Maoj K. haudhar, Sachi Malik, Rajesh Sigh Departmet of Statistics, Baaras Hidu Uiversit Varaasi-005, Idia Floreti Smaradache Uiversit of New Mexico, Gallup, USA Use of Auxiliar Iformatio for Estimatig Populatio Mea i Sstematic Samplig uder No- Respose Published i: Rajesh Sigh, Floreti Smaradache (Editors ON IMPROVEMENT IN ESTIMATING POPULATION PARAMETER(S USING AUILIAR INFORMATION Educatioal Publishig (olumbus & Joural of Matter Regularit (Beijig, USA - hia, 03 ISBN: 978--59973-30-5 pp. 5-6
Abstract I this paper we have adapted Sigh ad Shukla (987 estimator i sstematic samplig usig auxiliar iformatio i the presece of o-respose. The properties of the suggested famil have bee discussed. Expressios for the bias ad mea square error (MSE of the suggested famil have bee derived. The comparative stud of the optimum estimator of the famil with ratio, product, dual to ratio ad sample mea estimators i sstematic samplig uder o-respose has also bee doe. Oe umerical illustratio is carried out to verif the theoretical results. Kewords: Auxiliar variable, sstematic samplig, factor-tpe estimator, mea square error, o-respose. 5
. Itroductio There are some atural populatios like forests etc., where it is ot possible to appl easil the simple radom samplig or other samplig schemes for estimatig the populatio characteristics. I such situatios, oe ca easil implemet the method of sstematic samplig for selectig a sample from the populatio. I this samplig scheme, ol the first uit is selected at radom, the rest beig automaticall selected accordig to a predetermied patter. Sstematic samplig has bee cosidered i detail b Madow ad Madow (944, ochra (946 ad Lahiri (954. The applicatio of sstematic samplig to forest surves has bee illustrated b Hasel (94, Fie (948 ad Nair ad Bhargava (95. The use of auxiliar iformatio has bee permeated the importat role to improve the efficiec of the estimators i sstematic samplig. Kushwaha ad Sigh (989 suggested a class of almost ubiased ratio ad product tpe estimators for estimatig the populatio mea usig jack-kife techique iitiated b Queouille (956. Later Baarasi et al. (993, Sigh ad Sigh (998, Sigh et al. (0, Sigh et al. (0 ad Sigh ad Solaki (0 have made a attempt to improve the estimators of populatio mea usig auxiliar iformatio i sstematic samplig. The problem of o-respose is ver commo i surves ad cosequetl the estimators ma produce bias results. Hase ad Hurwitz (946 cosidered the problem of estimatio of populatio mea uder o-respose. The proposed a samplig pla that ivolves takig a subsample of o-respodets after the first mail attempt ad the eumeratig the subsample b persoal iterview. El-Badr (956 exteded Hase ad Hurwitz (946 techique. Hase ad Hurwitz (946 techique i simple radom samplig is described as: From a populatio U = (U, U, ---, UN, a large first phase sample of size is selected b simple radom samplig without replacemet (SRSWOR. A smaller secod phase sample of size is selected from b SRSWOR. No-respose occurs o the secod phase of size i which uits respod ad uits do ot. From the o-respodets, b SRSWOR a sample of r = / k; k > uits is selected. It is assumed that all the r uits respod this time roud. (see Sigh ad Kumar (0009. Several authors such as ochra (977, Sodipo ad Obisesa (007, Rao (987, Khare ad Srivastava ( 997 ad Okafor ad Lee (000 have studied the problem of o-respose uder SRS. 6
