STATISTICA, ao LXV,. 3, 2005 ESTIMATION OF FINITE POPULATION MEAN WITH KNOWN COEFFICIENT OF VARIATION OF AN AUXILIAR CHARACTER H.P. Sigh, R. Tail 1. INTRODUCTION AND THE SUGGESTED ESTIMATOR It is well kow that the use of auiliar ifmatio at the estimatio stage provides efficiet estimats of the parameter (s) of the stud character. Whe the populatio mea X of the auiliar character is kow, a large umber of estimats such as ratio, product ad regressio estimats ad their modificatios, have bee suggested b various auths. Das ad Tripathi (1980) have advocated that the coefficiet of variatio C of the auiliar character is also available i ma practical situatios. Keepig this i view, Sisodia ad Dwivedi (1981), Sigh ad Upadhaa (1986) ad Pade ad Dube (1988) have made the use of coefficiet of variatio C alogwith the populatio mea X i estimatig the populatio mea of. Suppose pairs ( i, i ) (i =1,2,...,) observatios are take o uits sampled from N populatio uits usig simple radom samplig without replacemet (SRSWOR) scheme. The classical ratio ad product estimats f are respectivel defied b ad R P X X (1) (2) where /, i 1 i / ad i 1 i N X / N is the kow populatio i 1 mea of the auiliar character. Whe the populatio coefficiet of variatio i C alogwith the populatio
302 H.P. Sigh, R. Tail mea X of is also kow, Sisodia ad Dwivedi (1981) suggested a ratio-tpe estimat f as MR ( X C ) ( C ) (3) ad Pade ad Dube (1988) proposed a product tpe estimat f as ( C ) MP ( X C ) (4) Motivated b Rao ad Mudholkar (1967) ad Sigh ad Ruiz Espejo (2003), we suggest a ratio - product estimat f as X C C (1 ), (5) C X C where is a suitabl chose scalar. We ote that f =1, reduces to the estimat MR suggested b Sisodia ad Dwivedi (1981) while f = 0 it reduces to the estimat MP repted b Pade ad Dube (1988). 2. BIAS OF To obtai the bias of (1 e ) 0 X(1 e ) such that 1, we write E( e0 ) E( e1 ) 0 2 (1 f ) 2 E( e0 ) C 2 (1 f ) 2 E( e1 ) C (1 f ) 2 E( e0e1 ) C C (6)
Estimatio of fiite populatio mea with kow coefficiet of variatio of a auiliar character 303 where C S /, C S / X, S / S S, C C / C, N N 2 2 2 2 i i i 1 i 1 S ( X ) /( N 1), S ( ) /( N 1), N S ( X )( )/( N 1). i i i 1 Epressig (5) i terms of e s we have (1 e )[ (1 e ) (1 )(1 e )], (7) where X ( X C ) 1 0 1 1 We ow assume that e1 1 so that we ma epad 1 1 (1 e ) as a series i powers of e1. Epadig, multiplig out ad retaiig terms of e s to the secod degree, we obtai [ (1 e e e e e ) (1 )(1 e e e e )] 2 2 0 1 1 0 1 0 1 0 1 2 2 ( ) e0 e1 e0e1 ( e1 2 e0e1 2 e1 ) (8) 0( Takig epectatios of both sides of (8), we obtai the bias of to der 1 ) as (1 f ) 2 B( ) C [ C ( 2 C )] (9) which vaishes if C /(2 C ) (10) Thus the estimat with C /(2 C ) is almost ubiased. We also ote from (9) that the bias of is egligible if the sample size is sufficietl large. To the first degree of approimatio, the biases of R, respectivel give b P, MR ad MP are (1 f ) 2 B( R ) C (1 C ) (11)
304 H.P. Sigh, R. Tail (1 f ) 2 B( P ) C C (12) (1 f ) 2 B( MR ) C ( C ) (13) (1 f ) 2 B( MP ) C C (14) From (9) ad (11) we ote that B( ) B( ) if { C ( 2 C )} (1 C ) i.e. if R {1 C(1 )} {1 C(1 )} ( 2 C ) ( 2 C ) {1 C(1 )} {1 C(1 )} ( 2 C ) ( 2 C ) (15) From (9) ad (12) we ote that B( ) B( ) if { C ( 2 C )} C i.e. if P C(1 ) C(1 ) ( 2 C ) ( 2 C ) C(1 ) C(1 ) ( 2 C ) ( 2 C ) (16) From (9) ad (13) it follows that B( ) B( ) if MR { C ( 2 C )} ( C ) i.e. if 1 1 (17)
