Estimating a Finite Population Mean under Random Non-Response in Two Stage Cluster Sampling with Replacement

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1 Open Journal of Statistics, 07, 7, ISS Online: ISS Print: 6-78X Estimating a Finite Population ean under Random on-response in Two Stage Cluster Sampling with Replacement elson Kiprono Bii, Christopher Ouma Onyango, John Odhiamo Institute of athematical Sciences, Strathmore University, airoi, Kenya Department of Statistics, Kenyatta University, airoi, Kenya How to cite this paper: Bii,.K., Onyango, C.O. and Odhiamo, J. (07 Estimating a Finite Population ean under Random on-response in Two Stage Cluster Sampling with Replacement. Open Journal of Statistics, 7, Received: Septemer, 07 Accepted: Octoer 4, 07 Pulished: Octoer 7, 07 Copyright 07 y authors and Scientific Research Pulishing Inc. This work is licensed under the Creative Commons Attriution International License (CC BY Open Access Astract on-response is a regular occurrence in Sample Surveys. Developing estimators when non-response eists may result in large iases when estimating population parameters. In this paper, a finite population mean is estimated when non-response eists randomly under two stage cluster sampling with replacement. It is assumed that non-response arises in the survey variale in the second stage of cluster sampling. Weighting method of compensating for non-response is applied. Asymptotic properties of the proposed estimator of the population mean are derived. Under mild assumptions, the estimator is shown to e asymptotically consistent. Keywords on-response, adaraya-watson Estimation, Two Stage Cluster Sampling. Introduction In survey sampling, non-response is one source of errors in data analysis. onresponse introduces ias into the estimation of population characteristics. It also causes samples to fail to follow the distriutions determined y the original sampling design. This paper seeks to reduce the non-response ias in the estimation of a finite population mean in two stage cluster sampling. Use of regression models is recognized as one of the procedures for reducing ias due to non-response using auiliary information. In practice, information on the variales of interest is not availale for non-respondents ut information on auiliary variales may e availale for non-respondents. It is therefore desirale to model the response ehavior and incorporate the auiliary data into DOI: 0.436/ojs Oct. 7, Open Journal of Statistics

2 the estimation so that the ias arising from non-response can e reduced. If the auiliary variales are correlated with the response ehavior, then the regression estimators would e more precise in estimation of population parameters, given the auiliary information is known. any authors have developed estimators of population mean where non-response eists in the study and auiliary variales. But there eist cases that do not ehiit non-response in the auiliary variales, such as: numer of people in a family, duration one takes to go through education. Imputation techniques have een used to account for non-response in the study variale. For instance, [] applied compromised method of imputation to estimate a finite population mean under two stage cluster sampling, this method however produced a large ias. In this study, the adaraya-watson regression technique is applied in deriving the estimator for the finite population mean. Kernel weights are used to compensate for non-response. Reweighting ethod on-response causes loss of oservations and therefore reweighting means that the weights are increased for all or almost all of the elements that fail to respond in a survey. The population mean, Y, is estimated y selecting a sample of size n at random with replacement. If responding units to item y are independent so that the proaility of unit j responding in cluster i is p ( i=,,, n; j =,,, m then an imputed estimator, y I, for Y, is given y yi = wy + wy (.0 w i, j s i, j sr i, j sm where w = gives sample survey weight tied to unit j in cluster i and π [, ] π = pi j s is its second order proaility of inclusion, s r, is the set of r units responding to item y and s m is the set of m units that failed to respond to item y so that r+ m= n and y is the imputed value generated so that the missing value y is compensated for, [].. The Proposed Estimator of Finite Population ean Consider a finite population of size consisting of clusters with i elements in the th i cluster. A sample of n clusters is selected so that n units respond and n units fail to respond. Let y denote the value of the survey variale Y for unit j in cluster i, for i=,,,, j =,,, i and let population mean e given y i Y i i = j = Y = (. Let an estimator of the finite population mean e defined y Y Y = δ + Yδ n i s j sπ n i s j s π Y as follows: (. DOI: 0.436/ojs Open Journal of Statistics

