Research Article A Two-Parameter Ratio-Product-Ratio Estimator Using Auxiliary Information

Iteratioal Scholarly Research Network ISRN Probability ad Statistics Volume, Article ID 386, 5 pages doi:.54//386 Research Article A Two-Parameter Ratio-Product-Ratio Estimator Usig Auxiliary Iformatio Peter S. Chami,, Berd Sig, ad Doeal Thomas The Warre Alpert Medical School of Brow Uiversity, Box G-A, Providece, RI 9, USA Departmet of Computer Sciece, Mathematics ad Physics, Faculty of Sciece ad Techology, The Uiversity of the West Idiesat Cave Hill, P.O. Box 64, Cave Hill, Bridgetow, St. Michael BB, Barbados Correspodece should be addressed to Doeal Thomas, doealt@gmail.com Received 5 August ; Accepted September Academic Editors: P. E. Jorgese ad V. Makis Copyright q Peter S. Chami et al. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. We propose a two-parameter ratio-product-ratio estimator for a fiite populatio mea i a simple radom sample without replacemet followig the methodology i the studies of Ray ad Sahai 98, Sahai ad Ray 98, A. Sahai ad A. Sahai 985, ad Sigh ad Espejo 3.The bias ad mea squared error of our proposed estimator are obtaied to the first degree of approximatio. We derive coditios for the parameters uder which the proposed estimator has smaller mea squared error tha the sample mea, ratio, ad product estimators. We carry out a applicatio showig that the proposed estimator outperforms the traditioal estimators usig groudwater data take from a geological site i the state of Florida.. Itroductio We cosider the followig settig. For a fiite populatio of size N, we are iterested i estimatig the populatio mea Y of the mai variable y takig values y i for i,...,n from a simple radom sample of size where <N draw without replacemet. We also kow the populatio mea X for the auxiliary variable x takig values x i for i,...,n. Weuse the otatio y ad x for the sample meas, which are ubiased estimators of the populatio meas Y ad X, respectively. We deote the populatio variaces of Y ad X by S Y V Y N N ), Y i Y S X V X N i N, X i X). i

ISRN Probability ad Statistics respectively. Furthermore, we defie the coefficiet of variatio of Y ad X as C Y S Y Y, C X S X X,. respectively, ad the coefficiet of correlatio C betwee the two variables as C ρ CY C X,.3 where ρ S XY /S X S Y deotes the populatio Pearso correlatio coefficiet. As estimators of the populatio mea Y, the usual sample mea y, theratio estimator y R y/x X,adtheproduct estimator y P x y /X are used. Murthy ad Sahai ad Ray compared the relative precisio of these estimators ad showed that the ratio estimator, sample mea, ad product estimator are most efficiet whe C>/, / C / ad C < /, respectively. I other words, whe the study variate y ad the auxiliary variate x show high positive correlatio, the the ratio estimator shows the highest efficiecy; whe they are highly egative correlated, the the product estimator has the highest efficiecy; whe the variables show a weak correlatio oly, the the sample mea is preferred. I the paper, we say a estimator is most efficiet or has the highest efficiecy, if it has the lowest mea squared error MSE of all the estimators cosidered. For estimatig the populatio mea Y of the mai variable, we proposed the followig two-parameter ratio-product-ratio estimator: [ ) ] [ ) β x βx βx β X α βx β ) y α ) ]y,.4 X β x βx where α, β are real costats. Our goal i this paper is to derive values for these costats α, β such that the bias ad/or the mea squared error MSE of is miimal. I fact, i Sectio 5 we are able to use the two parameters α ad β to obtai a estimator y C that is up to first degree of approximatio both ubiased ad has miimal MSE N / N S Y ρ ; it was Srivastava 3, 4 who showed that this is the miimal possible MSE up to first degree of approximatio for a large class of estimators to which the oe i.4 also belogs. The estimator y C thus corrects the limitatios of the traditioal estimators y, y R,ady P which are to be used for a specific rage of the parameter C ad, i additio, outperforms the traditioal estimators by havig the least MSE. Note that y α, β, that is, the estimator is ivariat uder a poit reflectio through the poit α, β /, /. I the poit of symmetry α, β /, /, the estimator reduces to the sample mea; that is, we have y /,/ y. I fact, o the whole lie β / our proposed estimator reduces to the sample mea estimator, that is, y α,/ y. Similarly, we get y, y, xy /X y P product estimator ad y, y, yx /x y R ratio estimator. Its simplicity essetially just usig covex combiatios ad/or a ratio of covex combiatios ad that all three traditioal estimators sample mea, product, ad ratio estimators ca be obtaied from it by choosig appropriate parameters are the reasos why we study the estimator i.4 ad compare it to the three traditioal estimators.

