ID 505 Ratio Estimator for Population Variations Using Additional

Home , Ratio estimator

Proceedings of the 5th NA International Conference on Industrial Engineering and Operations Management Detroit, Michigan, USA, August 10 - 14, 2020 Ratio Estimator for Population Variations Using Additional Information on Simple Random Sampling

Haposan Sirait, Stepi Karolin Mathematics Study Program, Faculty of Mathematics and Natural Sciences, Universitas Riau, Pekanbaru, Indonesia. [email protected]

Sukono Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Padjadjaran, Bandung, Indonesia. [email protected]

Kalfin Doctor Program of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Padjadjaran, Indonesia [email protected]

Abdul Talib Bon Department of Production and Operations, Universiti Tun Hussein Onn Malaysia (UTHM) Parit Raja 86400, Johor, Malaysia [email protected]

Abstract

Variance is a parameter that is often estimated in a research, because variance can give a picture of the degree of homogeneity of a population. To estimate the variance for rare and population has been the main problem in survey sampling Variance obtained based on simple random sampling, there further modified by utilizing additional characters that are assumed to be related to the character under study. This article discusses the population mean estimator by utilizing additional character variances, and then the modified estimator will be compared to the mean square error to obtain an efficient estimator

Keywords: Simple random sampling, Mena square error, efficient estimator

1. Introduction In manufacturing industries and pharmaceutical laboratories sometimes researchers are interested in the variation of their products. To measure the variations within the values of study variable y, the problem of estimating the population variance of of study variable Y received a considerable attention of the statistician in survey 2 sampling including Isaki (1983),𝑦𝑦 Singh and Singh (2001, 2003), Jhajj et al. (2005), Kadliar and Cingi (2006), Singh et al. (2008), Grover (2010),𝑆𝑆 Singh et al. (2011) and Singh and Solanki (2012) have suggested improved estimator for estimation of . 2 Simple random sampling𝑦𝑦 is a method for taking n units of population of size N, where each element has the same opportunity to be 𝑆𝑆taken as a sample member. Sampling can be done with a refund (with replacement) or without refund (without replacement). Cochran (1977) states that in simple random sampling without returning, the number of samples to be formed is . Let us consider a finite population𝑵𝑵 = ( , , ,…, ) having N units and let Yand X are the study and 𝒏𝒏 auxiliary variable with means𝑪𝑪 X and Y respectively. Let us suppose that a sample of size n is drawn from the 𝑈𝑈 𝑈𝑈1 𝑈𝑈2 𝑈𝑈3 𝑈𝑈𝑁𝑁

population using simple random sampling without replacement (SRSWOR) method. Let For a simple random sample 1 = 𝑛𝑛 ( ) , 1 2 2 𝑦𝑦 𝑖𝑖 is an unbiased estimate of 𝑠𝑠 � 𝑦𝑦 − 𝑦𝑦� 𝑛𝑛 − 𝑖𝑖=1 1 = 𝑛𝑛 ( ) , 1 2 2 𝑆𝑆𝑦𝑦 � 𝑦𝑦𝑖𝑖 − 𝑌𝑌� 𝑁𝑁 − 𝑖𝑖=1 In other words, The usual sample variance estimator of the population variance was defined as:

= 2 2 ̂𝑦𝑦 𝑦𝑦 The variance of the usual unbiased estimator is given𝑆𝑆 by𝑠𝑠 : 𝟐𝟐 �𝒚𝒚 𝑺𝑺 = ( 1) 2 𝑁𝑁−𝑛𝑛 4 𝜇𝜇40 𝑦𝑦 𝑦𝑦 2 where = ( ) 𝑉𝑉�𝑆𝑆̂ � 𝑁𝑁𝑁𝑁 𝑆𝑆 𝜇𝜇20 − 1 𝑟𝑟 Ratio-type estimators𝑁𝑁 take advantage of the𝑠𝑠 correlation between the auxiliary variable, , Z and the study variable, 𝜇𝜇𝑟𝑟𝑟𝑟 𝑁𝑁−1 ∑𝑖𝑖=1�𝑦𝑦𝑖𝑖 − 𝑌𝑌⃐�� 𝑥𝑥𝑖𝑖 − 𝑋𝑋⃐ . When information is available on the auxiliary variable that is positively correlated with the study variable, the ratio estimator is a suitable estimator to estimate the population variance. For ratio estimators𝑋𝑋 in sampling theory, population𝑌𝑌 information of the auxiliary variable, such as the variansi , is often used to increase the efficiency of the estimation for a population variance. Abid et al.(2016) suggested a number of new ratio estimators that were modified in simple random samples Yadav et al.(2013) proposed an estimator for variance with one additional variable, According to Perri (2007), additional variables are usually used in sample survey practices to get better designs and to achieve higher accuracy in estimating population parameters. Ismail et al.(2018) modified the estimator proposed by Isaki (1983) using two additional variables. Furthermore, it is shown that the new ratio is biased so that a square error (MSE) comparison is performed. Getting smaller. The more MSE obtained the more efficient the estimator. This method can be used to increase accuracy in searching for data security. The author is interested in reviewing a new method contained in the article Ismail et al.(2018). The existing ratio estimator for estimating the population variance Y , of the study variable Y is defined as : = 2 2 𝑠𝑠𝑦𝑦 2 2 Bias and Mean square error of ratio type variancê𝑅𝑅 estimator𝑠𝑠𝑥𝑥 𝑥𝑥 was : 𝑆𝑆 1 𝑆𝑆 ( ) 1 ( 1) and 2 2 𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵�𝑆𝑆̂𝑦𝑦 � ≈ 𝑆𝑆𝑦𝑦 ��𝛽𝛽2 𝑥𝑥 − � − 𝜆𝜆22 − � 1 𝑛𝑛 ( ) 1 2( 1) + ( ) 1 2 4 Where = , 𝑀𝑀( )𝑀𝑀𝑀𝑀=�𝑆𝑆 ̂𝑦𝑦 � ≈ 𝑆𝑆𝑦𝑦 ��𝛽𝛽2 𝑥𝑥 − � − 𝜆𝜆22 − �𝛽𝛽2 𝑥𝑥 − �� 𝜇𝜇𝑟𝑟𝑟𝑟 𝜇𝜇40 𝑛𝑛 𝑟𝑟� 𝑠𝑠� 2 𝑟𝑟𝑟𝑟 2 2 2 𝑥𝑥 𝜇𝜇02 𝜆𝜆 𝜇𝜇20 𝜇𝜇02 𝛽𝛽

2. Proposed Modified Ratio Estimators Estimators for the population variance in the simple random sampling using some known auxiliary information on variance . They showed that their suggested estimators are more efficient than traditional ratio estimator in the estimation of the population variance. Ismail et al.(2018) estimators are given as :

= + + (1 ) + . 2 𝜏𝜏 2 𝜒𝜒 2 𝜙𝜙 2 2 𝑆𝑆𝑢𝑢 𝑠𝑠𝑢𝑢 𝑆𝑆𝑤𝑤 and 𝑆𝑆̂𝑟𝑟𝑟𝑟𝑟𝑟1 �𝑠𝑠𝑦𝑦 𝑑𝑑� �𝜔𝜔� 2� − 𝜔𝜔 −𝛾𝛾 � 2� 𝜓𝜓 � 2 � � − 𝑑𝑑 𝑠𝑠𝑢𝑢 𝑆𝑆𝑢𝑢 𝑠𝑠𝑤𝑤 = + + (1 ) 2 𝜏𝜏 2 𝜒𝜒 2 2 𝑆𝑆𝑢𝑢 𝑠𝑠𝑢𝑢 𝑆𝑆̂𝑟𝑟𝑟𝑟𝑟𝑟2 �𝑠𝑠𝑦𝑦 𝑑𝑑� �𝜔𝜔� 2� − 𝜔𝜔 � 2� � − 𝑑𝑑 𝑠𝑠𝑢𝑢 𝑆𝑆𝑢𝑢

