Enhancing the Mean Ratio Estimators for Estimating Population Mean Using Non-Conventional Location Parameters

Transcription

1 evista Colombiaa de Estadística Jauary 016, Volume 39, Issue 1, pp. 63 to 79 DOI: Ehacig the Mea atio Estimators for Estimatig Populatio Mea Usig No-Covetioal Locatio Parameters Mejoras a los estimadores de razó de medias co el fi de estimar la media poblacioal usado parámetros de localizació o covecioales Muhammad Abid 1,,a, Nasir Abbas,b, Hafiz Zafar Nazir 3,c, Zhegya Li 1,d 1 Departmet of Mathematics, Faculty of Basic ad Applied Scieces, Istitute of Statistics, Zhejiag Uiversity, Hagzhou, Chia Departmet of Statistics, Faculty of Sciece Ad Techology, Govermet College Uiversity, Faisalabad, Pakista 3 Departmet of Statistics, Faculty of Sciece, Uiversity of Sargodha, Sargodha, Pakista Abstract Covetioal measures of locatio are commoly used to develop ratio estimators. However, i this article, we attempt to use some o-covetioal locatio measures. We have icorporated tri-mea, Hodges-Lehma, ad mid-rage of the auxiliary variable for this purpose. To ehace the efficiecy of the proposed mea ratio estimators, populatio correlatio coefficiet, coefficiet of variatio ad the liear combiatios of auxiliary variable have also bee exploited. The properties associated with the proposed estimators are evaluated through bias ad mea square errors. We also provide a empirical study for illustratio ad verificatio. Key words: Bias, Correlatio Coefficiet, Coefficiet of Variatio, Hodges- Lehma Estimator, Mea Square Error, Tri-Mea. a Lecturer. mabid@gcuf.edu.pk b Visitig Lecturer. asirbhatti9181@yahoo.com c Assistat Professor. hafizzafarazir@yahoo.com d Professor. zli@zju.edu.c 63

2 64 Muhammad Abid, Nasir Abbas, Hafiz Zafar Nazir & Zhegya Li esume Las medidas covecioales de localizació so a meudo usadas co el fi de desarrollar estimatores de raz ó. Si embargo, e este artículo, se hace u iteto por usar alguas medidas de localizació o covecioales. Se icorpora la trimea, el estimador de Hodges-Lehma y el rago medio de la variable auxiliar co este propósito. Para mejorar la eficiecia de los estimadores de razó de medias propuestos, el coeficiete de correlació poblacioal, el coeficiete de variació y combiacioes lieales de variables auxiliares tambié ha sido explotados. Las propiedades asociadas co los estimadores propuestos so evaluadas a través del sesgo y el error cuadrático medio. U studio empírico es presetado co fies de ilustració y verificació. Palabras clave: coeficiete de correlació poblacioal, coeficiete de variació, error cuadrático medio, estimador de Hodges-Lehma, sesgo, trimea. 1. Itroductio A mechaism, i statistical aalysis, i which a predetermied umber of observatios are take from a statistical populatio is called samplig. Samplig plays a vital role i all kids of disciplies. The purpose is to reduce the cost ad/or the amout of work that it would take to survey the etire target populatio. The variable of iterest or the variable about which we wat to draw some iferece is called a study variable. I survey research, there are situatios i which whe the iformatio is available o every uit i the populatio. If a variable, that is kow for every uit of the populatio, is ot a variable of direct iterest but istead employed to improve the samplig pla or to ehace estimatio of the variables of iterest, the it is called a auxiliary variable. The auxiliary iformatio is commoly associated with the use of ratio, product ad regressio estimatio methods ad to improve the efficiecy of the estimators i survey samplig. The ratio estimator is most effective for estimatig populatio mea whe there is a liear relatioship betwee study variable ad auxiliary variable ad they have a positive correlatio. Cosider a fiite populatio U = {U 1, U,..., U N } of N distict ad idetifiable uits. Let be the study variable with value i measured o U i, i = 1,,..., N givig a vector = { 1,,..., N }. The objective is to estimate populatio mea = 1 N N i=1 i, o the basis of a radom sample. Whe the populatio parameters of the auxiliary variable X such as populatio mea, coefficiet of variatio, co-efficiet of kurtosis, co-efficiet of skewess, media, quartiles, populatio correlatio coefficiet, deciles etc., are kow the a umber of estimators available i the literature which performs better tha the usual simple radom mea uder certai coditios. Below we provide here the complete list of otatios that are used i this paper: evista Colombiaa de Estadística ) 63 79

