Joural of Moder Applied Statistical Methods Volume Issue Article 3 --03 A Geeralized Class of Estimators for Fiite Populatio Variace i Presece of Measuremet Errors Praas Sharma Baaras Hidu Uiversit, Varaasi, Idia, praassharma0@gmail.com Rajesh Sigh Baaras Hidu Uiversit, Varaasi, Idia, rsighstat@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 Sharma, Praas ad Sigh, Rajesh (03) "A Geeralized Class of Estimators for Fiite Populatio Variace i Presece of Measuremet Errors," Joural of Moder Applied Statistical Methods: Vol. : Iss., Article 3. DOI: 0.37/jmasm/383790 Available at: http://digitalcommos.wae.edu/jmasm/vol/iss/3 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.
Joural of Moder Applied Statistical Methods November 03, Vol., No., 3-. Copright 03 JMASM, Ic. ISSN 538 97 A Geeralized Class of Estimators for Fiite Populatio Variace i Presece of Measuremet Errors Praas Sharma Baaras Hidu Uiversit Varaasi, Idia Rajesh Sigh Baaras Hidu Uiversit Varaasi, Idia The problem of estimatig the populatio variace is preseted usig auiliar iformatio i the presece of measuremet errors. The estimators i this article use auiliar iformatio to improve efficiec ad assume that measuremet error is preset both i stud ad auiliar variable. A umerical stud is carried out to compare the performace of the proposed estimator with other estimators ad the variace per uit estimator i the presece of measuremet errors. Kewords: Populatio mea, stud variate, auiliar variates, mea squared error, measuremet errors, efficiec. Itroductio Over the past several decades, statisticias are paig their attetio towards the problem of estimatio of parameters i the presece of measuremet errors. I surve samplig, the properties of 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 o-respose errors, 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 ofte lead to uepected, udesirable ad ufortuate cosequeces (see Srivastava ad Shalabh, 00). Some importat sources of measuremet errors i Praas Sharma is a Research Fellow i the Departmet of Statistics. Email him at praassharma0@gmail.com. Rajesh Sigh is Assistat professor i the Departmet of Statistics. Email at: rsighstat@gmail.com. 3
A CLASS OF ESTIMATORS FOR FINITE POPULATION VARIANCE surve data are discussed i Cochra (968), Shalabh (997), ad Sud ad Srivastva (000). Sigh ad Karpe (008, 00), Kumar et al. (0a, b) studied some estimators of populatio mea uder measuremet error. Ma authors, icludig Das ad Tripathi (978), Srivastava ad Jhajj (980), Sigh ad Karpe (009) ad Diaa ad Giorda (0), studied the estimatio of populatio Variace of the stud variable usig auiliar iformatio i the presece of measuremet errors. The problem of estimatig the populatio variace ad its properties are studied here i the presece of measuremet errors. 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 a set of paired observatios are obtaied through simple radom samplig procedure o two characteristics X ad Y. Further assume that i ad i for the i th samplig uits are observed with measuremet error as opposed to 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 () v X () i i i where u i ad v i are associated measuremet errors which are stochastic i ature with mea zero ad variaces u ad v, respectivel. Further, let the u i s ad 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)). Notatios Let i, i, be the ubiased estimator of populatio meas X i i ad Y, respectivel but s ( ) i ad s ( ) i are ot i i ubiased estimator of (, ), respectivel. The epected values of s ad s i the presece of measuremet error are, give b, 3
SHARMA & SINGH ( ) E s + v ( ) E s + u Whe the error variace v is kow, the ubiased estimator of, is ˆ s v > 0, ad whe u is kow, the the ubiased estimator of is ˆ s u 0 Defie >. ( 0 ) ( e ) ˆ + e µ + such that E( e 0 ) ( ) E e 0, Ee ( ) C v + C θ, ad to the first degree of approimatio (whe fiite populatio correctio factor is igored) A 0, ( ) E e where, E( ee) 0 λc. A θ u u γ + γu + + + u µ ( ) ( ) EY i µ µ (, ) λ, C, µ, θ + v, γ ( ) 3 β, ( ) 3 µ ( u) µ ( ) γu β u, β ( u), β ( ), µ ( u) µ ( ), ( ) ( ) u Eu i µ., 33