Where = + , = ( + ) , = + , = ( + ) . 𝟐𝟐 𝟐𝟐 𝟐𝟐 𝟐𝟐 𝟐𝟐 𝟐𝟐 𝟐𝟐 𝟐𝟐 𝟐𝟐 𝟐𝟐 Bias and𝒔𝒔𝒘𝒘 Mean𝒎𝒎𝒔𝒔 Square𝒛𝒛 𝒑𝒑𝒑𝒑 𝒚𝒚Error𝑺𝑺𝒘𝒘 of Ratio𝒎𝒎 Type𝒑𝒑 𝑺𝑺𝒛𝒛 Variance𝒔𝒔𝒖𝒖 𝒃𝒃𝒔𝒔 𝒙𝒙Estimator𝒄𝒄𝑺𝑺𝒙𝒙 𝐝𝐝𝐝𝐝𝐝𝐝 𝑺𝑺𝒖𝒖 𝒃𝒃 𝒄𝒄 𝑺𝑺𝒙𝒙 In order to drive bias of ratio type variance estimator by using 2, = , = ( ) 𝑆𝑆̂𝑟𝑟𝑟𝑟𝑟𝑟1 = ( ) was written as: 2 −2 2 2 −2 2 2 ̂𝑟𝑟𝑟𝑟𝑟𝑟1 0 𝑦𝑦 𝑦𝑦 𝑦𝑦 1 𝑥𝑥 𝑥𝑥 𝑥𝑥 −2 2 2 𝑆𝑆 (1 + 𝑒𝑒) +𝑆𝑆 �𝑠𝑠 − 𝑆𝑆 � 𝑒𝑒 𝑆𝑆 𝑠𝑠 − 𝑆𝑆 2 𝑧𝑧 𝑧𝑧 𝑧𝑧 = ( (1 + ) + ) 𝑒𝑒 𝑆𝑆 𝑠𝑠 − 𝑆𝑆 ( + )2 2 −𝜏𝜏 2 𝑏𝑏 𝑒𝑒1 𝑆𝑆𝑥𝑥 𝑐𝑐𝑆𝑆𝑥𝑥 𝑟𝑟𝑟𝑟𝐺𝐺1 𝑦𝑦 0 2 𝑆𝑆̂ 𝑆𝑆 𝑒𝑒 𝑑𝑑 �𝜔𝜔 � 𝑥𝑥 � (1 + ) 𝑏𝑏+ 𝑐𝑐 𝑆𝑆 +(1 ) ( + )2 2 𝜒𝜒 𝑏𝑏 𝑒𝑒1 𝑆𝑆𝑥𝑥 𝑐𝑐𝑆𝑆𝑥𝑥 2 − 𝜔𝜔 −𝜓𝜓 � 𝑥𝑥 � (1 + 𝑏𝑏) 𝑐𝑐+𝑆𝑆 + . ( + )2 2 −𝜙𝜙 𝑚𝑚 𝑒𝑒2 𝑆𝑆𝑧𝑧 𝑝𝑝𝑆𝑆𝑧𝑧 By suppose, /( + ) = and /( + ) =𝜓𝜓 obtained � 2 � � − 𝑑𝑑 𝑚𝑚 𝑝𝑝 𝑆𝑆𝑧𝑧 1 2 𝑏𝑏 𝑏𝑏 𝑐𝑐 𝜃𝜃 𝑚𝑚= (𝑚𝑚 +𝑝𝑝 + 𝜃𝜃 )( (1 + ) + (1 ) 2 2 −𝜏𝜏 ̂𝑟𝑟𝑟𝑟𝐺𝐺1 𝑦𝑦 𝑦𝑦 0 1 1 𝑆𝑆 (𝑆𝑆1 + 𝑑𝑑 )𝑆𝑆 +𝑒𝑒 (𝜔𝜔1 + 𝜃𝜃 𝑒𝑒) . − 𝜔𝜔 −𝜓𝜓 By suppose ( ) = , let. 𝜒𝜒 −𝜙𝜙 𝜃𝜃1𝑒𝑒1 𝜓𝜓 𝜃𝜃2𝑒𝑒21 � − 𝑑𝑑 1 1 ( ) = , 𝜃𝜃 𝑒𝑒 𝑥𝑥 (1 + ) 1 𝜏𝜏 𝑓𝑓 𝑥𝑥( ) = 1 1 , (1 +𝜃𝜃 𝑒𝑒) Furthermore, using the MacLaurin series approximation𝑓𝑓 𝑥𝑥 is obtained𝜏𝜏 𝑥𝑥 ( ) = (0) + (0) + (0) + + (0), 2!2 𝑛𝑛! 1 ′ ( +𝑥𝑥 1) ′′ 𝑥𝑥 𝑛𝑛 𝑓𝑓 𝑥𝑥 = 1𝑓𝑓 + 𝑥𝑥𝑓𝑓 𝑓𝑓( ) ⋯ 𝑓𝑓 (0). (1 + ) 2 𝑛𝑛 !𝑛𝑛 𝜏𝜏 𝜏𝜏 2 𝜃𝜃1𝑒𝑒1 𝑛𝑛 Then, the same way is done for𝜏𝜏 1/(1 + 1 1 ) , obtained1 1 1 1 𝜏𝜏𝜃𝜃 𝑒𝑒 − 𝜃𝜃 𝑒𝑒 − ⋯ − 𝑓𝑓 1 𝜃𝜃 𝑒𝑒 −𝜒𝜒( + 1) 𝑛𝑛 = 1 + 1 1 ( ) (0). (1 + ) 𝜃𝜃 𝑒𝑒 2 !𝑛𝑛 𝜒𝜒 𝜒𝜒 2 𝜃𝜃1𝑒𝑒1 𝑛𝑛 and for 1/(1 + ) obtained−𝜒𝜒 1 1 1 1 1 1 𝜒𝜒𝜃𝜃 𝑒𝑒 − 𝜃𝜃 𝑒𝑒 − ⋯ − 𝑓𝑓 𝜃𝜃 𝜙𝜙𝑒𝑒1 ( + 1) 𝑛𝑛 1 1 = 1 + ( ) (0). 𝜃𝜃(1𝑒𝑒 + ) 2 !𝑛𝑛 𝜙𝜙 𝜙𝜙 2 𝜃𝜃2𝑒𝑒2 𝑛𝑛 assumption | | < 1 and |𝜙𝜙 | < 1𝜙𝜙𝜃𝜃, so2𝑒𝑒 that2 − 𝜃𝜃2𝑒𝑒2 − ⋯ − 𝑓𝑓 𝜃𝜃2𝑒𝑒2 ( + 1)𝑛𝑛 1 1 ( 2 +2 )(1 + ) 1 ( ) 𝜃𝜃 𝑒𝑒 𝜃𝜃 𝑒𝑒 2 2 𝜏𝜏 𝜏𝜏 2 �𝑆𝑆̂𝑟𝑟𝑟𝑟𝐺𝐺1� ≈ � 𝑆𝑆𝑦𝑦 𝑑𝑑 𝜃𝜃0𝑒𝑒0 � −( 𝜔𝜔+ 1)1 �𝜏𝜏𝜃𝜃 𝑒𝑒1 − 𝜃𝜃1𝑒𝑒1 � +(1 ) ( ) 2 𝜒𝜒 𝜒𝜒 2 − 𝜔𝜔 −𝜓𝜓 �𝜒𝜒𝜃𝜃( 1+𝑒𝑒1 1−) 𝜃𝜃1𝑒𝑒1 � ( ) . 2 𝜙𝜙 𝜙𝜙 2 −𝜓𝜓2 �𝜙𝜙𝜃𝜃 𝑒𝑒2 − 𝜃𝜃2𝑒𝑒2 � − 𝑑𝑑� Apply expectation on both side of ,so that , ( + 1) ̂𝑟𝑟𝑟𝑟𝐺𝐺1 ( + )(1 + ) 1 ( ) 𝑆𝑆 2 2 𝜏𝜏 𝜏𝜏 2 𝐸𝐸�𝑆𝑆̂𝑟𝑟𝑟𝑟𝐺𝐺1� ≈ 𝐸𝐸 � 𝑆𝑆𝑦𝑦 𝑑𝑑 𝜃𝜃0𝑒𝑒0 � − 𝜔𝜔1 �𝜏𝜏𝜃𝜃 𝑒𝑒1 − 𝜃𝜃1𝑒𝑒1 � ( + 1) +(1 ) ( ) 2 𝜒𝜒 𝜒𝜒 2 − 𝜔𝜔 −𝜓𝜓 �𝜒𝜒𝜃𝜃1𝑒𝑒1 − 𝜃𝜃1𝑒𝑒1 �

