Hacettepe Journal of Mathematics and Statistics Volume, 1 Generalized Multi-Phase Regression-type Estimators Under the Effect of Measuemnent Error to Estimate the Population Variance Saadia Masood, Javid Shabbir Abstract In this article, we suggest some regression-type estimators for the estimation of finite population variance using multi-variate auxiliary information under multi-phase sampling schemes when measurement error ME contaminates the study variable An empirical study is also carried out to judge the merits of proposed estimators Keywords: Multi-phase sampling, measurement error, mean square error, efficiency 000 AMS Classification: Mathematics Subject Classification 6D05 1 Introduction In sample surveys, it is customary to exploit the auxiliary information to enhance the precision of estimators Ratio and regression estimators provide one type of example Sometimes the sample units are chosen with probability proportionate to some measure of size based on the auxiliary variate In all these cases it is information on just one auxiliary variate that is used for reasons of sample selection or estimation Pretty often we take information on several variates and it may be considered important to make use of the whole of the available material to improve the precision of at least some of the key items in the survey see Raj 10 Isaki 7 has discussed multi-variate ratio estimators to estimate finite population variance S y Singh and Solanki 16, 17 and Solanki and Singh 19 Department of Statistics Quaid-i-Azam University, Islamabad, Pakistan, Email: saadiamasood@yahoocom Corresponding Author: Saadia Masood Department of Statistics Quaid-i-Azam University, Islamabad, Pakistan, Email: javidshabbir@gmailcom
proposed the procedure for variance estimation using auxiliary information under simple random sampling Two-phase sampling of a finite population occurs when a sample from the population is itself sampled, with the goal of determining variates in the sub-sample not already available in the sample An important example is the regression estimator for means or totals, which uses values of an auxiliary variable from the full sample to estimate the mean of a variable of interest that is available only on the subsample Multi-phase sampling is not widely discussed in literature Mukerjee et al 9 considered mainly three phases Singh 18 proposed a class of estimators for population variance under two-phase sampling, whose composition was partially defined for the single auxiliary variable Dorfman 5 proposed regression estimator for estimation of population variance under two-phase sampling scheme Allen et al 1 proposed a family of estimators of population mean using multiauxiliary information in presence of measurement errors In most of the statistical studies, it is one of the common believes that the data are error-free but usually in realistic circumstances this statement is not absolutely met and the data are infected by errors The consequences made for the error free data become invalid for the measurement error situation Some important sources of measurement error are discussed in Cochran 3 In sampling theory, the use of suitable auxiliary information results in considerable reduction in mean square error Shukla et al 11,1 contributed by suggesting a mean estimator as well as classes of factor-type estimators in the presence of measurement error Singh and Karpe 13 have paid attention towards the estimation of population mean of the study variable y using the auxiliary information in presence of measurement error Singh and Karpe 14 considered the problem of estimation of population variance Sy under the assumptions: i when the study variable y is measured without error and auxiliary variable x is affected by error with known error variance Sv, ii when the study variable y is affected by error with known error variance Su and the auxiliary variable x is free from error Furthermore, under the assumption of measurement error in study variabley, Singh and Karpe 15 paid attention towards the estimation of finite population variance Sy Bhushan et al proposed two-phase generalized class of regression-type estimators using auxiliary information Diana and Giordan 4 have proposed a family of estimators for the population variance Sy by assuming error in both variables yand x under the regression approach In practical application, let a psychiatrist wants