A CLASS OF ESTIMATORS FOR FINITE POPULATION VARIANCE θ ad θ are the reliabilit ratios of X ad Y, respectivel, lig betwee 0 ad. Estimator of populatio variace uder measuremet error Accordig to Koucu ad Kadilar (00), a regressio tpe estimator t is defied as t w ˆ + w ( µ ) (3) where w ad w are costats that have o restrictio. Epressio (3) ca be writte as t ( w ) + w e w µ e () 0 Takig epectatio both sides of (), results i Bias( t ) ( w ) (5) Squarig both sides of () ( ) ( ) t w + w e0 wµ e (6) or ( ) t ( w ) + w e + w µ e + ( w ) w e 0 0 ( w ) w µ e ww µ e0e) (7) Simplifig equatio (7), takig epectatios ad usig otatios, results i the mea square error of t up to first order of approimatio, as A C ww µ λc MSE( t) w( + ) + ( w) + w µ θ (8) 3
SHARMA & SINGH I the case, whe the measuremet error is zero, MSE of t without measuremet error is give b, * C C MSE ( t) { γ + + } + ( w) + w µ ww µ λ (9) ad M u u u C v t γ u + w + + µ (0) is the cotributio of measuremet errors i the MSE of estimator t. Differetiatig (8) with respect to w ad w partiall, equatig them to zero ad after simplificatio, results i the optimum values of w ad w, respectivel as w B C * *, w C AB C AB () A µ C µ Cλ where, A ( + ), B ad C. θ * * Usig the values of ω ad ω from equatio () ito equatio (8), gives the miimum MSE of the estimator t i terms of A, B ad C as ( C AB) MSE( t) mi 3BC AB BC + () ( C AB) Aother estimator uder measuremet error Based o Solaki ad Sigh (0), a estimator t 3 is defied as 35
A CLASS OF ESTIMATORS FOR FINITE POPULATION VARIANCE t ( ) ( + ) α β µ ˆ - ep µ µ (3) where α ad β are suitabl chose costats. Epressig the estimator t, i terms of e s is t ˆ ( e) α ep ( βe ) e + + () Epadig equatio () ad simplifig results i k e ( t ) e0 ( e+ ee 0 ) ( k k) 8 (5) where k ( β α) +. O takig epectatios of both sides of (5), the bias of the estimator t 3 up to the first order of approimatio is obtaied as Bias t ( ) λc k - 8 k k C θ (6) Squarig both sides of (5) ad after simplificatio, k t e0 + e ke0e ( ) (7) Takig epectatios of (7) ad usig otatios, the MSE of estimator t is calculated as MSE( t) A + C k C θ k θ λ θ (8) 36
SHARMA & SINGH Differetiatig equatio (8) with respect to k ad equatig to zero ad after simplificatio the optimum value of k is k * λθ (9) C Puttig the optimum value of k from (9) to (8), results i the miimum MSE of estimator t as MSE( t) mi A λθ (0) Remark: Sigh ad Karpe (009) defied a class of estimator for as t d ˆ db ( ) () where, d(b) is a fuctio of b such that d(), ad certai other coditios, similar to those give i Srivastava (97). The miimum MSE of t d is give b, MSE( t ) mi d A λθ () which is the same as the miimum MSE of estimator t, give i equatio (0). A Geeral Class of Estimators A geeral class of estimator t 3 is proposed as ( ) ( ) α β µ t ˆ 3 m + m( µ ) - ep µ + µ (3) Where m ad m are costats chose so as to miimize the mea squared error of the estimator t 3. Equatio (3) ca be epressed i terms of e s as 37
A CLASS OF ESTIMATORS FOR FINITE POPULATION VARIANCE ( k k) k t3 m + m e0 mµ e e e 8 () Epadig equatio () ad subtractig from both sides, results i k t m m e + m e m µ e ( ) ( ) 3 0 e mk k m( k k) ee 0 + mµ e 8 (5) O takig epectatios of both sides of (5) the bias of the estimator t 3 up to the first order approimatio is obtaied as mk Bias t m m k k + m (6) C λc k C 3 µ 8 θ θ ( ) ( ) ( ) Squarig both sides of (5), results i k t3 m m e+ m e0 mµ e ( ) ( ) (7) Simplifig equatio (7) ad takig epectatios both sides the MSE of estimator t 3 up to the first order of approimatio is obtaied as ( ) MSE( t3) m + mp + mq mm R (8) where A kc k P C θ λ + +, µ C Q ad θ C R k + λc θ µ. Miimizig MSE t 3 with respect to m ad m the optimum values of m ad m is 38