( + 1) ( ) 2 𝜙𝜙 𝜙𝜙 2 or it can be written in form, −𝜓𝜓2 �𝜙𝜙𝜃𝜃 𝑒𝑒2 − 𝜃𝜃2𝑒𝑒2 � − 𝑑𝑑� = ( + )(1 + ) (1 + ) + (1 ) 2 (1 + ) ) −𝜏𝜏 ̂𝑟𝑟𝑟𝑟𝐺𝐺1 𝑦𝑦 0 0 1 1 𝐸𝐸�𝑆𝑆 � 𝐸𝐸� 𝑆𝑆 𝑑𝑑 𝜃𝜃 𝑒𝑒𝜒𝜒 𝜔𝜔 𝜃𝜃 𝑒𝑒 − 𝜔𝜔 1 1 and using ( ) = 0, ( ) = 0 ,E( ) = 0 we obtained,𝜃𝜃 𝑒𝑒 − 𝑑𝑑

0 1 2 𝐸𝐸 𝑒𝑒 𝐸𝐸 𝑒𝑒 𝑒𝑒 ( ( ) +(1 ) 2 ′ 2 2 ̂𝑟𝑟𝑟𝑟𝐺𝐺1 𝑦𝑦 2 1 𝐸𝐸 �𝑆𝑆 � ≈ 𝑆𝑆+ −( 𝛽𝛽)( 𝑥𝑥 𝑤𝑤+ (1 − 𝜔𝜔) −𝜓𝜓 )𝜒𝜒 ( 2 )) ′ 2 ′ 2 1 1 Finally, 𝛽𝛽 𝑥𝑥 𝜔𝜔𝜏𝜏 − 𝜔𝜔 −𝜓𝜓 𝜒𝜒 𝑤𝑤 𝑤𝑤 − 𝑣𝑣 ( ( ) +(1 ) ′ 2 2 ̂𝑟𝑟𝑟𝑟𝐺𝐺1 2 1 𝐵𝐵�𝑆𝑆 � + ≈ 𝛽𝛽( )(𝑥𝑥 𝑤𝑤 + (1− 𝜔𝜔 −𝜓𝜓 )𝜒𝜒 ) ( 2 )) In order to drive mean square error of , ′ 2 ′ 𝛽𝛽2 𝑥𝑥 𝜔𝜔𝜏𝜏 − 𝜔𝜔 −𝜓𝜓 𝜒𝜒 𝑤𝑤1 𝑤𝑤1 − 𝑣𝑣 𝑆𝑆̂𝑟𝑟𝑟𝑟𝐺𝐺1 = . 2 2 ̂𝑟𝑟𝑟𝑟𝐺𝐺1 ̂𝑟𝑟𝑟𝑟𝐺𝐺1 𝑦𝑦 The ratio estimator is , expressed𝑀𝑀𝑀𝑀𝑀𝑀�𝑆𝑆 as a� function𝐸𝐸�𝑆𝑆 (− 𝑆𝑆, � , ) and the estimated parameter is 2 2 2 2 𝑆𝑆̂𝑟𝑟𝑟𝑟𝐺𝐺1 𝑦𝑦 𝑥𝑥 𝑧𝑧 𝑆𝑆𝑦𝑦 ( , , ), so that : 𝑓𝑓 𝑠𝑠 𝑠𝑠 𝑠𝑠 2 2 2 𝑓𝑓 𝑆𝑆𝑦𝑦 𝑆𝑆𝑥𝑥 𝑆𝑆𝑧𝑧 ( , , ) = , , , , . 2 2 2 2 2 2 2 2 2 2 𝑦𝑦 𝑥𝑥 𝑧𝑧 𝑦𝑦 𝑥𝑥 𝑧𝑧 𝑦𝑦 𝑥𝑥 𝑧𝑧 by approximating to 𝑀𝑀( 𝑀𝑀𝑀𝑀, �𝑓𝑓, 𝑠𝑠 ) 𝑠𝑠 with𝑠𝑠 the� Taylor𝐸𝐸 �𝑓𝑓�𝑠𝑠 series𝑠𝑠 and𝑠𝑠 � ignoring − 𝑓𝑓�𝑆𝑆 𝑆𝑆degrees𝑆𝑆 �� greater than one, obtained : 2 2 2 𝑦𝑦 𝑥𝑥 𝑧𝑧 𝑓𝑓 𝑠𝑠 𝑠𝑠 𝑠𝑠 = = ( , , ) , , = , , 2 = 2 + ( ) 2 = 2 𝑠𝑠𝑦𝑦 𝑆𝑆𝑦𝑦 2 2 2 𝑠𝑠𝑦𝑦 𝑆𝑆𝑦𝑦 2 2 2 2 2 2 2 = 2 2 2 𝜕𝜕𝜕𝜕 𝑠𝑠𝑦𝑦 𝑠𝑠𝑥𝑥 𝑠𝑠𝑧𝑧 2 = 2 𝑓𝑓�𝑠𝑠𝑦𝑦 𝑠𝑠𝑥𝑥 𝑠𝑠𝑧𝑧 � 𝑓𝑓�𝑠𝑠𝑦𝑦 𝑠𝑠𝑥𝑥 𝑠𝑠𝑧𝑧 � �𝑠𝑠𝑥𝑥 𝑆𝑆𝑥𝑥 � 𝑠𝑠𝑦𝑦 − 𝑆𝑆𝑦𝑦 2 �𝑠𝑠𝑥𝑥 𝑆𝑆𝑥𝑥 2 2 𝜕𝜕𝑠𝑠𝑦𝑦 2 2 𝑠𝑠𝑧𝑧 𝑆𝑆𝑧𝑧 =𝑠𝑠𝑧𝑧 𝑆𝑆𝑧𝑧 ( , , ) +( ) 2 = 2 2 2 2 𝑠𝑠𝑦𝑦 𝑆𝑆𝑦𝑦 2 2 𝜕𝜕𝜕𝜕 𝑠𝑠𝑦𝑦 𝑠𝑠𝑥𝑥 𝑠𝑠𝑧𝑧 2 = 2 𝑥𝑥 𝑥𝑥 2 𝑥𝑥 𝑥𝑥 𝑠𝑠 − 𝑆𝑆 𝑥𝑥 �𝑠𝑠2 𝑆𝑆2 𝜕𝜕𝑠𝑠 𝑧𝑧 𝑧𝑧 𝑠𝑠 = 𝑆𝑆 , , +( ) 2 = 2 2. 2 2 2 𝑠𝑠𝑦𝑦 𝑆𝑆𝑦𝑦 2 2 𝜕𝜕𝜕𝜕�𝑠𝑠𝑦𝑦 𝑠𝑠𝑥𝑥 𝑠𝑠𝑧𝑧 � 2 = 2 𝑧𝑧 𝑧𝑧 2 𝑥𝑥 𝑥𝑥 𝑠𝑠 − 𝑆𝑆 𝑧𝑧 �𝑠𝑠2 𝑆𝑆2� 𝜕𝜕𝑠𝑠 𝑧𝑧 𝑧𝑧 Equivalent to 𝑠𝑠 𝑆𝑆

= ( , , ) , , = ( , , ) + ( ) 2 = 2 2 2 2 𝑠𝑠𝑦𝑦 𝑆𝑆𝑦𝑦 2 2 2 2 2 2 2 2 𝜕𝜕𝜕𝜕 𝑠𝑠𝑦𝑦 𝑠𝑠𝑥𝑥 𝑠𝑠𝑧𝑧 2 = 2 𝑓𝑓�𝑠𝑠𝑦𝑦 𝑠𝑠𝑥𝑥 𝑠𝑠𝑧𝑧 � 𝑓𝑓 𝑆𝑆𝑦𝑦 𝑆𝑆𝑥𝑥 𝑆𝑆𝑧𝑧 � 𝑠𝑠𝑦𝑦 − 𝑆𝑆𝑦𝑦 2 �𝑠𝑠𝑥𝑥 𝑆𝑆𝑥𝑥 𝜕𝜕𝑠𝑠𝑦𝑦 2 2 𝑠𝑠𝑧𝑧 = 𝑆𝑆𝑧𝑧 ( , , ) +( ) 2 = 2 2 2 2 𝑠𝑠𝑦𝑦 𝑆𝑆𝑦𝑦 2 2 𝜕𝜕𝜕𝜕 𝑠𝑠𝑦𝑦 𝑠𝑠𝑥𝑥 𝑠𝑠𝑧𝑧 2 = 2 𝑠𝑠𝑥𝑥 − 𝑆𝑆𝑥𝑥 2 �𝑠𝑠𝑥𝑥 𝑆𝑆𝑥𝑥 𝜕𝜕𝑠𝑠𝑥𝑥 2 2 𝑠𝑠𝑧𝑧 𝑆𝑆𝑧𝑧

