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9(4), April, 2020 ISSN: 2277-9655 IJESRT: 9(4), April, 2020 ISSN: 2277-9655 I International Journal of Engineering Sciences & Research X Technology (A Peer Reviewed Online Journal) Impact Factor: 5.164 IJESRT Chief Editor Executive Editor Dr. J.B. Helonde Mr. Somil Mayur Shah Website: www.ijesrt.com Mail: [email protected] O ISSN: 2277-9655 [Kumari et al., 9(4): April, 2020] Impact Factor: 5.164 IC™ Value: 3.00 CODEN: IJESS7 IJESRT INTERNATIONAL JOURNAL OF ENGINEERING SCIENCES & RESEARCH TECHNOLOGY AN IMPROVED ESTIMATION OF POPULATION VARIANCE USING THE COEFFICIENT OF KURTOSIS AND MEDIAN OF AN AUXILIARY VARIABLE Chandni Kumari*1 & Ratan Kumar Thakur2 *1&2Department of Statistics, Babasaheb Bhimrao Ambedkar University (A Central University), Lucknow-226025, India DOI: 10.5281/zenodo.3782484 ABSTRACT 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. KEYWORDS: Ratio Type Variance Estimator, Auxiliary Variable, Kurtosis, Median, Bias, Mean Squared Error (MSE), Percentage Relative Efficiency (PRE), Simple Random Sampling. 1. INTRODUCTION Consider a finite population of size 푁 identifiable and non-overlapping units. From this population size, a sample of size 푛 is drawn by simple random sampling. Let, the variable under study is denoted by 푦 and the variable which contains the auxiliary information about the study variable is denoted by 푥. The population mean of study and auxiliary variable is denoted by 푌̅ and ̅푋. Further, the population and sample variances are given as under 2 1 푁 2 2 1 푁 2 푆 = ∑ (푌 − 푌̅) , 푆 = ∑ (푋 − 푋̅) and 푦 푁−1 푖=1 푖 푥 푁−1 푖=1 푖 2 1 푁 2 2 1 푁 2 푠 = ∑ (푦 − 푦̅) , 푠 = ∑ (푥 − 푥̅) 푦 푁−1 푖=1 푖 푥 푁−1 푖=1 푖 In general, we define the following parameters 1 푁 푟 푠 µ22 µ푟푠 = ∑푖=1(푦푖 − 푦̅) (푥푖 − 푥̅) , 휆22 = , 퐶푦 and 퐶푥 be the coefficient of variation of the study and 푁−1 µ02 µ20 auxiliary variable. µ04 µ40 퐾푥 = 휆04 = 2 and 퐾푦 = 휆40 = 2 be the coefficient of kurtosis of the study and auxiliary variable. µ02 µ20 푆푥푦 휌푥푦 = be the coefficient of correlation between y and x. 푆푦푆푥 푀푥 be the population median of 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 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 퐸(푥 − µ)4 퐾 = (퐸(푥 − µ)2)2 http: // www.ijesrt.com© International Journal of Engineering Sciences & Research Technology [168] IJESRT is licensed under a Creative Commons Attribution 4.0 International License. ISSN: 2277-9655 [Kumari et al., 9(4): April, 2020] Impact Factor: 5.164 IC™ Value: 3.00 CODEN: IJESS7 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. 2. 