I the sequece of improvig the estimator, Sigh ad Shukla (987 proposed a famil of factor-tpe estimators for estimatig the populatio mea i simple radom samplig usig a auxiliar variable, as ( A + ( A + fb + fbx T = (. + x where ad x are the sample meas of the populatio meas ad respectivel. A, B ad are the fuctios of, which is a scalar ad chose so as the MSE of the estimator T is miimum. Where, A = ( (, = ( ( 4 ( ( 3( 4 B, = ; > 0 ad f =. N Remark : If we take =,, 3 ad 4, the resultig estimators will be ratio, product, dual to ratio ad sample mea estimators of populatio mea i simple radom samplig respectivel (for details see Sigh ad Shukla (987. I this paper, we have proposed a famil of factor-tpe estimators for estimatig the populatio mea i sstematic samplig i the presece of o-respose adaptig Sigh ad Shukla (987 estimator. The properties of the proposed famil have bee discussed with the help of empirical stud.. Samplig Strateg ad Estimatio Procedure Let us assume that a populatio cosists of N uits umbered from to N i some order. If N = k, where k is a positive iteger, the there will be k possible samples each cosistig of uits. We select a sample at radom ad collect the iformatio from the uits of the selected sample. Let uits i the sample respoded ad uits did ot respod, so that + =. The uits ma be regarded as a sample from the respose class ad uits as a sample from the o-respose class belogig to the populatio. Let us assume that N ad N be the umber of uits i the respose class ad o-respose 7
class respectivel i the populatio. Obviousl, N ad N are ot kow but their ubiased estimates ca be obtaied from the sample as N ˆ = / ; N ˆ = N /. Further, usig Hase ad Hurwitz (946 techique we select a sub-sample of size h from the o-respodet uits such that = hl ( L > ad gather the iformatio o all the uits selected i the sub-sample (for details o Hase ad Hurwitz (946 techique see Sigh ad Kumar (009. Let ad be the stud ad auxiliar variables with respective populatio meas ad. Let ij ( x ij be the observatio o the th j uit i the i th sstematic sample uder stud (auxiliar variable ( i =... k : j =....Let us cosider the situatio i which orespose is observed o stud variable ad auxiliar variable is free from o-respose. The Hase-Hurwitz (946 estimator of populatio mea ad sample mea estimator of based o a sstematic sample of size, are respectivel give b = + h ad x = x ij j= where ad h are respectivel the meas based o respodet uits ad h orespodet uits. Obviousl, respective variaces of ad x are ubiased estimators of ad respectivel. The ad x are expressed as V ad N ( + L { } S + W S = ρ (. ( x N V = { + ρ } S (. 8
where ρ ad ρ are the correlatio coefficiets betwee a pair of uits withi the sstematic sample for the stud ad auxiliar variables respectivel. S ad respectivel the mea squares of the etire group for stud ad auxiliar variables. S are S be the populatio mea square of o-respose group uder stud variable ad W is the orespose rate i the populatio. Assumig populatio mea of auxiliar variable is kow, the usual ratio, product ad dual to ratio estimators based o a sstematic sample uder o-respose are respectivel give b R =, (.3 x ad P = x (.4 ( N x D = ( N. (.5 Obviousl, all the above estimators R, ad P are biased. To derive the biases D ad mea square errors (MSE of the estimators approximatio, let R, P ad D uder large sample = ( + e 0 x = ( + e such that E ( e 0 = ( e V ( E = ( e 0 V E ( ( x = e ad E = 0, N L S ρ, (.6 = { + } + W N = { + ρ } 9 (.7