= , , +( ) 2 = 2 2 2 2 2 𝑠𝑠𝑦𝑦 𝑆𝑆𝑦𝑦 2 2 𝜕𝜕𝜕𝜕�𝑠𝑠𝑦𝑦 𝑠𝑠𝑥𝑥 𝑠𝑠𝑧𝑧 � 2 = 2 𝑧𝑧 𝑧𝑧 2 𝑥𝑥 𝑥𝑥 𝑠𝑠 − 𝑆𝑆 𝑧𝑧 �𝑠𝑠2 𝑆𝑆2� 𝜕𝜕𝑠𝑠 𝑧𝑧 𝑧𝑧 Thus. 𝑠𝑠 𝑆𝑆

= ( , , ) 2 2 ( , , ) ( , , ) = ( ) 2 2 2 𝑦𝑦 = 𝑦𝑦 2 𝑠𝑠 𝑆𝑆 2 2 2 2 2 2 2 2 𝜕𝜕𝜕𝜕 𝑠𝑠𝑦𝑦 𝑠𝑠𝑥𝑥 𝑠𝑠𝑧𝑧 2 = 2 �𝑓𝑓 𝑆𝑆𝑦𝑦 𝑆𝑆𝑥𝑥 𝑆𝑆𝑧𝑧 − 𝑓𝑓 𝑆𝑆𝑦𝑦 𝑆𝑆𝑥𝑥 𝑆𝑆𝑧𝑧 � � 𝑠𝑠𝑦𝑦 − 𝑆𝑆𝑦𝑦 2 �𝑠𝑠𝑥𝑥 𝑆𝑆𝑥𝑥 𝜕𝜕𝑠𝑠𝑦𝑦 2 2 𝑠𝑠𝑧𝑧= 𝑆𝑆𝑧𝑧 ( , , ) +( ) 2 = 2 2 2 2 𝑠𝑠𝑦𝑦 𝑆𝑆𝑦𝑦 2 2 𝜕𝜕𝜕𝜕 𝑠𝑠𝑦𝑦 𝑠𝑠𝑥𝑥 𝑠𝑠𝑧𝑧 2 = 2 𝑥𝑥 𝑥𝑥 2 𝑥𝑥 𝑥𝑥 𝑠𝑠 − 𝑆𝑆 𝑥𝑥 �𝑠𝑠2 𝑆𝑆2 𝜕𝜕𝑠𝑠 𝑧𝑧 𝑧𝑧 𝑠𝑠 = 𝑆𝑆 , , +( ) 2 = 2 2 2 2 2 𝑠𝑠𝑦𝑦 𝑆𝑆𝑦𝑦 2 2 𝜕𝜕𝜕𝜕�𝑠𝑠𝑦𝑦 𝑠𝑠𝑥𝑥 𝑠𝑠𝑧𝑧 � 2 = 2 𝑧𝑧 𝑧𝑧 2 𝑥𝑥 𝑥𝑥 𝑠𝑠 − 𝑆𝑆 𝑧𝑧 �𝑠𝑠2 𝑆𝑆2� 𝜕𝜕𝑠𝑠 𝑧𝑧 𝑧𝑧 Apply expectation on both side, so that, 𝑠𝑠 𝑆𝑆

= , , ( , , ) = 2 = 2 2 ( ) 2 2 2 𝑠𝑠𝑦𝑦 𝑆𝑆𝑦𝑦 2 2 2 𝜕𝜕𝜕𝜕�𝑠𝑠𝑦𝑦 𝑠𝑠𝑥𝑥 𝑠𝑠𝑧𝑧 � 2 = 2 2 2 2 𝑀𝑀𝑀𝑀𝑀𝑀�𝑓𝑓 𝑠𝑠𝑦𝑦 𝑠𝑠𝑥𝑥 𝑠𝑠𝑧𝑧 � 𝐸𝐸 �� 2 �𝑠𝑠𝑥𝑥 𝑆𝑆𝑥𝑥 � 𝑠𝑠𝑦𝑦 − 𝑆𝑆𝑦𝑦 � 𝜕𝜕𝑠𝑠𝑦𝑦 2 2 𝑠𝑠𝑧𝑧 𝑆𝑆𝑧𝑧 = , , + 2 = 2 2 ( ) 2 2 2 𝑠𝑠𝑦𝑦 𝑆𝑆𝑦𝑦 𝜕𝜕𝜕𝜕�𝑠𝑠𝑦𝑦 𝑠𝑠𝑥𝑥 𝑠𝑠𝑧𝑧 � 2 = 2 2 2 2 𝐸𝐸 �� 2 �𝑠𝑠𝑥𝑥 𝑆𝑆𝑥𝑥 � 𝑠𝑠𝑥𝑥 − 𝑆𝑆𝑥𝑥 � 𝜕𝜕𝑠𝑠𝑥𝑥 2 2 𝑠𝑠𝑧𝑧 𝑆𝑆𝑧𝑧 = , , + 2 = 2 2 ( ) 2 2 2 𝑠𝑠𝑦𝑦 𝑆𝑆𝑦𝑦 𝜕𝜕𝜕𝜕�𝑠𝑠𝑦𝑦 𝑠𝑠𝑥𝑥 𝑠𝑠𝑧𝑧 � 2 = 2 2 2 2 𝐸𝐸 �� 2 �𝑠𝑠𝑥𝑥 𝑆𝑆𝑥𝑥 � 𝑠𝑠𝑧𝑧 − 𝑆𝑆𝑧𝑧 � 𝜕𝜕𝑠𝑠𝑧𝑧 2 2 𝑠𝑠𝑧𝑧 𝑆𝑆𝑧𝑧 = , , +2 2 = 2 2 2 2 𝑠𝑠𝑦𝑦 𝑆𝑆𝑦𝑦 𝜕𝜕𝜕𝜕�𝑠𝑠𝑦𝑦 𝑠𝑠𝑥𝑥 𝑠𝑠𝑧𝑧 � 2 = 2 2 𝑥𝑥 𝑥𝑥 𝐸𝐸 �� 𝑦𝑦 �𝑠𝑠2 𝑆𝑆2� 𝜕𝜕𝑠𝑠 𝑧𝑧 𝑧𝑧 = 𝑠𝑠 𝑆𝑆 , , 2 = 2 ( ) 2 2 2 𝑠𝑠𝑦𝑦 𝑆𝑆𝑦𝑦 𝜕𝜕𝜕𝜕�𝑠𝑠𝑦𝑦 𝑠𝑠𝑥𝑥 𝑠𝑠𝑧𝑧 � 2 = 2 2 2 2 𝑥𝑥 𝑥𝑥 𝑦𝑦 𝑦𝑦 � 𝑦𝑦 �𝑠𝑠2 𝑆𝑆2� � 𝑠𝑠 − 𝑆𝑆 𝜕𝜕𝑠𝑠 𝑧𝑧 𝑧𝑧 𝑠𝑠 𝑆𝑆 = , , ( )) + 2 2 = 2 2 2 2 𝑠𝑠𝑦𝑦 𝑆𝑆𝑦𝑦 2 2 𝜕𝜕𝜕𝜕�𝑠𝑠𝑦𝑦 𝑠𝑠𝑥𝑥 𝑠𝑠𝑧𝑧 � 2 = 2 𝑥𝑥 𝑥𝑥 2 𝑥𝑥 𝑥𝑥 𝑠𝑠 − 𝑆𝑆 � 𝐸𝐸 �� 𝑥𝑥 �𝑠𝑠2 𝑆𝑆2� 𝜕𝜕𝑠𝑠 𝑧𝑧 𝑧𝑧 = 𝑠𝑠 𝑆𝑆 , , 2 = 2 (( ) 2 2 2 𝑠𝑠𝑦𝑦 𝑆𝑆𝑦𝑦 𝜕𝜕𝜕𝜕�𝑠𝑠𝑦𝑦 𝑠𝑠𝑥𝑥 𝑠𝑠𝑧𝑧 � 2 = 2 2 2 � 2 �𝑠𝑠𝑥𝑥 𝑆𝑆𝑥𝑥 � 𝑠𝑠𝑥𝑥 − 𝑆𝑆𝑥𝑥 𝜕𝜕𝑠𝑠𝑥𝑥 2 2 𝑠𝑠𝑧𝑧 𝑆𝑆𝑧𝑧