3 where δ is an indicator variale defined y j δ = 0, elsewhere th th, if unit in the cluster responds and n and n are the numer of units that respond and those that fail to respond respectively. π is the proaility of selecting the th j unit in the th i cluster into the sample. Let w( = to e the inverse of the second order inclusion proailities π and that is the th i auiliary random variale from the th j cluster. It follows that Equation (. ecomes Suppose success Y = w Y + w Y n i s j s n i s j s i ( δ ( ( δ (.3 δ is known to e Bernoulli random variales with proaility of δ, then, E( δ pr ( δ δ = = = and ( δ δ ( δ the epected value of the estimator of population mean is given y =, [3]. Thus, ( E Y = E w Y + E w Y n i s j s n i s j s ( ( δ ( ( δ (.4 Assuming non-response in the second stage of sampling, the prolem is therefore to estimate the values of Y. To do this, a linear regression model applied y [4] and [5] given elow is used; ( Y = m + e (.5 where m (. is a smooth function of the auiliary variales and e is the residual term with mean zero and variance which is strictly positive, Sustituting Equation (.5 in Equation (.4 the following result is otained: (( ( ( E Y = E m + e w δ n i sj s E w m e δ ( ( ( ( + + n i sj s (.6 Assuming that n = n = n, and simplifying Equation (.6 we otain the following (( ( ( E Y = E m + e w n i sj s ( ( ( ( E w m e + + i sj s A detailed work done y [5] proved that ( reduces to E Y = E m E w n i sj s δ δ (.7 E e = 0. Therefore Equation (.7 ( ( ( ( ( ( ( ( E w E m e + + i sj s δ δ (.8 DOI: 0.436/ojs Open Journal of Statistics

4 The second term in Equation (.8 is simplified as follows: E w E m e n i s j s = E( w( ( m δ n i s j s + E( w( eδ n i s j s ( ( ( ( + δ * But E( m( ( m m( E( w( E( m( + e = =, [6]. Thus we get the following: δ n i sj s = δm( w( δm( n i = m+ j= n+ + E e E w e n i = m+ j= n+ ( δ ( ( ( δ ( ( ( ( + n = + + n E w E m e δ i sj s {( ( m ( ( n ( δ m( w( δm( } ( ( m ( n δ E e E e δw { ( ( ( ( } n But E( e = 0, for details see [5]. On simplification, Equation (. reduces to Recall w( n E( w( ( ( E m + e δ i sj s ( ( m+ ( ( n+ δm( ( w( { } = n = π so that Equation (. may e re-written as follows: E( w( ( ( E m + e δ n i sj s ( ( m+ ( ( n+ π = δ m( n π (.9 (.0 (. (. (.3 Assume the sample sizes are large i.e. as n and m, Equation (.3 simplifies to E w E m e n i sj s π = δ m( n π ( ( ( ( + δ (.4 DOI: 0.436/ojs Open Journal of Statistics

5 Comining Equation (.4 with the first term in Equation (.08 ecomes; δ π E Y = E m E + m n i s j s π i s j s π ( ( ( ( δ (.5 Since the first term represents the response units, their values are all known. The prolem is to estimate the non-response units in the second term. Let the indicator variale δ =, the prolem now reduces to that of estimating the m, which is a function of the auiliary variales,. Hence the function ( epected value of the estimator of the finite population mean under non-response is given as; π E Y = Y + m n i s j s i s j s π ( ( δ (.6 In order to derive the asymptotic properties of the epected value of the proposed estimator in.6, first a review of adaraya-watson estimator is given elow. 3. Review of adaraya-watson Estimator Given a random sample of ivariate data ( i, yi,,( n, yn g( y, with the regression model given y Y = m + e as in Equation (.5, where (. ( term satisfy the following conditions: ( ( σ ( i j having a joint pdf m is unknown. Let the error E e = 0, Var e =, cov e, e = 0 for i j (3.0 Furthermore, let K (. denote a symmetric kernel density function which is twice continuously differentiale with: d d = 0 < wk w w= d k k( w w= wk ( w w k ( w dw ( d k( w = k( w In addition, let the smoothing weights e defined y ( =, =,,, ; =,,, X K i s i s (3. X K w i n j m (3. where is a smoothing parameter, normally referred to as the andwidth such that, w ( =. i j Using Equation (3., the adaraya-watson estimator of m( is given y: DOI: 0.436/ojs Open Journal of Statistics

