Abstract. Ranked set sampling, auxiliary variable, variance.

Transcription

1 Hacettepe Joural of Mathematics ad Statistics Volume (), 1 A class of Hartley-Ross type Ubiased estimators for Populatio Mea usig Raked Set Samplig Lakhkar Kha ad Javid Shabbir Abstract I this paper, we propose a class of Hartley-Ross type ubiased estimators for estimatig the fiite populatio mea of the study variable uder raked set samplig (RSS), whe populatio mea of the auxiliary variable is kow. The variaces of the proposed class of ubiased estimators are obtaied to first degree of approximatio. Both theoretically ad umerically, the proposed estimators are compared with some competitor estimators, usig three differet data sets. It is idetified through umerical study that the proposed estimators are more efficiet as compared to all other competitor estimators. Keywords: Raked set samplig, auxiliary variable, variace. 1. Itroductio I applicatios, there might be a situatio whe the variable of iterest caot be easily measured or is very expesive to do so, but it ca be raked easily at o cost or at very little cost. I view of this, Mcltyre [5] was the first who proposed the cocept of raked set samplig (RSS) i the cotext of obtaiig reliable farm yield estimates based o samplig of pastures ad crop yield. He provided a clear ad isightful itroductio to the basic framework of RSS ad laid out the ratioale for how it ca be lead to improved estimatio relative to simple radom samplig (SRS). Takahasi ad Wakimoto [11] have provided the ecessary mathematical theory of RSS ad showed that the sample mea uder RSS is a ubiased estimator of the fiite populatio mea ad more precise tha the sample mea estimator uder SRS. The auxiliary iformatio plays a importat role i icreasig efficiecy of the estimator. Samawi ad Muttlak [6] have suggested a estimator for populatio ratio i RSS ad showed that it has less variace as compared to usual ratio estimator i SRS. I RSS, perfect rakig of elemets was cosidered by Mcltyre [5] ad Takahasi ad Wakimoto [11] for estimatio of populatio mea. I some situatios, rakig may ot be perfect. Dell ad Clutter [2] have studied the case i which there are errors i rakig. They poited out that a loss i efficiecy would be Departmet of Statistics Quaid-i-Azam Uiversity, Islamabad, Pakista, lakhkarkha.stat@gmail.com Correspodig Author. Departmet of Statistics Quaid-i-Azam Uiversity, Islamabad, Pakista, javidshabbir@gmail.com

2 2 caused by the errors i rakig. The sample mea i RSS is a ubiased estimator of the populatio mea regardless of errors i rakig of the elemets. To reduce the error i rakig, several modificatios of the RSS method had bee suggested. Stokes [10] has proposed use of the cocomitat variable to aid i the rakig process to obtai raked set data. She has also studied the raked set sample approach for makig ifereces about the populatio variace ad correlatio coefficiet. Here, the rakig of elemets was doe o basis of the auxiliary variable istead of judgmet. Sigh et al. [7] have proposed a estimator for populatio mea ad rakig of the elemets is observed o basis of the auxiliary variable. Sigh et al.[9] have also proposed the ratio ad the product type estimators for populatio mea uder stratified raked set samplig (SRSS). Hartley ad Ross [3] proposed a ubiased ratio estimators for fiite populatio mea i SRS. Motivated by Sigh et al. [8], we suggest a class of Hartley-Ross type ubiased estimators based o RSS for populatio mea, usig some kow populatio parameters of the auxiliary variable. It is show that the proposed estimators outperform as compared to some existig estimators i RSS. 2. RSS procedure To create raked sets, we must partitio the selected first phase sample ito sets of equal size. I order to pla RSS desig, we must therefore choose a set of size m that is typically small, aroud three or four, to miimize rakig error. Here m is the umber of sample uits allocated to each set. The RSS procedure ca be summarized as follows: Step 1:Radomly select m 2 bivariate sample uits from the populatio. Step 2:Allocate m 2 selected uits radomly as possible ito m sets, each of size m. Step 3:Each sample is raked with respect to oe of the variables Y or X. Here, we assume that the perfect rakig is doe o basis of the auxiliary variable X while the rakig of Y is with error. Step 4:A actual measuremet from the first sample is the take of the uit with the smallest rak of X, together with variable Y associated with smallest rak of X. From secod sample of size m, the variable Y associated with the secod smallest rak of X is measured. The process is cotiued util from the mth sample, the Y associated with the highest rak of X is measured. Step 5:Repeat Steps 1 through 4 for r cycles util the desired sample size = mr, is obtaied for aalysis. As a illustratio, we select a sample of size 36 from a populatio by simple radom samplig with replacemet (SRSWR). These data are grouped ito 3 sets each of size 3 ad we repeat this process 4 times. Accordig to RSS methodology, we order the X values from smaller to larger ad assume that there is o judgmet error i this orderig. The, the smallest uit is selected from the first ordered set, the secod smallest uit is selected from the secod ordered set ad so o. Similarly from the third ordered set, the third smallest uit is selected. By this

