On The Estimation of Population Mean in Current Occasion in Two- Occasion Rotation Patterns

Transcription

1 J. Stat. Appl. Pro. 4 No (05) 305 Journal of Statstcs Applcatons & Probablt An Internatonal Journal On The stmaton of Populaton Mean n Current Occason n Two- Occason Rotaton Patterns Housla P. Sngh and Sura K. Pal *. School of Studes n Statstcs Vkram Unverst Ujjan M.P Inda. Receved: 9 Feb. 05 Revsed: 0 Apr. 05 Accepted: Apr. 05. Publshed onlne: Jul. 05. Abstract: The present work s an effort to develop some estmators of current populaton mean n two-occason successve samplng utlng the known populaton mean Z alongwth known correlaton coeffcent of the and coeffcent of kurtoss ( ) aular varable on both occasons n successve samplng. Optmum replacement polc relevant to the proposed estmators has been dscussed. Numercal llustraton s carred out and approprate recommendatons are made. Kewords: Aular varable Stud varable Bas Mean square error ffcenc Comparson. Introducton The problem of samplng on two successve occasons was frst consdered b Jessen [] who ntroduced the dea of samplng on two occasons b usng the nformaton gathered on the prevous occasons to mprove the precson of the current estmate. Later several authors ncludng Patterson [] Naran [3] ckler [4] Rao and Graham [5] Gordon [6] Arnab and Okafor [7] and Sngh et al. [8] among others have developed the theor of successve samplng. Sen [9] appled ths theor n desgnng the strateges for estmatng the populaton mean on the current occason usng nformaton on two aular varables. Sen [0 ] etended hs work for multple aular varables. Fen and Zou [] and Bradar and Sngh [3] used the aular nformaton on both occasons for estmatng the current populaton mean n the successve samplng. Sngh [4] Sngh and Vshwakarma [5 6 7] have used the aular nformaton on both occasons and envsaged several estmators for estmatng the populaton mean on current (second) occason n two - occason successve (rotaton) samplng. Recentl Sngh and Pal [8 9] have suggested some estmators utlng the known populaton mean Z alongwth known coeffcent of varaton C and Standard devaton S of the aular varable on both occasons n successve samplng for estmatng the current (second) populaton mean n two occason successve samplng. Motvated b Sngh and Pal [8 9] Sngh and Talor [0] and Sngh et al. [] we have suggested some estmators utlng the known populaton mean Z alongwth known correlaton coeffcent and coeffcent of kurtoss ( ) of the aular varable on both occasons n successve samplng for estmatng the current (second) populaton mean n two occason successve samplng. The Suggested Class of stmators Let U= (U U U N ) be the fnte populaton of se N unts whch has been sampled over two occasons. The character under stud be denoted b () on the frst (second) occason respectvel. It s assumed that nformaton on populaton coeffcent of Kurtoss ( ) of an aular varable along wth known populaton mean Z s avalable for the both the occasons. As ponted b Sngh and Talor [0] and Saha and Saha [] the populaton correlaton coeffcent between the stud varable and aular varable can also be made known from the post data or eperence gathered n due to course of tme. So we have assumed that s also known along wth the populaton mean Z of an aular * Correspondng author e-mal: surakantpal6676@gmal.com 05 NSP Natural Scences Publshng Cor.