ISRN Probability ad Statistics 3 However, i the outlook, Sectio 6.5, we also compare this estimator to more sophisticated estimators for the applicatio i the groudwater data cosidered here.. First-Degree Approximatio to the Bias I order to derive the bias of up to O /,weset e y Y Y, e x X X.. Thus, we have y Y e ad x X e, ad the relative estimators are give by ŷ y Y e, x x X e.. Thus, the expectatio value of the e i s is E e i for i,,.3 ad uder a simple radom sample without replacemet, the relative variaces are ) V y ) ) V rel y E e Y V e f ) SY, Y V rel x V x ) E e X V e f ) SX, X.4 where f /N is the samplig fractio. Also, we have E e e f ρc Y C X,.5 see, 5, 6. Furthermore, we ote that E e e O /,ade e i ej whe i j is a odd iteger. Now reexpressig.4 i terms of e i s ad by substitutig x ad y, we have [ ] [ e βe α Y e α βe βe e βe ] Y e..6 I the followig, we assume that e < mi{/ β, / β }, ad therefore we ca expad βe ad β e as a series i powers of e. We ote that mi{/ β, / β } attais its maximal value at β /. We get up to O e 3 e Y [ α β ) e α β ) β ) )] e O e 3..7

4 ISRN Probability ad Statistics We assume that the sample is large eough to make e so small that cotributios from powers of e of degree higher tha two are egligible; compare 6. By retaiig powers up to e,weget Y Y { [ e e α β ) e α β ) β ) ]} e..8 Takig expectatios o both sides of.8 ad substitutig C ρ C Y /C X, we obtai the bias of to order O as ) ) B E Y f ) [ ] ) C Y β α β α ρ C X C Y X f )[ ] β α β α C C X Y..9 Equatig.9 to zero, we obtai β or β α C αc.. The proposed ratio-product-ratio estimator, substituted with the values of β from., becomes a approximately ubiased estimator for the populatio mea Y. I the threedimesioal parameter space α, β, C R 3, these ubiased estimators lie o a plae i the case β / ad o a saddle-shaped surface, see Figure a. Furthermore, as the sample size approaches the populatio size N, the bias of teds to zero, sice the factor f / clearly teds to zero. 3. Mea Squared Error of We calculate the mea squared error of up to order O by squarig.8, retaiig terms up to squares i e ad e, ad the takig the expectatio. This yields the first-degree approximatio of the MSE MSE ) f Y { C Y C X α β )[ α β ) C ]}. 3. Takig the gradiet / α, / β of 3., weget MSE ) 4 f Y C [ ) ] ) X α β C β, α. 3.

ISRN Probability ad Statistics 5 β β C C α α a b Figure : Surface of bias-free estimators defied by.9 i the parameter space α, β, C R 3. b Surface of AOE parameters defied by 3.4. The poits of itersectio of the two surfaces see Sectio 5 are draw as black curves. Settig 3. to zero to obtai the critical poits, we obtai the followig solutios: α, β 3.3 or C α β ). 3.4 Oe ca check that the critical poit i 3.3 is a saddle poit uless C, i which case we get a local miimum. However, the critical poits determied by 3.4 are always local miima; for a give C, 3.4 is the equatio of a hyperbola symmetric through α, β /, /. Thus, i the three-dimesioal parameter space α, β, C R 3, the estimators with miimal MSE or better, miimal first approximatio to the MSE; see calculatio i 3.6 lie o a saddle-shaped surface, see Figure b. We ow calculate the miimal value of the MSE. Substitutig 3.3 ito the estimator yields the ubiased estimator y sample mea of the populatio mea Y. Thus, we arrive at the mea squared error of the sample mea: ) MSE y /,/ MSE y ) f Y C Y f S Y. 3.5 By substitutig 3.4 ito the estimator, a asymptotically optimum estimator AOE y o α,β is foud. For the first-degree approximatio of the MSE, we fid idepedet of α ad β MSE y o α,β ) f Y ) C Y C C X f S Y ρ ), 3.6