= , , ( )) + 2 2 = 2 2 2 2 𝑠𝑠𝑦𝑦 𝑆𝑆𝑦𝑦 2 2 𝜕𝜕𝜕𝜕�𝑠𝑠𝑦𝑦 𝑠𝑠𝑥𝑥 𝑠𝑠𝑧𝑧 � 2 = 2 𝑠𝑠𝑥𝑥 − 𝑆𝑆𝑥𝑥 � 𝐸𝐸 �� 2 �𝑠𝑠𝑥𝑥 𝑆𝑆𝑥𝑥 � 𝜕𝜕𝑠𝑠𝑥𝑥 2 2 𝑠𝑠𝑧𝑧 𝑆𝑆𝑧𝑧 ( )( ) 2 2 2 2 𝑦𝑦 𝑦𝑦 𝑥𝑥 𝑥𝑥 In connection ( , , ) = , then : � 𝑠𝑠 − 𝑆𝑆 𝑠𝑠 − 𝑆𝑆 �

2 2 2 𝑟𝑟𝑟𝑟𝐺𝐺1 𝑦𝑦 𝑥𝑥 𝑧𝑧 𝑆𝑆̂ + 𝑓𝑓 𝑠𝑠 𝑠𝑠 𝑠𝑠 ( + +2 ) 2 2 2 2 −𝑆𝑆𝑦𝑦 𝑟𝑟𝑟𝑟𝐺𝐺1 𝑦𝑦 𝑦𝑦 2 2 2 𝑀𝑀𝑀𝑀𝑀𝑀�𝑆𝑆̂ � ≈ 𝐸𝐸 ��𝑠𝑠 − 𝑆𝑆 � � 𝐸𝐸 �� 𝑥𝑥 𝑧𝑧 � ( ) ) 𝑆𝑆+ 𝑆𝑆 𝑘𝑘𝑘𝑘 ( + +2 ) 2 2 2 2 −𝑆𝑆𝑦𝑦 𝑥𝑥 𝑥𝑥 2 2 2 𝑠𝑠 − 𝑆𝑆 𝐸𝐸 �� 𝑥𝑥 𝑧𝑧 � ( ) ) + 2 𝑆𝑆 𝑆𝑆 𝑘𝑘𝑘𝑘 ( + +2 ) 2 2 2 2 −𝑆𝑆𝑦𝑦 𝑠𝑠𝑧𝑧 − 𝑆𝑆𝑧𝑧 𝐸𝐸 �� 2 2 2 � ( ) +𝑆𝑆𝑥𝑥 2 𝑆𝑆𝑧𝑧 𝑘𝑘𝑘𝑘 2 2 2 2 𝑦𝑦 𝑦𝑦 𝑥𝑥 𝑥𝑥 �𝑠𝑠 − 𝑆𝑆 � 𝑠𝑠 − 𝑆𝑆 � 𝐸𝐸 ( + +2 ) ( + +2 ) −𝑆𝑆𝑦𝑦 −𝑆𝑆𝑦𝑦 2 2 2 2 2 2 �� 𝑥𝑥 𝑧𝑧 � � 𝑥𝑥 𝑧𝑧 � ( 𝑆𝑆 )(𝑆𝑆 𝑘𝑘𝑘𝑘 ) + 2𝑆𝑆 𝑆𝑆 𝑘𝑘𝑘𝑘 ( + +2 ) 2 2 2 2 −𝑆𝑆𝑦𝑦 𝑠𝑠𝑥𝑥 − 𝑆𝑆𝑥𝑥 𝑠𝑠𝑧𝑧 − 𝑆𝑆𝑧𝑧 � 𝐸𝐸 �� 2 2 2 � ( ) 𝑆𝑆𝑥𝑥 𝑆𝑆𝑧𝑧 𝑘𝑘𝑘𝑘 Therefore, 2 2 2 2 �𝑠𝑠𝑦𝑦 − 𝑆𝑆𝑦𝑦 � 𝑠𝑠𝑧𝑧 − 𝑆𝑆𝑧𝑧 � 1 + ( ) ( + 2+ ) 2 4 𝑆𝑆𝑦𝑦 𝑟𝑟𝑟𝑟𝐺𝐺1 𝑦𝑦 2 𝑦𝑦 2 2 2 𝑀𝑀𝑀𝑀𝑀𝑀�𝑆𝑆̂ � ≈ 𝛼𝛼𝑆𝑆 �𝛽𝛽 − � � 𝑥𝑥 𝑧𝑧 � 1 + 𝑆𝑆 𝑆𝑆 𝑘𝑘𝑘𝑘 ( ) ( + 2+ ) 2 4 𝑆𝑆𝑦𝑦 𝛼𝛼𝑆𝑆𝑦𝑦 �𝛽𝛽2 𝑥𝑥 − � � 2 2 2 � 1 2 𝑥𝑥 𝑧𝑧 ( ) 𝑆𝑆( +𝑆𝑆 2+𝑘𝑘𝑘𝑘 ) 4 𝑆𝑆𝑦𝑦 𝑦𝑦 2 𝑧𝑧 2 2 2 𝛼𝛼𝑆𝑆 �𝛽𝛽 − � − � 𝑥𝑥 𝑧𝑧 � ( 1) + 2 𝑆𝑆 𝑆𝑆 𝑘𝑘𝑘𝑘 ( + 2+ ) 2 2 2 𝑆𝑆𝑦𝑦 or it can be written in form 𝛼𝛼𝑠𝑠𝑦𝑦 𝑠𝑠𝑥𝑥 𝜆𝜆22 − � 2 2 2 � 𝑆𝑆𝑥𝑥 𝑆𝑆𝑧𝑧 𝑘𝑘𝑘𝑘 = ( ) 1 + ( ) 1 + ( ) 1 + 2( 1) 4( 1), 2 4 𝑟𝑟𝑝𝑝𝑝𝑝1 𝑦𝑦 2 𝑦𝑦 1 1 2 𝑥𝑥 2 𝑧𝑧 22 22 𝑀𝑀𝑀𝑀𝑀𝑀�𝑆𝑆̂ � 𝛼𝛼𝑆𝑆 =��𝛽𝛽 − (� ) 𝐴𝐴1�𝐴𝐴+ �𝛽𝛽 − ��( ) �𝛽𝛽1 +− �( ) 1𝜆𝜆 +− 2( � − 1)𝜆𝜆 , − 2 4 Where = 𝑀𝑀/(𝑀𝑀𝑀𝑀�𝑆𝑆̂+𝑟𝑟𝑟𝑟𝑟𝑟2�+ 𝛼𝛼𝑆𝑆)𝑦𝑦 and��𝛽𝛽 2 𝑦𝑦 =− �/( 𝐴𝐴2+�𝐴𝐴2�𝛽𝛽). 2 𝑥𝑥 − �� 𝛽𝛽2 𝑧𝑧 − � 𝜆𝜆22 − � 𝟐𝟐 𝟐𝟐 𝟐𝟐 𝟐𝟐 𝟐𝟐 𝑨𝑨𝟏𝟏 𝑺𝑺𝒚𝒚 𝑺𝑺𝒙𝒙 𝑺𝑺𝒛𝒛 𝒌𝒌𝒌𝒌 𝑨𝑨𝟐𝟐 𝑺𝑺𝒚𝒚 𝑺𝑺𝒚𝒚 𝒌𝒌𝒌𝒌 Bias and Mean Square Error of Ratio Type Variance Estimator In order to drive bias of ratio type variance estimator 2 𝑆𝑆̂𝑟𝑟𝑟𝑟𝑟𝑟2 ( + )2 + = ( + ) 𝑆𝑆̂𝑟𝑟𝑟𝑟𝑟𝑟2 + (1 ) , + 2 𝜏𝜏 ( 2+ ) 2 𝜒𝜒 2 𝑏𝑏 𝑐𝑐 𝑆𝑆𝑥𝑥 𝑏𝑏𝑠𝑠𝑥𝑥 𝑐𝑐𝑆𝑆𝑥𝑥 𝑟𝑟𝑟𝑟𝐺𝐺2 𝑦𝑦 2 2 2 𝑆𝑆̂ 𝑠𝑠 𝑑𝑑 �𝜔𝜔 � 𝑥𝑥 𝑥𝑥 � − 𝜔𝜔 � 𝑥𝑥 � � − 𝑑𝑑 by using , = , = ( 𝑏𝑏𝑠𝑠 ) 𝑐𝑐𝑆𝑆 = ( ) 𝑏𝑏 was𝑐𝑐 written𝑆𝑆 as: −2 2 2 −2 2 2 −2 (12 + 2) + 𝑒𝑒0 𝑆𝑆𝑦𝑦 �𝑠𝑠𝑦𝑦 − 𝑆𝑆𝑦𝑦 � 𝑒𝑒1 =𝑆𝑆𝑥𝑥 ( (1𝑠𝑠𝑥𝑥 − + 𝑆𝑆𝑥𝑥) +𝑒𝑒2 ) 𝑆𝑆𝑧𝑧 𝑠𝑠𝑧𝑧 − 𝑆𝑆𝑧𝑧 ( + )2 2 −𝜏𝜏 2 𝑏𝑏 𝑒𝑒1 𝑆𝑆𝑥𝑥 𝑐𝑐𝑆𝑆𝑥𝑥 𝑆𝑆̂𝑟𝑟𝑟𝑟𝐺𝐺2 𝑆𝑆𝑦𝑦 𝑒𝑒0 𝑑𝑑 �𝜔𝜔 � 2 � 𝑏𝑏 𝑐𝑐 𝑆𝑆𝑥𝑥

