Variance Estimation Using Quartiles and their Functions of an Auxiliary Variable

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2 68 J. Subramani et al.: Variance Estimation Using Quartiles and their Functions of an Auiliary Variable Q 3 Third (upper) quartile Q r Inter-quartile range Q d Semi-quartile range Q a Semi-quartile average B(. ) Bias of the estimator MSE(. ) Mean squared error of the estimator S R Traditional ratio type variance estimator of S y S KCi Eisting modified ratio type variance estimator of S y S JGi Proposed modified ratio type variance estimator of S y Isaki[10] suggested a ratio type variance estimator for the population variance S y when the population variance S of the auiliary variable X is known together with its bias and mean squared error and are as given below: S R = s S (1) y s BS R = γs y β () 1 (λ 1) MSES R = γs y 4 β (y) 1 + β () 1 (λ 1) where β (y) = μ 40 μ, β () = μ 04 0 μ, λ = μ 0 μ rs = 1 N (y N i=1 i Y) r ( i X) s μ 0 μ 0 and The Ratio type variance estimator given in (1) is used to improve the precision of the estimate of the population variance compared to simple random sampling when there eists a positive correlation between X and Y. Further improvements are also achieved on the classical ratio estimator by introducing a number of modified ratio estimators with the use of known parameters like Co-efficient of Variat ion, Co-efficient of Kurtosis. The problem of constructing efficient estimators fo r the population variance has been widely discussed by various authors such as Agarwal and Sithapit[1]hmed et al.[], Al-Jararha and Al-Haj Eb rahem[3]rcos et al.[4], Bhushan[5], Cochran[6], Das and Tripathi[7], Garcia and Cebrain[8], Gupta and Shabbir[9], Isaki[10], Kad ilar and Cingi[11,1], Murthy[13], Prasad and Singh[14], Reddy[15], Singh and Chaudhary[16], Singh et al.[17,19], Upadhyaya and Singh[3] and Wolter[4]. Motivated by Singh et al.[18], Sisodia and Dwivedi[0], and Upadhyaya and Singh[], suggested four ratio type variance estimators using known values of Co-efficient of variation C X and Co-efficient of Kurtosis β () of an auiliary variable X together with their biases and mean squared errors as given in the Table 1. The eisting modified ratio type variance estimators discussed above are biased but have minimum mean squared errors compared to the traditional ratio type variance estimator suggested by Isaki[10]. The list of estimators given in Table 1 uses the known values of the parameters like S, C, β and their linear combinations. Subramani and Kumarapandiyan[1] used Quartiles and their functions of the auiliary variable like Inter-quartile range, Semi-quartile range and Semi-quartile average to improve the ratio estimators in estimation of population mean. Further we know that the value of quartiles and their functions are unaffected by the etreme values or the presence of outliers in the population values unlike the other parameters like the variance, coefficient of variation and coefficient of kurtosis. The points discussed above have motivated us to introduce a modified ratio type variance estimators using the known values of the quartiles and their functions of the auiliary variable. Now briefly we will discuss about quartiles and its functions. The median divides the data into two equal sets. The first (lower) quartile is the middle value of the first set, where 5% of the values are smaller than Q 1 and 75% are larger. The third (upper) quartile is the middle value of the second set, where 75% of the values are smaller than the third quartile Q 3 and 5% are larger. It should be noted that the median will be denoted by the notation Q, the second quartile. The inter-quartile range is another range used as a measure of the spread. The difference between upper and lower quartiles, which is called as the inter-quartile range, also indicates the dispersion of a data set. The formula for inter-quartile range is: Q r = (Q 3 Q 1 ) () The semi-quartile range is another measure of