A General Class of Estimators of Population Median Using Two Auxiliary Variables in Double Sampling

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1 ohammad Khoshnevisan School o Accountin and inance riith University Australia Housila P. Sinh School o Studies in Statistics ikram University Ujjain P. India Sarjinder Sinh Departament o athematics and Statistics University o Saskatchewan anada lorentin Smarandache University o New eico allup USA A eneral lass o Estimators o Population edian Usin wo Auiliary ariables in Double Samplin Published in:. Khoshnevisan S. Saena H. P. Sinh S. Sinh. Smarandache Editors RANDONESS AND OPIAL ESIAION IN DAA SAPLIN second edition American Research Press Rehoboth USA ISBN: pp. 6 -

2 Abstract: In this paper we have suested two classes o estimators or population median o the study character usin inormation on two auiliary characters and in double samplin. It has been shown that the suested classes o estimators are more eicient than the one suested by Sinh et al. Estimators based on estimated optimum values have been also considered with their properties. he optimum values o the irst phase and second phase sample sizes are also obtained or the ied cost o survey. Keywords: edian estimation hain ratio and reression estimators Study variate Auiliary variate lasses o estimators ean squared errors ost Double samplin. S: 6E99. INRODUION In survey samplin statisticians oten come across the study o variables which have hihly skewed distributions such as income ependiture etc. In such situations the estimation o median deserves special attention. Kuk and ak 989 are the irst to introduce the estimation o population median o the study variate usin auiliary inormation in survey samplin. rancisco and uller 99 have also considered the problem o estimation o the median as part o the estimation o a inite population distribution unction. Later Sinh et al have dealt etensively with the problem o estimation o median usin auiliary inormation on an auiliary variate in two phase samplin. onsider a inite population U{ i...n}. Let and be the variable or study and auiliary variable takin values i and i respectively or the i-th unit. When the two variables are stronly related but no inormation is available on the population median o we seek to estimate the population median o rom a sample S m obtained throuh a two-phase selection. Permittin simple random samplin without replacement SRSWOR desin in each phase the two-phase samplin scheme will be as ollows: i ii he irst phase sample S n S n U o ied size n is drawn to observe only in order to urnish an estimate o. iven S n the second phase sample S m S m S n o ied size m is drawn to observe only. Assumin that the median o the variable is known Kuk and ak 989 suested a ratio estimator or the population median o as 6

3 . and are the sample estimators o and respectively based on a sample S m o size m. Suppose that y y y m are the y values o sample units in ascendin order. urther let t be an inteer such that t t and let pt/m be the proportion o values in the sample that are less than or equal to the median value an unknown population parameter. I p is a predictor o p the sample median can be written in terms o quantities as Q p p. 5. Kuk and ak 989 deine a matri o proportions P ij y as and a position estimator o y iven by > otal P y P y P y > P y P y P y otal P y P y p Q p. p m p y m m p y m p y p y m p y m m p y m with p ij y bein the sample analoues o the P ij y obtained rom the population and m the number o units in S m with. ~ ~ Let A y and B y denote the proportion o units in the sample S m with and > respectively that have values less than or equal to y. hen or estimatin Kuk and ak 989 suested the 'stratiication estimator' as ~ [ y ~ y y A St ~ y in { y :.5}. B It is to be noted that the estimators deined in.. and. are based on prior knowlede o the median o the auiliary character. In many situations o practical importance the population median o may not be known. his led Sinh et al to discuss the problem o estimatin the population median in double samplin and suested an analoous ratio estimator as d. 7

4 is sample median based on irst phase sample S n. Sometimes even i is unknown inormation on a second auiliary variable closely related to but compared remotely related to is available on all units o the population. his type o situation has been briely discussed by amon others hand 975 Kireyera 98 8 Srivenkataramana and racy 989 Sahoo and Sahoo 99 and Sinh 99. Let be the known population median o. Deinin e e e e and e such that Ee k and e k < or k; and are the sample median estimators based on second phase sample S m and irst phase sample S n. Let us deine the ollowin two new matrices as and > otal P z P z P z > P z P z P z otal P z P z > otal P yz P yz P yz > P yz P yz P yz otal P yz P yz Usin results iven in the Appendi- to the irst order o approimation we have Ee N-m N m- { } - Ee N-m N m- { } - Ee N-n N n- { } - Ee N-m N m- { } - Ee N-n N n- { } - Ee e N-m N m- {P y-}{ } - Ee e N-n N n- {P y-}{ } - Ee e N-m N m- {P yz-}{ } - Ee e N-n N n- {P yz-}{ } - Ee e N-n N n- { } - Ee e N-m N m- {P z-}{ } - 8