3 Ehacig the Mea atio Estimators Usig No-Covetioal Measures 65 Nomeclature oma N Populatio size Sample size X Auxiliary variable f = 1 N Samplig fractio x, y Sample meas S x, S y Populatio stadard deviatios x, y Sample totals C x, C y Coefficiet of variatio B.) Bias of the Estimator MSE.) Mea squared error of the estimator i Existig estimators j Proposed estimators M d Media of X HL = media X j + X k )/, 1 j k N) Hodges-Lehma estimator M = X 1) +X N) Mid-rage T M = Q 1+Q +Q 3 4 Tri-mea QD = Q 3 Q 1 Quartile Deviatio Subscript i For existig estimators j For proposed estimators Greek ρ β 1 = N N i=1 X i X) 3 N 1)N )S 3 β = NN+1) N i=1 X i X) 4 i pj N 1)N )N 3)S 4 3N 1) N )N 3) Coefficiet of correlatio Coeff. of skewess of auxiliary variable Coeff. of kurtosis of auxiliary variable Existig modified ratio estimator of Proposed modified ratio estimator of Based o the above metioed otatios, the mea ratio estimator for estimatig the populatio mea of the study variable is defied as r = y x X = X, 1) where = y x = y x is the estimate of = X = X. The bias, costat ad the mea square error of the mea ratio estimator is give by: BŶ r) = = X. 1 X S x ρs x S y ) evista Colombiaa de Estadística ) 63 79

4 66 Muhammad Abid, Nasir Abbas, Hafiz Zafar Nazir & Zhegya Li MSEŶ r) = S y + S x ρs x S y ) The mea ratio estimator give i 1) is used to improve the precisio of the estimate of the populatio mea i compariso with the sample mea estimator wheever a positive correlatio exists betwee the study variable ad the auxiliary variable. Moreover, we mode improvemets by itroducig a large umber of modified ratio estimators with the use of the kow coefficiet of variatio, coefficiet of kurtosis, co- efficiet of skewess, deciles etc. Cochra 1940) suggested a classical ratio type estimator for the estimatio of a fiite populatio mea usig oe auxiliary variable uder a simple radom samplig scheme. Murthy 1967) proposed a product type estimator to estimate the populatio mea or total of study variable y by usig auxiliary iformatio whe the coefficiet of a correlatio is egative. ao 1991) itroduced a differece type estimator that outperforms covetioal ratio ad liear regressio estimators. Sigh & Tailor 003) proposed a family of estimators usig kow values of some parameters by usig SSWO to estimate the populatio mea of the study variable. Sigh, Tailor, Tailor & Kakra 004) ad Sisodia & Dwivedi 01) used a coefficiet of variatio of the auxiliary variate. Upadhyaya & Sigh 1999) derived ratio type estimators usig a coefficiet of variatio ad coefficiet of kurtosis of the auxiliary variate. The orgaizatio of the rest of the article is as follows: Sectio provides a descriptio of the existig estimators. The structure of the proposed mea ratio estimator ad the efficiecy compariso of the proposed estimator with the existig estimator are preseted i Sectio 3. Sectio 4 cosists of a empirical study of proposed estimators. Fially, Sectio 5 summarizes the fidigs of this study.. Existig Modified atio Estimators Kadilar & Cigi 004) suggested the followig ratio estimators for the populatio mea i simple radom samplig usig some auxiliary iformatio. 1 = y + b X x ) X x = y + b X x ) ) X + Cx x + C x ) 3 = y + b X x ) ) X + β x + β ) 4 = y + b X x ) ) Xβ + C x xβ + C x ) 5 = y + b X x ) ) XCx + β xc x + β ) evista Colombiaa de Estadística ) 63 79

