Joural of Moder Applied Statistical Methods Volume 4 Issue Article 5--05 Method of Estimatio i the Presece of Norespose ad Measuremet Errors Simultaeousl Rajesh Sigh Sigh Baaras Hidu Uiversit, Varaasi, Idia, rsighstat@gmail.com Praas Sharma Baaras Hidu Uiversit, Varaasi, Idia, praassharma0@gmail.com Follow this ad additioal works at: http://digitalcommos.wae.edu/jmasm Part of the Applied Statistics Commos, Social ad Behavioral Scieces Commos, ad the Statistical Theor Commos Recommeded Citatio Sigh, Rajesh Sigh ad Sharma, Praas (05) "Method of Estimatio i the Presece of No-respose ad Measuremet Errors Simultaeousl," Joural of Moder Applied Statistical Methods: Vol. 4 : Iss., Article. DOI: 0.37/jmasm/430453460 Available at: http://digitalcommos.wae.edu/jmasm/vol4/iss/ This Regular Article is brought to ou for free ad ope access b the Ope Access Jourals at DigitalCommos@WaeState. It has bee accepted for iclusio i Joural of Moder Applied Statistical Methods b a authorized editor of DigitalCommos@WaeState.
Method of Estimatio i the Presece of No-respose ad Measuremet Errors Simultaeousl Cover Page Footote The authors are ver idebted to the aomous referees for their valuable suggestios leadig to improvemet of the qualit of cotets ad presetatio of the origial mauscript. This regular article is available i Joural of Moder Applied Statistical Methods: http://digitalcommos.wae.edu/jmasm/vol4/ iss/
Joural of Moder Applied Statistical Methods Ma 05, Vol. 4, No., 07-. Copright 05 JMASM, Ic. ISSN 538 947 Method of Estimatio i the Presece of No-respose ad Measuremet Errors Simultaeousl Rajesh Sigh Sigh Baaras Hidu Uiversit Varaasi, Idia Praas Sharma Baaras Hidu Uiversit Varaasi, Idia The problem of estimatig the fiite populatio mea of i simple radom samplig i the presece of o-respose ad respose error was cosidered. The estimators use auiliar iformatio to improve efficiec, assumig o respose ad measuremet error are preset i both the stud ad auiliar variables. A class of estimators was proposed ad its properties studied i the simultaeous presece of o-respose ad respose errors. It was show that the proposed class of estimators is more efficiet tha the usual ubiased estimator, ratio ad product estimators uder o-respose ad respose error together. A umerical stud was carried out to compare its performace. Kewords: Populatio mea, Stud variable, Auiliar variable, Mea squared error, Measuremet errors, No-respose. Itroductio Over the past several decades, statisticias were iterested i the problem of estimatig the parameters of iterest i the presece of respose error (measuremet errors). I surve samplig the properties of the estimators based o data usuall presuppose that the observatios are the correct measuremets o characteristics beig studied. However, this assumptio is ot satisfied i ma applicatios ad data is cotamiated with measuremet errors, such as reportig errors ad computig errors. These measuremet errors make the result ivalid, which are meat for o measuremet error case. If measuremet errors are ver small ad we ca eglect it, the the statistical ifereces based o observed data cotiue to remai valid. O the cotrar, whe the are ot appreciabl small ad egligible, the ifereces ma ot be simpl ivalid ad iaccurate but ma Rajesh Sigh is Assistat Professor i Departmet of Statistics. Email him at rsighstat@gmail.com. Praas Sharma is Facult of Sciece i the Departmet of Statistics. Email at praassharma0@gmail.com. 07