3 3 way, we select = mr = 12 observatios. A raked set sample desig with set size m = 3 ad umber of samplig cycles r = 4 is illustrated i Figure 1. Although 36 sample uits have bee selected from the populatio, oly the 12 circled uits are actually icluded i the fial sample for quatitative aalysis. Figure 1. Illustratio of raked set samplig. 3. Symbols ad Notatios We cosider a situatio whe rak the elemets o the auxiliary variable. Let (y [i]j, x (i)j ) be the ith judgmet orderig i the ith set for the study variable Y based o the ith order of the ith set of the auxiliary variable X i the jth cycle. To obtai bias ad variace of the estimators, we defie: ȳ [i] = Ȳ (1 + e 0), x (i) = X(1 + e 1 ), r (i) = R(1 + e 2 ), x (i) = X (1 + e 1), r (i) = R (1 + e 2), such that E(e i ) = 0, i=0,1,2. E(e i ) = 0, i=1,2. ad E(e 2 0) = γcy 2 Wy[i] 2, E(e2 1) = γcx 2 Wx(i) 2, E(e 0e 1 ) = γc yx W yx(i), E(e 2 1 ) = γcx 2 W 2 x (i), E(e 0e 1) = γc yx W yx (i), E(e 1e 2) = γc r x W r x (i), where 1 m W yx(i) = m 2 r XȲ τ yx(i), Wx(i) 2 = 1 m m 2 r X τ 2 2 x(i), W y[i] 2 = 1 m τ 2 m 2 rȳ 2 y[i], W yx (i) = 1 m 2 r X Ȳ i=1 m i=1 i=1 τ yx (i), Wx 2 (i) = 1 m 2 rx 2 i=1 i=1 m τx 2 (i), W 1 r x (i) = m 2 rx R τ x(i) = (µ x(i) X), τ y[i] = (µ y([i] Ȳ ), τ yx(i) = (µ y[i] Ȳ )(µ x(i) X), m i=1 τ r x (i),

