Keywords: Auxiliary variable, Bias, Exponential estimator, Mean Squared Error, Precision.

Transcription

1 IN: IO 9:8 Ctifid Intnational Jounal of Engining cinc and Innovativ Tchnolog (IJEIT) Volum 4, Issu 3, Ma 5 Imovd Exonntial Ratio Poduct T Estimato fo finit Poulation Man Ran Vija Kuma ingh and Ahmd Audu nio Lctu & Had, Datmnt of Mathmatics, Kbbi tat Univsit of ci. and Tch. Alio, Nigia Datmnt of Mathmatics, Usmanu Danfodio Univsit okoto, Nigia Abstact- In this a, an xonntial atio-oduct t stimato has bn oosd fo oulation und siml andom samling schm. Exssion fo th bias and ME of th oosd stimato hav bn divd u to fist od aoximation. Th otimum ME of th oosd stimato has bn obtaind. Th fficinc of th oosd stimato has bn comad thoticall and miicall with xisting stimatos. Th miical comaison shows that th oosd stimato is mo fficint than oths fo both whn th colations btwn th stud and auxilia vaiabls a ositiv and ngativ. Kwods: Auxilia vaiabl, Bias, Exonntial stimato, Man quad Eo, Pcision. I. INTRODUCTION It is wll known fact that th us of auxilia infomation at th stimation stag imovs th cision of stimats of th oulation man of chaactistic und stud. Classical atio, oduct and lina gssion stimatos a good xamls in this contxt. If th stud vaiabl Y is ositivl colatd with auxilia vaiabl, th atio mthod of stimation intoducd b [] is mo alicabl in actic whil th oduct stimato intoducd b [] is mo usful whn th stud vaiabl Y is ngativl colatd with auxilia vaiabl. Lat on, statisticians concntatd thi attntion to dvlo modifid atio and oduct t stimatos. uch modifid stimatos a gnall dvlod ith using on o mo unknown constants o intoducing a convx lina combination of saml and oulation mans of auxilia chaactistic with unknown wights. In both th cass, otimum choics of unknown aamts a mad b minimizing th man squa o of modifid stimatos so that th bcom suio than th convntional on. In th squnc of suggsting modification ov classical atio and oduct stimatos [4] - [9] considd a atio t stimatos with th us of wightd man of and in lac of in classical atio and oduct stimatos. []- [] do som oth makabl woks in this diction.[3]oosd nw xonntial atio and oduct ts stimatos fo stimating th man of th finit oulation using infomation on singl auxilia vaiabl.[4] hav suggstd imovd stimatos fo stimating unknown oulation man of stud vaiabl Y. In this a, a xonntial atio-oduct t stimato has bn suggstd fo stimating finit oulation man of chaactistic und stud. Lt II. NOTATION dnots th valu of chaactistic und stud fo th i th unit in oulation of siz N (i=, N). and Th valu of auxilia chaactistic fo th i th unit in oulation. Thn = ; Th oulation man of chaactistic und stud. = ; Th oulation man of auxilia chaactistic. ;Th atio of oulation mans = ; Th oulation man squa of chaactistic und stud. = ; Th oulation man squa of auxilia chaactistic. 37

2 IN: IO 9:8 Ctifid Intnational Jounal of Engining cinc and Innovativ Tchnolog (IJEIT) Volum 4, Issu 3, Ma 5 = ; Th cofficint of vaiation of chaactistic und stud. = ; Th cofficint of vaiation of auxilia chaactistic. ρ = ; Th Colation cofficint btwn th valu of auxilia vaiabl and valu of chaactistic und stud. Lt a saml of siz n has bn dawn b mthod of siml andom samling without lacmnt. Futh lt and dnot th valus of chaactistic und stud and auxilia chaactistic sctivl which is includd in th saml at i th daw (i=,, 3,...,n). Now, = ; Th saml man of stud chaactistic. = ; Th saml man of auxilia chaactistic. III. EITING ETIMATOR Th saml man stimato is an unbiasd stimato of oulation man Y, Th classical atio and oduct stimatos of oulation man of th stud vaiabl Y a sctivl, dfind as x and () x [3] suggstd xonntial atio-t and oduct t stimatos fo oulation man Y, sctivl, as R x x x and x P x x [5] suggstd a class of atio t stimatos fo oulation man Y as g x x x x To th fist dg of aoximation, th vaiancs/man squa os of th stimatos and g sctivl, givn as () (3),,,, R P (5) (6) (7) (8) () (9) (4) 38

3 IN: IO 9:8 Ctifid Intnational Jounal of Engining cinc and Innovativ Tchnolog (IJEIT) Volum 4, Issu 3, Ma 5 IV. PROPOED ETIMATOR Motivatd b wok don b [3] and oths xonntial atio-oduct t stimato fo oulation man has bn oosd as: x x k x k x x x Wh k is constant. Whn k=, th oosd stimato ducs to an stimato which is modifid ov xonntial atio t stimato suggstd b [3]. Whn k=, th oosd stimato ducs to an stimato which is modifid ov xonntial oduct t stimato suggstd b [3]. V. BIA AND MEAN QUARE ERROR (ME) OF To obtain th bias and man squa o, lt us suos and x Y thus, () = (3) Fom () and () w hav, Y Y k k k Th bias of th oosd stimato and substituting sults obtaind in (3) as () (4) to tms of od n can b obtaind b taking th xctation of (4) B k R n N 4 4 (5) quaing both sids of (4), thn taking xctation and using sults in (3), w obtain th ME of th stimato to tms of od n as ME k k R n N 4 Fom (6) th otimum valu of k is obtaind as k R R Y c B substituting th otimum valu of k in (6), w gt th minimum ME of as ME min Y n N (8) This is th ME of th lina gssion stimato fo oulation man. Hnc fo otimum valu of k th oosd stimato is quall fficint as th lina gssion stimato. VI. EFFICIENCY COMPARION In this sction fficinc of th oosd stimato is comad with that of xisting stimatos and conditions a obtaind und which th oosd stimato is mo fficint. (6) (7) 39