6 ISRN Probability ad Statistics that is, the same miimal mea squared error as foud i, 5 7. I fact, Srivastava 3, 4 showed that this is the miimal possible mea squared error up to first degree of approximatio for a large class of estimators to which the estimator.4 also belogs, for example, for estimators of the form y g y g x/x where g is a C -fuctio with g. I 8 it was show that icorporatig sample ad populatio variaces of the auxiliary variable might yield a estimator that has a lower MSE tha f / S Y ρ especially whe the relatioship betwee the study variate y ad the auxiliary variate x is markedly oliear. Thus, whatever value C has, we are always able to select a AOE y o from the α,β two-parameter family i.4. 4. Compariso of MSEs ad Choice of Parameters Here we compare MSE i 3. with the MSE of the product, ratio, ad sample mea estimators, respectively. It is kow see, 5 that MSE y ) V y ) f Y C Y, 4. MSE yr ) f Y { C Y C X C }, 4. MSE yp ) f Y { C Y C X C }. 4.3 4.. Comparig the MSE of the Product Estimator to Our Proposed Estimator From, 5 7, we kow that, for C< /, the product estimator is preferred to the sample mea ad ratio estimators. Therefore, we seek a rage of α ad β values where our proposed estimator has smaller MSE tha the product estimator. From 4.3 ad 3., the followig expressio ca be verified: MSE yp ) MSE ) which is positive if 4 f Y C [ ][ )] X αβ α β C αβ α β, 4.4 [ αβ α β ][ C αβ α β )] >. 4.5 We obtai the followig two cases: i C>αβ α β> if both factors i 4.5 are positive or ii C<αβ α β< if both factors i 4.5 are egative. Notig that we are oly iterested i the case C< /, we get from i >C>αβ α β>. 4.6

ISRN Probability ad Statistics 7 We ote that this implies <C< /, ad the rage for α ad β where these iequalities hold are explicitly give by the followig two cases. i If β</, the β C / β <α< β / β. ii If β>/, the β / β <α< β C / β. For ay give C, we agai ote that the two regios determied here are symmetric through α, β /, /. We also ote that the parameters α, β which give a AOE see 3.4, which for a fixed C lie o a hyperbola, are cotaied i these regios. I case ii, where C< ad therefore automatically C< /, the followig rage for α ad β ca be foud. i If β</, the β / β <α< β C / β. ii If β>/, the β C / β <α< β / β. The same remark as i the previous case applies. Furthermore, ote that, for C, the product estimator attais the same miimal MSE as our proposed estimator o the hyperbola give by 3.6.IFigure a we show the regio i parameter space α, β, C R 3 calculated here ad i the ext two sectios where the proposed estimator works better tha the three traditioal estimators. 4.. Comparig the MSE of the Ratio Estimator to Our Proposed Estimator For C>/, the ratio estimator is used istead of the sample mea or product estimator; compare, 5 7. As a result, we are cocered with a rage of plausible values for α ad β, where works better tha the ratio estimator. Takig the differece of 4. ad 3., we have MSE yr ) MSE ) 4 f Y C [ ][ )] X αβ α β C αβ α β 4.7 which is positive if [ αβ α β ][ C αβ α β )] >. 4.8 Therefore, i C > αβ α β>or ii C < αβ α β<. Hece, from solutio i, where C>, we have the followig. i If β</, the β C / β <α<β/ β. ii If β>/, the β/ β <α< β C / β. Also, from solutio ii, where / <C<, we obtai the followig. i If β</, the β/ β <α< β C / β. ii If β>/, the β C / β <α<β/ β.

