ALMOST UNBIASED RATIO AND PRODUCT TYPE EXPONENTIAL ESTIMATORS

Transcription

1 STATISTIS IN TANSITION-new series, December STATISTIS IN TANSITION-new series, December 0 Vol. 3, No. 3, pp ALMOST UNBIASED ATIO AND ODUT TYE EXONENTIAL ESTIMATOS ohini Yadav, Lakshmi N. Upadhaa, Housila. Singh, S. haerjee ABSTAT This paper considers he problem of esimaing he populaion mean Y of he sud variae using informaion on auiliar variae. We have suggesed a generalized version of Bahl and Tueja (99) esimaor and is properies are sudied. I is found ha asmpoic opimum esimaor (AOE) in he proposed generalized version of Bahl and Tueja (99) esimaor is biased. In some applicaions, biasedness of an esimaor is disadvanageous. So appling he procedure of Singh and Singh (993) we derived an almos unbiased version of AOE. A numerical illusraion is given in he suppor of he presen sud. e words: sud variable, auiliar variable, almos unbiased raio-pe and produc-pe eponenial esimaors, bias, mean squared error.. Inroducion I is well known ha he use of auiliar informaion provides efficien esimaes of he populaion parameers. aio, regression and produc mehods of esimaion are good illusraions in his cone. Le here be N unis in populaion and informaion be available on auiliar variable. We draw a sample of size n using simple random sampling wihou replacemen (SSWO) scheme and on he basis of which we can esimae he populaion mean of he characer undersud. Le (, ) be he sample means of (, ) respecivel. Deparmen of Applied Mahemaics, Indian School of Mines, Dhanbad-86004, India. rohiniadav.ism@gmail.com, lnupadhaa@ahoo.com, scha_@ahoo.co.in. School of Sudies in Saisics, Vikram Universi, Ujjain , India. hpsujn@gmail.com.

2 538. Yadav, L. N. Upadhaa, H.. Singh, S. haerjee: Almos unbiased Assume ha we have informaion abou he populaion mean X of he auiliar characer and using X i is desired o esimae he populaion mean Y of he sud characer. When he correlaion beween he sud variable and he auiliar variable is posiive, he classical raio esimaor for populaion mean Y is defined b X = (.) In such a siuaion, Bahl and Tueja (99) suggesed a raio-pe eponenial esimaor for he populaion mean Y as X- =ep (.) X+ To he firs degree of approimaion, he biases and mean squared errors (MSEs) of and are respecivel given b ( ) ( ) B =f Y - (.3) B =fy ( ) ( ) ( ) ( ) (.4) MSE =f Y + - (.5) MSE ( ) =fy + ( -4) 4 (.6) where f= - n N, S S =, =, =ρ, Y X N S = ( i ) ( ) N -Y, S = ( i ) N- i= ( ) -X, N- i= N S = ( i-x)( i-y) N- ( ) and ρ is he correlaion coefficien beween and. i= I is observed from (.3) and (.4) ha he esimaor due o Bahl and Tueja (99) is less biased han he classical raio esimaor if B( ) < B( )

3 STATISTIS IN TANSITION-new series, December i.e. if ( 3-4 ) < ( - ) 8 i.e. if ( ) > 0 (.7) From (.5) and (.6) i follows ha he esimaor due o Bahl and Tueja (99) is more efficien han he classical raio esimaor if if MSE( ) < MSE( ) 3 3 < or ρ < 4 4 (.8) Murh (964) suggesed a produc-pe esimaor = (.9) X for he populaion mean Y which is useful in he siuaion where he correlaion beween he sud variable and he auiliar variable is negaive (high). In negaive correlaion siuaion, Bahl and Tueja (99) suggesed a producpe eponenial esimaor for he populaion mean Y is defined as -X = ep (.0) +X To he firs degree of approimaion, he biases and mean squared errors (MSEs) of and are respecivel given b ( ) B =f Y (.) B( ) =fy ( 4-) (.) 8 MSE ( ) =fy +( +) (.3) MSE ( ) =fy + ( +4) 4 (.4) I is observed from (.) and (.) ha he produc-pe eponenial esimaor is less biased han he usual produc esimaor B( ) < B( ) if

4 540. Yadav, L. N. Upadhaa, H.. Singh, S. haerjee: Almos unbiased i.e. if ( 4- ) < ( ) 8 i.e. if ( ) > 0 (.5) From (.3) and (.4), i follows ha he Bahl and Tueja (99) producpe eponenial esimaor is more efficien han he produc esimaor if MSE( ) < MSE( ) 3 3 > - or ρ > if (.6). Generalized version of and We define a generalized version of and as X- c = ep c (.) X+ where c is non-zero consan. For c=, c reduces o while for c=- i reduces o. To obain he bias and mean squared error (MSE) of he esimaor c up o he firs degree of approimaion, we wrie such ha E ( e 0) =E ( e ) =0, =Y( +e 0) and =X ( +e ) ( ) ( ) ( ) E e =f, E e =f and E e e =f ρ. 0 Y X 0 Y X Epressing c in erms of e s, we have -e c=y( +e0) ep c +e Now, epanding he righ hand side of he above, mulipling ou and neglecing erms of e s having power greaer han wo, we have c c c c c=y +e0- e- e0e + e + e 4 8