spread. It is calculated as one half of the differences between the quartiles Q 3 and Q 1. The formula for semi-quartile range is: Q d = (Q 3 Q 1 )/ (3) Subramani and Kumarapandiyan[1] suggested another new measure called as Semi-quartile average denoted by the notation Q a and defined as: Q a = (Q 3 + Q 1 )/ (4) Table 1. Eisting Modified ratio type variance estimators with their biases and mean squared errors Estimator Bias - BB(. ) Mean squared error MMMMMM(. ) S KC 1 = s y S + C s + C S KC = s y S + β () s + β () S KC 3 = s y S β () + C s β () + C S KC 4 = s y S C + β () s C + β () γ S y A KC 1 A KC 1 β () 1 (λ 1) γ S 4 y β (y) 1 + A KC 1 β () 1 A KC 1 (λ 1) γ S y A KC A KC β () 1 (λ 1) γ S 4 y β (y) 1 + A KC β () 1 A KC (λ 1) γ S y A KC 3 A KC 3 β () 1 (λ 1) γ S 4 y β (y) 1 + A KC 3 β () 1 A KC 3 (λ 1) γ S y A KC 4 A KC 4 β () 1 (λ 1) γ S 4 y β (y) 1 + A KC 4 β () 1 A KC 4 (λ 1)

3 International Journal of Statistics and Applications 01, (5): where A KC1 = S S +C KC = S S +β KC3 = ( ) S β ( ) S β ( ) +C and A KC4 = S C S C +β ( ). Proposed Estimators Using Quartiles and Their Functions In this section we have suggested a class of modified ratio type variance estimators using the quartiles and their functions of the auiliary variable like Inter-quartile range, Semi-quartile range and Semi-quartile average The proposed class of modified ratio type variance estimators S JGi, ii = 1,,,5 for estimating the population variance S y together with the first degree of approimation, the biases and mean squared errors and the constants are given below: Table. Proposed modified ratio type variance estimators with their biases and mean squared errors Est imator Bias - BB(. ) Mean squared error MMMMMM(. ) S JG 1 = s y S + Q 1 s γs + Q y A JG 1 A JG 1 β () 1 (λ 1) γs y4 β (y) 1 + A JG 1 β () 1 A JG 1 (λ 1) 1 S JG = s y S + Q 3 s + Q 3 γs y A JG A JG β () 1 (λ 1) γs y4 β (y) 1 + A JG β () 1 A JG (λ 1) S JG 3 = s y S + Q r s γs + Q y A JG 3 A JG 3 β () 1 (λ 1) γs y4 β (y) 1 + A JG 3 β () 1 A JG 3 (λ 1) r S JG 4 = s y S + Q d s γs + Q y A JG 4 A JG 4 β () 1 (λ 1) γs y4 β (y) 1 + A JG 4 β () 1 A JG 4 (λ 1) d S JG 5 = s y S + Q a s γs + Q y A JG 5 A JG 5 β () 1 (λ 1) γs y4 β (y) 1 + A JG 5 β () 1 A JG 5 (λ 1) a where A JG1 = S S S + Q JG = 1 S + Q JG3 = S 3 S + Q JG4 = S r S and A + Q JG5 = S d S + Q a 3. Efficiency of the Proposed Estimators As we mentioned earlier the bias and mean squared error of the traditional ratio type variance estimator are given below: BS R = γs y β () 1 (λ 1) (5) MSES R = γs y 4 β (y) 1 + β () 1 (λ 1) (6) For want of space; for the sake of convenience to the readers and for the ease of comparisons, the biases and mean squared errors of the eisting modified ratio type variance estimators given in Table 1 are represented in single class as given below: BS KCi = γs y A KCi A KCi β () 1 (λ 1); i = 1,,3 and 4 (7) MSES KCi = γs 4 y β (y) 1 + A KCi β () 1 A KCi (λ 1); i = 1,,3 and 4 (8) where A KC1 = S S β ( ) S +C KC = S +β KC3 = S and A ( ) S β ( ) +C KC4 = S C S C +β ( ) In the same way, the biases and mean squared errors of the proposed modified ratio type variance estimators given in Table are represented in a single class as given below: BS JGj = γs y A JGj A JGj β () 1 (λ 1); j = 1,,3,4 and 5 (9) MSES JGj = γs 4 y β (y) 1 + A JGj β () 1 A JGj (λ 1); j = 1,,3,4 and 5 (10) where A JG1 = S S +Q JG = S 1 S +Q JG3 = S 3 S +Q JG4 = S and A r S +Q JG5 = S d S +Q a From the epressions given in (6) and (10) we have derived the condition for which the proposed estimators S JGj ; j = 1,,3,4 and 5 are more efficient than the traditional ratio type variance estimator and it is given below: MSES JGj < MSES R if λ > 1 + A JGj +1β ( ) 1 ; j = 1,,3,4 and 5 (11) From the epressions given in (8) and (10) we have derived the conditions for which the proposed estimators S JGj ; j = 1,,3,4 and 5 are more efficient than the eisting modified ratio type variance estimators given in Table 1, S KCi ; i = 1,, 3 and 4 and are given below: MSES JGj < MSES KCi if λ > 1 + A JGj +A i β ( ) 1 ; i = 1,, 3 and 4 ; j = 1,,3,4 and 5 (1)