2 306 H. Sngh S. Pal: On The stmaton of Populaton Mean varable on both occasons. In such stuaton we have to proposed two combned classes of estmators of the populaton mean Y on the current (second) occason one s based on ( Z ) and second on ( ( ) Z ). A smple random sample of n unts s drawn wthout replacement (WOR) on the frst occason. A random sub sample of m (= n λ) unts s retaned (matched) from the sample drawn on the frst occason for ts use on the current (second) occason whle a fresh sample of se u= (n-m) = n unts s drawn on the current (second) occason from the entre populaton b smple random samplng wthout replacement (SRSWOR) method so that the sample se on the current (second) s also n. The fractons of the matched and fresh samples are respectvel desgnated b λ and such that λ+ =. The followng notatons have been used throughout the paper. X Y Z : The populaton means of the varables and respectvel. m n u suffces. C C m u n : The sample means of the respectve varables based on C : The coeffcents of varaton of the varables and respectvel the sample ses ndcated n : The correlaton coeffcents between the varables shown n suffces. S N ( N ) ( X ) S N ) N ( ( Y ) S ( N ) ( Z ) are The populaton mean squares of and respectvel f n / N : The samplng fracton. For estmatng the populaton mean Y on the second (current) occason two dfferent sets of estmators are proposed. One set of estmators } based on sample se u( n) drawn afresh on the second occason and the second set u { u u { m m of estmators m } based on the sample se m( n) common wth both occasons. stmators of sets u and are gven below: m Z u u u (.) Z ( ) u u u ( ) (.) And Z n m m m n (.3) n Z ( ) m m. m n ( ) (.4) Where ( ( ) ) are known correlaton coeffcent between and and coeffcent of kurtoss. Combnng the estmators of sets (current) occason: () when s known along wth populaton mean Z : u and m we have the followng estmators of the populaton mean Y at the second (.5) u ( ) m N 05 NSP Natural Scences Publshng Cor.

3 J. Stat. Appl. Pro. 4 No (05) / () when ( ) s known along wth populaton mean Z : u ( ) m In short we can defne the estmators (.5) and (.6) as: Where. (.6) ( ) (=) ; (.7) u (= ) are unknown constants to be determned under certan crteron. 3 Bas and Mean Squared rrors of Suggested stmators (= ) m The bases and mean squared errors of the class of estmators Theorem 3. Bases of the proposed estmators Where B( m Proof s smple so omtted. ( = ) are gven n the followng Theorems 3. and 3.. ( = ) to the frst order of appromaton are gven b B ) B( ) ( ) B( ) (3.) ( u m B( ) Y ((/ u) (/ N)) ( C C C ) (=) (3.) u ) Y[((/ m) (/ N))( C C C ) ((/ n) (/ N)) ( C Z /( Z ) And Z /( Z ( )). C C )] (3.3) Theorem 3. The mean squared errors of the suggested combned class of estmators ( = ) to frst degree of appromaton are gven b M( ) [ M( ) ( ) M( ) ( ) Cov( )](=) (3.4) Where u m M ( ) Y ((/ u) (/ N))[ C ( C C C )] u M ( Cov( u m m u m Y ((/ u) (/ N))( ) (3.5) C ) Y [(/ m) (/ n)( C ) / N( C )] * C Y [(/ m) (/ n) / N( )] (3.6) ) ( Y / N)( C ) (=) (3.7) ( C C C C ) (3.8) ( C C C ) (3.9) * ( C ) (3.0) C C C ( C C C ) ; (= ). 05 NSP Natural Scences Publshng Cor.

4 308 H. Sngh S. Pal: On The stmaton of Populaton Mean 4 Mnmum Mean Squared rror of Proposed stmators ( = ) Dfferentatng M ( ) at (3.4) wth respect to and equatng to ero we get the optmum value of [ M ( ) Cov( m u m opt [ M ( u ) M ( m ) Cov( u m )] Where A ( )} { )] as ( A B ) (4.) ( A B ) and B ( )}. { Thus the resultng mnmum MS of ( = ) s gven b mn. M ( ) [ M ( u ) M ( m ) { Cov( u m )} ] [ M ( ) M ( 5 Optmum Replacement Polc u m ) Cov( u m )] S A [( f ) A B f B ]. (4.) n ( A B ) To obtan the optmum value of (fracton of a sample to be drawn afresh on the second occason) so that the populaton mean Y ma be estmated wth mamum precson. Dfferentatng mn. M( ) at (4.) wth respect to and equatng t to ero we get B A A 0 (= ) (5.) Soluton of the above equaton s gven b A ( )A o B (= ). (5.) From (5.) two values of 0 are possble hence to choose a value of 0 t should be recommended that 0 0 all other values of 0 are nadmssble. Substtutng the value of 0 from equaton (5.) nto equaton (4.) we have 6 ffcenc Comparson S A [( f ) A 0B f B ] 0 mn. M ( ) opt (= ) (5.3) n ( A B ) The percent relatve effcences(prs) of the suggested estmators wth respect to usual unbased estmator n when there s no matchng and the estmator Y ˆ u ( ) m when no aular nformaton s used at an occason where m m ( n m ) and s the known populaton regresson coeffcent; have been obtaned for varous choces and. 0 Followng Sukhatme et al. [3] the varance of usual unbased estmator gven b n and optmum varance of Yˆ are respectvel S V ( n ) ( f ) (6.) n 05 NSP Natural Scences Publshng Cor.

