ESTIMATION OF POPULATION VARIANCE USING KNOWN COEFFICIENT OF KURTOSIS AND MEDIAN OF AN AUXILIARY VARIABLE

Chandni Kumari and Ratan Kumar Thakur Department of Statistics, Babasaheb Bhimrao Ambedkar University (A Central University), Lucknow-226025, India. ...................................................................................................................... Manuscript Info Abstract ......................... ........................................................................ Manuscript History Received: 16 February 2020 Final Accepted: 18 March 2020 Published: April 2020

We have suggested an improved ratio type log estimator for population variance by using coefficient of kurtosis and median of an auxiliary variable x. The properties of proposed estimator have been derived up to first order of Taylor's series expansion. The efficiency conditions derived theoretically under which the proposed estimator performs better than existing estimators. The proposed estimator as illustrated by the empirical studies using real populations performs better than the existing estimators i.e. it has the smallest mean squared error and the highest Percentage Relative Efficiency.

…………………………………………………………………………………………………….... Introduction:-
Consider a finite population of size N identifiable and non-overlapping units. From this population size, a sample of size n is drawn by simple random sampling. Let, the variable under study is denoted by y and the variable which contains the auxiliary information about the study variable is denoted by x. The population mean of study and auxiliary variable is denoted by Y and X . Further, the population and sample variances are given as under In general, we define the following parameters µ rs = 1 N−1 µ 02 µ 20 , C y and C x be the coefficient of variation of the study and auxiliary variable.  Many authors have come up with more precise estimators by employing prior knowledge of certain population parameter(s). [2] for example attempted use of the coefficient of variation of study variable but prove inadequate for in practice, this parameter is unknown. Motivated by [2] work, [3] [4] and [5] used the known coefficient of variation but now that of the auxiliary variable for estimating population mean of study variable. Reasoning along the same path [6] used the prior value of coefficient of kurtosis of an auxiliary variable in estimating the population variance of the study variable y. Kurtosis in most cases is not reported or used in many research articles, in spite of 804 the fact that fundamentally speaking every statistical package provides a measure of kurtosis. This may be attributed to the likelihood that kurtosis is not well understood or its importance in various aspects of statistical analysis has not been explored fully. Kurtosis can simply be expressed as where E denotes the expectation, µ is the mean. Moreover, one of the measure of central tendency which we use in this article is Median. Median is the middle term of the series of the distribution under study when the values are arranged in ascending or descending order. Inspite of many applications of median, the most important is advantage of median is that it is less affected by the outliers and skewed data, thus is preferred to the mean especially when the distribution is not symmetrical. We can therefore utilize the median and the coefficient of kurtosis of the auxiliary variable to derive a more precise ratio type log estimator for population variance.

Estimators Available in Literature:
In this section we have reviewed some finite population variance estimators existing in literature which will help in the construction and development of the proposed estimator. Notably, when auxiliary information is not available the usual unbiased estimator to the population variance is t 1 = s y 2 (1) The bias and MSE of t 1 is Population variance, estimation using auxiliary information was considered by [7], and proposed ratio type population variance estimator, given by The bias and Mean Squared Error of Isaki's estimator, [6] initiated the use of coefficient of kurtosis in estimating population variance of a study variable y. Later, the coefficient of kurtosis was used by [3] [5] [8] in the estimating the population mean. [9] using the known information on both S x 2 and k x suggested modified ratio type population variance estimator for S y 2 as The estimator, bias and MSE obtained as [10] suggested four modified ratio type variance estimators using known values of C x and k x , 805 The The bias and MSE of their estimator The biases and MSE of their estimators 9 and 10 as follows Motivated by [10] and [11] [13] considered the estimation of finite population variance using known coefficient of variation and median of an auxiliary variable, proposed an estimator.
The bias and MSE obtained to be, 11 [25] and [26] in the improvement of the performance of the population variance estimator of the study variable using known population parameters of an auxiliary variable. We introduce the following improved ratio type log estimator for population variance using a known value of population coefficient of kurtosis and median of an auxiliary variable.

Theoretical Comparisons:
The theoretical conditions under which the proposed modified ratio type estimators 12 is more efficient than the other existing estimator , j=1, 2, … , 11, 12 from MSE of , j=1, 2, … , 11, 12; given to the first degree of approximation in general as Using the above summary values we have the results in Table 1 below. From the table, Mean Squared Errors it is clear that our proposed ratio type log estimator 13 for population variance has the least Mean Squared Error (MSE). The efficiency of our proposed estimator 13 is examined numerically by its Percentage Relative Efficiency (PRE(s)) in comparison with those of existing  x 100 Using formula (40) and (43) we compute the Percent Relative Efficiencies and tabulate the results in Table 2.
Percentage Relative efficiency being a robust statistical tool that is used to ascertain the efficiency of suggested estimator over conventional estimator. From the findings summarized in the table above it is clear that our proposed estimator 13 performed best, that is it has the highest PRE among all the other estimators. This therefore implies that we can apply our proposed estimator to appropriate practical situations and obtain better and more efficient results than the traditional and other existing population variance estimators.

Conclusion:-
In this article, we have proposed an improved log type ratio estimator for population variance estimator using known coefficient of kurtosis and the median of the auxiliary variable x. We have analyzed the performance of our proposed estimator against the usual unbiased variance estimator and existing estimators using two natural populations by comparing their PRE(s). Based on the results of our studies, it is proved that our proposed estimator works better than the other existing estimators having the highest Percentage Relative Efficiency. Hence can be applied to practical applications, where knowledge of population parameters of auxiliary variable is available. We