Estimatio of fiite populatio mea with kow coefficiet of variatio of a auiliar character 305 Further from (9) ad (14) we see that B( ) B( ) if MP { C ( 2 C )} C i.e. if 2C 0 ( 2 C ) 2C 0 ( 2 C ) (18) 3. VARIANCE OF Squarig both sides of (8) ad eglectig terms of e s havig power greater tha two we have ( ) [ e (1 2 ) {(1 2 ) e 2 e e }] (19) 2 2 2 2 0 1 0 1 Takig epectatios both sides i (19), we get the variace of to the first degree of approimatio as (1 f ) 2 2 2 V( ) [ C (1 2 ) C {(1 2 ) 2 C}] (20) which is miimized f ( C ) 0 (sa) (21) 2 Substitutio of (21) i (5) ields the asmptoticall optimum estimat (AOE) f as (0) X C C ( C ) ( C ) 2 C X C (22) Puttig (21) i (9) ad (20) we get the bias ad variace of as (0) respectivel
306 H.P. Sigh, R. Tail ad (0) (1 f ) 2 B( ) C ( C )( C ) (23) 2 (0) (1 f ) 2 2 V( ) S (1 ) (24) It is to be metioed here that the variace of (0) at (24) is same as that of the approimate variace of the usual liear regressio estimat ( X ), where is the sample regressio coefficiet of o. lr Remark 3.1. I practice, with a good guess of C obtaied through pilot surves, past data eperiece gathered i due course of time, a optimum value of fairel close to its true value 0 ca be obtaied. This problem has bee also discussed amog others b Murth (1967, pp. 96-99), Redd (1978) ad Srivekataramaa ad Trac (1980). Further if a good guess of the iterval cotaiig C (i.e. C1 C C 2 ) which is me realistic tha a specific guess about C, ca be made o the, accumulated eperiece ad/ a scatter diagram f at least a part of curret data the it is also advisable to use the suggested estimat i practice. 4. EFFICIENC COMPARISONS It is well kow uder SRSWOR that (1 f ) 2 2 V( ) C (25) ad the variace of R, are respectivel give b P, MR ad MP to the first degree of approimatio (1 f ) 2 2 2 V( R ) [ C C (1 2 C )] (26) (1 f ) 2 2 2 V( P ) [ C C (1 2 C )] (27) (1 f ) 2 2 2 V( MR ) [ C C ( 2 C )] (28)
Estimatio of fiite populatio mea with kow coefficiet of variatio of a auiliar character 307 (1 f ) 2 2 2 V( MP ) [ C C ( 2 C )] (29) From (20) ad (25) we have (1 f ) 2 2 V( ) V( ) C (1 2 ){ (1 2 ) 2 C} which is egative if 1 1 C 2 2 1 C 1 2 2 (30) From (20) ad (26) we have (1 f ) 2 2 V( ) V( R ) C {(1 2 ) 1}{(1 2 ) 2C 1} which is egative if (1 ) 2C 1 2 2 2C 1 (1 ) 2 2 (31) From (20) ad (27) we have (1 f ) 2 2 V( ) V( P ) C {(1 2 ) 1}{(1 2 ) 2C 1} which is egative if 2C 1 ( 1) 2 2 ( 1) ( 2C 1) 2 2 (32) From (20) ad (28) we have (1 f ) 2 2 V( ) V( MR ) 4 C ( 1)( C ) which is egative if
308 H.P. Sigh, R. Tail C 1 C 1 (33) Further from (20) ad (29) we have (1 f ) 2 2 V( ) V( MP ) 4 C { ( 1) C} which is egative if C 0 1 C 1 0 (34) 5. ESTIMATOR BASED ON ESTIMATED OPTIMUM If eact good guess of C is ot available, we ca replace C b the sample estimate Ĉ i (22) ad get the estimat (based o estimated optimum) as (0) ( X C ) ( C C C ) 2 C X C (35) where 2 ( / ){1/ } C s XC where, we recall, X ad C are kow, ad s ( )( )/( 1). i i i 1 To obtai the variace of C C (1 e ) 2 (0) we write with 1 E( C ) C o( ). Epressig (0) i terms of e ' s we have (0) 1 (1 e0 )[{ C(1 e2 )}(1 e1 ) { C(1 e2 )}(1 e1 )] 2 where e 0 ad e 1 are same as defied i sectio 2. The variace of (0) is
Estimatio of fiite populatio mea with kow coefficiet of variatio of a auiliar character 309 V( ) E( ) (0) (0) 2 2 (1 e0 ) [{ (1 1 2 )}(1 1 ) E C e e { C (1 e2 )}(1 e1 )] 1 2 (36) Epadig the terms o the right had side of (36) ad eglectig power of ' e s that are greater tha two we have V( ) E( e C e ) E( e 2 C e e C e ) (0) 2 2 2 2 2 2 0 1 0 0 1 1 (1 f ) 2 2 2 2 (1 f ) 2 2 [ C C C 2 C C C )] S (1 ) 2 which is same as that of i. e. V( ) V ( ). Thus it is established (0) (0) (0) that the variace of the estimat to terms of der (0) 1, is the same as that of i (35) based o the estimated optimum, (0) i (22). 6. EMPIRICAL STUD To eamie the merits of the suggested estimat we have cosidered five atural populatio data sets. The descriptio of the populatio are give below. Populatio I: Murth (1967, p. 228) N= 80, : Output = 20, : Fied Capital 51.8264, X 11.2646, C 0.3542, C 0.7507, 0.9413, C 0.4441 f 0.25, 0.9375., Populatio II: Murth (1967, p. 228) N= 80, : Output = 20, : Number of Wkers 51.8264, X 2.8513, C 0.3542, C 0.9484 0.9150, C 0.3417 f 0.25, 0.7504., Populatio III: Das (1988) N= 278, : Number of agricultural labourers f 1971 = 30, : Number of agricultural labourers f 1961 39.0680, X 25.1110, C 1.4451, C 1.6198, 0.7213, C 0.6435 f 0.1079, 0.9394.,