(1 + ) + . ( + )2 2 𝜒𝜒 𝑏𝑏 𝑒𝑒1 𝑆𝑆𝑥𝑥 𝑐𝑐𝑆𝑆𝑥𝑥 By suppose, /( + ) = , obtained � 2 � � − 𝑑𝑑 𝑏𝑏 𝑐𝑐 𝑆𝑆𝑥𝑥 𝑏𝑏 𝑏𝑏 𝑐𝑐 𝜃𝜃1 = (1 + ) + ( (1 + ) + (1 ) 2 (1 + ) ) . −𝜏𝜏 ̂𝑟𝑟𝑟𝑟𝐺𝐺2 𝑦𝑦 0 1 1 𝑆𝑆 �𝑆𝑆 𝑒𝑒 𝑑𝑑� 𝜔𝜔𝜒𝜒 𝜃𝜃 𝑒𝑒 − 𝜔𝜔 Furthermore, using the MacLaurin series approximation is obtained𝜃𝜃1𝑒𝑒1 − 𝑑𝑑

Assumption | | < 1 and | | < 1, so that ( + 1) 𝜃𝜃1𝑒𝑒1 𝜃𝜃2=𝑒𝑒2 + (1 + )1 ( ) 2 2 𝜏𝜏 𝜏𝜏 2 𝑆𝑆̂𝑟𝑟𝑟𝑟𝐺𝐺2 ��𝑆𝑆𝑦𝑦 𝑑𝑑� 𝜃𝜃0𝑒𝑒0 ( −+ 𝜔𝜔 1) 1 �𝜏𝜏𝜃𝜃 𝑒𝑒1 − 𝜃𝜃1𝑒𝑒1 � (1 ) ( ) . 2 𝜒𝜒 𝜒𝜒 2 2 Apply expectation on both side of − 𝜔𝜔, �𝜒𝜒𝜃𝜃1𝑒𝑒1 − 𝜃𝜃1𝑒𝑒1 � − 𝑑𝑑𝑦𝑦 −𝑆𝑆 �

2 ( + 1) = 𝑆𝑆̂𝑟𝑟𝑟𝑟𝐺𝐺 + (1 + )1 ( ) 2 2 𝜏𝜏 𝜏𝜏 2 𝐸𝐸�𝑆𝑆̂𝑟𝑟𝑟𝑟𝐺𝐺2� 𝐸𝐸 ��𝑆𝑆𝑦𝑦 𝑑𝑑� 𝜃𝜃0(𝑒𝑒0 +− 1) 𝜔𝜔1 �𝜏𝜏𝜃𝜃 𝑒𝑒1 − 𝜃𝜃1𝑒𝑒1 � (1 ) ( ) . 2 𝜒𝜒 𝜒𝜒 2 2 − 𝜔𝜔 �𝜒𝜒𝜃𝜃1𝑒𝑒1 − 𝜃𝜃1𝑒𝑒(1 +� 1−) 𝑑𝑑𝑦𝑦 −𝑆𝑆 � = + ( ) 2 2 𝜏𝜏 𝜏𝜏 2 𝐸𝐸 ��𝑆𝑆𝑦𝑦 𝑑𝑑�𝜃𝜃0𝑒𝑒0 − 𝜔𝜔1 �𝜏𝜏𝜃𝜃 𝑒𝑒1 − 𝜃𝜃1𝑒𝑒1 � ( + 1) +(1 ) ( ) 2 𝜒𝜒 𝜒𝜒 2 − 𝜔𝜔 �𝜒𝜒𝜃𝜃1𝑒𝑒1 − 𝜃𝜃1𝑒𝑒1 � − 𝜔𝜔𝜔𝜔𝜃𝜃0𝜃𝜃1𝑒𝑒0𝑒𝑒1 +(1 ) , 2 − 𝜔𝜔 𝜒𝜒𝜃𝜃0𝜃𝜃1𝑒𝑒0𝑒𝑒1 − 𝑑𝑑𝑦𝑦 −𝑆𝑆 � And using ( ) = 0, ( ) = 0 , E( ) = 0 we obtained,

0 1 2 𝐸𝐸 𝑒𝑒 𝐸𝐸 𝑒𝑒 = 𝑒𝑒 ( ) ( + (1 ) ) ( 2 ) 2 ′ 2 2 ′ ̂𝑟𝑟𝑟𝑟𝐺𝐺2 𝑦𝑦 2 1 1 1 Finally, 𝐸𝐸�𝑆𝑆 � 𝑆𝑆 − �𝛽𝛽 𝑥𝑥 𝑤𝑤 𝜔𝜔𝜏𝜏 − 𝜔𝜔 𝜒𝜒 𝑤𝑤 𝑤𝑤 𝑣𝑣 �

