Generalized Family of Efficient Estimators of Population Median Using Two-Phase Sampling Design

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1 International Journal of athematics and tatistics Invention (IJI) E-IN: IN: Volume 3 Issue February Generalized Family of Efficient Estimators of opulation edian Using Two-hase ampling Design, H.. Jhajj,, Harpreet Kaur, 3, uneet Jhajj ABTRACT : For estimating the population median of variable under study, a generalized family of efficient estimators has been proposed by using prior information of population parameters based upon auxiliary variables under two-phase simple random sampling design. The comparison of proposed family of estimators has been made with the existing ones with respect to their mean square errors and biases. It has been shown that efficient estimators can be obtained from the family under the given practical situations which will have smaller mean square error than the linear regression type estimator, ingh et al estimator (006), Gupta et al (008) estimator and Jhajj et al estimator (04). Effort has been made to illustrate the results numerically as well as graphically by taking some empirical populations considered in the literature which also show that bias of efficient estimators obtained from the proposed family of estimators is smaller than other considered estimators. KEWORD: Auxiliary variable; Bias; ean square error; edian; Two-phase sampling. I. INTRODUCTION In survey sampling, statisticians have given more attention to the estimation of population mean, total, variance etc. but median is regarded as a more appropriate measure of location than mean when the distribution of variables such as income, expenditure etc. is highly skewed. In such situations, it is necessary to estimate median. ome authors such as Gross(980), Kuk and ak (989), ingh et al (00,006), Jhajj et al (03) etc. have considered the problem of estimating the median. ometimes in a trivariate distribution consisting of study variable and auxiliary variables (, ), the correlation between the variables and is only because of their high correlation with the variable but are not directly correlated to each other. For example:. In agriculture labour (say ) and crop production (say ) are highly correlated with the area under crop (say ) but not directly correlated to each other.. In any repetitive survey, the values of a variables of interest corresponding to both the last to last year (say ) and current year (say ) are highly correlated with the values of some variable corresponding to the last year (say ), whereas the values corresponding to the last to last year () and the values corresponding to the current year () are correlated with each other due to only their correlation with values of the some variable () corresponding to the last year. uppose the prior information about population median of variable is available where as the population median the two- phase sampling design. of variable is not known. uch unknown information is generally predicted by using Let u consider a finite population U= (,, i,...n). Let i and i, i be the values of study th variable and auxiliary variables, respectively on the i unit of the population. Corresponding small letters indicate the value in the samples. Let and, be the population medians of study variable and auxiliary variables respectively. Under two- phase sampling design, first phase sample of size n is selected using simple random sampling without replacement and observations on, and are obtained on the sample units. Then second phase sample of size m under simple random sampling without replacement is drawn from the first phase sample and observations on the variables, and are taken on selected units. Corresponding, and are sample medians are denoted by, and for second phase sample while sample medians for the first phase sample. uppose that y (), y (),,y (m) are the values of variable on sample units in ascending order such that y (t) y (t +) for some integral value of t. Let p = t/m be the proportion of values in the sample that are less than or equal to the value of median (an unknown population parameter). If p is an estimator of p, 39 a g e

2 with p = 0.5. Let the correlation coefficients between the estimators in, the sample median can be written in terms of quantiles as Q p, denoted by, and respectively which are defined as, and, 4 x, y, where, x, y 4 y, z, where, y, z 4 x, z, where, x, z are Assuming that as N, the distribution of the trivariate variable (,, ) approaches a continuous distribution with marginal densities, and of variables, and respectively. This assumption holds in particular under a superpopulation model framework, treating the values of (,, ) in the population as a realization of N independent observations from a continuous distribution. Under these assumptions, Gross (980) has shown that conventional sample median is consistent and asymptotically normal with median and variance Var m N 4 ( ) f For the case of trivariate distribution of, and and using two phase sampling design, linear regression type estimator of population median is defined as where and are constants. (.) (.) lr Up to the first order of approximation, E lr is minimized for f and f f f and its minimum is given by E lr 4 f( ) m N m n n N ingh et al (006) defined a ratio type estimator of median under the two phase sampling design as (.3) 3 (.4) where i; i,,3 are constants. Up to first order of approximation, they minimized E ( ) for: f f f f, 40 a g e