5 STATISTIS IN TANSITION-new series, December 0 54 or c c c c ( c-y ) =Y e0- e- e0e + e + e (.) 4 8 Taking epecaions of boh sides of (.), we ge he bias of he esimaor c up o he firs order of approimaion as c B( c) =fy ( +c-4 ) (.3) Squaring boh sides of (.) and neglecing erms of e s having power greaer han wo, we have c ( c-y ) =Y e0- e (.4) Taking epecaion of boh sides of (.4), we ge mean squared error (MSE) of he esimaor c up o he firs order of approimaion as c MSE ( c) =fy +c - 4 (.5) Differeniaing (.5) w.r.. c and equaing i o zero, we ge he opimum value of c as c= (.6) Thus, he subsiuion of opimum value c= in (.) ields he asmpoic opimum esimaor (AOE) in he class of esimaor c as X- = ep (.7) X+ The bias and mean squared error (MSE) of he esimaor are respecivel given b B =f Y - / (.8) { } ( ) Y( ) ( ) ( ) MSE =f Y -ρ (.9) I is observed from (.9) ha he MSE of he AOE is equal o he =+βˆ X-, which is approimae variance of he regression esimaor lr ( ) biased, where ˆβ is sample esimae of he populaion regression coefficien β. Epression (.8) clearl indicaes ha he AOE is a biased esimaor. So our objecive is o obain an almos unbiased esimaor for he populaion mean Y. In he following secion, we mee wih our objecive using Singh and Singh (993) approach.

6 54. Yadav, L. N. Upadhaa, H.. Singh, S. haerjee: Almos unbiased 3. Almos unbiased eponenial esimaor We consider he esimaors X- = ep X+ X- = ep 4 X+ (3.) (3.) X- 3= ep 6 (3.3) X+ such ha,, 3 H, where H denoes he se of all possible esimaors for esimaing he populaion mean Y. To he firs degree of approimaion, he biases and mean squared errors (MSEs) of he esimaors, and 3 are respecivel given b B( ) =fy ( /) ( - ). (3.4) B( ) =fy (3.5) B( ) =f Y( 3/) ( - ) (3.6) 3 ( ) ( ) MSE =f Y -ρ (3.7) MSE ( ) =fy (3.8) MSE ( ) =f Y +3 (3.9) 3 Now, considering he esimaors (3.), (3.) and (3.3), we sugges a class of eponenial esimaors for Y as 3 h = h j j H (3.0) wih j= 3 h j=, h j, (3.) j= where h j (j=,, 3) denoes he saisical consans and denoes he se of real numbers.

7 STATISTIS IN TANSITION-new series, December Epressing h in erms of e s, we have - - e e h ( 0 ) + ( 0 ) = h Y +e ep -e + h Y +e ep -e + - e +h3y ( +e0) ep -3e e e e =Y h( +e0) -e e e + h( +e0) -e + + e e 9 e +h3( +e0) -3e + + e +... e e e =Y h( +e0) -e ( -e +... ) e e + h( +e0) -e e ( -e +... ) e e 9 +h3( +e0) -3e e ( -e +... ) e e e e e =Y h +e0-e ( -e +... )-e0e e e e e +h +e0-e e ( -e +... )-e0e e e 9 e e +h3 +e0-3e e ( -e +... )-3e0e

8 544. Yadav, L. N. Upadhaa, H.. Singh, S. haerjee: Almos unbiased Neglecing he erms of e s having power greaer han wo, we have e h =Y +e 0-h ( e +e0e ) + { h+ ( h +4h +9h3 ) } or e ( h -Y) =Y e0-h ( e +e0e ) + { h+ ( h +4h +9h3 ) } (3.) (3.3) where ( h +h +3h 3) =h (a consan). (3.4) Taking epecaion of boh sides of (3.3), we ge he bias of h o he firs degree of approimaion as B( h ) =fy ( -) h+ ( h +4h +9h3 ) (3.5) Squaring boh sides of (3.3) and neglecing erms of e s having power greaer han wo, we have ( ) h -Y =Y e 0 +h e -he0e (3.6) Taking epecaion of boh sides of (3.6), we ge he MSE of h o he firs degree of approimaion as MSE ( h ) =fy +h -hkρ (3.7) Minimizing (3.7) wih respec o h, we ge he opimum value of h as h +h +3h =h= (3.8) ( ) 3 Subsiuion of (3.8) in (3.7) ields minimum MSE of h as min. MSE =f Y -ρ (3.9) ( h ) Y ( ) In order o ge unique soluion of j j=,, 3, we shall impose he linear resricion as we have onl wo equaions in hree unknowns. 3 h jb( j) = 0, (3.0) j= where B( j ) represens he bias of he h s ( ) h j esimaor.