310 H.P. Sigh, R. Tail Populatio IV: Steel ad Trie (1960, p. 282) N= 30, : Log of leaf bur i secs = 6, : Clie percetage 0.6860, X 0.8077, C 0.700123, C 0.7493, 0.4996, C 0.3202 0.5188 f 0.20, Populatio V: Maddala (1977) N= 16, : Cosumptio per capita = 4, : Deflated prices of veal 7.6375, X 75.4313, C 0.2278, C 0.0986, 0.6823 C 1.5761 0.9987 f 0.25 We have computed the rages of f which the proposed estimat is R P MR MP better tha,,, ad, optimum value of ad commo rage of ad displaed i Table 1. Table 2 shows the percet relative efficiecies of with respect to, R, P, MR ad MP. Populatio TABLE 1 Rage of i which is better tha, Rage of i which R, is better tha P, MR ad MP Optimum value of Commo rage of i which is better R tha,, R P MR MP 0 ad MP. I (0.50,0.9737) (0.4404,1.0333) (-0.033, 1.5070) (0.4737, 1.00) (0.00, 1.4737) 0.73685 (0.50, 0.9737) II (0.50,0.9554) (0.2891,1.1663) (-0.1663,1.6217) (0.4554, 1.00) (0.00, 1.4554) 0.72768 (0.50, 0.9554) III (0.50,1.1850) (0.6528,1.0323) (-0.0323,1.7173) (0.6850, 1.00) (0.00, 1.6850) 0.84251 (0.6850, 1.00) IV (-0.1172,0.50) (-1.0810,1.4638) (-0.4638,0.8466) (-0.6172, 1.00) (0.00, 0.3828) 0.19140 (0.00, 0.3828) V (-1.0782,0.50) (-1.5788,1.0007) (-0.577,-0.0007) (-1.5782, 1.00) (-0.5782, 0.00) -0.28908 (-0.5775,0.0007) P, MR Percet relative efficiecies of TABLE 2 with respect to, (0) (0) R, P, MR ad MP Populatio Percet relative efficiecies of R P with respect to: I 877.62 1318.18 * 1059.44 * II 614.40 2008.96 * 835.86 * III 208.46 133.29 * 122.94 * IV 133.26 * 249.90 * 112.80 V 187.37 * 111.91 * 111.88 * Data ot applicable. MR MP
Estimatio of fiite populatio mea with kow coefficiet of variatio of a auiliar character 311 Table 1 ehibits that there is eough scope of selectig the scalar i to get better estimats. It is observed that eve if slides awa from its true optimum value, the efficiec of the suggested estimat ca be icreased (0) (0) cosiderabl. Table 2 clearl idicates that the suggested estimat is me efficiet (with substatial gai) tha the usual ubiased estimat, classical ratio estimat R ad product estimat P, ad the modified estimats MR ad MP suggested b Sisodia ad Dwivedi (1981) ad Pade ad Dube (0) is to be preferred i prac- (1988) respectivel. Thus the proposed estimat tice. 7. CONCLUSION This article is cocered with estimatig the populatio mea of the stud variate usig auiliar ifmatio at the estimatio stage. Whe the populatio mea X ad coefficiet of variatio C of a auiliar variable is kow, a class of estimats f estimatig is suggested. Optimum estimat i the class is idetified with its approimate variace fmula. Estimat based o estimated optimum values is also proposed with its approimate variace fmula. It is iterestig to ote that the estimats based o optimum value ad estimated optimum value have the same approimate variace fmula. Thus we coclude that the studies carried out i the preset article ca be used fruitfull eve if the optimum values are ot kow. A empirical stud is carried out to throw light o the perfmace of the suggested estimat over alread eistig estimats. Further empirical studies carried out i this article clearl reflect the usefuless of the proposed estimats i practice. School of Studies i Statistics Vikram Uiversit, Ujjai, Idia H.P. SINGH RITESH TAILOR ACKNOWLEGDEMENTS Auths are thakful to the referees f makig valuable suggestio towards improvig the presetatio of the material.