5 Ehacig the Mea atio Estimators Usig No-Covetioal Measures 67 The biases, costats ad the mea squared errors of estimators of Kadilar & Cigi 004) are give by: B 1) = B ) = B 3) = B 4) = B 5) = Sx Sx Sx Sx 1 = 3 3 = 1 = X X+C x) X+β ) β Xβ +C x) MSE 1) = MSE ) = MSE 3) = 4 4 = MSE 4) = Sx 5 5 = Cx MSE XC x+β 5) = 1 Sx + Sy )) Sx + Sy )) 3 Sx + Sy )) 4 Sx + Sy )) 5 Sx + Sy )) They showed that the above ratio estimators are more efficiet tha traditioal ratio estimators to estimate the populatio mea. Kadilar & Cigi 006) has developed some modified ratio estimators usig the correlatio coefficiet, which are show below: 6 = y + b X x ) ) X + ρ x 7 = y + b X x ) XCx + ρ ) xc x + ρ) 8 = y + b X x ) ) Xρ + Cx xρ + C x ) 9 = y + b X x ) Xβ + ρ ) xβ + ρ) 10 = y + b X x ) ) Xρ + β xρ + β ) The biases, costats ad the mea squared errors i Kadilar & Cigi 006) are give by: B 6) = B 7) = B 8) = B 9) = B 10) = Sx 6 6 = X+ρ S x 7 7 = Cx XC x+ρ S x MSE 6) = MSE 7) = 8 8 = ρ MSE Xρ+C x 8) = Sx 9 9 = β MSE Xβ +ρ 9) = S x = ρ Xρ+β MSE 10) = 6 Sx + Sy )) 7 Sx + Sy )) 8 Sx + Sy )) 9 Sx + Sy )) 10 Sx + Sy )) These five estimators proved to be more efficiet tha all the existig ratio estimators, as is evidet from their applicatio to some atural populatios. evista Colombiaa de Estadística ) 63 79

6 68 Muhammad Abid, Nasir Abbas, Hafiz Zafar Nazir & Zhegya Li a & Tia 010) proposed the followig two modified ratio estimators usig coefficiet of skewess ad kurtosis; 11 = y + b X x ) ) X + β1 x + β 1 ) 1 = y + b X x ) ) Xβ1 + β xβ 1 + β ) The biases, costats ad the mea squared errors for a & Tia 010) are give by: B 11 ) = B 1 ) = Sx = Sx 1 1 = X+β 1) β 1 Xβ 1 +β ) MSE 11 ) = MSE 1 ) = 11 Sx + Sy )) 1 Sx + Sy )) The above-defied estimators that use the skewess ad kurtosis coefficiet respectively provide better estimates of the populatio mea i compariso with traditioal ratio estimators. Subramai & Kumarapadiya 01a, 01b, 01c) have itroduced estimators with the use of populatio media, skewess, kurtosis ad coefficiet of variatio for auxiliary iformatio i simple radom samplig to estimate populatio mea. 13 = y + b X x ) ) X + Md x + M d ) 14 = y + b X x ) ) Cx X + M d C x x + M d ) 15 = y + b X x ) ) ) B 1x) X + M d B1x) x + M d 16 = y + b X x ) ) ) B x) X + M d Bx) x + M d The biases, costats ad the mea squared errors for Subramai & Kumarapadiya 01a, 01b, 01c) are give by; B 13 ) = B 14 ) = B 15 ) = B 16 ) = Sx = X+M d MSE 13 ) = Sx = Cx XC x+m d MSE 14 ) = Sx = B 1x) XB 1x) +M d MSE 15 ) = S x = B x) XB x) +M d MSE 16 ) = 13 Sx + Sy )) 14 Sx + Sy )) 15 Sx + Sy )) 16 Sx + Sy )) Jeelai, Maqbool & Mir 013) suggested a estimator with the use of the coefficiet of skweess ad quartile deviatio of auxiliary iformatio i simple radom samplig to estimate populatio mea. evista Colombiaa de Estadística ) 63 79