= ( ) ( + (1 ) ) ( 2 ). ′ 2 2 ′ ̂𝑟𝑟𝑟𝑟𝐺𝐺2 2 1 1 1 In order to drive mean square error of𝐵𝐵𝐵𝐵 𝑎𝑎𝑠𝑠�𝑆𝑆 , The� ratio𝛽𝛽 estimator𝑥𝑥 𝑤𝑤 𝜔𝜔𝜏𝜏 is −, 𝜔𝜔 expressed𝜒𝜒 𝑤𝑤 𝑤𝑤 as a𝑣𝑣 function ( , , ) and the estimated parameter is ( , , ), so that in the same way, it is earned, 2 2 2 ̂𝑟𝑟𝑟𝑟𝐺𝐺2 ̂𝑟𝑟𝑟𝑟𝐺𝐺2 𝑦𝑦 𝑥𝑥 𝑧𝑧 2 2𝑆𝑆 2 2 𝑆𝑆 𝑓𝑓 𝑠𝑠 𝑠𝑠 𝑠𝑠 𝑦𝑦 𝑦𝑦 𝑥𝑥 𝑧𝑧 𝑆𝑆 𝑓𝑓 𝑆𝑆 𝑆𝑆 𝑆𝑆, , = + ( + 2 ) 2 2 2 2 2 2 2 −𝑆𝑆𝑦𝑦 𝑦𝑦 𝑥𝑥 𝑧𝑧 𝑦𝑦 𝑦𝑦 2 2 𝑀𝑀𝑀𝑀𝑀𝑀 �𝑓𝑓�𝑠𝑠 𝑠𝑠 𝑠𝑠 �� 𝐸𝐸 ��𝑠𝑠 − 𝑆𝑆 � � 𝐸𝐸 �� 𝑥𝑥 � ( ) ) + 2 𝑆𝑆 𝑘𝑘𝑘𝑘 ( + 2 ) 2 2 2 2 −𝑆𝑆𝑦𝑦 𝑠𝑠𝑥𝑥 − 𝑆𝑆𝑥𝑥 𝐸𝐸 �� 2 2 � 𝑆𝑆𝑥𝑥 𝑘𝑘𝑘𝑘 ( ) 2 2 2 2 𝑦𝑦 𝑦𝑦 𝑥𝑥 𝑥𝑥 �𝑠𝑠 1−+ 𝑆𝑆 � 𝑠𝑠 − 𝑆𝑆 � ( ) ( + 2 ) 2 4 −𝑆𝑆𝑦𝑦 𝑀𝑀𝑆𝑆𝐸𝐸�𝑆𝑆̂𝑟𝑟𝑟𝑟𝐺𝐺2� ≈ 𝛼𝛼𝑆𝑆𝑦𝑦 �𝛽𝛽2 𝑦𝑦 − � � 2 2 � 1 𝑥𝑥 2 ( ) 𝑆𝑆 𝑘𝑘(𝑘𝑘 + 2 ) 4 −𝑆𝑆𝑦𝑦 𝑦𝑦 2 𝑥𝑥 ( 1)2 2 𝛼𝛼𝑆𝑆 �𝛽𝛽 − � − � 𝑥𝑥 � 2 2 𝑆𝑆 𝑘𝑘𝑘𝑘 𝛼𝛼𝑠𝑠𝑦𝑦 𝑠𝑠𝑥𝑥 𝜆𝜆22 −

© IEOM Society International 2518 Proceedings of the 5th NA International Conference on Industrial Engineering and Operations Management Detroit, Michigan, USA, August 10 - 14, 2020 or it can be written in form

( ( ) 1) + ( ) 1 2 ( 1) Where /( + ) = , = 1/ 4 2 𝑀𝑀𝑀𝑀𝑀𝑀�𝑆𝑆̂𝑟𝑟𝑟𝑟𝐺𝐺2� ≈ 𝛼𝛼𝑆𝑆𝑦𝑦 � 𝛽𝛽2 𝑦𝑦 − 𝐴𝐴2�𝛽𝛽2 𝑥𝑥 − � − 𝐴𝐴2 𝜆𝜆22 − � 2 2 𝑆𝑆𝑦𝑦 𝑆𝑆𝑥𝑥 𝑘𝑘𝑘𝑘 𝐴𝐴2 𝛼𝛼 𝑛𝑛 3. Comparison of the estimators . We now compare the proposed estimator with other estimators considered here. These comparisons lead to the following obvious conditions. From the expressions of the MSE of the proposed estimators and the usually estimators, we have derived the conditions for which the proposed estimators are more efficient than the variance ratio estimators as follows :

(1). Comparison between estimator and estimator 𝟐𝟐 𝟐𝟐 �𝒓𝒓𝒓𝒓𝒓𝒓𝟏𝟏 𝒚𝒚 𝑺𝑺 <𝒔𝒔 2 2 1 ( ) 1 + ( ) 1 +𝑀𝑀𝑀𝑀𝑀𝑀(�𝑆𝑆)̂𝑟𝑟𝑟𝑟𝑟𝑟1�+ 2(𝑉𝑉�𝑆𝑆̂𝑦𝑦 � 1) 4( 1) < ( 1) 4 𝑁𝑁 −𝑛𝑛 4 𝜇𝜇40 𝛼𝛼𝑆𝑆𝑦𝑦 ��𝛽𝛽2 𝑦𝑦 − � 𝐴𝐴1 �𝐴𝐴1�𝛽𝛽2 𝑥𝑥 − �� 𝛽𝛽2 𝑧𝑧 − � 𝜆𝜆22 − � − 𝜆𝜆22 − 𝑆𝑆𝑦𝑦 2 − ( ) 1 + ( ) 1 + ( ) 1 + 2( 1) 4( 1) 𝑁𝑁𝑁𝑁 𝜇𝜇20 1 < 0 4 𝑁𝑁 −𝑛𝑛 4 𝜇𝜇40 By resolving𝑦𝑦 2 𝑦𝑦 this inequality1 1it is2 obtained𝑥𝑥 that e2stimator𝑧𝑧 22is more efficient22 than if 𝑦𝑦 2 𝛼𝛼𝑆𝑆 ��𝛽𝛽 − � 𝐴𝐴 �𝐴𝐴 �𝛽𝛽 − �� 𝛽𝛽 − � 𝜆𝜆 − � − 𝜆𝜆 − − 𝑆𝑆 � 20 − � 2( ) ( ) 2 𝑁𝑁𝑁𝑁 𝜇𝜇 ( ) 1 + 2 𝑟𝑟𝑟𝑟𝑟𝑟1 1 4 1 𝑦𝑦 ( ) > 𝑆𝑆̂ . 𝑠𝑠 �𝛽𝛽2 𝑧𝑧 − � 𝐴𝐴1 𝜆𝜆22 − − 𝜆𝜆22 − (2).Comparison between estimator2 𝑥𝑥 and estimator 𝛽𝛽 1 𝟐𝟐 < 𝐴𝐴𝟐𝟐 �𝒓𝒓𝒓𝒓𝒓𝒓𝟐𝟐 𝒚𝒚 𝑺𝑺 2 2 𝒔𝒔 2 ( ) 1 + 𝑀𝑀𝑀𝑀𝑀𝑀�𝑆𝑆(̂𝑟𝑟𝑟𝑟𝑟𝑟) �1 +𝑉𝑉�𝑆𝑆̂𝑦𝑦 �( ) 1 + 2( 1) < ( 1) 4 𝑁𝑁 −𝑛𝑛 4 𝜇𝜇40 𝛼𝛼𝑆𝑆𝑦𝑦 ��𝛽𝛽2 𝑦𝑦 − � 𝐴𝐴2 �𝐴𝐴2�𝛽𝛽2 𝑥𝑥 − �� 𝛽𝛽2 𝑧𝑧 − � 𝜆𝜆22 − � 𝑆𝑆𝑦𝑦 2 − ( ) 1 + ( ) 1 + ( ) 1 + 2( 1) 𝑁𝑁𝑁𝑁 𝜇𝜇20 1 < 0 4 𝑁𝑁 −𝑛𝑛 4 𝜇𝜇40 𝑦𝑦 2 𝑦𝑦 2 2 2 𝑥𝑥 2 𝑧𝑧 22 𝑦𝑦 2 𝛼𝛼𝑆𝑆 ��𝛽𝛽 − � 𝐴𝐴 �𝐴𝐴 �𝛽𝛽 − �� 𝛽𝛽 − � 𝜆𝜆 − � − 𝑆𝑆 � 20 − � By resolving this inequality it is obtained that estimator is more efficient𝑁𝑁𝑁𝑁 than𝜇𝜇 if 𝟐𝟐 2 ( ) 1 2( 1) > �𝑺𝑺�𝒓𝒓𝒓𝒓𝒓𝒓𝟐𝟐 � . 𝑠𝑠𝒚𝒚 2 2 𝑥𝑥 22 𝒙𝒙 �𝛽𝛽 − � − 𝜆𝜆 − 𝑆𝑆 2 (3).Comparison between estimator and estimator 𝐴𝐴 𝟐𝟐 𝟐𝟐 �𝒓𝒓𝒓𝒓𝒓𝒓𝟏𝟏 �𝒓𝒓𝒓𝒓𝒓𝒓𝟐𝟐 𝑺𝑺 𝑺𝑺< 2 2 ( ) 1 + ( ) 𝑀𝑀1𝑀𝑀𝑀𝑀�𝑆𝑆+̂𝑟𝑟𝑟𝑟𝑟𝑟1(�) 𝑀𝑀1𝑀𝑀𝑀𝑀+�𝑆𝑆 2̂𝑟𝑟𝑟𝑟𝑟𝑟( 2� 1) 4( 1) 4 𝑦𝑦 2 𝑦𝑦 1 1 2 𝑥𝑥 2 𝑧𝑧 22 22 𝛼𝛼𝑆𝑆 ��𝛽𝛽 − � < 𝐴𝐴 �𝐴𝐴 �𝛽𝛽( ) −1 ��+ �𝛽𝛽 −( )� 1 𝜆𝜆+ − ( �) −1 𝜆𝜆+ 2−( 1) 4 𝑦𝑦 2 𝑦𝑦 2 2 2 𝑥𝑥 2 𝑧𝑧 22 ( ) 1 + 𝛼𝛼𝑆𝑆 ��𝛽𝛽( ) 1− �+ 𝐴𝐴 ( �𝐴𝐴) �𝛽𝛽1 + 2−( �� 1�𝛽𝛽) 4−( � 1)𝜆𝜆 − � 4 𝑦𝑦 2 𝑦𝑦 1 1 2 𝑥𝑥 2 𝑧𝑧 22 22 𝛼𝛼𝑆𝑆 ��𝛽𝛽 − � 𝐴𝐴 �𝐴𝐴 �𝛽𝛽 ( ) − 1��+ �𝛽𝛽 − (�) 1𝜆𝜆 +− (�) − 1𝜆𝜆 +− 2( 1) < 0 4 By resolving this inequality− it �𝛼𝛼𝑆𝑆 is obtained𝑦𝑦 ��𝛽𝛽2 𝑦𝑦 that− e�stimator𝐴𝐴2 �𝐴𝐴 2�𝛽𝛽2 𝑥𝑥 is −more�� efficient�𝛽𝛽2 𝑧𝑧 than− � 𝜆𝜆 22if − �� <2 2 ̂𝑟𝑟𝑟𝑟𝑟𝑟1 ̂𝑟𝑟𝑟𝑟𝑟𝑟2 and 𝑆𝑆 𝑆𝑆 𝐴𝐴1 𝐴𝐴2 2( 1) ( ) 1 4( 1) > ( ) ( + ) 𝜆𝜆22 − − 𝐴𝐴1 �𝐴𝐴1�𝛽𝛽2 𝑧𝑧 − � − 𝜆𝜆22 − � 2 𝑥𝑥 𝛽𝛽 1 2 𝐴𝐴 𝐴𝐴