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 2 푡1 = 푠푦 (1) The bias and MSE of t1 is 1−푓 2 휆22−1 퐵푖푎푠(푡1) = 푆푦 {(퐾푥 − 1) 훹1 (훹1 − )} = 0 (2) 푛 퐾푥−1 1−푓 4 휆22−1 푀푆퐸(푡1) = 푉(푡1) = 푆푦 {(퐾푦 − 1) + (퐾푥 − 1) 훹1 (훹1 − 2 ( ))} (3) 푛 퐾푥−1 1−푓 = 푆4 (퐾 − 1) ; where 훹 = 0 푛 푦 푦 1 Population variance, estimation using auxiliary information was considered by [7], and proposed ratio type population variance estimator, given by 2 푆푥 2 푡2 = 2 푠푦 푠푥 (4) The bias and Mean Squared Error of Isaki’s estimator, 1 − 푓 2 휆22 − 1 1 − 푓 2 퐵푖푎푠(푡2) = 푆푦 {(퐾푥 − 1) 훹2 (훹2 − )} = 푆푦 {(퐾푥 − 1) − (휆22 − 1) } 푛 퐾푥 − 1 푛 1−푓 4 휆22−1 푀푆퐸(푡2) = 푆푦 {(퐾푦 − 1) + (퐾푥 − 1) 훹2 (훹2 − 2 ( ))} (5) 푛 퐾푥−1 1−푓 4 = 푆 {(퐾 − 1) + (퐾 − 1) − 2(휆 − 1) } ; where 훹 = 1 푛 푦 푦 푥 22 2 [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 2 2 information on both 푆푥 and 푘푥 suggested modified ratio type population variance estimator for 푆푦 as 2 푆푥+ 푘푥 2 푡3 = [ 2 ] 푠푦 (6) 푠푥+ 푘푥 The estimator, bias and MSE obtained as 1−푓 2 휆22−1 퐵푖푎푠(푡3) = 푆푦 {(퐾푥 − 1) 훹3 (훹3 − )} (7) 푛 퐾푥−1 1−푓 4 휆22−1 푀푆퐸(푡2) = 푆푦 {(퐾푦 − 1) + (퐾푥 − 1) 훹3 (훹3 − 2 ( ))} (8) 푛 퐾푥−1 2 푆푥 where 훹3 = [ 2 ] 푆푥 + 푘푥 [10] suggested four modified ratio type variance estimators using known values of 퐶푥 and 푘푥 , 2 푆푥− 퐶푥 2 푡4 = [ 2 ] 푠푦 (9) 푠푥− 퐶푥 2 푆푥− 푘푥 2 푡5 = [ 2 ] 푠푦 (10) 푠푥− 푘푥 2 푆푥 푘푥− 퐶푥 2 푡6 = [ 2 ] 푠푦 (11) 푠푥 푘푥− 퐶푥 2 푆푥 퐶푥− 푘푥 2 푡7 = [ 2 ] 푠푦 (12) 푠푥 퐶푥− 푘푥 http: // www.ijesrt.com© International Journal of Engineering Sciences & Research Technology [169] IJESRT is licensed under a Creative Commons Attribution 4.0 International License. ISSN: 2277-9655 [Kumari et al., 9(4): April, 2020] Impact Factor: 5.164 IC™ Value: 3.00 CODEN: IJESS7 The biases and MSE of their estimators, 1−푓 2 휆22−1 퐵푖푎푠(푡4) = 푆푦 (퐾푥 − 1) { 훹4 (훹4 − )} (13) 푛 퐾푥−1 1−푓 4 휆22−1 푀푆퐸(푡4) = 푆푦 {(퐾푦 − 1) + (퐾푥 − 1) 훹4 (훹4 − 2 ( ))} (14) 푛 퐾푥−1 1−푓 2 휆22−1 퐵푖푎푠(푡5) = 푆푦 (퐾푥 − 1) { 훹5 (훹5 − )} (15) 푛 퐾푥−1 1−푓 4 휆22−1 푀푆퐸(푡5) = 푆푦 {(퐾푦 − 1) + (퐾푥 − 1) 훹5 (훹5 − 2 ( ))} (16) 푛 퐾푥−1 1−푓 2 휆22−1 퐵푖푎푠(푡6) = 푆푦 (퐾푥 − 1) { 훹6 (훹6 − )} (17) 푛 퐾푥−1 1−푓 4 휆22−1 푀푆퐸(푡6) = 푆푦 {(퐾푦 − 1) + (퐾푥 − 1) 훹6 (훹6 − 2 ( ))} (18) 푛 퐾푥−1 1−푓 2 휆22−1 퐵푖푎푠(푡7) = 푆푦 (퐾푥 − 1) { 훹7 (훹7 − )} (19) 푛 퐾푥−1 1−푓 4 휆22−1 푀푆퐸(푡7) = 푆푦 {(퐾푦 − 1) + (퐾푥 − 1) 훹7 (훹7 − 2 ( ))} (20) 푛 퐾푥−1 Where 2 2 2 2 푆푥 푆푥 푆푥 퐾푥 푆푥 퐶푥 훹4 = [ 2 ] , 훹5 = [ 2 ], 훹6 = [ 2 ] , 훹7 = [ 2 ] 푆푥− 퐶푥 푆푥− 퐾푥 푆푥 퐾푥− 퐶푥 푆푥퐶푥− 퐾푥 [11] utilizing population median 푀푥 came up with a modified ratio type population variance estimator as 2 푆푥 − 푀푥 2 푡8 = [ 2 ] 푠푦 (21) 푠푥 − 푀푥 The bias and MSE of their estimator 1−푓 2 휆22−1 퐵푖푎푠(푡8) = 푆푦 (퐾푥 − 1) { 훹8 (훹8 − )} (22) 푛 퐾푥−1 1−푓 4 휆22−1 푀푆퐸(푡8) = 푆푦 {(퐾푦 − 1) + (퐾푥 − 1) 훹8 (훹8 − 2 ( ))} (23) 푛 퐾푥−1 2 푆푥 Where 훹8 = [ 2 ] 푆푥+ 푀푥 [12] using the known quartiles (upper and lower quartile 푄3 and 푄1 respectively) of the auxiliary variable x suggested 2 푆푥 + 푄1 2 푡9 = [ 2 ] 푠푦 (24) 푠푥 + 푄1 2 푆푥+ 푄3 2 푡10 = [ 2 ] 푠푦 (25) 푠푥 + 푄3 The biases and MSE of their estimators 푡9and 푡10 as follows 1−푓 2 휆22−1 퐵푖푎푠(푡9) = 푆푦 (퐾푥 − 1) { 훹9 (훹9 − )} (26) 푛 퐾푥−1 1−푓 4 휆22−1 푀푆퐸(푡9) = 푆푦 {(퐾푦 − 1) + (퐾푥 − 1) 훹9 (훹9 − 2 ( ))} (27) 푛 퐾푥−1 1−푓 2 휆22−1 퐵푖푎푠(푡10) = 푆푦 (퐾푥 − 1) { 훹10 (훹10 − )} (28) 푛 퐾푥−1 1−푓 4 휆22−1 푀푆퐸(푡10) = 푆푦 {(퐾푦 − 1) + (퐾푥 − 1) 훹10 (훹10 − 2 ( ))} (29) 푛 퐾푥−1 2 2 푆푥 푆푥 Where 훹9 = [ 2 ] and 훹10 = [ 2 ] 푆푥 + 푄1 푆푥+ 푄3 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.
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