5 Ee e N-n N n- {P z-}{ } - Ee e N-n N n- {P z-}{ } - Ee e N-n N n- {P z-}{ } - Ee e N-n N n- - it is assumed that as N the distribution o the trivariate variable approaches a continuous distribution with marinal densities y and z or and respectively. his assumption holds in particular under a superpopulation model ramework treatin the values o in the population as a realization o N independent observations rom a continuous distribution. We also assume that and are positive. Under these conditions the sample median mean and variance is consistent and asymptotically normal ross 98 with N m N m { } In this paper we have suested a class o estimators or usin inormation on two auiliary variables and in double samplin and analyzes its properties.. SUESED LASS O ESIAORS otivated by Srivastava 97 we suest a class o estimators o o as { : u v }. u v and uv is a unction o u and v such that and such that it satisies the ollowin conditions.. Whatever be the samples S n and S m chosen let uv assume values in a closed conve subspace P o the two dimensional real space containin the point.. he unction uv is continuous in P such that.. he irst and second order partial derivatives o uv eist and are also continuous in P. Epandin uv about the point in a second order aylor's series and takin epectations it is ound that E n so the bias is o order n. Usin a irst order aylor's series epansion around the point and notin that we have 9

6 [ e e e e n or [ e e e e. and denote irst order partial derivatives o uv with respect to u and v respectively around the point. Squarin both sides in. and then takin epectations we et the variance o o approimation as to the irst deree ar A B. m N m n n N A y P B z P y..5 he variance o in. is minimized or P y P y z.6 hus the resultin minimum variance o is iven by min. ar P y P y z m N m n n N.7 Now we proved the ollowin theorem. heorem. - Up to terms o order n - ar m N m n n N with equality holdin i y P y P y z

7 z z z P y P y z It is interestin to note that the lower bound o the variance o y at. is the variance o the linear reression estimator d d y p y p y z l d d.8 with p y and p y z bein the sample analoues o the p y and p y z and and can be obtained by ollowin Silverman 986. respectively Any parametric unction uv satisyin the conditions and can enerate an asymptotically acceptable estimator. he class o such estimators are lare. he ollowin simple unctions uv ive even estimators o the class β u u v u v u v β v { u v u β v u v u β v } 5 β u v wu wv w w 6 β 7 u v u v u v ep{ u β v } Let the seven estimators enerated by i uv be denoted by i u v i i to7. It is easily seen that the optimum values o the parameters βw i i- are iven by the riht hand sides o.6.. A WIDER LASS O ESIAORS he class o estimators. does not include the estimator d d d d d bein constants.

8 However it is easily shown that i we consider a class o estimators wider than. deined by o is a unction o u v. u and v such that denotin the irst partial derivative o with respect to and.. Proceedin as in Section it is easily seen that the bias o terms the variance o is iven by is o the order n and up to this order o ar n N [ m N m n P y P y z and denote the irst partial derivatives o u and v respectively around the point.. he variance o is minimized or P y P y z. Substitution o. in. yields the minimum variance o as min. ar min.ar [ m N m n P y P y z. hus we established the ollowin theorem. heorem. - Up to terms o order n - n N

9 ar m N m n n N with equality holdin i P y P y z P y P y z I the inormation on second auiliary variable z is not used then the class o estimators reduces to the class o estimators o as.5 H H u H u is a unction o u H such that H and H H H. he estimator is reported by Sinh et al. he minimum variance o H to the irst deree o approimation is iven by min.ar H m N m n P y.6 rom. and.6 we have minar H min.ar P y z n N.7 which is always positive. hus the proposed class o estimators H considered by Sinh et al. is more eicient than the estimator. ESIAOR BASED ON ESIAED OPIU ALUES We denote P y P y z.

10 In practice the optimum values o - and - are not known. hen we use to ind out their sample estimates rom the data at hand. Estimators o optimum value o and are iven as. p y p y z. Now ollowin the procedure discussed in Sinh and Sinh 9 and Srivastava and Jhajj 98 we deine the ollowin class o estimators o based on estimated optimum as is a unction o u v such that u v and such that it satisies the ollowin conditions: u v.. Whatever be the samples S n and S m chosen let u v assume values in a closed conve subspace S o the our dimensional real space containin the point.. he unction uv continuous in S.. he irst and second order partial derivatives o u v S. Under the above conditions it can be shown that E n est. and are also continuous in

11 and to the irst deree o approimation the variance o is iven by min.ar is iven in.7. min.ar ar.5 A wider class o estimators o based on estimated optimum values is deined by u v.6 p y p y z.7 are the estimates o and is a unction o u v such that P y P y z.8 u v 5

12 5 Under these conditions it can be easily shown that E n and to the irst deree o approimation the variance o is iven by min.ar is iven in.. ar min.ar.9 It is to be mentioned that a lare number o estimators can be enerated rom the classes based on estimated optimum values. and 5. EIIEN O HE SUESED LASS O ESIAORS OR IED OS he appropriate estimator based on on sinle-phase samplin without usin any auiliary variable is whose variance is iven by ar 5. m N In case when we do not use any auiliary character then the cost unction is o the orm -m and are total cost and cost per unit o collectin inormation on the character. he optimum value o the variance or the ied cost is iven by Opt. ar 5. N 5. When we use one auiliary character then the cost unction is iven by m 5. n is the cost per unit o collectin inormation on the auiliary character. H he optimum sample sizes under 5. or which the minimum variance o is optimum are 6