4 4 τ x (i) = (µ x (i) X ), τ yx (i) = (µ y[i] Ȳ )(µ x (i) X ), τ r x (i) = (µ r (i) R )(µ x (i) X ). Here γ = ( 1 mr ) ad C yx = ρc y C x, where C y ad C x are the coefficiets of variatio of Y ad X respectively. Also Ȳ ad X are the populatio meas of Y ad X respectively. The values of µ y[i] ad µ x(i) deped o order statistics from some specific distributios (see Arold et al.[1]). The followig otatios will be used through out this paper. ȳ [i] = (1/) y [i]j, x (i) = (1/) x (i)j, r (i) = R = E( r (i) ), r (i)j, r (i)j = y [i]j x (i)j, r (i) = r (i)j, r(i)j = y [i]j x, x (i)j = (ax (i)j + b), (i)j x (i) = (a x (i) + b), X = (a X + b), R = E( r (i) ), r (i) = r (i)j, r (i)j = y [i]j, x x (i)j = (x (i)jc x + ρ), x (i) = ( x (i)c x + ρ), X = ( XC x + ρ), (i)j R = E( r (i)), r (i) = r (i)j, r (i)j = y [i]j x (i)j, x (i)j = (x (i)jβ 2 (x) + C x ), x (i) = ( x (i)β 2 (x) + C x ), X = ( Xβ R 2 (x) + C x ), = E( r (i)), r r (i)j = y [i]j x (i)j, x (i)j = (x (i)jc x + β 2 (x)), x (i) = ( x (i)c x + β 2 (x)), (i) = r (i)j, X = ( XC x + β 2 (x)) ad R = E( r (i) ), where a ad b are kow populatio parameters, which ca be coefficiet of variatio, coefficiet of skewess ad coefficiet of kurtosis ad the coefficiet of correlatio of the auxiliary variable. Followig Sigh [8], the variace of the Hartley-Ross type ubiased estimator based o Upadhyaya ad Sigh [12] estimator i SRS, is give by (3.1) V ( US2(SRS) ) = γ (Ȳ 2 Cy 2 + X 2 R 2 C 2 2 R Ȳ X C x yx Uder RSS scheme, the variace of ȳ RSS = ȳ [i] = (1/) y [i]j, is give by ( ) (3.2) V (ȳ RSS ) = Ȳ 2 γcy 2 Wy[i] Proposed Hartley-Ross ubiased estimator i RSS Followig Sigh et al. [8], we cosider the followig ratio estimator: (4.1) ȳ H(RSS) = r (i) X. The bias of ȳ H(RSS), is give by (N 1) B(ȳ H(RSS) ) = N S r (i) x (i), where S r(i) x (i) = 1 N N (r (i)j R)(x (i)j X) ad a ubiased estimator of S r(i) x (i) is give by s r(i) x (i) = 1 (r (i)j r (i) )(x (i)j x (i) ) 1 = 1 (ȳ [i] r (i) x (i) ). ).

5 5 So bias of ȳ H(RSS) becomes (4.2) (N 1) B(ȳ H(RSS) ) = N( 1) (ȳ [i] r (i) x (i) ). Thus a ubiased Hartley-Ross type estimator of populatio mea based o RSS is give by (4.3) H(RSS) = r X (N 1) (i) + N( 1) (ȳ [i] r (i) x (i) ). I terms of e s, we have H(RSS) = X R(1 (N 1) [Ȳ + e 2 ) + (1 + e0 ) N( 1) X R(1 + e 1 )(1 + e 2 ) ]. Uder the assumptio (N 1) N( 1) = 1, we ca write ( H(RSS) Ȳ ) = (Ȳ e 0 X Re 1 ). Takig square ad the expectatio, the variace of H(RSS), is give by (4.4) V ( H(RSS) ) = Ȳ 2 (γc 2 y W 2 y[i] ) + X 2 R2 (γc 2 x W 2 x(i) ) 2 RȲ X(γC yx W yx(i) ). 5. Proposed class of Hartley-Ross type ubiased estimators i RSS Cosider the followig ratio estimator: (5.1) ȳ P (RSS) = r (i) X. The bias of ȳ P (RSS), is give by where S r (i) x (i) S r (i) x (i) (N 1) B(ȳ P (RSS) ) = N S r, (i) x (i) = 1 N N (r (i)j R )(x (i)j X ) ad a ubiased estimator of is give by s r (i) x (i) = 1 1 (r(i)j r (i) )(x (i)j x (i) ) = 1 (ȳ [i] r (i) x (i) ). We give the followig theorem Theorem. A ubiased estimator of S r (i) x (i) is give by s r = 1 (i) x (i) 1 (r (i)j r (i) )(x (i)j x (i) ). = 1 N N (r (i)j R )(x (i)j X )