7 Ehacig the Mea atio Estimators Usig No-Covetioal Measures = y + b X x ) Xβ1 + QD ) xβ 1 + QD) The bias, costat ad the mea square error for Jeelai et al. 013) is as follows; B 17 ) = Sx = β 1 Xβ 1 +QD) MSE 17 ) = 17 Sx + S y )) 3. The Proposed Modified atio Estimators I this sectio, we propose differet modified ratio type estimators usig the populatio tri-mea, mid-rage, Hodges-Lehma, coefficiet of variatio ad populatio correlatio coefficiet. The midrage is defied as M = X 1)+X N), where X 1) ad X N) are the lowest ad highest order statistics i a populatio of size N. It is highly sesitive to outliers as its desig structure is based o oly extreme values of data cf. Ferrell 1953) for more details). We also iclude the measure based o the media of the pairwise Walsh averages which is defied as: HL = media X j + X k )/, 1 j k N). The mai advatage of the HL is that it is robust agaist outliers. For more properties of HL see Hettmasperger & McKea 1988). The HL is also kow as the Hodges-Lehma estimator. The ext measure icluded i this study is the trimea, which is the weighted average of the populatio media ad two quartiles. It is defied as: T M = Q1+Q+Q3 4, where Q p p = 1,, 3) deote oe of the three quartiles i a populatio. For detailed properties of trimea T M) see Wag, Li & Cui 007) ad Nazir, iaz, oald & Abbas 013). The proposed estimators are give below: p1 = y + b X x ) X + T M) x + T M) p = y + b X x ) XCx + T M ) xc x + T M) p3 = y + b X x ) ) Xρ + T M xρ + T M) p4 = y + b X x ) ) X + M x + M) p5 = y + b X x ) XCx + M ) xc x + M) p6 = y + b X x ) Xρ + M) xρ + M) evista Colombiaa de Estadística ) 63 79

8 70 Muhammad Abid, Nasir Abbas, Hafiz Zafar Nazir & Zhegya Li p7 = y + b X x ) X + HL) x + HL) p8 = y + b X x ) XCx + HL ) xc x + HL) p9 = y + b X x ) ) Xρ + HL xρ + HL) The biases, costats ad MSEs of proposed estimators are give below. BŶ BŶ BŶ BŶ BŶ BŶ BŶ BŶ BŶ Sx p1) = Sx p) = Sx p3) = Sx p4) = Sx p5) = Sx p6) = Sx p7) = Sx p8) = p9) = 1 1 = = 3 3 = 4 4 = 5 5 = 6 6 = 7 7 = 8 8 = Sx 9 9 = X+T M) XC x+t M) Xρ+T M) X+M) XC x+m) Xρ+M) X+HL) XC x+hl) Xρ+HL) MSEŶ p1) = 1 Sx + S y )) MSEŶ p) = MSEŶ p3) = MSEŶ p4) = MSEŶ p5) = MSEŶ p6) = MSEŶ p7) = MSEŶ p8) = MSEŶ p9) = Sx + Sy )) 3 Sx + Sy )) 4 Sx + Sy )) 5 Sx + Sy )) 6 Sx + Sy )) 7 Sx + Sy )) 8 Sx + Sy )) 9 Sx + Sy )) 4. Efficiecy Comparisos I this sectio, the coditio uder which the proposed modified ratio estimators will have miimum mea square error compared to usual ratio estimators ad existig modified ratio estimators to estimate the fiite populatio mea have bee derived algebraically Compariso with Usual Mea atio Estimator From the expressio of proposed ad usual ratio estimator MSEs, we have derived the coditios for which the proposed estimators are more efficiet tha the ratio estimator as: ) ) MSE Ŷ pj MSE Ŷ r 1 f pj Sx + Sy )) S y + S x ρs x S y ) evista Colombiaa de Estadística ) 63 79

9 Ehacig the Mea atio Estimators Usig No-Covetioal Measures 71 where j = 1,,..., 9. pjs x ρ S y S x ρs x S y, 4.. Compariso with Existig Modified atio Estimators From the expressio of proposed ad existig MSEs, we have derived the coditios for which the proposed estimators are more efficiet tha the existig modified ratio estimators as: ) ) MSE Ŷ pj MSE Ŷ i 1 f pj S x + S y )) 1 f pjs x i S x pj i where j = 1,,..., 9 ad i = 1,,..., 17. i Sx + Sy )) 5. Empirical Study The performace of the proposed modified mea ratio estimators ad the existig modified ratio estimators are evaluated by usig 5 atural populatios. The populatio 1 ad are take from Sigh & Chaudhary 1986) page 177, populatio 3 is take from Cochra 1940) page 15 ad populatio 4 ad populatio 5 are take from Murthy 1967) page 8. The characteristics of the five populatios are give below i Table 1, whereas the costats ad the biases are give i Tables -5. The MSEs of the existig ad proposed modified ratio estimators are give i Tables 6 ad 7. The percetage relative efficiecies PEs) of the proposed estimators p), with respect to the existig estimators e), are computed as P Ee, p) = MSEe) MSEp) 100 ad are give i Tables 8-1. Tables -5 reveals that the costats ad bias for the proposed ratio estimators are smaller compared to the existig ratio estimators. It is evidet i Table 6 ad 7 that the MSEs of the proposed ratio estimators with regard to the existig oes are much smaller, which idicates that proposed modified mea ratio estimators are more efficiet. It ca be see that the proposed ratio estimators perform well i compariso with the existig estimators i terms of PEs cf. Tables 8-1). evista Colombiaa de Estadística ) 63 79