9 STATISTIS IN TANSITION-new series, December So, we have hree equaions (3.), (3.8) and (3.0) wih hree unknowns. These can be wrien in he mari form as 3 B B B ( ) ( ) ( ) 3 h h = h 3 0 (3.) Using (3.), we ge he unique values of h j s (j=,, 3) as 3 h = h = - h 3= (3.) Using hese h j s (j=,, 3), we can remove he bias of he esimaor c up o - he erms of order o(n ). Thus, an almos unbiased eponenial esimaor for populaion mean Y is defined as X- X- X- = ep + ep 4 + ep 6 4 X+ X+ 4 X+ (3.3) ( 3+) ( -) ( - ) I can be shown o he firs degree of approimaion ha he mean squared error of is MSE =f Y -ρ (3.4) 4. Efficienc comparison ( ) ( ) Y I is well known under SSWO ha he variance of usual unbiased esimaor is Var ( ) =MSE ( ) =fy (4.)

10 546. Yadav, L. N. Upadhaa, H.. Singh, S. haerjee: Almos unbiased From (.5), (.6), (.3), (.4), (3.4) and (4.), we have (i) ( ) ( ) MSE MSE > 0 if ρ > 0 (4.) (ii) ( ) ( ) MSE MSE > 0 if ( -+) > 0 > or ρ > (4.3) (iii) ( ) ( ) if MSE MSE > 0 ( 4-4+) > 0 > or ρ > (iv) ( ) ( ) ( ++ ) > 0 MSE -MSE >0 if (4.4) > - or ρ > - (4.5) (v) ( ) ( ) MSE MSE > 0 ( ) > 0 > - or ρ > - (4.6) 5. Empirical sud To see he performance of he esimaors,,, and over, we consider wo populaion daa ses. Using he formula ( ) MSE E (., ) = 00 MSE., ( ) ( ). =,,,, and

11 STATISTIS IN TANSITION-new series, December We have compued he percen relaive efficiencies (Es) of he esimaors,,, and over and compiled in Table. The values of scalars h j s (j=,, 3) of he almos unbiased eponenial esimaor are calculaed for differen populaion daa ses and compiled in Table. The descripion of he populaions is given below: osiive correlaed variables: opulaion- I: [Source: Murh (967, pp. 8)] I consiss of 80 facories in a region, he characers and being fied capial and oupu respecivel. The variaes are defined as follows: Y: oupu X: he number of fied capial = 0.359, = , ρ = 0.943, = Negaive correlaed variables: opulaion- II: [Source: Seel and Torrie (960, pp. 8)] Y: log of leaf burn in secs, X: chlorine percenage = , = , ρ = , = -0.3 Table. ercen relaive efficiencies (Es) of he esimaors,,,, and wih respec o S. No. Esimaor opulaion I E (., ) opulaion II

12 548. Yadav, L. N. Upadhaa, H.. Singh, S. haerjee: Almos unbiased Table. Values of h j's (j=,, 3) for almos unbiased eponenial esimaor Scalars opulaion I opulaion II h h h From Table, i is observed ha he suggesed almos unbiased eponenial esimaor, is more efficien han he usual unbiased esimaor, classical raio esimaor classical produc esimaor produc-pe esimaor respecivel. and Bahl and Tueja (99) raio-pe esimaor and From Table, we can sa ha b using hese values of scalars h j's (j=,, 3), one can reduce he bias of he esimaor up o he firs degree of approimaion. 6. onclusions I is observed from (.3) and (.) ha he classical raio esimaor and he produc esimaor are biased. In some applicaions, biasedness of an esimaor is disadvanageous. So keeping his in view, firs we have suggesed a generalized version of Bahl and Tueja (99) raio-pe and produc-pe esimaors. I is observed ha he suggesed generalized esimaor is also biased. So using he echnique as adoped b Singh and Singh (993), we have suggesed an almos unbiased esimaor for he populaion mean Y wih is variance formula. From Table and Table, we have observed ha he suggesed almos unbiased eponenial esimaor is more efficien han,,,, and. We shall see ha he suggesed almos unbiased =ρ /, he value esimaor depends onl on he well known parameer ( ) Y X of which can be obained quie accurael from some earlier surve or a pilo sud.

13 STATISTIS IN TANSITION-new series, December Acknowledgemens The auhors acknowledge he Universi Grans ommission, New Delhi, India for financial suppor in he projec number F. No /008(S). The auhors are also hankful o Indian School of Mines, Dhanbad and Vikram Universi, Ujjain for providing he faciliies o carr ou he research work. The auhors are also graeful o he referees for valuable suggesions regarding improvemen of he paper.

14 550. Yadav, L. N. Upadhaa, H.. Singh, S. haerjee: Almos unbiased EFEENES BAHL, S. and TUTEJA,.. (99). aio and produc pe eponenial esimaor, Informaion and Opimizaion Sciences, vol. (), MUTHY, M. N. (964). roduc mehod of esimaion. Sankha, A, 6, MUTHY, M. N. (967). Sampling Theor and Mehods. Saisical ublishing Socie, alcua, India. SINGH, S. and SINGH,. (993). A new mehod: Almos separaion of bias precipiaes in sample surves. Jour. Ind. Sa. Assoc., 3, STEEL,. G. D. and TOIE, J. H. (960). rinciples and rocedures of Saisics, Mc Graw Hill, New York.