6 6 Proof. We have to prove that E(s r ) = S (i) x r. Here for fixed i, j = 1, 2,...,, (i) (i) x (i) r(i)j ad x (i)j are simple radom samples of size. E(s r ) =E 1 (r (i) x (i) (i)j 1 r (i) )(x (i)j x (i) ), = 1 1 E r(i)j x (i)j r (i) x (i), (5.2) = 1 1 = 1 1 = 1 = 1 E(r(i)j x (i)j ) E( r (i) x (i) ), N 1 N ( =S r (i) x (i). So bias of ȳ P (RSS) becomes S r N ( r(i)j x (i)j Cov( r (i), x (i) ) + R ) X, N r(i)j x (i)j R X S r (i) x (i), ) (i) x (i) S r (i) x (i) (N 1) B(ȳ KP (RSS) ) = N( 1) (ȳ [i] r (i) x (i) ). Thus a ubiased class of Hartley-Ross type estimators of populatio mea based o RSS is give by (5.3) P (RSS) = r X (i) (N 1) + N( 1) (ȳ [i] r (i) x (i) ). I terms of e s, we have P (RSS) = X R (1 + e (N 1) (Ȳ 2) + (1 + e0 ) N( 1) X R (1 + e 1)(1 + e 2) ). Uder the assumptio (N 1) N( 1) = 1, we have ( P (RSS) Ȳ ) = (Ȳ e 0 X R e 1). Takig square ad the expectatio, the variace of P (RSS), is give by (5.4) V ( P (RSS) ) =Ȳ 2 (γc 2 y W 2 y[i] ) + X 2 R 2 (γc 2 x W 2 x (i) ) 2 R Ȳ X (γc yx W yx (i)).,

7 7 Note: (i). If a = C x ad b = ρ, the from Equatio (5.3), we get the Hartley-Ross type ubiased estimator based o Kadilar ad Cigi [4] estimator KC(RSS), as: (5.5) KC(RSS) = X (N 1) r (i) + N( 1) (ȳ [i] r (i) x (i) ). The variace of ȳ KC(RSS), is give by (5.6) V ( KC(RSS) ) =Ȳ 2 (γc 2 y W 2 y[i] ) + X 2 R 2 (γc 2 x W 2 x (i) ) 2 R Ȳ X (γc yx W yx (i) ). (ii). If a = β 2 (x) ad b = C x, the Equatio (5.3) becomes the Hartley-Ross type ubiased estimator based o Upadhyaya ad Sigh [12] estimator US1(RSS) ad is give by (5.7) US1(RSS) = X (N 1) r (i) + N( 1) (ȳ [i] r (i) x (i) ). The variace of ȳ US1(RSS), is give by (5.8) V ( US1(RSS) ) =Ȳ 2 (γc 2 y W 2 y[i] ) + X 2 R 2 (γc 2 x W 2 x (i) ) 2 R Ȳ X (γc yx W yx (i) ). (iii). If a = C x ad b = β 2 (x), the Equatio (5.3) becomes the Hartley-Ross type ubiased estimator based o Upadhyaya ad Sigh [12] estimator US2(RSS) ad is give by (5.9) US2(RSS) = r X (N 1) (i) + N( 1) (ȳ [i] r (i) x (i) ). The variace of ȳ US2(RSS), is give by (5.10) V ( US2(RSS) ) =Ȳ 2 (γc 2 y W 2 y[i] ) + X 2 R 2 (γc 2 x W 2 x (i) ) 6. Efficiecy compariso 2 R Ȳ X (γc yx W yx (i) ). The proposed estimator US2(RSS) is more efficiet tha ȳ(u) US2(SRS), ȳ (RSS), H(RSS), ȳ(u) KC(RSS) ad ȳ(u) US1(RSS) respectively if the followig coditios hold: (i). (Ȳ W y[i] X R W x (i) )2 < 0 (ii). X R (γc 2 x W 2 x (i) ) 2Ȳ (γc yx W yx (i) ) < 0 (iii). X 2 R 2 (γc 2 x W 2 x (i) ) 2 X R Ȳ (γc yx W yx (i) ) X 2 R2 (γc 2 x W 2 x(i) ) + 2 R XȲ (γc yx W yx(i) ) < 0 (iv). X 2 R 2 (γc 2 x W 2 x (i) ) 2 X R Ȳ (γc yx W yx (i) ) X 2 R 2 (γc 2 x W 2 x (i) ) + 2 R X Ȳ (γc yx W yx (i) ) < 0.