SHARMA & SINGH Q * R * m ad m R PQ R PQ Puttig the optimum values of m ad m i equatio (8) results i the miimum MSE of estimator t 3 as MSE t 3 ( ) Q ( PQ R ) (9) Empirical Stud Data Statistics: The data used for empirical stud was take from Gujrati ad Sageetha (007) - pg, 539., where, Y i True cosumptio epediture, X i True icome, i Measured cosumptio epediture, Measured icome. i From the data give we get the followig parameter values: Table. Parameter values from empirical data N µ µ ρ 0 7 70 78 3300 0.96 36.0 36.0 u v Table. Showig the MSE of the estimators with ad without measuremet errors Estimators MSE without meas. Error Cotributio of meas. Errors i MSE MSE with meas. Errors ˆ 5670 3558 88 t 973 3035 60088 39
A CLASS OF ESTIMATORS FOR FINITE POPULATION VARIANCE Table cotiued. Estimators MSE without meas. Error Cotributio of meas. Errors i MSE MSE with meas. Errors t mi 5 356 8087 t ( α, β 0) 70 30 7786 3mi ( α 0, β ) ( α, β ) ( α, β ) ( α 0, β ) ( α 0.9, β ) 30 30555 67957 68 309 98363 30 33555 67957 3969 30600 6569 95 30365 5950 Coclusio Table shows that the MSE of proposed estimator t 3 (for α 0.9, β ) is miimum amog all other estimators cosidered. It is also observed that the effect due to measuremet error o the estimator t ad usual estimators is less tha the effect o the estimator t uder measuremet error for this give data set. Refereces Alle, J., Sigh, H. P., & Smaradache, F. (003). A famil of estimators Of populatio mea usig multi auiliar iformatio i presece of measuremet errors. Iteratioal Joural of Social Ecoomics 30(7), 837 89. Cochra, W. G. (968). Errors of Measuremet i statistics. Techometrics 0, 637-666 Das, A. K., & Tripathi, T. P. (978). Use of auiliar iformatio i estimatig the Fiite populatio variace. Sakha C, 39-8 Diaa, G., & Giorda, M. (0). Fiite Populatio Variace Estimatio i Presece of Measuremet Errors. Commuicatio i Statistics Theor ad Methods,, 30-3. Gujarati, D. N., & Sageetha (007). Basic ecoometrics. McGraw Hill. Koucu, N., & Kadilar, C. (00). O the famil of estimators of Populatio mea i stratified samplig.pakista Joural of Statistics, 6, 7-3. 0
SHARMA & SINGH Kumar, M., Sigh, R., Sigh, A. K., & Smaradache, F. (0a). Some ratio Tpe estimators uder measuremet errors. World Applied Scieces Joural, (), 7-76. Kumar, M., Sigh, R., Sawa, N., & Chauha, P. (0b). Epoetial ratio method Of estimators i the presece of measuremet errors. Iteratioal Joural of Agricultural ad Statistical Scieces 7(), 57-6. Maisha, M., & Sigh, R. K. (00). Role of regressio estimator ivolvig Measuremet errors. Brazilia Joural of Probabilit ad Statistics 6, 39-6. Shalabh. (997). Ratio method of estimatio i the presece of measuremet errors. Joural of Idia Societ of Agricultural Statistics 50(), 50 55. Sigh, H. P. & Karpe, N. (008). Ratio product estimator for populatio mea i presece of measuremet errors. Joural of Applied Statistical Scieces, 6(), 9-6. Sigh, H. P. & Karpe, N. (009). Class of estimators usig auiliar Iformatio for estimatig fiite populatio variace i presece of measuremet errors. Commuicatio i Statistics Theor ad Methods, 38, 73-7. Sigh, H. P. & 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(), 338-0. Solaki R., Sigh H.P., & Rathour A. (0). A alterative estimator for estimatig the fiite populatio mea usig auiliar iformatio i sample surves. ISRN Probabilit ad Statistics, doi:0.50/0/65768. Srivastava, M. S. (97). O Fied-Width Cofidece Bouds for Regressio Parameters, Aals of Mathematical Statistics,, 03-. Srivastava, A., K., & Shalabh. (00). Effect of Measuremet Errors O the Regressio Method of Estimatio i Surve Samplig. Joural of Statistical Research, 35(), 35-. Srivastava, S. K., & Jhajj, H.S. (980) A class of estimators usig auiliar iformatio for estimatig fiite populatio variace. Sakha Ser. C, 87-96. Sud, U. C., & Srivastava, S. K. (000). Estimatio of populatio mea i repeat surves i the presece of measuremet errors. Joural of the Idia Societ of Agricultural Statistics, 53(), 5-33.