13 m opt / 5.5 [ n opt [ / P y-. Puttin these optimum values o m and n in the minimum variance epression o H the optimum min.ar as H in.6 we et [ H Opt. min.ar 5.7 N Similarly when we use an additional character then the cost unction is iven by m n 5.8 is the cost per unit o collectin inormation on character. It is assumed that > >. he optimum values o m and n or ied cost which minimizes the minimum variance o or.7 or. are iven by m opt [ 5.9 n opt [ 5. P yz-. he optimum variance o Opt or correspondin to optimal two-phase samplin stratey is [ min.ar or min.ar [ Assumin lare N the proposed two phase samplin stratey would be proitable over sinle phase samplin so lon as [ Opt.ar > Opt. [ min.ar or min.ar N 5. 7

14 8 < i.e. 5. When N is lare the proposed two phase samplin is more eicient than that Sinh et al stratey i [ [ H min.ar Opt or min.ar min.ar Opt < i.e. < ENERALIED LASS O ESIAORS We suest a class o estimators o as { } w v u : I 6. w v u / / / and the unction assumes a value in a bounded closed conve subset W R which contains the point and is such that denotin the irst order partial derivative o with respect to around the point. Usin a irst order aylor's series epansion around the point we et n w v u 6. and denote the irst order partial derivatives o w v u with respect to u v and w around the point respectively. Under the assumption that and we have the ollowin theorem. heorem 6.. Any estimator in I is asymptotically unbiased and normal. Proo: ollowin Kuk and ak 989 let P P and P denote the proportion o and values respectively or which and ; then we have n P p n P p n P p n P p z

15 9 and n P p z Usin these epressions in 6. we et the required results. Epression 6. can be rewritten as w v u or e e e e e 6. Squarin both sides o 6. and then takin epectation we et the variance o to the irst deree o approimation as ar A N n A n m A N m 6. z y P A z P y P A z z y P A he ar at 6. is minimized or

16 [ a a P y P z P y z [ P z say P z [ P y P y z P z [ P z [ a say P y z P y P z [ P z say 6.5 hus the resultin minimum variance o is iven by minar min.ar m N m n n N m n P z P y z D [ P z [ P y z P y P. D P y 6.6 D z 6.7 and min.ar is iven in. Epression 6.6 clearly indicates that the proposed class o estimators is more eicient than the class o estimator H or and hence the class o estimators suested by Sinh et al and the estimator at its optimum conditions. he estimator based on estimated optimum values is deined by { : u v w a a } p a 6.8

17 a a a [ p y p z p y z [ p z p z [ p y p y z p z p z [ [ p y z p y p z p z [ 6.9 are the sample estimates o a a and a iven in 6.5 respectively is a unction o u v w a a a such that a u a v a w a 5 a 6 a 7 a a a Under these conditions it can easily be shown that E n

18 and to the irst deree o approimation the variance o is iven by ar min.ar is iven in 6.6. min.ar 6. Under the cost unction 5.8 the optimum values o m and n which minimizes the minimum variance o is 6.6 are iven by m opt [ / 6. n opt [ / D [ P z or lare N the optimum value o min.ar is iven by Opt. [ 6. [ min.ar 6. he proposed two-phase samplin stratey would be proitable over sinle phase-samplin so lon as Opt. ar > Opt. min.ar [ [ i.e. c < 6. It ollows rom 5.7 and 6. that Opt. [ min.ar < Opt. min.ar H [ i > 6.5 or lare N. urther we note rom 5. and 6. that

19 [ min.ar Opt. min.ar Opt. < [ or i < 6.6 REERENES hand L. 975: Some ratio-type estimators based on two or more auiliary variables. Unpublished Ph.D. dissertation Iowa State University Ames Iowa. rancisco.a. and uller W.A. 99: Quntile estimation with a comple survey desin. Ann. Statist Kireyera B. 98: A chain ratio-type estimator in inite population double samplin usin two auiliary variables. etrika Kireyera B. 98: Reression-type estimators usin two auiliary variables and the model o double samplin rom inite populations. etrika 5-6. Kuk..A. and ak.k. 989: edian estimation in the presence o auiliary inormation. J.R. Statist. Soc. B Sahoo J. and Sahoo L.N. 99: A class o estimators in two-phase samplin usin two auiliary variables. Jour. Ind. Statist. Assoc. 7-. Sinh S. Joarder A.H. and racy D.S. : edian estimation usin double samplin. Aust. N.. J. Statist. -6. Sinh H.P. 99: A chain ratio-cum-dierence estimator usin two auiliary variates in double samplin. Journal o Raishankar University 6 B Science Srivenkataramana. and racy D.S. 989: wo-phase samplin or selection with probability proportional to size in sample surveys. Biometrika Srivastava S.K. 97: A eneralized estimator or the mean o a inite population usin multiauiliary inormation. Jour. Amer. Statist. Assoc Srivastava S.K. and Jhajj H.S. 98: A class o estimators o the population mean usin multi-auiliary inormation. al. Statist. Assoc. Bull