3 and f 3 f where E R R. min y 4 f( ) m N n N m n y. Using the knowledge of range R of variable along with its population median, Gupta et al (008) defined the estimator of population median under the same sampling design considered by ingh et al (006) as where i,,3 are constants. i R R R R 3 Up to the first order of approximation, they minimized E of for (.5) (.6) and, f, f f f R f 3 f R The bias and E of optimum estimator of obtained by them are: 4 a g e

4 ( Bias ) 8 f ( ) ( ) m n f f m N n N f f R f f R (.7) and E R. min y 4 f( ) m N n N m n (.8) = E min Recently Jhajj et al (04) defined an efficient family of estimators of median using the same information used by ingh et al (006) under same sampling design as here, and 3 are constants. 3 H e (.9) Up to the first order of approximation, they minimized E of H for and f f f 3 f, f f 4 a g e

5 The bias and E of optimum estimator of H obtained by them are: and ( Bias H ) 8 f ( ) ( ) m n f f m N f f n N f f (.0) E R H. min y 4 f( ) m N n N m n (.) = E min II. ROOED ETIATOR AND IT REULT Assuming that is known for a trivariate distribution of variables, and, in which variables and are highly correlated with variable and correlated among themselves through only. Then we proposed a family of estimators of population median under two phase sampling design described in ection as HL exp 3 where,, 3 are constants and 0.,, 3, 4, 5 To find the bias and E of the estimator HL, we define 0, such that E( i ) 0 for i = 0,,, 3, 4, 5 (.) 43 a g e

6 Assuming i < i, we expand HL in terms of s and retain the terms up to second degree of the derivation of bias and mean square error for obtaining the expressions up to first order of approximation Bias( ) f HL f 4 n N f m n 3 3 f f f f f 3 3 f f f f E( HL ) f 3 f 3 4 m N n N f f f f m n f f f ( ) f f f For any fixed value of, Bias HL and HL and f E are minimized for f, f 3 f f f s in (.) and their corresponding minimum values are given by Bias( HL ) 8 f m n f f f f f n N f E (.4) R. 4 f m N n N m n ( HL ) y (.3) (.5) 44 a g e

7 III. COARION We compare the proposed family of estimators with respect to its mean square error with ingh et al (006) estimator as well as linear regression type estimator lr because it is generally considered best in literature under the same conditions. Using the expressions (.3) and (.5), we have E lr E HL Ry. 4 f m n where K R y. 0 for K K (3.) Using the expressions (.5) and (.5), we have E E R. HL min y 4 f m n 0 for 0 (3.) From (3.) and (3.), we note that the proposed family of estimators is always better than the linear regression type estimator as well as ingh et al (006) estimator for 0. Note: From the expressions of the biases obtained, we noted that no concrete conclusion can be obtained theoretically by comparing proposed estimator with the others with respect to their biases. o, the biases are compared numerically in ection 4. IV. NUERICAL ILLUTRATION For illustration of results numerically and graphically, we take the following data Data (ource: ingh, 003). : the number of fish caught by marine recreational fishermen in 995; : The number of fish caught by marine recreational fishermen in 994; : The number of fish caught by marine recreational fishermen in 993. N = 69, = 0.505, =068, f = n = 4, = 0.366, =0, f = m =7, = 0.43, =307, f = Data (ource: Aezel and ounderpandian, 004). : The U.. exports to ingapore in billions of ingapore dollars; : The money supply figures in billions of ingapore dollars; : The local supply in U.. dollors. N = 67, = 0.664, =4.8, f = n = 3, = 0.864, =7.0, f = m =5, = 0.759, =5, f = Data 3. (ource: FA, 004). : District-wise tomato production (tones) in 003; : District-wise tomato production (tones) in 00; : District-wise tomato production (tones) in 00. N = 97, = 0.096, =4, f = n = 46, = 0.33, =33, f = m =33, = 0.496, =07, f = a g e