8 8 (v). X 2 R 2 (γc 2 x W 2 x (i) ) 2 X R Ȳ (γc yx W yx (i) ) X 2 R 2 (γc 2 x W 2 x (i) ) + 2 R X Ȳ (γc yx W yx (i) ) < Numerical Illustratio To observe performaces of the estimators, we use the followig three data sets. The descriptios of these populatios are give below. Populatio I [source: Valliat et al.[13]] The summary statistics are: y : Breast cacer mortality i , x : Adult female populatio i N = 301, = 12, m = 3, r = 4, X = , Ȳ = , ρ = , β 2 (x) = 10.79, R = , R R R = , = , = , X = , X = , X = , C y = , C x = , C x = , C x = , C x = , C yx = , C yx = , C yx = , C yx = , Wy[i] 2 = , W x(i) 2 = , W yx(i) = , W 2 = , x (i) W yx (i) = , W 2 = , x (i) W yx (i) = , W 2 = , x (i) W yx (i) = Populatio II [source: Valliat et al. [13]] The summary statistics are: y : Number of patiets discharged, x : Number of beds. N = 393, = 15, m = 3, r = 5, X = , Ȳ = , ρ = , β 2 (x) = , R = , R R R = , = , = , X = , X = , X = , C y = , C x = , C x = , C x = , C x = , C yx = , C yx = , C yx = , C yx = , Wy[i] 2 = , W x(i) 2 = , W yx(i) = , W 2 = , x (i) W yx (i) = , W 2 = , x (i) W yx (i) = , W 2 = , x (i) W yx (i) =

9 9 Populatio III [source: Valliat et al. [13]] The summary statistics are: y : Populatio, excludig residets of group quarters i 1960, x : Number of households i N = 304, = 12, m = 3, r = 4, X = , Ȳ = , ρ = , β 2 (x) = , R = , R R R = , = , = , X = , X = , X = , C y = , C x = , C x = , C x = , C x = , C yx = , C yx = , C yx = , C yx = , Wy[i] 2 = , W x(i) 2 = , W yx(i) = , W 2 = , x (i) W yx (i) = , W 2 = , x (i) W yx (i) = , W 2 = , x (i) W yx (i) = Table 1. Compariso values Populatio V ( US2(RSS) ) V (ȳ(u) US2(RSS) ) V (ȳ(u) US2(RSS) ) < V ( US2(SRS) ) < V (ȳ RSS) < V ( H(RSS) ) V (ȳ(u) US2(RSS) ) < V (ȳ(u) KC(RSS) ) V (ȳ(u) US2(RSS) ) < V (ȳ(u) US1(RSS) ) I < < < < < 0 II < < < < < 0 III < < < < < 0 We ivestigate the percet relative efficiecy (P RE) of Hartley-Ross ubiased estimator H(RSS) = ˆθ 1 (say), Hartley-Ross type ubiased estimator based o Kadilar ad Cigi [4] estimator KC(RSS) = ˆθ 2, Hartley-Ross type ubiased estimator based o Upadhyaya ad Sigh [12] estimator US1(RSS) = ˆθ 3 ad US2(RSS) = ˆθ 4 with respect to covetioal estimator ȳ RSS = ˆθ 0 (say). The P RE of proposed estimators ˆθ j, j = 1, 2, 3, 4, with respect to covetioal estimator ȳ RSS = ˆθ 0, is defied as: (7.1) P RE(ˆθ 0, ˆθ j ) = V (ˆθ 0 ) 100, j = 1, 2, 3, 4. V (ˆθ j ) The P RE s of our proposed estimators ad other existig estimators for Populatios I, II ad III are give i Tables 2, 3 ad 4 respectively.