4 70 J. Subramani et al.: Variance Estimation Using Quartiles and their Functions of an Auiliary Variable 4. Numerical Study The performance of the proposed modified ratio type variance estimators are assessed with that of traditional ratio type estimator and eisting modified ratio type variance estimators listed in Table 1 for certain natural populations. The populations 1 and are the real data set taken from the Report on Waste 004 drew up by the Italian bureau for the environment protection-apat. Data and reports are available in the website In the data set, Ta ble 3. Paramet ers and Const ant s of t he Populat ions for each of the Italian provinces, three variables are considered: the total amount (tons) of recyclable-waste collection in Italy in 003 (Y), the total amount of recyclable-waste collection in Italy in 00 (X 1 ) and the number of inhabitants in 003 (X ). The population 3 is taken from Murthy[13] given in page 8 and population 4 is taken from Singh and Chaudhary[16] given in page 108. The population parameters and the constants computed from the above populations are given below: Paramet ers Population 1 Population Population 3 Population 4 N n Y X ρ S y C y S C β () β (y) λ Q Q Q r Q d Q a A KC A KC A KC A KC A JG A JG A JG A JG A JG The biases and mean squared errors of the eisting and proposed modified ratio type variance estimators for the populations given above are given in the following Tables:

5 International Journal of Statistics and Applications 01, (5): Est imator Ta ble 4. Biases of the eisting and proposed modified ratio type variance estimators Bias BB(. ) Population 1 Population Population 3 Population 4 S R Isaki[10] S KC S KC S KC S KC S JG 1 Proposed Estimator S JG Proposed Estimator S JG 3 Proposed Estimator S JG 4 Proposed Estimator S JG 5 Proposed Estimator Ta ble 5. Mean squared errors of the eisting and proposed modified ratio type variance estimators Est imator Mean Squared Error MMMMMM(. ) Population 1 Population Population 3 Population 4 S R Isaki[10] S KC S KC S KC S KC S JG 1 Proposed Estimator Proposed Estimator S JG S JG 3 S JG 4 S JG 5 Proposed Estimator Proposed Estimator Proposed Estimator From the values of Table 4, it is observed that the biases of the proposed modified ratio type variance estimators are less than the biases of the traditional and eisting modified ratio type variance estimators. Similarly from the values of Table 5, it is observed that the mean squared errors of the proposed modified ratio type variance estimators are less than the mean squared errors of the traditional and eisting modified ratio type variance estimators. 5. Conclusions Estimating the finite population variance has great significance in various fields such as Industrygriculture, Medical and Biological sciences, etc. In this paper we have proposed a class of modified ratio type variance estimators using the quartiles and their functions of the auiliary variable like Inter-quartile range, Semi-quartile range and Se mi-quartile average. The biases and mean squared errors of the proposed modified ratio type variance estimators are obtained and compared with that of traditional ratio type variance estimator and eisting modified ratio type variance estimators. Further we have derived the conditions for which the proposed estimators are more efficient than the traditional and eisting estimators. We have also assessed the performance of the proposed estimators for some known natural populations. It is observed that the biases and mean squared errors of the proposed estimators are less than the biases and mean squared errors of the traditional and eisting modified estimators for certain known populations. Hence we strongly recommend that the proposed modified ratio type variance estimator may be preferred over the traditional ratio type variance estimator and eisting modified ratio type variance estimators for the use of practical applications. ACKNOWLEDGEMENTS The second author wishes to acknowledge his sincere thanks to the Vice Chancellor, Pondicherry University for the financial support to carry out this research work through the University Fellowship. REFERENCES [1] Agarwal, M.C. and Sithapit.B. (1995). Unbiased ratio type estimation, Statistics and Probability Letters 5, [] Ahmed, M.S., Raman, M.S. and Hossain, M.I. (000). Some competitive estimators of finite population variance Multivariate Auiliary Information, Information and Management Sciences, Volume11 (1), 49-54