312 H.P. Sigh, R. Tail REFERENCES A.K. DAS (1988), Cotributio to the the of samplig strategies based o auiliar ifmatio Ph.D. thesis submitted to BCKV; Mohapur, Nadia, West Begal, Idia. A.K. DAS, T.P. TRIPATHI (1980), Samplig strategies f populatio mea whe the coefficiet of variatio of a auiliar character is kow, Sakha, C, 42, pp. 76-86. G.S. MADDALA (1977), Ecoometrics, McGraw Hills pub.co. New k. M.N. MURTH (1967), Samplig the ad methods, Statistical Publishig Societ, Calcutta, Idia. B.N. PANDE, V. DUBE (1988), Modified product estimat usig coefficiet of variatio of auiliar variate. Assam Statistical Review, 2, part 2, pp. 64-66. P.S.R.S. RAO, G.S. MUDHOLKAR (1967), Geeralized multivariate estimats f thee mea of a fiite populatio. Joural. America. Statistical. Associatio., 62, pp. 1009-1012. V. N. REDD (1978), A stud o the use of pri kowledge o certai populatio parameters i estimatio. Sakha, C, 40, pp. 29-37. H.P. SINGH, M.R. ESPEJO (2003), O liear regressio ad ratio product estimatio of a fiite populatio mea. Statisticia, 52, part 1, pp. 59-67. H. P. SINGH, L.N. UPADHAA (1986), A dual to modified ratio estimat usig coefficiet of ariatio of auiliar variable. Proceedigs Natioal Academ of Scieces, Idia, 56, A, part 4, pp. 336-340. B. V. SISODIA, V. K. DWIVEDI (1981), A modified ratio estimat usig coefficiet of variatio of auiliar variable. Joural Idia Societ of Agricultural Statistics, New Delhi, 33, pp. 13-18. T. SRIVENKATARAMANA, D.S. TRAC (1980), A alterative to ratio method i sample surves. Aals of the Istitute of Statistical Mathematice, 32, A, pp. 111-120. R.G.D. STEEL, J.H. TORRIE (1960), Priciples ad procedures of Statistics, McGraw Hill Book Co. RIASSUNTO Stima della media di u popolazioe fiita co coefficiete di variazioe di u carattere ausiliario oto Il cotributo si occupa del problema della stima di ua media di popolazioe di ua variabile oggetto di studio utilizzado l ifmazioe sulla media di popolazioe X e sul coefficiete di variazioe C di u carattere ausiliario. Viee suggerito uo stima- te per il parametro e e vegoo studiate le proprietà el cotesto di sigoli campioi. Si dimostra che lo stimate proposto, sotto alcue codizioi realistiche, è più efficiete degli stimati proposti da Sisodia e Dwivedi (1981) e da Pade e Dube (1988). Tramite ua aalisi empirica vegoo esamiati i meriti dello stimate costruito rispetto agli atagoisti. SUMMAR Estimatio of fiite populatio mea with kow coefficiet of variatio of a auiliar character This paper deals with the problem of estimatig populatio mea of the stud variate usig ifmatio o populatio mea X ad coefficiet of variatio C of a
Estimatio of fiite populatio mea with kow coefficiet of variatio of a auiliar character 313 auiliar character. We have suggested a estimat f ad its properties are studied i the cotet of sigle samplig. It is show that the proposed estimat is me efficiet tha Sisodia ad Dwivedi (1981) estimat ad Pade ad Dube (1988) estimat uder some realistic coditios. A empirical stud is carried out to eamie the merits of the costructed estimat over others.