3 to estimate the population variance of level of pathology in certain class of patients which depends upon the thinking disturbance, aggressive attitude, number of major miss-haps in life, ect In literature, the work on estimation of finite population variance using multiauxiliary variables under multi-phase sampling is lacking especially when the study variable y is assumed to be contaminated with measurement error, so the present article is one of the steps to the solution of such situation Proposed Set-up: In the present study, we consider the following set-up: 1 Complete Information Case CIC: When information on all q auxiliary variables is known we use single phase sampling Incomplete Information Case IIC: When information on some auxiliary variables is known 3 No Information Case NIC: When information on all q auxiliary variables is unknown In Section, the symbols and notations used in this article are discussed Section 3 presents the generalized regression-type variance estimator based on complete information of the multi-auxiliary variables about population variance, when the study variable is contaminated with measurement error Section 4 present the generalized regression-type variance estimator when population variance of few auxiliary variables is known In Section 5, the generalized regression-type variance estimator is proposed when the population variance of all multi-auxiliary variable is unknown Sections 6, 7 and 8 present the efficiency comparison, numerical analysis and concluding remarks respectively Symbols and notations Let U = {1,,, j,, N} be a finite population of N distinct and identifiable units Let y and x i i = 1,,, r, r 1,, q, be the study and the q auxiliary variables respectively, taking values y j and x ij for the j -th population unit We are interested in estimating the finite population variance S y under multi-phase sampling schemes Specifically we assume that a preliminary large sample n 1 is drawn with simple random sampling without replacement SRSWOR from a population and information on the auxiliary variable x 1 is taken In second phase,
4 a relatively small sample of size n is drawn from n 1 n < n 1 and information on both auxiliary variables x 1 and x is taken This procedure goes up to the last phase when the smallest sample of size n q1 nq1 < n q < < n 1 is drawn At this phase, all the q auxiliary variables as well as the variable of interest y are also observed According to assumption, the measurement error is present in the variable of interest y denoted by y with known variance S u Moreover, let Sx i and s x il denote the known population variance and sample variance of the i-th auxiliary variable i = 1,,, r, r 1,, q at l-th phase l = i,, q, q 1, respectively We limit our numerical study to two-phase sampling using three auxiliary variables The observational or measurement errors are defined as u jl = y jl y jl and v ijl = x ijl x ijl i = 1,,, r, r 1,, q, where u jl and v ijl are assumed to be stochastic with zero mean and constant variances S u and S v i As ȳ l and x il are unbiased estimators but s y l and s x il are biased estimators Let Ȳ, X i and S y, S x i be the population means and population variances of the true values of y jl and x ijl respectively with corresponding sample means ȳl, x il and sample variances s y l, s x il at l-th phase We know that ȳ l = 1 nl n l j=1 y il is unbiased estimator but s y l = 1 nl n l 1 j=1 estimator of Sy due to measurement error Similarly x ijl = 1 is un- x ijl il x i = 1,,, r, r 1,, q biased estimator but s x i = 1 n l 1 is biased estimator of Sx i nl j=1 y jl ȳ l is biased due to measurement error at l-th phase nl n l j=1 x ijl The expected values of s y l and s x i are given by E s y l = Sy Su and E s x il = Sx i Sv i, where Su and Sv j are known, then the unbiased estimators of Sy and Sx j are Ŝ y l = s y l Su and Ŝ x il = s x il Sv i i = 1,,, r, r 1,, q To obtain the properties of proposed estimators, we use the following approximations For l-th and l 1-th phase, we define the notations as Let s y l = Sy 1 e yl, s y l1 = Sy 1 e yl1, s y l = Sy 1 eyl, s x il = S x i 1 e x il, s x il = S x i 1 exil, s xil1 = S x i 1 exil1, such that E e y l = ϕ l SyA 4 yy, E e y l1 = ϕ l1 SyA 4 yy, E e yl = ϕ l Syλ 4 yy, E e xil = ϕ l Sx 4 i λ x ix i, E e xil1 = E e xil e xil1 = ϕl1 Sx 4 i λ x ix i,