SINGH & SHARMA ofte lead to uepected, udesirable ad ufortuate cosequeces (see Srivastava & Shalabh 00). Some importat sources of measuremet errors i surve data are discussed i Cochra (968), Shalabh (997), Sud ad Srivastva (000). Sigh ad Karpe (008, 00), Kumar, Sigh, ad Smaradache (0), Kumar, Sigh, Sawa, ad Chauha (0) ad Sharma ad Sigh (03) studied the properties of some estimators of populatio parameters uder measuremet error. Cosider a fiite populatio U = (U, U,..., U N ) of N uits. Let Y ad X be the stud variate ad auiliar variate, respectivel. Suppose that we have a set of paired observatios obtaied through simple radom samplig procedure o two characteristics X ad Y. Further it is assumed that i ad i for the i th samplig uits are observed with measuremet error istead of their true values (X i, Y i ). For a simple radom samplig scheme, let ( i, i ) be observed values istead of the true values (X i, Y i ) for i th (i =.,, ) uit, as ui i Yi () vi i Xi () where u i ad v i are associated measuremet errors which are stochastic i ature with mea zero ad variaces ad respectivel. Further, let the u i s ad u, v v i s are ucorrelated although X i s ad Y i s are correlated. Let the populatio meas of X ad Y characteristics be μ ad μ, populatio variaces of (, ) be (, ) ad let ρ be the populatio correlatio coefficiet betwee ad respectivel (see Maisha ad Sigh 00). I sample surves, the problem of o-respose is commo ad is more widespread i mail surves tha i persoal iterviews. The usual approach to overcome o-respose problem is to cotact the o-respodet ad obtai the iformatio as much as possible. Hase ad Hurwitz (946) were the first to deal with the problem of o-respose. The proposed a samplig scheme that ivolves takig a subsample of o-respods after the first mail attempt ad the obtai the iformatio b persoal iterview. For a fiite populatio U = {U, U,, U N } of size N ad a radom sample of size is draw without replacemet. Let the characteristics uder stud, sa, takes value i o the uit U i (I =,,, N). I surve o huma populatio it is ofte the case that uit respod o the first attempt while (= - ) uits do ot provide a respose. I the case of o-respose of at iitial stage Hase 08
ESTIMATION WITH NON-RESPONSE AND MEASUREMENT ERRORS ad Hurwitz (946) suggested a double samplig pla for estimatig the populatio mea comprisig the followig steps: i. A simple radom sample of size is draw ad the questioaire is mailed to the sample uits; ii. A sub-sample of size r = ( / k), (k > ) from the o respodig uits i the iitial step attempt is cotacted through persoal iterviews. Note that Hase ad Hurwitz (946) cosidered the mail surves at the first attempt ad the persoal iterviews at the secod attempt. I the Hase ad Hurwitz method the populatio is supposed to be cosistig of respose stratum of size N ad the o-respose stratum of size N = (N - N ). Let ad S Y N N Y i N i N i deote the mea ad the populatio variace of the i stud variable. Let N ad S Y N Y N i i mea ad variace of respose group. Similarl, let N i N i deote the i N Y N i i S i Y N deote the mea ad variace of the o-respose group. The populatio mea ca be writte as Y WY WY, where W = (N / N) ad W = (N / N). The sample mea for Y, but has a bias equal to W Y Y The sample mea r r i i i i ad is a ubiased i estimatig the populatio mea Y. r is ubiased for the mea for the uits. Hase ad Hurwitz (946) suggested a ubiased estimator for the populatio mea Y is give b w w. r Where w = ( / ) ad w = ( / ) are respodig ad o-respodig proportios i the sample. The variace of is give b f W k V S S ; where f = ( / N). 09
SINGH & SHARMA I the samplig literature, it is kow that efficiec of the estimator of populatio mea of a stud variable ca be icreased b the use of auiliar iformatio related to which is highl correlated with stud variable. Cochra (977) suggested the ratio ad regressio estimator of the populatio mea Y of stud variable i which iformatio o the auiliar variable is obtaied from all sample uits, ad the populatio mea of auiliar variable is kow, while some uits do ot provide a iformatio o stud variable. Rao (986), Khare ad Srivastava (995,997), Okafor ad Lee (000) ad Sigh ad Kumar (008, 009, 00) have suggested some estimator for populatio mea of the stud variable usig auiliar iformatio