8 ISRN Probability ad Statistics β.5 C Y Oct 9 Sep ).5.5 α.5.5 X Oct 8 Sep 9).5 a b Figure : a Regio of the parameter space α, β, C R 3 where our proposed estimator has lower MSE tha the traditioal estimators. b Scatterplot of study ad auxiliary variables for the groudwater data studied i Sectio 6. 4.3. Comparig the MSE of the Sample Mea to Our Proposed Estimator Fially, we compare the MSE y to our proposed estimator, MSE.From, 5 7, we kow that sample mea estimator is preferred for / C /. Takig the differece of 4. ad 3., weget MSE y ) ) MSE f Y C X α β ){ C α β )} 4.9 which is positive if α β ){ C α β )} >. 4. Therefore, either i α>/, β>/ adc>/ α β, ii α</, β>/ adc</ α β, iii α>/, β</ adc</ α β,or iv α</, β</ adc>/ α β. Combiig these with the coditio / C /, we get the followig explicit rages. i If <C / adβ>/, the / <α< β C / β from i. ii If <C / adβ</, the β C / β <α</ from iv. iii If / C<adβ>/, the β C / β <α</ from ii. iv If / C<adβ</, the / <α< β C / β from iii.

ISRN Probability ad Statistics 9 We ote that the case C implies r, ad thus the sample mea estimator is the estimator with miimal MSE ad, as already oted, y y /,/. I Figure a we show the regio i parameter space α, β, C R 3 where the proposed estimator works better tha the three traditioal estimators. Note that the surface of AOE parameters i Figure b is a subset of this regio, except for the values C, C, ad C for which our proposed estimator oly works as well as the sample mea, product, ad ratio estimator, respectively. We also remark that the poits,, ad,, ote that y, y, y R,,, ad,, ote that y, y, y R as well as the lie α, /, ote that y α,/ y belog to the surface of AOE parameters i Figure b. 5. Ubiased AOE Combiig. ad 3.4, we ca calculate the parameters α ad β where our proposed estimator becomes at least up to first approximatio a ubiased AOE. We obtai a lie with recall that o this lie our estimator always reduces to the sample mea estimator β, C 5. or a curve α C,β C,C R 3 i the parameter space with α C ± C, β C C ) ± C C. 5. We ote that the parametric curve i 5. is oly defied for C orc>/ i fact, this parametric curve is three hyperbolas. The surface of bias-free estimator parameters i Figure a ad the surface of AOE parameters i Figure b oly itersect i these three hyperbolas ad the lie β / adc. I the regio <C / of the parameter space α, β, C R 3, we have the commo situatio where miimisig MSE comes with a trade-off i bias. The curves of itersectio are icluded i Figure. Explicitly, our proposed estimator usig the values i 5. is give by y C y α C,β C C X C x C C ) ) X x ) y. 4Xx C C X x 5.3 At first it might seem surprisig that this estimator y C is also defied i the regio <C /. The deomiator vaishes if C ± 9 3Xx/ X x /4. However, oe ca also let the parameters α, β i the defiitio of our proposed estimator i.4 be complex umbers but such that we still get a real estimator. Oe ca check that α C ad β C i 5. for <C</have this property.