10 10 Table 2. P RE s of various estimators for Populatio I. m r ȳ RSS H(RSS) KC(RSS) US1(RSS) US2(RSS) Table 3. P RE s of various estimators for Populatio II. m r ȳ RSS H(RSS) KC(RSS) US1(RSS) US2(RSS)

11 11 Table 4. P RE s of various estimators for Populatio III. m r ȳ RSS H(RSS) KC(RSS) US1(RSS) US2(RSS) From Tables 2, 3 ad 4, we see that the proposed Hartley-Ross type ubiased estimators are more efficiet tha usual covetioal estimator i RSS. Thus, if populatio coefficiet of variatio, populatio coefficiet of kurtosis ad populatio correlatio coefficiet are kow i advace, the our proposed estimators ca be used i practice. 8. Coclusio Table 1 has established the coditios obtaied i Sectio 6 umerically. It is show that all coditios are satisfied for all cosidered populatios. O the basis of results give i Tables 2, 3 ad 4, we coclude that the proposed class of Hartley-Ross type ubiased estimators are preferable over its competitive estimators uder RSS. It is also observed that the proposed ubiased estimator US2(RSS) has highest P RE i compariso to all other cosidered estimators i all three populatios. Ackowledgmets The authors are thakful to the editor ad the aoymous referees for their valuable suggestios which helped to improve the research paper. Refereces [1] Arold, B. C., Balakrisha, N. ad Nagaraja, H. N. A First Course i Order Statistics, Vol. 54, Siam, [2] Dell, T. ad Clutter, J. Raked set samplig theory with order statistics backgroud, Biometrics, , [3] Hartley, H.O. ad Ross, A. Ubiased ratio estimators, Nature, 174, , 1954.

12 12 [4] Kadilar, C ad Cigi, H. New ratio estimators usig correlatio coefficiet, Iterstat, 4(March), 1-11, [5] Mcltyre, G. A method for ubiased selective samplig usig raked sets, Crop ad Pasture Sciece, 3 (4), , [6] Samawi, H. M. ad Muttlak, H. A. Estimatio of ratio usig raked set samplig, Biometrical Joural, 38 (6), , [7] Sigh, H. P., Tailor, R. ad Sigh, S. Geeral procedure for estimatig the populatio mea usig raked set samplig, Joural of Statistical Computatio ad Simulatio, 84 (5), , [8] Sigh, H. P., Sharma, B. ad Tailor, R. Hartley-Ross type estimators for populatio mea usig kow parameters of auxiliary variate, Commuicatios i Statistics-Theory ad Methods, 43 (3), , [9] Sigh, H. P., Mehta, V. ad Pal, S.K. Dual to ratio ad product type estimators usig stratified raked set samplig, Joural of Basic ad Applied Egieerig Research, 1 (13), 7-12, [10] Stokes, S. L. Raked set samplig with cocomitat variables, Commuicatio i Statistics- Theory ad Methods, 6 (12), , [11] Takahasi, K. ad Wakimoto, K. O ubiased estimates of the populatio mea based o the sample stratified by meas of orderig, Aals of the Istitute of Statistical Mathematics, 20 (1), 1-31, [12] Upadhyaya, L. N. ad Sigh, H. P. Use of trasformed auxiliary variable i estimatig the fiite populatio mea, Biometrical Joural, 41 (5), , [13] Valliat, R., Dorfma, A. ad Royall, R. Fiite Populatio Samplig ad Iferece: A Predictio Approach, (Wiley series i probability ad statistics: Survey methodology sectio, Joh Wiley ad Sos, New York, 2000).