i presece of o-respose. Let i, (i =,,, N) deote a auiliar characteristics correlated with the stud variable i, (i =,,, N) the populatio mea of auiliar variable is N i. Let i X N o-respose groups. Let X ad X deote the populatio meas of the respose ad r deote the,, r i i r i i i i meas of the respodig uits, o-respodig uits, ad r = ( / k) subsampled uits respectivel. I this paper we have merged two major cocepts for improvemet of estimatio techiques that is cosideratio of measuremet error ad o-respose i the estimatio procedure ad proposed a class of estimators. Notatios Let, i, be the ubiased estimator of populatio meas X i i i ad Y, respectivel but s i ad s i ubiased estimator of (, ), respectivel. The epected values of i the presece of measuremet error are, give b, E s ad for o-respose group E s v u are ot i s ad s i 0
ESTIMATION WITH NON-RESPONSE AND MEASUREMENT ERRORS Whe the error variace ˆ s v 0 ˆ s u 0, ad whe. E s E s v v. u is kow, the ubiased estimator of, is u is kow, the the ubiased estimator of is Similarl, for the o-respose group the ubiased estimator of, is ˆ s 0, ad whe u is kow, the the ubiased estimator of is ˆ 0. v s u Defie such that E s E s v. u e 0 e E e E e 0 0, ad up to the first degree of approimatio (whe fiite populatio correctio factor is igored)
SINGH & SHARMA E e S u u 0 C S S C W k S C S W k S v v E e C S S CC W k E e0e C C C S Y, C S X, C S Y, C S X, S S S Adapted estimator A traditioal estimator for estimatig populatio mea i the simultaeous presece of respose ad o-respose error is give b, t (3) Epressio (3) ca be writte as 0 t Y Y e (4) Takig epectatio both sides of (4), we get bias of estimator t give as Squarig both sides of (4) Bias t 0 (5) t Y Y e0 (6) ad takig epectatio ad usig otatio, mea square error of t is obtaied up to first order of approimatio, as MSE t u u AS S S S (7)
ESTIMATION WITH NON-RESPONSE AND MEASUREMENT ERRORS or kw MSE t M (8) S u u where, A ad M AS. S S I the case, whe the measuremet error is zero or egligible, MSE of estimator t is give b, S MSE t AS (9) u where, Mt A u is the cotributio of measuremet errors i t. Whe there is o-respose ad respose error both are preset, a ratio tpe estimator for estimatig populatio mea is, give b t r X (0) Epressig the estimator t r i terms of e s Epadig equatio () ad simplifig, r t Y e e () 0 tr Y Y e0 e e0e e () ad takig epectatio both sides of (), the bias of estimator t r is S v v Bias tr AS SS A SS S S (3) Squarig both sides of (), 3
SINGH & SHARMA r t Y e0 e e0e (4) Takig epectatios of (4) ad usig otatios, we get the MSE of estimator t r as u v MSE tr S S SS S S AS S S S u v S S (5) S u u S v AS S S S v AS SS A SS S M N O (6) where, u u M S AS S S v v N S AS S S O SS A SS. A regressio estimator uder measuremet error ad o-respose is defied as t b X Epressig the estimator t r i terms of e s, ad epadig equatio (7) ad simplifig, lr (7) t Y e bxe, lr 0 4
ESTIMATION WITH NON-RESPONSE AND MEASUREMENT ERRORS tlr Y Ye0 bxe Squarig both sides of (8) ad after simplificatio, (8) lr t Y Y e b X e bxye e 0 0 (9) Takig epectatios both sides of (9) the MSE of estimator t lr is obtaied as MSE t M b R N bro (0) lr The optimum value of b is obtaied b miimizig (0) ad is give b O b R N () Substitutig the optimal value of b i equatio (0), the miimum MSE of the estimator t lr is obtaied as MSE t lr M mi O MN () I the case, whe the measuremet error is zero or egligible, MSE of estimator t is give b kw MSE tlr S S b S b SS (3) Proposed class of estimator A proposed class of estimators give b t p m m X (4) 5