ISRN Probability ad Statistics Furthermore, we ca check that the first degree of approximatio of the bias ad MSE of y C are give by B y C ), MSE y C ) f S Y ρ ) 5.4 compare.9 ad 3.6. Thus, the estimator y C of 5.3 is a ubiased AOE. Oe might also ask whether iside <C / there is a choice of real parameters α, β R such that we get a AOE with small bias. Usig 3.4 i.9, wegetthefirstdegree approximatio of the bias of a AOE B y o α,β ) f C X Y [ C C β ) ]. 5.5 From this expressio ad the costrait 3.4 it is clear that the bias ca oly be made zero if C orc /. Otherwise, there is always a positive cotributio comig from the term C C that does ot vaish o matter what we choose for β R. I fact, it looks as if the choice β / always yields the least possible bias; however two remarks are i order here. Firstly, give 3.4 ad uless C, we ca oly let β be close to / ad choose α accordigly the absolute value α is the large. Secodly, we already oted that y α,/ y, ad the MSE for the sample mea estimator is f / S Y,ot f / S Y ρ as for a AOE. We have arrived here at a poit where the first-degree approximatio to bias ad MSE breaks dow. To fid a choice of real parameters for give C with miimal MSE ad least bias, higher degrees of approximatio would have to be cosidered. 6. Applicatio ad Coclusio Usig data take from the Departmet of the Iterior, Uited States Geology Survey 9, site umber 9895 located i Florida, a compariso of our proposed estimator to the traditioal estimators was carried out. The study variables deoted by Y are take to be the maximum daily values i feet of groudwater at the site for the period from October 9 to September. The auxiliary variables deoted X are take as the maximum daily values i feet of groudwater for the period from October 8 to September 9. Our goal is to estimate the true average maximum daily groudwater Y for the period from October 9 to September. The questios we ask are as the follows. How may uits of groudwater must be take from the populatio Y to estimate the populatio mea Y withi d % at a 9% cofidece level α.? Ad how well do the estimators perform give this data set with auxiliary iformatio for the calculated sample size? Usig the etire data set, we calculate the followig parameters: Y.583, X.677, S Y.448, S X.7, ρ.95, C Y.768, C X.54, ad C.69. A scatterplot of the data set is show i Figure b, which adds emphasis to the positive measure of associatio betwee the study variable Y ad the auxiliary variable X. Oe should ote that the value of C.69 lies i the iterval /,, sowe choose values of α ad β from Sectio 4. resp., from Sectio 5. Ideed, we use 5. ad choose β β.69.376 ad α α.69.3349. Note that β.376 yields

ISRN Probability ad Statistics.874 <α<.573 i Sectio 4.. Usig the otatio of Sectio 5, wealsootethat y.3349,.376 y.69. 6.. Calculatig the Sample Size To estimate the populatio mea amout of groudwater recorded for the state of Florida from October 9 to September, a sample of size is draw from the populatio of size N 365 accordig to the simple radom samplig without replacemet, see.afirst approximatio to this sample size eeded is the ifiite populatio value Z α/ σ, 6. d where d is the chose margi of error from the estimate of Y ad Z α/ is a stadard ormal variable with tail probability of α/. Accoutig for the fiite populatio size N, we obtai the sample size / /N. 6. I geeral, the true value of σ is ukow but ca be estimated usig its cosistet estimator s. However, i our case σ is calculated from the populatio ad is give as S Y.6. Therefore, with α., that is, Z.5.6449, ad d % of Y i.e., d.583, the sample size ca be calculated as follows: we have.6449.6 /.583 59.59 ad roudig up gives 6; so, we get / /6 /365.3 ad thus take. 6.. Relative Efficiecies Table shows the relative efficiecies of the traditioal estimators sample mea y, ratioy R ad product y P estimators ad our proposed two-parameter ratio-product-ratio estimator for the parameters α, β.3349,.376. We ote that, with this choice of parameters, the estimator is a ubiased AOE, amely, y.3349,.376 y.69. The table shows that our two-parameter ratio-product-ratio estimator domiates the traditioal estimators i the sese that it has the highest efficiecy. We ca also observe that, i the computatio of the relative efficiecy, the specificatio of the sample size is ot importat sice the fiite populatio correctio factor f / is caceled out however, this would ot be the case for higher degrees of approximatio. 6.3. Costructig a 9% Cofidece Iterval for Y Usig Costructig a 9% cofidece iterval, the followig formulatio ca be used similar formulae hold for all estimators discussed here, see : S y Y.69 ± Z.5 N. 6.3 N The factor N / N is the fiite populatio correctio.

