On the Estimation Of Population Mean Under Systematic Sampling Using Auxiliary Attributes

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1 Oriental Journal of Physical Sciences Vol 1 (1 & ) 17 (016) On the Estimation Of Poulation Mean Under Systematic Samling Using Auxiliary Attributes Usman Shahzad Deartment of Mathematics Statistics PMAS Arid Agriculture University Rawalindi Pakistan *Corresonding authors usmanstat@yahoocom (Received: August 016; Acceted: Setember 0 016) ABSTRACT aik Guta (1996) Singh et al (007) AbdElfattah et al (010) introduced some estimators for estimating oulation mean using available auxiliary attributes under simle rom samling scheme We adat these estimators under systematic rom samling scheme using available auxiliary attributes Further a new family of estimators is roosed for the estimation of oulation mean under systematic rom samling scheme The roerties such as bias mean square error of the roosed estimators are derived From numerical illustration it is shown that roosed estimators are more efficient than the reviewed ones Keywords: Mean square error attributes Study variable Systematic rom samling ITRODUCTIO Systematic rom samling is the simlest tye of samling scheme requires only one rom start It rovides good results in some situations like; forest regions for assessing the volume of the timber etc For details see Murthy (1967) Cochran (1977) When sulementary information is available Swain (1964) Shukla (1971) Singh Solanki (01) Singh et al (01) have develoed some estimators for using available sulementary information But none of these have aid their attention towards auxiliary attributes So in our work we utilize available auxiliary attributes In the theory of survey samling sulementary information lays a vital role for increasing the efficiency of oulation arameters A number of authors have develoed estimators based on auxiliary information Another way to enhance the efficiency of an estimator is to utilize auxiliary attributes aik Guta (1996) Singh et al (007) AbdElfattah et al (010) Solanki Singh (01) Koyuncu (01) introduced various estimators utilizing available auxiliary attributes under simle rom samling Taking motivation from these we are going to roose a family of estimators under systematic rom samling scheme using available auxiliary attributes Preliminaries Adated Estimators Let l be the finite oulation having units 1 to Further we consider =nk where n k are ositive whole numbers Hence there will be k samles of size n Let M be rom variable having range 1 to k The systematic rom samle is then selected by the following rom sequence as { l l+ k l ( 1 ) } + n k

2 18 Shahzad Orient J Phys Sciences Vol 1 (1 & ) 17 (016) Let denote the values of the study variable auxiliary attribute for (i=1 k) (j=1 n) ote that is the binary character so it can take only two ossible values ie =1 if the ith unit of the oulation ossesses attribute F =0 otherwise Let A= a= denote the total number of units in the oulation samle resectively ossessing an auxiliary attribute F Hence the corresonding samle oulation roortions are y j 1 ij Similarly y = j= 1 ij Y = y = n are the samle oulation means of Y For finding MSE Let we define n E E E E using these notations we have where is the intraclass correlation of P is the intraclass correlation of Y is the correlation between P Y is The variance of the traditional samle mean V Following aik Guta (1996) we roose the usual ratio roduct estimators utilizing available auxiliary attributes under systematic rom samling scheme Table 1: Some members of roosed class a + b ap + b t = m1y r 1( ) ( ) + b Φ t1 = m1y rp + b1 Φ r + C P t = m1y rp+ CP r ( ) ( ) t 3 = m1y rp + b Φ + b Φ + b ( Φ) ( Φ) t 4 = m1y P + b + C P t 5 = m1y P+ CP CP + b ( Φ) t 6 = m1y CPP+ b ( Φ) a b r b 1 (F) r r C b (F)w 1 b (F) 1 C C b (F)