5 J. Stat. Appl. Pro. 4 No (05) / And S V ( Y ˆ ) [(/ ) ( ) f ] (6.) n For N=000 n=00 and the varous choces Z and ( ) Table 6. and Table 6. depcts the optmum values of ( = ) and the percent relatve effcences (PRs) and 0 respect to And n andyˆ respectvel b usng the followng formulae: () V ( n ) mn. M ( ) opt 00 ( f )( A A [( f ) A B V ( Y ˆ ) opt mn. M ( ) opt 0 () of the suggested estmator ( = ) wth 0 00 B ) f 0 ( ) f ( 0 ) A B A [( f ) A B f B ] Fndngs are shown n Table 6. and 6.. To make the numercal values obtaned for (PRs) (6.3) B ] 0 00 (6.4) In Table 6. and 6. more comprehensble to the readers we have represented these through fgures 6. and 6. respectvel. Table 6.: The percent relatve effcences (PRs) of and Z. ( = ) wth respect to andyˆ for dfferent values of n Z 0 () Z 0 () Z 0 () NSP Natural Scences Publshng Cor.

6 30 H. Sngh S. Pal: On The stmaton of Populaton Mean Fgure 6.: The percent relatve effcences (PRs) of ( = ) wth respect to Yˆ for fed values of 0. 5 Z 0 and dfferent values of. n and Table 6.: The percent relatve effcences of (PRs) Z and ( ) ( ) Z 0 ( = ) wth respect to () Z 0 andyˆ for dfferent values of n () Z () 05 NSP Natural Scences Publshng Cor.

7 J. Stat. Appl. Pro. 4 No (05) / * Fgure 6.: The percent relatve effcences (PRs) of ( = ) wth respect to n and Yˆ for fed values of 0. 5 ( ) 5. 0 Z 0 and dfferent values of. From Table 6. t s observed that: () For fed values of ( ) the values of 0 ncreases as the value of Z ncreases whle the values of and () decrease. () for fed values of ( Z ) the values of 0 decreases as the value of ncreases Whle the values of and () ncrease. () For fed values of ( Z ) the values of 0 and () ncrease wth ncreasng value of. Ths behavor s n agreement wth Sukhatme et al. [3] results whch eplaned that more the value of fractons of fresh sample requred at the current occason. more the (v) mnmum value of 0 s ( 0. 30) whch shows that the fracton to be replaced at the current occason s as low as about 30 percent of the total sample se leadng to a reducton of consderable amount n the cost of the surve. Smlar trend and conclusons can be drawn from Table 6.. In addton to these t s observed from Tables 6. and 6. that there s apprecable gan n effcences b usng the proposed estmators ( = ) over usual unbased estmator n and the estmator Yˆ. Thus we nfer that the use of aular nformaton at the estmaton stage s hghl rewardng n terms of the proposed estmator ( = ). We also note from Table 6. (and Fgure 6.) and Table 6. (and Fgure 6.) that the proposed estmators and eld more gan over n as compared toyˆ. 05 NSP Natural Scences Publshng Cor.