4 IN: IO 9:8 Ctifid Intnational Jounal of Engining cinc and Innovativ Tchnolog (IJEIT) Volum 4, Issu 3, Ma 5 ( i) va ME R 4 Fo th ang of k givn in (9) th oosd stimato ME ( ii) ME 5 3 R if min, k max, R R (9) is btt than. if min, k max, R R () Fo th ang of k givn in () th oosd stimato ME ( iii) ME 3 5 R if min, k max, R R Fo th intval of k givn in () th oosd stimato R ME ( iv) ME 3 R if min, k max, R R Fo th intval of k givn in () th oosd stimato R suggstd b [3]. P ME ( v) ME 3 R 4 is btt than atio stimato. is btt than oduct stimato. () () is mo fficint than th atio t stimato 3

5 if IN: IO 9:8 Ctifid Intnational Jounal of Engining cinc and Innovativ Tchnolog (IJEIT) Volum 4, Issu 3, Ma x x min, k max, R R x x Fo th intval of k givn in (3) th oosd stimato P suggstd b [3]. (3) is mo fficint than th oduct t stimato if min, k max, R R Fo th intval of k givn in (4) th oosd stimato ( and suggstd b [5]. (4) is mo fficint than th atio t stimato VII. EMPIRICAL TUDY Th fficinc of th oosd stimato has bn comad ov saml man stimato, classical atio and oduct stimato, xonntial atio t stimato suggstd b [3] and class of atio-t stimato suggstd b [5] using th oulation data sts. Th Dscitions of th oulations a givn in Tabl I. Tabl I: Dscition and aamts of th oulations Paamts Poulation I Poulation II Poulation III ouc: Cochan 977, ag 86 Y: Total no. of Mmbs : No. of Childn ouc: Cochan 977, ag 34 Y: Food Cost : Famil iz ouc: U Envionmntal Potction Agnc 99 Y: Avag mils gallon :Engin hosow N 33 8 n x x x R a Y k k ha, Tabl II: Effctiv angs and otimum valu of k of Poulation Rangs of k fo which is btt than R P kot I (.5,.33) (.33,.5) (-.5,4.33) (.33,.5) (-.5, 3.33) (-.67,3.5).4 II (.5, 3.5) (.5,.5) (-.5,5.5) (.5,.5) (-.5,.5) (.5, 3.5).78 III (.5,.4) (.4,.5) (-.5,4.4) (.4,.5) (-.5, 3.4) (-.58,3.5).46 3

6 IN: IO 9:8 Ctifid Intnational Jounal of Engining cinc and Innovativ Tchnolog (IJEIT) Estimato Volum 4, Issu 3, Ma 5 Tabl III: Pcntag lativ fficincis of diffnt stimatos with sct to R P Poulation I Poulation II Poulation III VIII. CONCLUION Th conditions, und which th oosd stimato is mo fficint than th considd stimatos a givn in sction 6 but tabl II ovids th angs of k fo which is btt than oths. Th stud of Tabl 3 vals that th oosd stimato is mo fficint than th oth stimatos as,, c und otimal condition fo all th th st of Poulations., R, P and REFERENCE [] W.J. Cochan, Th stimation of th ilds of th cals ximnts b samling fo th atio of gain to total oduc Th jounal of agicultual scinc,3,6-75(94). [] M.N. Muth, Poduct mthod of stimation, ankha, 6: 94-37(964). [3] H. O. Hatl and A. Ross, Unbiasd atio stimatos, Natu,74,. 7 7(954). [4] J. K. Walsh, Gnalization of atio stimato fo oulation total, ankha, 3A, 99-6 (97). [5]. K. Ra, A. ahai and A. ahai, A not on atio and oduct stimatos, Annals of th institut of Mathmatical tatistics, 3,4-44,(979). [6] T. ivnkataamana and, D.. Tac, On atio and oduct mthods of stimation in samling statistica, Nlandica,, ,(979). [7] J. W. E. Vos, Mixing of dict atio and oduct stimatos, tatistics Nlandica, 34,9-8, (98). [8]. K. Ra and A. ahai, Efficint familis of atio and oduct stimatos, Biomica, 67, 5-7, (98). [9] T. ivnkataamana, A dual to atio stimatos in saml suvs, Biomtica 67(), 99 4, (98). [] R. V. K. ingh and B. K. ingh, stud of a Class of Ratio T Estimato und Polnomial Rgssion Modl Pocding Of Mathmatical ocit., BHU, Vol 3(7). [] C. T. Isaki, Vaianc stimation using auxilia infomation, Jounal of th Amican tatistical Association, (983). [] D. N. hah and M. R. Guta, An fficinc comaison of dual,atio and oduct stimatos Communication in tatistics: Tho and Mthods, 6(3),4-. (987). [3]. Bhal and R. K. Tutja, Ratio and oduct t xonntial stimato.infomation and Otimization cincs. : 59-63, (99). [4] L. N. Uadhaa, H.P. ingh,. Chattj and R. Yadav, Imovd atio and Poduct xonntial t stimatos, Jounal of tat. Tho. and Pac. 5(): 85-3, (). [5] R.. olanki, H. P. ingh and A. Rathou, An atnativ stimato fo stimating th finit oulation man using auxilia infomation in saml suvs, IRN Pobabilit and tatistics. Volum : doi:.54//65768, (). 3