10 7 Muhammad Abid, Nasir Abbas, Hafiz Zafar Nazir & Zhegya Li Table 1: Characteristics of the Populatios. Parameter Populatio 1 Populatio Populatio 3 Populatio4 Populatio5 N X ρ S y C y S x C x β β M d QD T M M HL Table : The costats of existig ratio estimators. Estimator Costat Populatio 1 Populatio Populatio 3 Populatio 4 Populatio 5 r evista Colombiaa de Estadística ) 63 79

11 Ehacig the Mea atio Estimators Usig No-Covetioal Measures 73 Table 3: The biases of existig ratio estimators. Estimator Bias Populatio 1 Populatio Populatio 3 Populatio 4 Populatio 5 r Table 4: The costats of proposed ratio estimators. Estimator Costat Populatio 1 Populatio Populatio 3 Populatio 4 Populatio 5 p p p p p p p p p Table 5: The biases of proposed ratio estimators. Estimator Bias Populatio 1 Populatio Populatio 3 Populatio 4 Populatio 5 p p p p p p p p p evista Colombiaa de Estadística ) 63 79

12 74 Muhammad Abid, Nasir Abbas, Hafiz Zafar Nazir & Zhegya Li Table 6: Mea square errors of existig ratio estimators. Estimator Mea Square Errors Populatio 1 Populatio Populatio 3 Populatio 4 Populatio 5 r Table 7: Mea square errors of proposed modified ratio estimators. Estimator Mea Square Errors Populatio 1 Populatio Populatio 3 Populatio 4 Populatio 5 p p p p p p p p p evista Colombiaa de Estadística ) 63 79

13 Ehacig the Mea atio Estimators Usig No-Covetioal Measures 75 Table 8: Percetage relative efficiecy of existig estimators with respect to the proposed estimators of populatio 1. Existig Proposed p1 Ŷ p Ŷ p3 Ŷ p4 Ŷ p5 Ŷ p6 Ŷ p7 Ŷ p8 Ŷ p9 r Table 9: Percetage relative efficiecy of existig estimators with respect to the proposed estimators of populatio. Existig Proposed p1 Ŷ p Ŷ p3 Ŷ p4 Ŷ p5 Ŷ p6 Ŷ p7 Ŷ p8 Ŷ p9 r evista Colombiaa de Estadística ) 63 79

14 76 Muhammad Abid, Nasir Abbas, Hafiz Zafar Nazir & Zhegya Li Table 10: Percetage relative efficiecy of existig estimators with respect to the proposed estimators of populatio 3. Existig Proposed p1 Ŷ p Ŷ p3 Ŷ p4 Ŷ p5 Ŷ p6 Ŷ p7 Ŷ p8 Ŷ p9 r Table 11: Percetage relative efficiecy of existig estimators with respect to the proposed estimators of populatio 4. Existig Proposed p1 Ŷ p Ŷ p3 Ŷ p4 Ŷ p5 Ŷ p6 Ŷ p7 Ŷ p8 Ŷ p9 r evista Colombiaa de Estadística ) 63 79