ISRN Probability ad Statistics Table : Relative efficiecies comparisos. MSE y /MSE y MSE y /MSE y R MSE y /MSE y P MSE y /MSE y.69 % 96.% 6.73% 597.8% Table : Compariso of the estimators accordig to the absolute deviatio from the populatio mea Y i simulatios. y.69 Y < y R Y < y.69 Y < y Y < y P Y < y Y < y Y < Deviatio from populatio mea y R Y < y Y < y.69 Y < y Y < y.69 Y < y Y < y.69 Y < y P Y < y Y < y R Y < y R Y < y.69 Y < y P Y < y.69 Y < Couts y P Y 3 73 y P Y 978 y P Y 58 y P Y 97 y R Y 766 y R Y 38 y R Y 75 Of course, by the choice of the sample size, we get a margi of error give by approximately. Y.583; more precisely, the calculatio usig the above formula yields y.69 ±.58. 6.4. Compariso of Estimators To compare the proposed estimator with the traditioal oes, we selected times a sample of size ad calculated the estimators from it. We ote that there are 365 ).5 96 possibilities to choose data poits out of a total 365 without replacemet. I Table we show the relative positio of the estimators with respect to the populatio mea Y. I the simulatios, our proposed estimator outperformed the traditioal estimators o 4 7 occasios. The ratio estimator, the suggested estimator for this value of C by, performs better tha our proposed estimator 978 times i these cases it is actually the best of the studied estimators; ote that the ratio estimator is the worst 349 times. I Table 3, we compare the estimators by lookig at the followig criteria. The coverage probability is the proportio of the 9% cofidece iterval coverig the populatio mea Y;as expected, the usual mea sample estimator yields aroud 9%, while the ratio estimator ad our proposed estimator yield much higher values i this simulatio, all itervals calculated from our proposed estimator cover Y. For those 9% cofidece itervals that do ot cover Y, we check whether they lie to the left egative bias or to the right positive bias of Y. We also state the statistical iformatio lower ad upper quartile ad media that we get from the simulatios; we also show violi plots for the estimators the dashed lie idicates the value Y; the dotted lies idicate the 9% cofidece iterval to get a visual cofirmatio of the umbers just preseted. I the violi plot, we see that the values obtaied by our proposed estimators yield a arrow ormal distributio aroud the true value skewess is.46; kurtosis is.996, while the product estimator gives a spread-out distributio ad

ISRN Probability ad Statistics 3 Table 3: Compariso of the estimators i simulatios. See text for details. Estimator Coverage Neg. bias Pos. bias Lo. quart. Media Up. quart. y 89.86% 5.4% 5.%.5583.58.66 y R 97.%.%.87%.567.583.67 y P 49.59% 5.44% 4.97%.54.589.64 y.69.%.%.%.573.589.598 y y R y P y C).3.4.5.6 Estimator MSE MSE y /MSE est y.7 % y R.68 86.68% y P.755 6.79% y.69. 598.93%.7.8.9 the traditioally preferred ratio estimator gives a skewed distributio skewess is.53; kurtosis is 3.453. Fially, we compare the values of the MSEs; the experimetal values obtaied agree with the theoretical values listed i Table. We ifer that our proposed estimator is more efficiet ad robust tha the traditioal sample mea, ratio, ad product estimators. 6.5. Outlook Several authors have proposed efficiet estimators usig auxiliary iformatio. For example, Srivastava ad Reddy cosider a geeralisatio to the product ad ratio estimator give by y k y x/x k ; Reddy also itroduces the estimator y k y X/ X k x X ; i Sahai ad Ray the estimator y kt y x/x k where t stads for trasformed is cosidered; Sigh ad Espejo 6 itroduce a certai class of ratio-product estimators havig the form y RP k y k X/x k x/x. Choosig appropriate parameters k for these estimators ad calculatig the first-degree approximatio of the MSE, oe ca show that MSE y C ) )) ) ) C MSE yc MSE yct MSE y RP f S Y ρ ). Thus, these estimators ad our proposed estimator see 3.6 are equally efficiet up to the first degree of approximatio, havig the miimal possible MSE for this type of estimators 3, 4 i.e., estimators which are give by a product of y ad a fuctio of x/x. Ideed, all 6.4