8 3 H. Sngh S. Pal: On The stmaton of Populaton Mean 7 Concluson In ths artcle an effcent estmaton procedure has been developed utlng the known populaton mean Z alongwth known correlaton coeffcent and coeffcent of kurtoss ( ) of the aular varable on both the occasons for estmatng the current (second) populaton mean n two occason successve samplng. From numercal llustratons t ma be concluded that the suggested classes of estmators s more useful n estmaton of the populaton mean of the stud varable at the current occason n two-occason successve samplng. Fnall lookng on the good performance of the envsaged estmator our suggeston s to use the proposed classes of estmators n practce. 8 Acknowledgements The authors are hghl grateful to the dtor-n-chef and learned referees for ther ecellent comments whch helped us to mprove the prevous draft of the paper. References [] R. J. Jessen Statstcal nvestgaton of a sample surve for obtanng form facts. Iowa Agrcultural perment Staton Road Bulletn no. 304: Ames USA (94). [] H.D. Patterson Samplng on successve occasons wth partal replacement of unts. Jour. Ro. Statst. Assoc B 4-55 (950). [3] R. D. Naran On the recurrence formula n samplng on successve occasons. Journal of the Indan socet of agrculture statstcs (953). [4] A.R. ckler Rotaton samplng. Ann. Math. Statst (955). [5] J.N.K. Rao and J.. Graham Rotaton desgn for samplng on repeated occasons. Jour. Amer. Statst. Assoc (964). [6] L. Gordon Successve samplng n fnte populaton. The Annals of statstcs () (983). [7] R. Arnab Okafor F.C. A note on double samplng over two occasons. Pakstan Journal of Statstcs (99). [8] H.P. Sngh H.P. Sngh and V.P. Sngh A generaled effcent class of estmators of populaton mean n two-phase and successve Samplng. Int. Jour. Manage. Sstems 8() (99). [9] A.R. Sen Successve samplng wth two aular varables. Sankha B (97). [0] A.R. Sen Successve samplng wth p (p ) aular varables Ann. Math (97). [] A.R. Sen Theor and applcaton of samplng on repeated occasons wth several aular varables. Bometrcs 9: (973). [] S. Feng and G. Zou Samplng rotaton method wth aular varable. Commun. Statst. Theo. Meth. 6(6): (997). [3] R. S. Bradar and H.P. Sngh Successve samplng usng aular nformaton on both the occasons. Calcutta Statst. Assoc. Bull (00). [4] G.N. Sngh on the use of chan tpe rato estmator n successve Samplng Statstcs n Transton - new seres 7-6 (005). [5] H. P. Sngh and G.K. Vshwakarma Modfed eponental rato product estmators for fnte populaton mean n double samplng. Austran Jour. Statst. 36(3) 7 5 (007a). [6] H.P. Sngh and G.K. Vshwakarma A general class of estmators n successve samplng. Metron 65() 0-7 (007b). [7] H.P. Sngh and G.K. Vshwakarma A general procedure for estmatng populaton mean n successve samplng. Commun. Statst. Theo. Meth (009). 05 NSP Natural Scences Publshng Cor.

9 J. Stat. Appl. Pro. 4 No (05) / 33 [8] H. P. Sngh and S. K. Pal on the estmaton of populaton mean n successve samplng. Int. Jour. Math. Sc. Applca (05a). [9] H. P. Sngh and S. K. Pal on the estmaton of populaton mean n rotaton samplng. Jour. Statst. Applca. Pro. Lett. () 3-36 (05b). [0] H.P. Sngh and R. Talor Use of known correlaton coeffcent n estmatng the fnte populaton mean. Statstcs n Transton 6 (4) (003). [] H.P. Sngh Talor Rajesh Talor Rtesh and M. S. Kakran An mproved estmator of populaton mean usng power transformaton. Journal of the Indan Socet of Agrcultural Statstcs 58 () 3-30 (004). [] A. Saha and A. Saha. An effcent use of aular nformaton. Jour. Statst. Plan.Inf. 03- (985). [3] P. V. Sukhatme B. V. Sukhatme and C. Ashok. Samplng theor of surves wth applcatons 3rd ed. Ames IA Iowa State Unverst Press (984). 05 NSP Natural Scences Publshng Cor.