6 7 J. Subramani et al.: Variance Estimation Using Quartiles and their Functions of an Auiliary Variable [3] Al-Jararha, J. and Al-Haj Ebrahem, M. (01). A ratio estimator under general sampling designustrian Journal of Statistics, Volume 41(), [4] Arcos., Rueda, M., Martinez, M.D., Gonzalez, S., Roman, Y. (005). Incorporating the auiliary information available in variance estimationpplied Mathematics and Computation 160, [5] Bhushan, S. (01). Some efficient sampling strategies based on ratio type, estimator, Electronic Journal of Applied Statistical Analysis, Volume 5(1), [6] Cochran, W. G. (1977). Sampling techniques, Third Edition, Wiley Eastern Limited [7] Das.K. and Tripathi, T.P. (1978). Use of auiliary information in estimating the finite population variance, Sankhya 40, [8] Garcia, M.K. and Cebrain.A. (1997). Variance estimation using auiliary information: An almost unbiased multivariate ratio estimator, Metrika 45, [9] Gupta, S. and Shabbir, J. (008). Variance estimation in simple random sampling using auiliary information, Hacettepe Journal of Mathematics and Statistics, Volume 37, [10] Isaki, C.T. (1983). Variance estimation using auiliary information, Journal of the American Statistical Association 78, [11] Kadilar, C. and Cingi, H. (006). Improvement in variance estimation using auiliary information, Hacettepe Journal of Mathematics and Statistics Volume 35 (1), [1] Kadilar, C. and Cingi, H. (006). Ratio estimators for population variance in simple and stratified sampling, Applied Mathematics and Computation 173, [13] M urthy, M.N. (1967). Sampling theory and methods, Statistical Publishing Society, Calcutta, India [14] Prasad, B. and Singh, H.P. (1990). Some improved ratio type estimators of finite population variance in sample surveys, Communication in Statistics: Theory and Methods 19, [15] Reddy, V.N. (1974). On a transformed ratio method of estimation, Sankhya, Volume C36, [16] Singh, D. and Chaudhary, F.S. (1986). Theory and analysis of sample survey designs, New Age International Publisher [17] Singh, H.P., Chandra, P. and Singh, S. (003). Variance estimation using multi-auiliary information for random non-response in survey sampling, STATISTICA, anno LXIII, n. 1, 3-40 [18] Singh, H.P., Tailor, R., Tailor, R. and Kakran, M.S. (004). An improved estimator of population mean using power transformation, Journal of the Indian Society of Agricultural Statistics 58(), 3-30 [19] Singh, H.P., Upadhyaya, U.D. and Namjoshi, U.D. (1988). Estimation of finite population variance, Current Science 57, [0] Sisodia, B.V.S. and Dwivedi, V.K. (1981). A modified ratio estimator using coefficient of variation of auiliary variable, Journal of the Indian Society of Agricultural Statistics 33(1), [1] Subramani, J. and Kumarapandiyan, G. (01). Modified ratio estimators for population mean using function of quartiles of auiliary variable, Bonfring International Journal of Industrial Engineering and Management Science, Vol. (), 19-3 [] Upadhyaya, L.N. and Singh, H.P. (1999). Use of transformed auiliary variable in estimating the finite population mean, Biometrical Journal 41 (5), [3] Upadhyaya, L. N. and Singh, H. P. (006). Almost unbiased ratio and product-type estimators of finite population variance in sample surveys, Statistics in Transition 7 (5), [4] Wolter, K.M. (1985). Introduction to Variance Estimation, Springer-Verlag [5]

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