ov (, x E = ( e 0 e N = { + ρ } { + ρ } ρ (.8 where ad are the coefficiets of variatio of stud ad auxiliar variables respectivel i the populatio (for proof see Sigh ad Sigh(998 ad Sigh (003, pg. o. 38. The biases ad MSE s of the estimators R, approximatio usig (.6-.8, are respectivel give b ( R N B = + ( R N { ρ }( Kρ MSE = + ( P N { ρ } ρ + ( Kρ B = { + ρ } Kρ ( P N MSE = + ( D N B = + ( D N ad P up to the first order of, (.9 [ ] { ρ } ρ + ( + Kρ { ρ }[ ] ρ K MSE = { + ρ } + D L WS, (.0, (. [ ] ρ + L WS, (., (.3 f f + f f ρ K L WS + ( (.4 where, { ρ } ( ρ = + { + ρ } ad K = ρ. ( for details of proof refer to Sigh et al.(0. The regressio estimator based o a sstematic sample uder o-respose is give b 0
lr = + b( x (.5 MSE of the estimator lr is give b N MSE ( lr = + ρ { }[ K ] 3. Adapted Famil of Estimators ρ L WS + ( (.6 Adaptig the estimator proposed b Sigh ad Shukla (987, a famil of factor-tpe estimators of populatio mea i sstematic samplig uder o-respose is writte as ( A + ( A + fb + fbx T =. (3. + x The costats A, B,, ad f are same as defied i (.. It ca easil be see that the proposed famil geerates the o-respose versios of some well kow estimators of populatio mea i sstematic samplig o puttig differet choices of. For example, if we take =,, 3 ad 4, the resultig estimators will be ratio, product, dual to ratio ad sample mea estimators of populatio mea i sstematic samplig uder o-respose respectivel. 3. Properties of T Obviousl, the proposed famil is biased for the populatio mea. I order to fid the bias ad MSE of i terms of i e s ( = 0, T, we use large sample approximatios. Expressig the equatio (3. i we have ( + e ( + De [( A + + fb( + e ] 0 T = (3. A + fb + where D =. A + fb + Sice D < ad e i <, eglectig the terms of i tha two, the equatio (3. ca be writte as e s ( = 0, i havig power greater
= ( A + { e0 De + D e De0e } T A + fb + [ { e ( D e + D( D e ( D e e }] + fb. (3.3 0 Takig expectatio of both sides of the equatio (3.3, we get 0 E [ T ] = ( fb ( ( A + fb + A + E e E e0e fb +. Let φ ( = A + fb fb + ad φ ( = A + fb + the φ ( = φ ( - ( φ = fb. A + fb + Thus, we have E [ T ] = ( [ φ ( E( e E( e ] φ. (3.4 Puttig the values of ( equatio (3.4, we get the bias of ( T N 0e E ad ( e T as B = φ( { + ρ } φ ( E e 0 e from equatios (.7 ad (.8 ito the [ ρ ] K. (3.5 Squarig both the sides of the equatio (3.3 ad the takig expectatio, we get E [ T ] = [ E( e φ ( E( e φ ( E( e e ] Substitutig the values of E (, ( 0 +. (3.6 e 0 E ad ( (.6, (.7 ad (.8 ito the equatio (3.6, we get the MSE of ( T N e 0 E e 0 e from the respective equatios T as [ { ρ K} ] MSE = { + ρ } ρ + φ ( φ ( + ( L WS. (3.7
3. Optimum hoice of I order to obtai the optimum choice of, we differetiate the equatio (3.7 with respect to ad equatig the derivative to zero, we get the ormal equatio as N [ ρ ] K { + ρ } φ ( φ ( φ ( = 0 (3.8 where φ ( is the first derivative of ( φ with respect to. Now from equatio (3.8, we get ( φ = ρ K (3.9 which is the cubic equatio i. Thus has three real roots for which the MSE of proposed famil would attai its miimum. Puttig the value of φ ( from equatio (3.9 ito equatio (3.7, we get ( T mi N MSE = + { ρ }[ K ] ρ L WS + ( (3.0 which is the MSE of the usual regressio estimator of populatio mea i sstematic samplig uder o-respose. 4. Empirical Stud I the support of theoretical results, we have cosidered the data give i Murth (967, p. 3-3. These data are related to the legth ad timber volume for te blocks of the blacks moutai experimetal forest. The value of itraclass correlatio coefficiets ρ ad ρ have bee give approximatel equal b Murth (967, p. 49 ad Kushwaha ad Sigh (989 for the sstematic sample of size 6 b eumeratig all possible sstematic samples after arragig the data i ascedig order of strip legth. The particulars of the populatio are give below: N = 76, = 6, = 8.636, = 6.9943, S = 44.6700, S = 8.7600, ρ = 0.870, 3