SINGH & SHARMA Note for (m, m ) = (, 0) t (usual ubiased estimator), ad for (m, m ) = (0, ) t X (usual ratio estimator). Thus, the proposed class of estimators is geeralized versio of usual ubiased estimator ad ratio estimator. Epressig the estimator t p i terms of e s Epadig equatio (5) ad simplifig, p t my e m Y e e (5) 0 0 t p Y Y e0 m e e e0e (6) Squarig both sides of (6) ad after simplificatio, p t Y Y e m e m e e 0 0 (7) Takig epectatios of (7) ad usig otatios, the MSE of estimator t r is obtaied as p MSE t M m R N m RO (8) The optimum value of m is obtaied b miimizig (8), give b m O R N (9) ad m m. Substitutig the optimal value of m i equatio (8) the miimum MSE of the estimator t p is obtaied as MSE t p mi O M MN (30) 6
ESTIMATION WITH NON-RESPONSE AND MEASUREMENT ERRORS The miimum MSE of proposed class of estimator t p give i (30) is same as the MSE of regressio estimator uder simultaeous presece of o-respose ad measuremet error, give i equatio (). Efficiec comparisos First, the efficiec of the proposed estimator t p is compared with usual ubiased estimator, MSE t MSE t P mi 0 O O If M M 0, 0 MN MN (3) The coditio listed i (3) shows that proposed famil of estimators is alwas better tha the usual estimator uder the o-respose ad measuremet error. Net, the ratio estimator is compared with proposed famil of estimators t p, O MSE t MSE t 0, 0 mi P M N O M mi MN N O 0 (3) Observe that the coditio (3) holds ad shows proposed famil of estimators is better tha the ratio estimator uder the o-respose ad measuremet error. Empirical stud Data statistics The data used for empirical stud was take from Gujarati ad Sageetha (007, pg, 539) where, Y i = True cosumptio epediture, X i = True icome, 7
SINGH & SHARMA i = Measured cosumptio epediture, i = Measured icome. From the data give we get the followig parameter values: Table. Value of the parameters μ μ S S ρ 70 98.9 755.53 63.66 406.3 0.778 36.00 36.00 μ μ S S ρ R W 597.9 00.4 44. 63.5 0.445 0.5589 0.5 u v Table. Showig the MSE of the estimators with ad without measuremet errors Estimators MSE Without Error Cotributio of meas. error i MSE Cotributio of orespose MSE icludig me. Errors & o-respose t 0759.39.03 553.840 333.58 t r 6967.35.35 4607.335 574.9 t lr 446.903 0.86 57.75 6775.036 t p 446.903 0.86 57.75 6775.036 Table ehibits that measuremet error ad o-respose plas a importat role i icreasig the MSE of a estimator. We also coclude that cotributio of measuremet error ad o-respose i usual estimator is less tha i compariso to the ratio estimator; these observatios have iterestig implicatio where the ratio estimator performs better tha sample mea uder the absece of a measuremet error i X characteristics. There ma be a case whe ratio estimator is poor tha sample mea uder the cosideratio of a measuremet error. It is observed from Table that the performace of our proposed estimator t p is better tha usual estimator t ad ratio estimator t r uder o-respose ad measuremet error. Further it is observed that cotributio of o-respose error is larger tha the respose error i icreasig the MSE of the estimators. 8
ESTIMATION WITH NON-RESPONSE AND MEASUREMENT ERRORS Coclusio A class of estimator of the populatio mea of stud variable was proposed usig auiliar iformatio. The estimators use auiliar iformatio to improve efficiecies, assumig o-respose ad measuremet error are preset i both the stud ad auiliar variables. I additio, some kow estimator of populatio mea such as usual ubiased estimator ad ratio estimator for populatio mea are foud to be members of the proposed class of estimators. The MSEs of the proposed class of estimators were obtaied up to the first order of approimatio i the simultaeous presece of o-respose ad respose error. The proposed class of estimators are advatageous i the sese that the properties of the estimators which are members of the proposed class of estimators ca be easil obtaied from the properties of the proposed class of estimators. Refereces Alle, J., Sigh, H. P. & Smaradache, F. (003). A famil of estimators of populatio mea usig multiauiliar iformatio i presece of measuremet errors. Iteratioal Joural of Social Ecoomics, 30(7), 