Variance Estimation Using Quartiles and their Functions of an Auxiliary Variable

In this paper we have proposed a class of modified rat io type variance estimators for estimation of population variance of the study variable using Quart iles and their functions of the auxiliary variable are known. The b iases and mean squared errors of the proposed estimators are obtained and also derived the conditions for which the proposed estimators perform better than the traditional rat io type variance estimator and existing modified ratio type variance estimators. Further we have compared the proposed estimators with that of trad itional ratio type variance estimator and existing modified ratio type variance estimators for certain known populations. From the numerical study it is observed that the proposed estimators perform better than the traditional ratio type variance estimator and existing modified ratio type variance estimators.


Introduction
Consider a finite population U = { U 1 , U 2 , … , U N } of N distinct and identifiable units. Let be a real variab le with value Y i measured on U i , i = 1,2,3, … , N giving a vector Y = { Y 1 , Y 2 , … , Y N } . The problem is to estimate the populations mean or population variance S y 2 = 1 ( N−1 ) on the basis of a random sample selected fro m the population U. Estimat ing the finite population variance has great significance in various fields such as Industry, Agriculture, Medical and Biological Sciences. For examp le in matters of health, variations in body temperature, pulse beat and blood pressure are the basic guides to diagnosis where prescribed treat ment is designed to control their variation. In this paper we intend to suggest some estimators for population variance. When there is no additional informat ion on the au xiliary variable available, the simp lest estimator of population variance is the simp le random sample variance without replacement. It is co mmon practice to use the auxiliary variable for imp roving the precision of the estimate of a parameter. In this paper, we consider the au xiliary informat ion to improve the efficiency of the estimation of population variance S y 2 = 1 ( N−1 ) in simp le random sampling.
When the information on an au xiliary variable X is known, a number of estimators such as ratio, product and linear regression estimators are proposed in the literature.
When the correlation between the study variable and the auxiliary variable is positive, ratio method of estimat ion is quite effective. On the other hand, when the correlation is negative, Product method of estimat ion can be employed effectively. A mong the estimators mentioned above, the ratio estimator and its modificat ions are wid ely used fo r the es timatio n o f th e v a r ian ce o f th e s tu d y v ariab le. Estimation of population variance is considered by Isaki [10] where ratio and regression estimators are proposed. Prasad and Singh [14] have considered a ratio type estimator for estimation of population variance by improving Isaki [10] estimator with respect to bias and precision. Arcos et al. [4] have introduced another ratio type estimator, wh ich has also improved the Isaki [10] estimator, which is almost unbiased and more precise than the other estimators.
Before discussing further about the traditional rat io type variance estimator, existing modified ratio type variance estimators and the proposed modified rat io type variance estimators, the notations to be used in this paper are described below: of the au xiliary variable X is known together with its bias and mean squared error and are as given below: The Ratio type variance estimator given in (1) is used to improve the precision of the estimate of the population variance co mpared to simple rando m sampling when there exists a positive correlation between X and Y . Further improvements are also achieved on the classical rat io estimator by introducing a number of modified rat io 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], Ah med et al. [2], Al-Jararha and Al-Haj Eb rahem [3], Arcos 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 [11,12], Murthy [13], Prasad and Singh [14], Reddy [15], Singh and Chaudhary [16], Singh et al. [17,19], Upadhyaya and Singh [23] and Wolter [24].
Motivated by Singh et al. [18], Sisodia and Dwivedi [20], and Upadhyaya and Singh [22], Kadilar and Cingi [11] suggested four ratio type variance estimators using known values of Co-efficient of variation C X and Co-efficient of Kurtosis β 2 ( x ) of an au xiliary variable X together with their biases and mean squared errors as given in the Table 1.
The existing modified ratio type variance estimators discussed above are biased but have min imu m mean squared errors co mpared to the tradit ional 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 x 2 , C x , β 2 and their linear co mbinations.
Subraman i and Ku marapandiyan [21] used Quart iles and their functions of the au xiliary variab le 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 extreme 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 auxiliary variable. Now briefly we will d iscuss about quartiles and its functions. The median div ides the data into two equal sets. The first (lower) quartile is the midd le value of the first set, where 25% 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 25% are larger. It should be noted that the median will be denoted by the notation Q 2 , the second quartile. The inter-quartile range is another range used as a measure of the spread. The d ifference between upper and lower quartiles, wh ich is called as the inter-quartile range, also indicates the dispersion of a data set. The formula for inter-quart ile range is: 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: (3) Subraman i and Ku marapandiyan [21] suggested another new measure called as Semi-quartile average denoted by the notation Q a and defined as: Kadilar and Cingi [11] γ S y

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 auxiliary variable like Inter-quartile range, Semi-quartile range and Semi-quartile average The proposed class of modified ratio type variance estimators S � JGi 2 , = 1,2, … ,5 for estimat ing the population variance S y 2 together with the first degree of appro ximation, the biases and mean squared errors and the constants are given below:

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: 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 existing modified rat io type variance estimators given in Table 1  where In the same way, the biases and mean squared errors of the proposed modified ratio type variance estimators given in Table 2  Fro m the expressions given in (6) and (10) we have derived the condition for which the proposed estimators S � JGj 2 ; j = 1,2,3,4 and 5 are more efficient than the traditional ratio type variance estimator and it is given below: ; j = 1,2,3,4 and 5 (11) Fro m the expressions given in (8) and (10) we have derived the conditions for which the proposed estimators S � JGj 2 ; j = 1,2,3,4 and 5 are more efficient than the existing modified rat io type variance estimators given in Table 1, S � KCi 2 ; i = 1, 2, 3 and 4 and are given below:

Numerical Study
The performance of the proposed modified ratio type variance estimators are assessed with that of tradit ional ratio type estimator and existing modified ratio type variance estimators listed in Table 1 for certain natural populations. The populations 1 and 2 are the real data set taken from the Report on Waste 2004 drew up by the Italian bureau for the environment protection-APAT. Data and reports are available in the website http://www.osservatorionazionalerifiuti.it [25]. In the data set, for each of the Italian provinces, three variab les are considered: the total amount (tons) of recyclable-waste collection in Italy in 2003 (Y) , the total amount of recyclable-waste collection in Italy in 2002 ( X 1 ) and the number of inhabitants in 2003 ( X 2 ) . The population 3 is taken fro m Murthy [13] given in page 228 and population 4 is taken fro m Singh and Chaudhary [16] given in page 108. The population parameters and the constants computed from the above populations are given below: The biases and mean squared errors of the existing and proposed modified ratio type variance estimators for the populations given above are given in the follo wing Tables: Fro m the values of Table 4, it is observed that the biases of the proposed modified rat io type variance estimators are less than the biases of the trad itional and existing modified rat io type variance estimators. Similarly fro m the values of Tab le 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 trad itional and existing modified ratio type variance estimators.

Conclusions
Estimating the fin ite population variance has great significance in various fields such as Industry, Agriculture, Medical and Biological sciences, etc. In this paper we have proposed a class of modified rat io type variance estimators using the quartiles and their functions of the auxiliary variable like Inter-quart ile range, Semi-quart ile range and Se mi-quartile average. The b iases 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 existing modified ratio type variance estimators. Further we have derived the conditions for which the proposed estimators are mo re efficient than the traditional and existing 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 o f the traditional and existing modified estimators for certain known populations. Hence we strongly recommend that the proposed modified rat io type variance estimator may be preferred over the traditional ratio type variance estimator and existing mod ified ratio type variance estimators for the use of practical applications.