E e yl e xil = ϕ l SyS x i λ yx i, E e yl e xil1 = ϕ l1 SyS x i λ yx i, 5 where A yy = γ y γu1 θy θ, θ y y = S y S, γ y y = β y 3 and γ u = β u 3, S u here β y and β u are the population co-efficients of kurtosis for the variable y and u Let λ x ix i = λ xix i 1, λ yx i = λ yxi 1, µ yx i = µ yxi µ y µ xi, µ y = S y, µ xi = S x i and ϕ l = 1 n l Also λ ts = where µ abt,s = µ t,s µ 0t,s µ 0t,s = µts µ tµ s or t = y, x i and s = y, x i i = 1,,, r, r 1,, q, N t i T a s i S b N 1 For a = and b = µ t,s = For a = 0 and b = µ 0t,s = For a = and b = 0 µ 0t,s = N t i T s i S N 1 N s i S N 1 N t i T N 1 3 Generalized Regression-Type Estimators Using Multi-Auxiliary Variables In this section, the estimators are formulated under the proposed setup 31 Generalized regression-type estimators using multi-auxiliary variables under multi-phase sampling in the presence of ME under CIC Let s y l and s x il be the sample variances of the study variable y under measurement error and the i-th auxiliary variable i = 1,,, r, r 1,, q respectively The population variance Sx i i = 1,,, r, r 1,, q of all multi-auxiliary variables is known We consider the following generalized multi-phase regression-type estimator for population variance S y using i = 1,,, r, r 1,, q as unknown constants 31 Ŝ y1 = s y l In terms of e s, we have 3 Ŝ y1 S y = S ye y l S x i s x il Sx i e xil
6 Squaring 3 and then taking expectation, we get M SE 33 MSE Ŝ y1 = E Sye y l Sx i e xil Ŝ y1 as For optimum value of = 1 i1 Λyx i y xq Λ xxq q i = 1,,, q, the resulting minimum MSE, to first order of approximation, is given by 34 MSE Let R s y s xq 35 MSE Ŝ y1 Ŝ y1 = ϕ l Sy 4 A yy min = q 1i1 Λyx i y xq Λ xxq q Ŝ y1 = ϕ l Sy 4 min A yy R s y s xq 1 Λ i1 yx i y xq µ xi λ yx i Λ xxq q µ y µ yx i µ, then 34 can be written as y Remark 311: Single-phase sampling using q auxiliary variables For full information case using q multi-auxiliary variables, we replace l by 1, which is the case of simple random sampling The estimator given in 31 becomes 36 Ŝ y1 = s y 1 S x i s x i1 For the optimum values of = 1 i1 i Λyx y xq Λ xxq q i = 1,,, q, the resulting minimum MSE, to first order of approximation, is given by 37 MSE Ŝ y1 Ŝ y1 min = ϕ 1 S 4 y A yy R s y s xq Here Λ yxi y xq is the determinant of matrix of population variances of the variables y, x 1,, x q and Λ xxq q is the determinant of matrix of population variances of the variables x 1,, x q Remark 31: Single-phase sampling using q auxiliary variables in the absence of measurement error Let the observations of variable of interest y be recorded without an error
7 Substituting Su = 0 in 35, we get A yy = λ yy, 38 MSE Ŝ y1 = ϕ 1 Sy 4 λ yy R s y s xq min 3 Generalized regression-type estimators using multi-auxiliary information under multi-phase sampling in the presence of ME under IIC Let s x il and s x il1 be sample variances of the auxiliary variables x i i = 1,,, r, r 1,, q at l- th and l 1-th phases respectively with the sample size n l and n l1 having the population variance Sx i Also s y l1 be the sample variances of the study variable y of size n l1 selected at l 1-th phase The population variance Sx i i = 1,,, r, r 1,, q on some auxiliary variables is known We formulate the generalized regression-type estimator for the estimation of unknown finite population variance S y using, δ i i = 1,,, r and γ i i = r 1, r,, q as unknown constants 39 Ŝ y =s y l1 i=r1 In terms of e s, we have 310 Ŝ y S y = S x i s x il γ i s x il s x il1 S ye y l1 Sx i e xil δ i S x i s x il1 δ i Sx i e xil1 Squaring 310 and then taking expectation, we get Ŝ MSE y =E Sye y l1 Sx i e xil 311 i=r1 γ i S x i exil e xil1 i=r1 δ i Sx i e xil1 For the optimum values of = 1 i1 Λyxi y xq Λ xxq q Λyx i y xr Λ xxr r γ i S x i exil e xil1, δ i = 1 i1 Λ yxi y xr Λ xxr r i = 1,,, r and γ i = 1 i1 Λyx i y xq Λ xxq q i = r 1, r,, q, the resulting