4. Numerical illustration We use the following data. Source: Basic Econometrics, Gujarati & Sangeetha (2007) Data collected from 64 countries regarding fertility and other factors: number of children died in a year under age 5 per one thousand live- births, literacy rate of females in percentage, and total fertility rate during 1980-1985.

Table 1. Data description

0.53695 0.50809 0.2719 5772.66672 676.40872 2.3277062 𝐶𝐶𝑦𝑦 𝐶𝐶𝑥𝑥 𝐶𝐶𝑧𝑧 𝑆𝑆𝑦𝑦 𝑆𝑆𝑥𝑥 𝑆𝑆𝑧𝑧 ( ) ( ) ( )

0.81829𝑦𝑦𝑦𝑦 0.67114𝑦𝑦𝑦𝑦 0.62595𝑥𝑥𝑥𝑥 2378332 𝑦𝑦 1.65751112 𝑥𝑥 2.816912 𝑧𝑧 𝜌𝜌 𝜌𝜌 𝜌𝜌 𝛽𝛽 𝛽𝛽 𝛽𝛽 Table 2: MSE and PRE of the estimators Estimators MSE PRE 11627.2 100 2 3927.17 296.0707 𝑆𝑆̂𝑦𝑦 2 3473.024 334.7860 𝑆𝑆̂𝑟𝑟𝑟𝑟𝑟𝑟1 𝑟𝑟𝑟𝑟𝐺𝐺2 𝑆𝑆̂ ( ) where Percentage Relative Efficiency (PRE) is computed as = 100% (2 ) 𝑉𝑉 𝑠𝑠𝑦𝑦 𝑃𝑃𝑃𝑃𝑃𝑃�𝜃𝜃�� 𝑀𝑀𝑀𝑀𝑀𝑀 𝜃𝜃� 𝑥𝑥

5. Conclusion From the discussion the results showed that the proposed ratio estimator was more efficient compared to the usual sample variance estimator, assuming given conditions. has a higher precision than the normal estimator, and the more additional information content that is included in the modifiers of the estimator makes the estimator more efficient, of course if it meets the efficiency requirements

References Isaki, C.T.(1983): Variance estimation using auxiliary information. Journal of American Statistical Association 78, 117–123. Jhajj, H .S., Sharma, M. K. and Grover, L. K. (2005) : An efficient class of chain estimators of population variance under sub-sampling scheme. J. Japan Stat. Soc., 35(2), 273-286 Kadilar, C. and Cingi, H. (2006) : Improvement in variance estimation using auxiliary information. Hacettepe Journal of Mathematics and Statistics 35 (1), 111–115. Kaur,(1985): An efficient regression type estimator in survey sampling, Biom. Journal 27 (1),107–110. Ray, S.K. and Sahai, A (1980) : Efficient families of ratio and product type estimators, Biometrika 67(1), 211–215 M. Abid, N. Abbas, R. A. K. Sherwani, dan H. Z. Nazir, Improved ratio estimator for the population mean using non-conventional measures of dispension, Pakistan Journal of Statistic and Operation Research, 12 (2016), 353-367. W. G. Cochran, Sampling Techniques, Third Edition, John Wiley, New York, 1977. Gujarati, D. N. & Sangeetha. (2007). Basic econometrics (4th Ed.). New Delhi: TATA McGRAW HILL Companies. Pp. 189. G. Diana, dan P. F. Perri, Estimation of finite population mean using multiauxilary information, International Journal of Statistics, 65 (2007), 99- 112. C. T. Isaki, Variance estimation using auxiliary information, Journal of the American Statistical Association, 78 (1983),117-123. M. Ismail, N. Kanwal, dan M. Q. Shahbaz, Generalized ratio-product-type estimator for variance using auxiliary information in simple random sampling, Kuwait Journal of Science, 45 (1) (2018), 79-88. R. Yadav, L. N. Upadhayaya, H. P. Singh, dan S. Chatterje, A generalized family of transformed ratio-product estimator for variance in sample surveys, Communications in Statistics-Theory and Methods, 42 (2013), 1839-1850.

Biographies

Haposan Sirait is a lecturer in the Study Program of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Riau, Pekanbaru, Indonesia. The field of applied mathematics, with a field of concentration of Statistics.

Stepi Karolin is a lecturer in the Study Program of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Riau, Pekanbaru, Indonesia. The field of applied mathematics, with a field of concentration of Statistics.

Sukono is a lecturer in the Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Padjadjaran, Bandung, Indonesia. Currently, as Chair of the Research Collaboration Community (RCC), the field of applied mathematics, with a field of concentration of financial mathematics and actuarial sciences.

Kalfin is currently a Student at the Doctoral Program at Padjadjaran University, Indonesia since 2019. He received his M.Mat in Mathematics from Padjadjaran University (UNPAD), Indonesia in 2019. His current research focuses on financial mathematics and actuarial sciences.

Abdul Talib Bon is a professor of Production and Operations Management in the Faculty of Technology Management and Business at the Universiti Tun Hussein Onn Malaysia since 1999. He has a Ph.D. in Computer Science, which he obtained from the Universite de La Rochelle, France in the year 2008. His doctoral thesis was on topic Process Quality Improvement on Beltline Moulding Manufacturing. He studied Business Administration in the Universiti Kebangsaan Malaysia for which he was awarded the MBA in the year 1998. He’s bachelor degree and diploma in Mechanical Engineering which he obtained from the Universiti Teknologi Malaysia. He received his postgraduate certificate in Mechatronics and Robotics from Carlisle, United Kingdom in 1997. He had published more 150 International Proceedings and International Journals and 8 books. He is a member of MSORSM, IIF, IEOM, IIE, INFORMS, TAM, and MIM.