6 X K Y i s j s m( = w( Y =, i =,,, n; j =,,, m (3.3 i s j s X K i s j s Given the model Y ( = m + e and the conditions of the error term as eplained in 3.0 aove, the epression for the survey variale Y relative to the auiliary variale g, y as follows: where g ( y X can e given as a joint pdf of ( ( ( yg, y dy m( = E ( Y X = = yg [ y ] dy = g y, dy (3.4, dy is the marginal density of X. The numerator and the denominator of Equation (3.4 can e estimated separately using kernel functions as follows: and g( y, is estimated y; X y Y g ( y, = K K mn i j (3.5 X y Y mn i j yg (, y dy = K K ydy (3.6 Using change of variales technique; let y Y w = y = w + Y dy= w d (3.7 So that X mn i j yg (, y dy = K ( w + Y K ( w dw (3.8 X = + mn i j K wk( ww d Y K( ww d (3.9 From the conditions specified in Equation (3., the following (3.9 simplifies to X yg (, y dy = K 0 + Y mn i j (3.0 which reduces to: X yg (, y dy = K Y (3. mn i j Following the same procedure, the denominator can e otained as follows: DOI: 0.436/ojs Open Journal of Statistics

7 X y Y g ( y, dy= K K dy mn i j n m X y Y = K K dy mn i= j= (3. Using change of variale technique as in Equation (3.7, Equation (3. can e re-written as follows: which yields Since ( n m X mn i= j= g ( y, dy= K K( ww d (3.3 n m X g ( y, dy= K (3.4 mn i= j= K ww d is a pdf and therefore integrates to. It follows from Equations ((3. and (3.4 that the estimator m( is as given in Equation (3.3. Thus the estimator of m( is a linear smoother since it is a linear function of the oservations, Y. Given a sample and a specified kernel function, then for a given auiliary value, the corresponding y-estimate is otained y the estimator outlined in Equation (3.3, which can e written as: where ( W ( ( y = m = W Y (3.5 W i j m is the adaraya-watson estimator for estimating the unknown function (. m, for details see [7] [8]. This provides a way of estimating for instance the non-response values of the survey variale Y, given the auiliary values, for a specified kernel function. 4. Asymptotic Bias of the ean Estimator Y Equation (.6 may e written as E Y = Y + m y i = j = i = n + j = m + n m ( W (4. Replacing y and re-writing Equation (3.5 using the property of symmetry associated with adaraya-watson estimator, then X K Y i s j s m ( W =, i =,,, n; j =,,, m (4. X K i s j s X = K Y g ( mn i j (4.3 DOI: 0.436/ojs Open Journal of Statistics

8 where (. K. Bii et al. g is the estimated marginal density of auiliary variales X. But for a finite population mean, the epected value of the estimator is given in Equation (4.. The ias is given y Bias Y E = Y Y (4.4 n m Bias Y = E Y ( + m i= j= i= n+ j= m+ n m Y + Y i= j= i= n+ j= m+ (4.5 Which reduces to Bias Y = m Y i = n + j = m + i = n + j = m + ( (4.6 = m( m( (4.7 i = n + j = m + i = n + j = m + Y = m X + e as Re-writing the regression model given y ( Y = m( + m( X m( + e (4.8 So that from Equation (4.3 the first term in Equation (4.7 efore taking the epectation is given as: X K Y i= n+ j= m+ mn g ( X = K m( g ( i= n+ j= m+ X + K m( X m( mn i= n+ j= m+ X + K e mn i= n+ j= m+ Simplifying Equation (4.9 the following is thus otained: X K Y mng ( i= n+ j= m+ g ( ( ( i n j m m m m( = + = = mng ( where X m ( = K m( X m( i= n+ j= m+ (4.9 (4.0 DOI: 0.436/ojs Open Journal of Statistics