8 minimum MSE 31 Let R s y s xr MSE 313 MSE Ŝ y, to first order of approximation, is given by Ŝ y =Sy 4 min ϕ l1 A yy ϕ l ϕ l ϕ l1 q = r 1i1 Λyx i y xq Λ xxq q Ŝ y = Sy 4 min µ yx i µ y 1 i1 Λyx i y x r µ yx i Λ xxq q µ y 1 i1 Λyx i y x q Λ xxq q µ yx i µ y then 31 can be written as ϕ l1 A yy ϕ l R s ϕ y s xr l ϕ l1 R s y s xq Remark 31: Two-phase sampling using q auxiliary variables For the case of two-phase sampling using q multi-auxiliary variables, we replace l by 1 The estimator given in 39 becomes 314 Ŝ y = s y S x i s x i1 δ i S x i s x i For optimum values of = 1 i1 Λyxi y xq Λyx i y xr Λ xxq q Λ xxr r i=r1 γ i s x i1 s x i, δ i = 1 i1 Λyx i y xq Λ xxq q i = 1,,, r and γ i = 1 i1 Λyx i y xq i = r 1, r,, q,the minimum MSE Λ xxq q to first order of approximation, is given by Ŝ y, 315 MSE Ŝ y = Sy ϕ 4 A yy R ϕ s min y s xr 1 ϕ R R s y s xq s y s xr Remark 3: Two-phase sampling using q auxiliary variables in the absence of measurement error Let the observations of variable of interest y be recorded without an error Substituting S u = 0 in 313, we get A yy = λ yy, so 316 MSE Ŝ y = Sy 4 min ϕ λ yy ϕ 1 R s ϕ y s xr 1 ϕ R s y s xq 33 Generalized regression-type estimators using multi-auxiliary information under multi-phase sampling in the presence of ME under NIC Let s y l1 and s x il1 be the sample variances of the study variable y under measurement error and the i-th auxiliary variable i = 1,,, r, r 1,, q respectively at l 1-th phase, whereas s x il be the sample variance of i-th auxiliary variable at l-th phase The population
9 variance S x i i = 1,,, r, r 1,, q of all multi-auxiliary variables is unknown We consider the following generalized regression-type estimator for population variance S y under no information case using i = 1,,, r, r 1,, q as unknown constant 317 Ŝ y3 = s y l1 s x il s x il1 To the first order of approximation, we write 317 as 318 Ŝ y3 S y = S ye y l1 S x i e xil e xil1 Squaring 318 and then taking expectation, we get MSE as Ŝ 319 MSE y3 = SyE 4 e y l1 e xil e xil1 For optimum value of MSE, to first order of approximation, is given by 30 Ŝ y3 MSE = 1 i1 Λyx i y xq i = 1,,, q,the resulting minimum Λ xxq q Ŝ y3 =Sy ϕ 4 l1 A yy min ϕ l We can write 30 in compact form as 31 MSE Ŝ y3 = Sy 4 min 1 i1 Λyx i y x q Λ xxq q µ yx i µ y 1 i1 Λyx i y x q Λ xxq q ϕ l1 A yy ϕ l ϕ l1 R s y s xq µ yx i µ y Remark 331: Two-phase sampling using q auxiliary variables For no information case using q auxiliary variables, we replace l by 1, which is the case of two-phase sampling The estimator given in 317 becomes 3 Ŝ y3 = s y s x i1 s x i For optimum value of MSE, to first order of approximation, is given by Ŝ y3 33 MSE = 1 i1 Λyx i y xq Λ xxq q i = 1,,, q, the resulting minimum Ŝ y3 = Sy 4 ϕ A yy ϕ 1 ϕ R s min y s xq
10 Remark 33: Two-phase sampling using q auxiliary variables in the absence of measurement error Let the observations of variable of interest y be recorded without an error Substituting S u = 0 in 31, we get A yy = λ yy, 34 MSE Ŝ y3 = Sy 4 min 4 Efficiency Comparison ϕ λ yy ϕ 1 ϕ R s y s xq To obtain the efficiency of proposed estimators, we compare the mean square errors of proposed multi-phase regression-type variance estimators under measurement error with the estimators assumed to be free of error By 37 and 38, 315 and 316, 33 and 34, it is evident that 41 λ yy < A yy Note: The Condition 41 is always true Remark: The numerical comparison is made under the efficiency conditions given above 5 Data Description Population 1: Source: Mukherjee et al 8 The fertility data is based on 64 countries Let y =Total fertility rate, 1980 1985, the average number of children born to a woman, using age specific fertility rates for a given year, x 1 = Child mortality, the number of deaths of children under age 5 in a year per 1000 live births, x =Female literacy rate, percent and x 3 =Per capita GNP in billions in 1980 N = 64, Sy = 77, Sx 1 = 577670, Sx = 676409, Sx 3 = 74941700, Ȳ = 5549, X 1 = 141500, X = 51188, X 3 = 140150, Su = 155, λ yy = 773, λ x1 x 1 = 341, λ x x = 1631, λ x3 x 3 = 34046, λ yx1 = 1458, λ yx = 1069, λ yx3 = 0540, λ x1 x = 1415, λ x1 x 3 = 191, λ x x 3 = 037, A yy = 134 Population :Source: Gujarati 6 The data is baesd on the demand for chicken in USA, 1960-198 Let y = Per capita consumption of chickens in pounds, x 1 = Real disposable income per capita in dollars, x = Real retail price of chicken per pound in cents and x 3 = Real retail price of pork per pound in cents