3 S = S = 8086.005. 4 Table depicts the MSE s ad variace of the estimators of proposed famil with respect to o-respose rate ( W. Table : MSE ad Variace of the Estimators for L =. W 0. 0. 0.3 0.4 (= R 37.37 484.4 597.45 70.48 (= P 908.8 0.85 34.89 47.93 3(= D 063. 76.6 89.30 40.33 4(= 40.69 53.3 366.7 479.05 opt = ( T 70.67 383.7 496.75 609.78 ( mi 5. oclusio I this paper, we have adapted Sigh ad Shukla (987 estimator i sstematic samplig i the presece of o-respose usig a auxiliar variable ad obtaied the optimum estimator of the proposed famil. It is observed that the proposed famil ca geerate the o-respose versios of a umber of estimators of populatio mea i sstematic samplig o differet choice of. From Table, we observe that the proposed famil uder optimum coditio has miimum MSE, which is equal to the MSE of the regressio estimator (most of the class of estimators i samplig literature uder optimum coditio attais MSE equal to the MSE of the regressio estimator. It is also see that the MSE or variace of the estimators icreases with icrease i o respose rate i the populatio. 4
Refereces. Baarasi, Kushwaha, S.N.S. ad Kushwaha, K.S. (993: A class of ratio, product ad differece (RPD estimators i sstematic samplig, Microelectro. Reliab., 33, 4, 455 457.. ochra, W. G. (946: Relative accurac of sstematic ad stratified radom samples for a certai class of populatio, AMS, 7, 64-77. 3. Fie, D.J. (948: Radom ad sstematic samplig i timber surves, Forestr,, 64-99. 4. Hase, M. H. ad Hurwitz, W. N. (946 : The problem of o-respose i sample surves, Jour. of The Amer. Stat. Assoc., 4, 57-59. 5. Hasel, A. A. (94: Estimatio of volume i timber stads b strip samplig, AMS, 3, 79-06. 6. Kushwaha, K. S. ad Sigh, H.P. (989: lass of almost ubiased ratio ad product estimators i sstematic samplig, Jour. Id. Soc. Ag. Statistics, 4,, 93 05. 7. Lahiri, D. B. (954: O the questio of bias of sstematic samplig, Proceedigs of World Populatio oferece, 6, 349-36. 8. Madow, W. G. ad Madow, L.H. (944: O the theor of sstematic samplig, I. A. Math. Statist., 5, -4. 9. Murth, M.N. (967: Samplig Theor ad Methods. Statistical Publishig Societ, alcutta. 0. Nair, K. R. ad Bhargava, R. P. (95: Statistical samplig i timber surves i Idia, Forest Research Istitute, Dehradu, Idia forest leaflet, 53.. Queouille, M. H. (956: Notes o bias i estimatio, Biometrika, 43, 353-360.. Sigh, R ad Sigh, H. P. (998: Almost ubiased ratio ad product tpe- estimators i sstematic samplig, Questiio,,3, 403-46. 3. Sigh, R., Malik, S., haudhar, M.K., Verma, H. ad Adewara, A. A. (0 : A geeral famil of ratio tpe- estimators i sstematic samplig. Jour. Reliab. ad Stat. Stud.,5(, 73-8. 4. Sigh, R., Malik, S., Sigh, V. K. (0 : A improved estimator i sstematic samplig. Jour. Of Scie. Res., 56, 77-8. 5
5. Sigh, H.P. ad Kumar, S. (009 : A geeral class of dss estimators of populatio ratio, product ad mea i the presece of o-respose based o the sub-samplig of the o-respodets. Pak J. Statist., 6(, 03-38. 6. Sigh, H.P. ad Solaki, R. S. (0 : A efficiet class of estimators for the populatio mea usig auxiliar iformatio i sstematic samplig. Jour. of Stat. Ther. ad Pract., 6(, 74-85. 7. Sigh, S. (003 : Advaced samplig theor with applicatios. Kluwer Academic Publishers. 8. Sigh, V. K. ad Shukla, D. (987: Oe parameter famil of factor-tpe ratio estimators, Metro, 45 (-, 73-83. 6