9 X m ( = K e i= n+ j= m+ Taking conditional epectation of Equation (4.0 we get E i= n+ j= m+ ( ( ( ( ( m m = E m( + + mn g g i= n+ j= m+ (4. To otain the relationship etween the conditional mean and the selected andwidth, the following theorem due to [6] is applied; Theorem: (Dorfman, 99 Let ( ( k w e a symmetric density function with wk ( wdw = 0 and wk wdw= k. Assume n and increase together such that n π with 0< π <. Besides, assume the sampled and non-sampled values of are in the interval [ cd, ] and are generated y densities d s and d p s respectively oth cd and assumed to have continuous second de- ounded away from zero on [, ] rivatives. If for any variale, E( U= u = Au ( + OB ( and Var ( U = u = O( C, then = Au ( + Op B+ C. Applying this theorem, we have Y mn k w dw = ( ( mn ( mn ( ( ( SE mng ( ( g( g m 4 + k ( k m ( + 4mn 4 + O( + O + mn mn (4. This theorem is stated without proof. To prove it, we partition it into the ias and variance terms and separately prove them as follows: = 0. From Equation (3.0 it follows that E( e X = 0. Therefore, E m ( Thus E m ( can e otained as follows: i= n+ j= m+ ( E m X = E K m( X m( mn i= n+ j= m+ Using sustitution and change of variale technique elow Equation (4.3 can simplify to: (4.3 V w = so that V = + w and dv = dw (4.4 DOI: 0.436/ojs Open Journal of Statistics

10 i= n+ j= m+ ( E m mn = k ( w m ( + w m ( g ( + w d w mn (4.5 mn = k ( w m ( + w m ( g ( w + d w mn (4.6 Using the Taylor s series epansion aout the point can e derived as follows:, the k th order kernel k k k g( + w = g( + g ( w+ g ( w + + g ( w + O( (4.7 k! Similarly, k k k m( + w = m( + m ( w+ m ( w + + m ( w + O( (4.8 k! Epanding up to the 3 rd order kernels, Equation (4.8 ecomes m( w m( m ( w m ( w m + = + + ( w 3! 3 3 (4.9 In a similar manner, the epansion of Equation (4.6 up to order O( is given y: i= n+ j= m+ ( E m ( ( ( ( ( ( mn = k w m w + m w g + g w d w mn Simplifying Equation (4.0 gives; mn E m ( g ( m = ( wk ( w dw i n j m mn = + = + mn + g ( m ( wk ( w d w mn mn + g( m ( wk( w dw+ O ( mn (4.0 (4. Using the conditions stated in Equation (3., the derivation in (4. can further e simplified to otain: i= n+ j= m+ ( E m mn = g ( m ( + g( m ( dk + O( mn (4. Hence the epected value of the second term in Equation (4. then ecomes: i= n+ j= m+ ( E m ( ( ( mn g m = m ( + d + O mn g ( k (4.3 DOI: 0.436/ojs Open Journal of Statistics

11 ( mn m = + g g m d + O mn mn = dc k ( + O ( mn where ( ( ( k ( (4.4 (4.5 C( = m ( + g( g ( m ( (4.6 and d k is as stated in Equation (3. Using equation of the ias given in (4.4 and the conditional epectation in Equation (4., we otain the following equation for the ias of the estimator: mn Bias Y = dc k ( + O ( mn mn = dc k ( + O ( mn ( Asymptotic Variance of the Estimator, Y From Equations ((4.9 and (4., n m X m ( = K e mn i= j= (5.0 Hence where Var n m m ( = Var ( D (5. ( mn mn i= n+ j= m+ i= j= X D = K e Epressing Equation (5. in terms of epectation we otain: Var ( ( mn { [ ] ( } m ( = E D E D i= n+ j= m+ mn (5. Using the fact that the conditional epectation ( 0 E e X =, the second term in Equation (4.3 reduces to zero. Therefore, where ( mn Var m ( = σ (5.3 ( ( i= n+ j= m+ mn ( = ( E e X σ Let X = X, and =, and making the following sustitutions DOI: 0.436/ojs Open Journal of Statistics