4 ISRN Probability ad Statistics Table 4: Compariso of AOEs i simulatios. See text for details. Estimator Coverage Neg. bias Pos. bias Lo. quart. Media Up. quart. y.69 % % %.573.589.598 y.69 % % %.5738.5835.5935 y.69 % % %.573.589.598 y.69t % % %.579.586.595 y RP.846 99.99% %.%.5749.585.5953 y C) y C) y C y Ct C + y RP.54.56.58.6.6.64 Estimator MSE MSE y /MSE est y.69. 598.93% y.69.4 593.9% y.69. 598.9% y.69t.5 59.5% y RP.846.8 555.9% these estimators give similar results as our proposed estimator i the above applicatio, see Table 4. Comparig the first degree of approximatio of the bias doig calculatios as i Sectio reveals why our ubiased AOE y C ad Reddy s y C behave similarly they are both ubiased AOEs: B yct ) f B y ) B y C ) B yc ), ) f B yp CC X y, B y C ) f )) C 3C C X y, B y RP C ) f B yr C C X y, C C C X y, f C C C X y. 6.5 With C.69, oly y Ct is egatively biased, compare the quartiles ad the box plot i Table 4. For our proposed estimator i.4 which cotais the three traditioal estimators, amely, sample mea, product, ad ratio estimators, weareabletousethe two parameters α ad β to obtai a estimator y C i 5.3 that is up to first degree of approximatio both ubiased ad has miimal MSE. While the idea behid creatig is simple, the form of the ubiased AOE y C derived from it is ot ad the above list shows that there are may AOEs, but they are ot ecessarily ubiased. A thorough compariso of estimators usig auxiliary iformatio e.g., the oe i.4 ad the oes metioed above ivolvig higher degrees of approximatio of MSE ad bias as well as accompayig simulatios might be desirable, for example, to fid the estimator

ISRN Probability ad Statistics 5 that behaves well if the parameter C is ukow i advace i which case it may be replace with its cosistet estimate, Ĉ r Ĉ Y /Ĉ X, where r is the sample Pearso correlatio coefficiet ad Ĉ Y ad Ĉ X are the estimates of the coefficiets of variatio of Y ad X, resp.. Recall that our aalysis i Sectio 5 shows that the first-degree approximatio to MSE ad bias for values of the parameter C close to zero breaks dow. Refereces M. N. Murthy, Product method of estimatio, SakhyāA, vol. 6, pp. 69 74, 964. A. Sahai ad S. K. Ray, A efficiet estimator usig auxiliary iformatio, Metrika, vol.7,o.4, pp. 7 75, 98. 3 S. K. Srivastava, A geeralized estimator for the mea of a fiite populatio usig multi-auxiliary iformatio, Joural of the America Statistical Associatio, vol. 66, o. 334, pp. 44 47, 97. 4 S. K. Srivastava, A class of estimators usig auxiliary iformatio i sample surveys, The Caadia Joural of Statistics, vol. 8, o., pp. 53 54, 98. 5 S. K. Ray ad A. Sahai, Efficiet families of ratio ad product-type estimators, Biometrika, vol. 67, o., pp. 5, 98. 6 H. P. Sigh ad M. R. Espejo, O liear regressio ad ratio-product estimatio of a fiite populatio mea, Joural of the Royal Statistical Society D, vol. 5, o., pp. 59 67, 3. 7 A. Sahai ad A. Sahai, O efficiet use of auxiliary iformatio, Joural of Statistical Plaig ad Iferece, vol., o., pp. 3, 985. 8 S. K. Srivastava ad H. S. Jhajj, A class of estimators of the populatio mea i survey samplig usig auxiliary iformatio, Biometrika, vol. 68, o., pp. 34 343, 98. 9 Uited States Geology Survey, Water resources of the Uited States aual water data report,, http://wdr.water.usgs.gov/. W. G. Cochra, Samplig Techiques, Joh Wiley & Sos, New York, NY, USA, 3rd editio, 977. S. K. Srivastava, A estimator usig auxiliary iformatio i sample surveys, Calcutta Statistical Associatio Bulleti, vol. 6, o. 6-63, pp. 3, 967. V. N. Reddy, O ratio ad product methods of estimatio, Sakhyā B, vol. 35, o. 3, pp. 37 36, 973.

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