837 849. doi:0.08/030689030478775 Cochra, W. G. (968). Errors of measuremet i statistics. Techometrics, 0(4), 637-666. doi:0.080/0040706.968.04906 Cochra, W. G. (977). Samplig Techiques (3rd ed.). New York: Joh Wile ad Sos. Das, A. K. & Tripathi, T. P. (978). Use of auiliar iformatio i estimatig the fiite populatio variace. Sakha, 40(C), 39-48. Diaa, G. ad Giorda, M. (0). Fiite Populatio Variace Estimatio i Presece of Measuremet Errors. Commuicatio i Statistics-Theor ad Methods, 4(3), 430-434. doi:0.080/036096.0.57365 Gujarati, D. N. & Sageetha (007). Basic ecoometrics. Tata McGraw-Hill. Hase, M.H. & Hurwitz, W.N. (946). The problem of o-respose i sample surves. Joural of the America Statistical Associatio, 4(36), 57 59. doi:0.080/06459.946.050894 Khare, B.B. & Srivastava, S. (995). Stud of covetioal ad alterative two-phase Samplig ratio, product ad regressio estimators i presece of orespose. Proceedigs of the Idia Natioal Sciece Academ, 65, 95 03. 9
SINGH & SHARMA Khare, B. B., & Srivastava, S. (997). Trasformed ratio tpe estimators for the populatio mea i the presece of orespose. Commuicatios i Statistics - Theor ad Methods, 6(7), 779-79. doi:0.080/036099708830 Koucu, N. & Kadilar, C. (00). O the famil of estimators of populatio mea i stratified radom samplig. Pakista Joural of Statistics, 6(), 47-443. Kumar, M., Sigh, R., Sigh, A. K. & Smaradache, F. (0). Some ratio tpe estimators uder measuremet errors. World Applied Scieces Joural, 4(), 7-76. Kumar, M., Sigh, R., Sawa, N. & Chauha, P. (0). Epoetial ratio method of estimators i the presece of measuremet errors. Iteratioal Joural of Agricultural ad Statistical Scieces, 7(), 457-46. Maisha & Sigh, R. K. (00). Role of regressio estimator ivolvig measuremet errors. Brazilia Joural of Probabilit ad Statistics, 6, 39-46. Okafor, F. C. & Lee, H. (000). Double samplig for ratio ad regressio estimatio with sub-samplig the o-respodets, Surve Methodolog, 6(), 83 88. Rao, P. S. R. S. (986). Ratio estimatio with sub samplig the orespodets. Surve Methodolog, (), 7 30. Shalabh (997). Ratio method of estimatio i the presece of measuremet errors. Joural of Idia Societ of Agricultural Statistics, 50(), 50 55. Sharma, P. & Sigh, R. (03). A geeralized class of estimator for populatio variace i presece of measuremet errors. Joural of Moder Applied Statistical Methods, (), 3-4. Retrieved from http://digitalcommos.wae.edu/jmasm/vol/iss/3 Sigh, H.P. & Karpe, N. (008). Ratio-product estimator for populatio mea i presece of measuremet errors. Joural of Applied Statistical Sciece, 6(4), 437-45. Sigh, H. P. ad Karpe, N. (009). A class of estimators usig auiliar iformatio for estimatig fiite populatio variace i presece of measuremet errors. Commuicatio i Statistics - Theor ad Methods, 38(5),734-74. doi:0.080/036090809073 Sigh, H. P. ad Karpe, N. (00). Effect of measuremet errors o the separate ad combied ratio ad product estimators i stratified radom samplig. Joural of Moder Applied Statistical Methods, 9(), 338-40. Retrieved from http://digitalcommos.wae.edu/jmasm/vol9/iss/8 0
ESTIMATION WITH NON-RESPONSE AND MEASUREMENT ERRORS Sigh, H. P. & Kumar, S. (008). Estimatio of mea i presece of orespose usig two phase samplig scheme. Statistical Papers, 50, 559 58. doi:0.007/s0036-008-040-5 Sigh, H. P. & Kumar, S. (009). A geeral class of estimators of the populatio mea i surve samplig usig auiliar iformatio with sub samplig the o-respodets, The Korea Joural of Applied Statistics, (), 387 40. doi:0.535/kjas.009...387 Srivastava, A., K., & Shalabh (00). Effect of measuremet errors o the regressio method of estimatio i surve samplig. Joural of Statistical Research, 35(), 35-44. Sud, U. C. & Srivastava, A. K. (000): Estimatio of populatio mea i repeated surves i the presece of measuremet errors. Joural of the Idia Societ of Agricultural Statistics, 53(), 5-33.