12 Var X w = X = w dx= w d ( mn X ( ( ( mn (5.4 m ( = K σ g X dx (5.5 i= n+ j= m+ which can e simplified to get: Var ( mn mn ( ( σ ( ( = K w g + w dw (5.6 ( mn mn( ( = σ d + (5.7 i= n+ j= m+ Var m K( w g( w O ( mn i= n+ j= m+ m ( n m X = Var K ( X m( ( i= n+ j= m+ mn i= j= ( mn mn ( X ( = Var ( ( (5.8 Var m K X m i= n+ j= m+ (5.9 Hence Var i= n+ j= m+ m ( mn ( ( mn X = E K ( X m( g( X dx where X = w + so that dx= w d. Changing variales and applying Taylor s series epansion then Var i= n+ j= m+ m ( mn mn ( ( ( ( ( ( = K w m + w m g + w dw ( mn mn ( ( ( ( ( ( ( ( (5.0 (5. = K w m + m w + m g + g w dw (5. which simplifies to ( ( Var mn m = O (5.3 i= n+ j= m+ mn For large samples, as n, m and for 0, then mn. Hence the variance in Equation (5. asymptotically tends to zero, that is, i= n+ j= m+ ( Var m 0 DOI: 0.436/ojs Open Journal of Statistics

13 ( mn ( ( + ( g ( m m Var Y = ( + mn = + = + On simplification, Var m (5.4 i n j m ( mn Var Y = Var m ( (5.5 mn( g i n j m ( = + = + Sustituting Equations ((5.7 into (5.5 yields the following: ( ( σ ( mn K w dw ( mn Var Y = O + + ( mn( g ( mn mn ( ( σ ( mn H w ( mn = O + + ( mn( g ( mn mn (5.6 (5.7 where, H( w = K( w dw It is notale that the variance term still depends on the marginal density function, g( of the auiliary variales X. It can also e oserved that the variance is inversely related to the smoothing parameter. This implies that an increase in results in a smaller variance. However, increasing the andwidth would give a larger ias. Therefore there is a trade-off etween the ias and variance of the estimated population mean. A andwidth that provides a compromise etween the two measures would therefore e desirale. 6. ean Squared Error (SE of the Finite Population ean Estimator Y The SE of is, Y comines the ias and the variance terms of this estimator that SE Y = E Y Y (6.0 SE Y = E Y E Y+ E Y Y (6. Epanding Equation (6. gives: SE Y E Y E Y E E Y Y = + + E Y E Y E Y Y = + + Var Y Bias 0 (6. (6.3 Comining the ias in Equation (4.7 and the variance in Equation (5.7 and conditioning on the auiliary values of the auiliary variales X then DOI: 0.436/ojs Open Journal of Statistics

14 SE Y X = ( ( mn H w σ ( ( mn = O + + ( mn( g ( mn mn mn dc + k ( + O ( mn SE Y X = ( mn H ( w σ ( = ( mn g ( ( ( ( g( ( mn g m 4 + d k m ( + 4( mn ( 4 mn + O( + O + mn mn where ( ( H w = K w dw, ( d = wk wdw, k (6.4 (6.5 C( = m ( + g( g ( m ( as used earlier in the rest of the derivations. 7. Conclusion If the sample size is large enough, that is as n and m the of Y in Equation (6.5 due to the kernel tends to zero for sufficiently a small andwidth. The estimator Y is therefore asymptotically consistent since its SE converges to zero. References [] Singh, S. and Horn, S. (000 Compromised Imputation in Survey Sampling. etrika, 5, [] Lee, H., Rancourt, E. and Sarndal, C. (00 Variance Estimation from Survey Data under Single Imputation. Survey onresponse, [3] Bethlehem, J.G. (0 Using Response Proailities for Assessing Representativity. Statistics etherlands, International Statistical Review, 80, [4] Ouma, C. and Wafula, C. (005 Bootstrap Confidence Intervals for odel-based Surveys. East African Journal of Statistics,, [5] Onyango, C.O., Otieno, R.O. and Orwa, G.O. (00 Generalized odel Based Confidence Intervals in Two Stage Cluster Sampling. Pakistan Journal of Statistics and Operation Research, 6. [6] Dorfman, A.H. (99 onparametric Regression for Estimating Totals in Finite Populations. In: Proceedings of the Section on Survey Research ethods, American DOI: 0.436/ojs Open Journal of Statistics

15 Statistical Association Aleandria, VA, [7] adaraya, E.A. (964 On Estimating Regression. Theory of Proaility & Its Applications, 9, [8] Watson, G.S. (964 Smooth Regression Analysis. Sankhya: The Indian Journal of Statistics, Series A, DOI: 0.436/ojs Open Journal of Statistics