3 19 Shahzad Orient J Phys Sciences Vol 1 (1 & ) 17 (016) On the lines of AbdElfattah et al (010) we roose the family of estimators utilizing available auxiliary attributes under systematic rom samling scheme The MSEs of are Motivated by Singh et al (007) we roose the exonential ratio roduct estimators utilizing available auxiliary attributes under systematic rom samling scheme The MSEs of are The minimum MSE of these estimators ( ) is equal to the MSE of regression estimator ie MSE = Where Solanki Singh (013) develoed the generalized estimator given below Where by utting (α=0 1 1) we get resectively The Proosed Family of Estimators Taking motivation from AbdElfattah et al (010) we roose the following family of Table : MSEs of Adated Proosed Estimators Est Po 1 Est Po 1 Est Po Est Po ˆt t ˆ 1 37 ˆt tˆ ˆt t ˆ 74 1ˆt t ˆ ˆt t ˆ ˆt t ˆ ˆt t ˆ ˆt t ˆ ˆt ˆ 5 t 6781 ˆt t ˆ t ˆreg 1084 t ˆ t ˆreg 5657 t ˆ

4 0 Shahzad Orient J Phys Sciences Vol 1 (1 & ) 17 (016) estimators utilizing available auxiliary attributes under systematic rom samling scheme equations equating to zero we have the following where a b be any known oulation characteristics or 1 Let we exress in terms of as follows ow by solving matrix inversion method we get the otimum values of ie where ow by simlifying we get Let we take exectation on both sides get the bias of as By utting in get minimum mean square error of ie ow squaring both sides of as ( tˆ Yˆ ) = Y + m Y 1 + e + m {1 + e } { 1 0} θ 1 + m m Y{ 1+ θe e} m Y m Y 1 o 1 1 The MSE of i s g i v e n b y ( ˆ ) Y 1 1 MSE t = + m A + m B + m m C m E m 1 F Efficiency Comarison In current section we find the efficiency conditions for the roosed estimators by looking at the mean square error of the existing estimators as given below Where Partially differentiating wrt

5 1 Shahzad Orient J Phys Sciences Vol 1 (1 & ) 17 (016) n=16 From the above mentioned conditions we can say that roosed estimators are more efficient as comare to adated estimators umerical Illustration The erformance of roosed existing estimators examined through two real data sets Poulation 1 Data is taken from Mur thy (1967) where Y=Volume of the timber F=length Descritives of the oulation are = Poulation Data is taken from Murthy (1967) where Y=Volume of the timber Y= Volume Descritives of the oulation are = Conclusion n=16 We have roosed a class of estimators for using available auxiliary attributes under systematic rom samling scheme obtained its bias minimum MSE equations All the adated estimators are comared with roosed estimators using MSE With the hel of these comarisons efficiency condition has been found where roosed estimators erform much better The theoretical conditions numerical illustrations show that roosed estimators are much better Hence it is advisable to use the roosed class of estimators References 1 aik V D Guta P C : A note on estimation of mean with known oulation roortion of an auxiliary character Jour Ind Soc Agr Stat 48(): (1996) Singh R Chouhan P Sawan Smarache F : Ratioroduct Tye Exonential Estimator for Estimating Finite Poulation Mean Using Information on Auxiliary Attribute Renaissance High Press USA 18 3 (007) 3 AbdElfattah A M ElSherieny E A Mohamed S M & Abdou O F : Imrovement in estimating the oulation mean in simle rom samling using information on auxiliary attribute Alied mathematics comutation 15(1): (010) 4 Cochran W G (1977): Samling techniques 34rd ed ew York Y John Wiley Sons 5 Shukla D: Systematic samling roduct method of estimation In Proceeding of all India Seminar on Demograhy Statistics BHU Varanasi India (1971) 6 Swain AKPC : The use of systematic samling in ratio estimate Jour Ind Stat Assoc (13) (1964) 7 Singh H P Solanki R S : An efficient class of estimators for the oulation mean using auxiliary information in systematic samling Journal of Statistical Theory Practice 6(): 7485 (01) 8 Singh R Malik S Chaudhary M K Verma H K & Adewara A A : A general family of ratiotye estimators in systematic samling Journal of Reliability Statistical Studies 5(1): 738 (01) 9 Koyuncu : Efficient estimators of oulation mean using auxiliary attributes Alied

6 Shahzad Orient J Phys Sciences Vol 1 (1 & ) 17 (016) Mathematics Comutation 18(): (01) 10 Murthy M Samling theory methods (1967) 11 Solanki R S Singh H P : Imroved estimation of oulation mean using oulation roortion of an auxiliary character Chilean journal of Statistics 4(1): 317