15 Ehacig the Mea atio Estimators Usig No-Covetioal Measures 77 Table 1: Percetage relative efficiecy of existig estimators with respect to the proposed estimators of populatio 5. Existig Proposed p1 Ŷ p Ŷ p3 Ŷ p4 Ŷ p5 Ŷ p6 Ŷ p7 Ŷ p8 Ŷ p9 r Coclusio Samplig plays a vital role i all kids of disciplies. The availability of auxiliary iformatio ehaces the efficiecy of the estimators. Mea ratio estimators have bee proposed usig kow values of populatio tri-mea, mid-rage, Hodges-Lehma estimator, coefficiet of variatio ad populatio correlatio coefficiet by usig the study variable ad auxiliary variable iformatio. It is observed that the mea squared errors of the suggested estimators based o the tri-mea, mid-rage, Hodges-Lehma, coefficiet of variatio ad populatio correlatio coefficiet of the auxiliary variable are smaller tha those for the existig modified ratio estimators for all the five kow populatios cosidered i the umerical study. Also, it is pertiet to ote that the parameters such as the mea, coefficiet of skewess ad coefficiet of kurtosis are affected by the extreme values i the populatio, whereas the tri-mea, mid-rage, Hodges-Lehma are robust to extreme values. Hece, the modified ratio estimators proposed i this study may be used for better ad more stable results, ad are preferred to the existig modified ratio estimators for practical applicatios. Moreover, empirical studies reveal that the bias ad mea square error for the proposed estimators are lower tha that of the existig methods i terms of various atural populatios. For the give populatios aloe, the proposed estimators perform better tha the exitig estimators. [ eceived: Jue 014 Accepted: March 015 ] evista Colombiaa de Estadística ) 63 79

16 78 Muhammad Abid, Nasir Abbas, Hafiz Zafar Nazir & Zhegya Li efereces Cochra, W. G. 1940), Samplig Techiques, 3 ed, Wiley Easter Limited, New ork. Ferrell, E. 1953), Cotrol charts usig midrages ad medias, Idustrial Quality Cotrol 95), Hettmasperger, T. & McKea, J. W. 1988), obust Noparametric Statistical, 1 ed, Arold, Lodo. Jeelai, M. I., Maqbool, S. & Mir, S. A. 013), Modified ratio estimators of populatio mea usig liear combiatio of coefficiet of skewess ad quartile deviatio, Iteratioal Joural of Moder Mathematical Scieces 63), Kadilar, C. & Cigi, H. 004), atio estimators i simple radom samplig, Applied Mathematics ad Computatio 151, Kadilar, C. & Cigi, H. 006), A improvemet i estimatig the populatio mea by usig the correlatio coefficiet, Hacettepe Joural of Mathematics ad Statistics 351), Murthy, M. 1967), Samplig Theory ad Methods, 1 ed, Statistical Publishig Society, Idia. Nazir, H., iaz, M., oald, J. D. & Abbas, N. 013), obust cusum cotrol chartig, Quality Egieerig 53), ao, T. J. 1991), O certai methods of improvig ratio ad regressio estimators, Commuicatios i Statistics-Theory ad Methods 010), Sigh, D. & Chaudhary, F. S. 1986), Theory ad Aalysis of Sample Survey Desigs, 1 ed, New Age Iteratioal Publisher, Idia. Sigh, H. P. & Tailor,. 003), Use of kow correlatio coefficiet i estimatig the fiite populatio meas, Statistics i Trasitio 64), Sigh, H. P., Tailor,., Tailor,. & Kakra, M. 004), A improved estimator of populatio mea usig power trasformatio, Joural of the Idia Society of Agricultural Statistics 58), Sisodia, B. V. S. & Dwivedi, V. K. 01), A modified ratio estimator usig coefficiet of variatio of auxiliary variable, Joural of the Idia Society of Agricultural Statistics 331), Subramai, J. & Kumarapadiya, G. 01a), Estimatio of populatio mea usig co-efficiet of variatio ad media of a auxiliary variable, Iteratioal Joural of Probability ad Statistics 14), evista Colombiaa de Estadística ) 63 79

17 Ehacig the Mea atio Estimators Usig No-Covetioal Measures 79 Subramai, J. & Kumarapadiya, G. 01b), Estimatio of populatio mea usig kow media ad co-efficiet of skewess, America Joural of Mathematics ad Statistics 5), Subramai, J. & Kumarapadiya, G. 01c), Modified ratio estimators usig kow media ad co-efficiet of kurtosis, America Joural of Mathematics ad Statistics 4), Upadhyaya, L. N. & Sigh, H. 1999), Use of trasformed auxiliary variable i estimatig the fiite populatio mea, Biometrical Joural 415), Wag, T., Li,. & Cui, H. 007), O weighted radomly trimmed meas, Joural of Systems Sciece ad Complexity 01), a, Z. & Tia, B. 010), atio method to the mea estimatio usig coefficiet of skewess of auxiliary variable, Iformatio Computig ad Applicatios 106, evista Colombiaa de Estadística ) 63 79