8 Efficiency For Data Table 4. : Bias and relative efficiency of estimators Bias Relative Efficiency of H HL GR lr E E E E E E E E E E E E-08 HL Figure roposed Estimator HL 30 0 Regression type Estimator lr ingh et al estimator Gross(980)Estimator GR a g e

9 Bias Figure ingh et al Estimator Jhajj et al Estimator H Gupta et al Estimator 0 0 roposed Estimator HL For Data Table 4. : Bias and relative efficiency of estimators Bias Relative Efficiency of Estimators H HL GR lr HL a g e

10 BIA Efficiency Figure roposed EstimatorHL ingh et al Estimator Regression type Estimator lr Gross(980) Estimator GR Figure 4.4 ingh et al estimator Gupta et al estimator roposed Estimator HL Jhajj et al estimator H a g e

11 For Data 3 Table 4.3 : Bias and relative efficiency of estimators Bias Relative Efficiency of Estimators H E E E E E E E E E E E E E-08 HL GR lr HL a g e

12 Bias Figure 4.5 Figure ingh et al Estimator Jhajj et al Estimator H 4 Gupta et al Estimator 3 0 roposed Estimator HL From tables (4.), (4.) and (4.3), we note that proposed family of estimators have smaller mean square error than Gross(980) estimator, linear regression type estimator, ingh et al (006) estimator, Gupta et al (008) estimator and Jhajj et al estimator (04) as well as have smaller bias than ingh et al (006) estimator, Gupta et al (008) estimator and Jhajj et al estimator (04) in all the three populations considered corresponding to 0, lr. The graphical representation also supports the numerical results. 50 a g e

13 V. CONCLUION The range of variation of involved in the proposed family of estimators of population median under two-phase sampling design has been obtained theoretically under which its estimators are efficient than the existing ones. It has been found theoretically that proposed family of estimators is efficient than the linear regression type estimator estimator (04) for 0, lr, ingh et al (006) estimator Gupta et al (008) estimator and Jhajj et al. Numerical results for all the three populations also show that optimum estimator of proposed family having smaller bias is efficient than them. Graphical representation also gives the same type of interpretation. Hence, we conclude that better estimators can be developed from the proposed 0,. o, it is strongly recommended that proposed family of family by choosing suitable values of estimators should be used for getting the accurate value of population median. REFERENCE []. Azel A.D., ounderpandian J. (004).Complete Business tatistics. 5 th ed. New ork: cgraw Hill, []. Gross.T. (980). edian estimation in sample roc. urv. Res. eth. ect. Amer. tatist Ass.,8-84. [3]. Gupta., habbir J. and Ahmad.(008). Estimation of median in two- phase sampling using two auxiliary variables. Communication in tatistics-theory and ethods, 37(), [4]. Jhajj H..and Harpreet Kaur (03),Generalized estimators of population median using auxiliary information, IJER, (4), [5]. Jhajj H.., Harpreet Kaur and Walia Gurjeet(03), Efficient family of ratio-product type estimators of median, AA, Vol 9, 04, [6]. Jhajj H.., Harpreet Kaur and Jhajj uneet (04), Efficient family of estimators of median using two phase sampling design, Communication in tatistics-theory and ethods, Accepted. [7]. Kuk.C.A. and ak T.K. (989). edian estimation in the presence of auxiliary information. J.R tatist.oc. B, (), [8]. FA (004). Crops Area roduction. inistery of food and Agriculture. Islamabad: akistan. ingh. (003).Advanced ampling Theory with Applications. Vol. &. London Kluwer Academic ublishers. [9]. ingh., Joarder A. H. and Tracy D..(00), edian estimation using double sampling. Austral. N.J. tatist. 43(), [0]. ingh., ingh H. and Upadhyaya L.N. (006). Chain ratio and regression type estimators for median estimation in survey sampling. tatist. ap. 48, a g e