) ad (.1.13) (5.3.1), we have S 1 S 0,. (5.3.1) C C C C C C C C S S S S 1 C C C C 1 S 1 S C C C C

10 666 O estmato of populato mea the presece of measuremet error (5.3.) s true f ad 1 S S 1 1 C 1 C 16 1 S S 1 1. (5.3.) C C S S 1 1 CC 1 0, (5.3.3a) 16 1 S S 1 1. (5.3.3b) C C 1 0 From (5.3.3a), we have 1 C S 1 1, (5.3.a) C S ad from (5.3.3b), we have 1 1 C S 1 1. (5.3.b) C S 6. NMERICAL STD We were uable to fd real lfe data from ay populato. Due to ths lmtato ad for the sake of emprcal comparso of proposed estmator topt wth Hase ad * Hurwtz (196) estmator y, Cochra (1977) rato estmator t C ad Sgh ad Kumar (008) estmator t SK, we geerated artfcal populatos. Hase ad Hurwtz (196), Cochra (1977) ad Sgh ad Kumar (008) have ot cosdered measuremet error whle suggestg ther estmators. For comparso purpose we have derved ther MSE ad provded above. It s obvous that creasg the orespose rate ad decreasg sze of re-cotractg sample from o-respodets results crease MSE of a estmator. So we oly cosdered oe case of orespose rate ad re-cotractg sample sze. We have geerated sx populatos whch the correlato coeffcet s low, hgh, postve ad egatve. The populato sze for each populato s assumed to be 5000; respodet s populato sze s 3750 ad sample sze 500. It s assumed that 70% are respodets ad further rec-cotactg sample sze s assumed to be 50% of orespodets (30%) of ma sample. The true auxlary varable s assumed such that N 10,. Further the measured auxlary varable s assumed such that 0,1 x N the x. The study varable s the geerated by a lear model

11 Azeem ad Haf 667 b N 0,1. The value of b s chaged to cotrol the correlato betwee study ad auxlary varable. The assumed values are 0.1, 0.3, 0.5, -0.1, -.03 ad Smlar to y N 0,1 the auxlary varable the measured study varable s geerated by y. The requred parameters for MSE s ad values of MSE of proposed ad other three estmators are gve the followg table 1. The last three rows of the table 1 cotas percet relatve effcecy (PRE) of Cochra (1977), Sgh ad Kumar (008) ad Proposed estmator w.r.t Hase ad Hurwtz (196). Table 1 Parameters of Populato ad Resultg MSEs ad PRE of Estmators Parameters Pop-1 Pop- Pop-3 Pop- Pop-5 Pop S S S S S S S S * MSE y C MSE t SK MSE t MSE t opt C PRE t 88% 87% 77% 9% 95% 93% SK PRE t 73% 60% % 88% 97% 91% PRE t opt 101% 10% 9% 98% 100% 96%

12 668 O estmato of populato mea the presece of measuremet error We also coducted smulato study cosderg above sx populatos. For ths purpose, we selected 5000 samples from each populato ad absolute bas, percet relatve bas (PRB), emprcal mea square error (EMSE) ad emprcal percet relatve effcecy (EPRE) for each estmator are computed. The results are gve the followg table. Table Absolute Bas, PRB ad Emprcal MSE ad EPRE Parameters Pop-1 Pop- Pop-3 Pop- Pop-5 Pop-6 * Bas y C Bas t SK Bas t Bas t opt C PRB t 106% 17% 67% 750% 387% 37% SK PRB t 11% 658% 39% 18% 6% 3% PRB t opt 10% 10% 98% 19% 5% 38% * EMSE y C EMSE t SK EMSE t EMSE t opt C EPRE t 93% 8% 8% 9% 9% 93% SK EPRE t 69% 9% 38% 85% 97% 88% EPRE t opt 10% 96% 9% 99% 99% 95% 7. CONCLSION O the bass of percet relatve effcecy gve table 1, t s cocluded that for all sx populatos the proposed estmator s better tha other three estmators. Table depcts that proposed estmator s less based as compared to other two based estmators. Also the proposed estmator has more emprcal PRE tha other three estmators.

13 Azeem ad Haf 669 ACKNOWLEDGEMENT The authors are thakful to Dr. Zahoor Ahmad, versty of Southampto, K for help durg the emprcal ad smulato study. REFERENCES 1. Alle, J., Sgh, H.P. ad Smaradache, F. (003). A famly of estmators of populato mea usg mult-auxlary formato presece of measuremet errors. It. J. Soc. Eco., 30(7), Cochra, W.G. (1968). Errors of measuremet statstcs. Techometrcs, 10(), Cochra, W.G. (1977). Samplg Techques, 3 rd Edto. New ork: Joh Wley & Sos, Ic.. Fuller, W.A. (1995). Estmato the presece of measuremet error. It. Statst. Revew, 63(), Hase, M.H. ad Hurwtz, W.N. (196). The problem of o-respose sample surveys.. J. Amer. Statst. Assoc., 1, Khare, B.B. ad Srvastava, S. (1993). Estmato of populato mea usg auxlary character presece of o-respose. Nat. Acad. Sc. Lett. Ida, 16, Khare, B.B. ad Srvastava, S. (1995). Study of covetoal ad alteratve twophase samplg rato, product ad regresso estmators presece of o-respose. Proc Ida Nat. Sc. Acad., 65, Khare, B.B. ad Srvastava, S. (1997). Trasformed rato type estmators for the populato mea presece of o-respose. Comm. Statst. Theory Methods, 6, Khare, B.B. ad Srvastava, S. (010). Geeralzed two phase estmators for the populato mea the presece of o-respose. Algarh J. Statst., 30, Kumar, M., Sgh, R., Sgh, A.K. ad Smaradache, F. (011). Some rato type estmators uder measuremet errors. World Appled Sceces Joural, 1(), Kumar, S. (01). Rato cum regresso estmator for estmatg a populato mea wth a sub samplg of o-respodets. Commucatos of the Korea Statstcal Socety, 19(5), Kumar, S. (01). Estmato of the rato, product ad mea usg mult auxlary varables the presece of o-respose. Chlea Joural of Statstcs, 5(1), Masha ad Sgh, R.K. (001). A estmato of populato mea the presece of measuremet errors. J. Id. Soc. Agr. Statst., 5(1), Masha ad Sgh, R.K. (00). Role of regresso estmator volvg measuremet errors. Brazla J. Probablty Statstcs, 16, Rao, P.S.R.S. (1986). Rato estmato wth sub samplg the o-respodets. Survey Methodology, 1(), Rao, P.S.R.S. (1987). Rato ad Regresso estmates wth sub samplg of orespodets. Paper preseted at a specal cotrbuted sesso of the teratoal statstcal assocato Meetgs Sept., -16, Tokyo, Japa.

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