11 N = 3, Sy = 54360, Sx 1 = 38173500, Sx = 1359, Sx 3 = 140710, Ȳ = 39669, X 1 = 1035065, X = 47995, X 3 = 90400, Su = 3987, λ yy = 03, λ x1 x 1 = 696, λ x x = 1756, λ x3 x 3 = 1951, λ yx1 = 094, λ yx = 1541, λ yx3 = 1758, λ x1 x = 1997, λ x1 x 3 = 145, λ x x 3 = 1755, A yy = 1033 Population 3:Source: Vandaele 0 The data is based on the crime rate data of USA in 1960 Let y =Number of offenses reported to police per million population, x 1 =Number of males of age 14-4 per 1000 population, x =Indicator variable for southern states and x 3 =Mean number of years of schooling times 10 for persons age 5 or older N = 47, Sy = 1495853, Sx 1 = 151516, Sx = 09, Sx 3 = 14076, Ȳ = 90508, X 1 = 137511, X = 0340, X 3 = 105406, Su = 148881, λ yy = 3859, λ x1 x 1 = 3684, λ x x = 143, λ x3 x 3 = 1896, λ yx1 = 0456, λ yx = 0743, λ yx3 = 1041, λ x1 x = 1354, λ x1 x 3 = 1356, λ x x 3 = 10, A yy = 703 Table 1 MSE of proposed ratio-type estimators Ŝ y1, Ŝ y and Ŝ y3 Estimators Pop1 Pop Pop3 Ŝ y1 0408 119840 17957 011 05 4106 Ŝ y 0709 4531 6581 0134 14781 1408 Ŝ y3 0554 8070 198 0379 050 4106 *The results written in Table 1 in bold format are the absolute values of measurement error
1 6 Conclusion In general, the presence of measurement error in the survey data invalidates the results The goal of this study was to show how measurement error is to be seperated in case of multi-phase sampling using multi-auxiliary variables for estimation of population variance Sy The values of absolute measurement error are shown in Table 1 It is also evident that the condition 41 holds for all the populations Hence, the use of proposed estimators are highly preferred in the cases of multi-phase sampling under CIC, IIC and NIC Acknowledgments The authors wish to thank the editor and the anonymous referees for their suggestions which led to improvement in the earlier version of the manuscript References 1 Allen, J, Singh, HP and Smarandache, F A family of estimators of population mean using multi-auxiliary information in presence of measurement error, International Journal of Social Economics, 30 7: 837-849, 003 Bhushan, S, Singh, R K and Pandey, A Some generalized classes of double sampling regression type estimators using auxiliary information, Science Vision, 11 1: -6, 011 3 Cochran WG Errors of measurement in statistics Technometrics, 10: 637-666, 1968 4 Diana, T and Giordan, M Finite population variance estimation in presence of measurement error, Communications in Statistics-Theory and Methods, 41: 430-431, 01 5 Dorfman, A H A note on variance estimation for regression estimator for double sampling, Journal of American Statistical Association, 89: 137-140, 1994 6 Gujarati, D M Basic Econometrics The McGraw-Hill Companies, 004 7 Isaki, CT Variance estimation using auxiliary information,journal of American Statistical Association, 381: 117 13, 1983 8 Mukherjee, C, White, H and Whyte, M Econometrics and Data Analysis for Developing Countries, Routledge, London, 1998 9 Mukerjee R, Rao TJ and Vijayan K Regression-type estimators using multiple auxiliary information, Australian Journal of Statistics, 9 3: 44 54, 1998 10 Raj, D On a method of using multi-auxiliary information in sample surveys, Journal of American Statistical Association, 60: 70-77, 1965 11 Shukla, D, Pathak, S and Thakur, N S An estimator for mean estimation in presence of measurement error, Research and Reviews: A Journal of Statistics, 1, 1-8, 01 a 1 Shukla, D, Pathak, S and Thakur, N S Classes of factor-type estimators in presence of measurement error, Journal of Applied Statistical Methods, 11, 336-347, 01 b 13 Singh, HP and Karpe, N Ratio-product estimator for population mean in the presence of measurement errors, Journal of Applied Statistical Science, 16: 49-64, 008 a 14 Singh, HP and Karpe, N Estimation of population variance using auxiliary information in the presence of measurement errors, Statistics in Transition, 9 3: 443-470, 008b
13 15 Singh, HP and Karpe, N A class of estimators using auxiliary information for estimating finite population variance in presence of measurement errors, Communications in Statistics- Theory and Methods, 38: 734-741, 009 16 Singh H P and Solanki, RS Improved estimation of finite population variance using auxiliary information, Communications in Statistics-Theory and Methods, 4 15, 718-730, 013 a 17 Singh H P and Solanki, RS A new procedure for variance estimation in simple random sampling using auxiliary information, Statistical Papers, 54, 479-497, 013 b 18 Singh, S Estimation of finite population variance using double sampling, Aligarh Journal of Statistics, 11: 53-56, 1991 19 Solanki R S and Singh H P An improved class of estimators for the population variance, Model Assisted Statistics and Applications, 8 3, 9-38, 013 0 Vandaele, W Participation in Illegitimate Activities: Erlich Revisted Deterrence and Incapacitation, National Academy of Sciences 70 335, 1978