# Estimation of a Finite Population Mean under Random Nonresponse Using Kernel Weights

Authors: Nelson Kiprono Bii; Christopher Ouma Onyango; John Odhiambo
Journal: Journal of Probability and Statistics
 (2020)

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## Abstract

Nonresponse is a potential source of errors in sample surveys. It introduces bias and large variance in the estimation of finite population parameters. Regression models have been recognized as one of the techniques of reducing bias and variance due to random nonresponse using auxiliary data. In this study, it is assumed that random nonresponse occurs in the survey variable in the second stage of cluster sampling, assuming full auxiliary information is available throughout. Auxiliary information is used at the estimation stage via a regression model to address the problem of random nonresponse. In particular, auxiliary information is used via an improved Nadaraya–Watson kernel regression technique to compensate for random nonresponse. The asymptotic bias and mean squared error of the estimator proposed are derived. Besides, a simulation study conducted indicates that the proposed estimator has smaller values of the bias and smaller mean squared error values compared to existing estimators of a finite population mean. The proposed estimator is also shown to have tighter confidence interval lengths at95% coverage rate. The results obtained in this study are useful for instance in choosing efficient estimators of a finite population mean in demographic sample surveys.


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## Body

## 1. Introduction

Many authors such as [1–4] have looked at estimation of a finite population mean in the presence of nonresponse using various assumptions. However, the estimators developed in these studies need improvements on the efficiency and the bias. In the sequence of improving estimation of a finite population mean in the presence of random nonresponse, an improved Nadaraya–Watson kernel regression estimator is proposed in this study. The improved Nadaraya–Watson kernel regression technique was first fronted by [5]. To compensate for random nonresponse, auxiliary information is used in this study via an improved Nadaraya–Watson kernel regression technique due to [5].An improvement of the Nadaraya–Watson estimator [6, 7] has been proposed by [5] using local bandwidth factor λij determined using [8] algorithm. The improved Nadaraya–Watson estimator is given by(1)mIMWx^=∑i∈s∑j∈sYij/λijKx−Xij/λijb∑i∈s∑j∈s1/λijKx−Xij/λijb,where b is a smoothing parameter while the local bandwidth factor λij is given by(2)λij=mX^ija−α,where a is an arithmetic mean given by a=∑i=1n∑j=1mmX^ij/mn while α is a sensitivity parameter which satisfies 0≤α≤1. It has been suggested by [8] that taking α=1/2 produces good results.


## 2. The Proposed Estimator of Finite Population Mean Using Improved Nadaraya–Watson Kernel Regression Technique

Consider a finite population of sizeN consisting of M clusters with Nj elements in the jth cluster. A sample of m clusters is selected so that n1i units respond and n2i units fail to respond. Let yij denote the value of the survey variable y for unit j in cluster i, for i=1,2,…,N,j=1,2,…,Ni, and let the population mean be given by(3)Y¯¯=1N∑i=1M∑j=1NiYij.The proposed estimator is given by(4)Y¯¯^INW=1M1n1i∑i∈s∑j∈syij+1n2i∑i∈s∑j∉sy^ij,where y^ij is an estimator of the nonresponse component of the sample. Assuming auxiliary information Xij is known throughout, y^ij can be obtained using the improved Nadaraya–Watson regression technique by(5)y^ij=mINWx^ij=∑i∈s∑j∈s1/λijKx−Xij/λijbYij∑i∈s∑j∈s1/λijKx−Xij/λijb,so that the estimator of the finite population mean can be rewritten as(6)Y¯¯^INW=1M1n1i∑i∈s∑j∈syij+1n2i∑i∈s∑j∉smINWx^ij.A special case wheren1i=n2i=n is assumed in this study. This simplifies mathematical computations so that equation (7) can be rewritten as(7)Y¯¯^INW=1Mn∑i∈s∑j∈syij+∑i∈s∑j∉smINWx^ij,where mINWx^ij is the improved Nadaraya–Watson kernel regression estimator given in equation (1), which is a weighted sum of the values of the survey variable Yij. Data are generated using a regression model given by(8)Y^ij=mx^ij+e^ij,where m⋅ is an unknown smooth function of auxiliary random variables Xij. It is assumed that the error term e^ij satisfies the following conditions:(9)Ee^ij=0,Vare^ij=σij2,Cove^i,e^j=0,fori≠j.Hence, the unspecified function of the auxiliary random variablesmx^ij is replaced by the improved Nadaraya–Watson kernel estimator mINWx^ij. The estimator can be rewritten as(10)mINWx^ij=∑i∈s∑j∈swxijYij,where wxij=1/λijKx−Xij/λijb/∑i∈s∑j∈s1/λijKx−Xij/λijb are the improved Nadaraya–Watson kernel weights, where K⋅ is a given kernel function assumed to be symmetrical. Since the choice of the kernel function is not critical for the performance of the kernel regression estimator, a simplified Gaussian kernel with mean 0 and variance 1 is used in this study. This is given by(11)Kw=12πe−w2/2=12πe−x−Xij/λijb2/2.In this case, the improved Nadaraya–Watson kernel estimation at any pointxij is given by(12)y^ij=mINWx^ij=∑i∑j1/2πe−x−Xij/λijb2/2Yij∑i∑j1/2πe−x−Xij/λijb2/2,where b is the bandwidth while λij is given in equation (2) due to [5].This provides a way of estimating the nonresponse values of the survey variableYij, in the ith cluster given the auxiliary values xij, for a specified kernel function.


### 2.1. The Asymptotic Bias of the Proposed Estimator

The expected value of the proposed estimator is given by(13)EY¯¯^INW=1Mn∑i=1n∑j=1mYij+∑i=n+1N∑j=m+1MmINWx^ij.Rewriting equation (5) using the property of symmetry associated with Nadaraya–Watson estimator,(14)mINWx^ij=∑i∈s∑j∈sKXij−xij/λijbYij∑i∈s∑j∈sKXij−xij/λijb,i=1,2,…,n;j=1,2,…,m.Following the procedure by [9], equation (14) can be rewritten as(15)mINWx^ij=1gx^ij1mnλijb∑i∑jKXij−xijλijbYij,where gx^ij is the estimated marginal density of auxiliary variables Xij. The bias of the estimator can be written as(16)BiasY¯¯^INW=EY¯¯^INW−Y¯¯,(17)BiasY¯¯^INW=E1Mn∑i=1n∑j=1mYij+∑i=n+1N∑j=m+1MmINWx^ij−1Mn∑i=1n∑j=1mYij+∑i=n+1N∑j=m+1MYij,which reduces to(18)BiasY¯¯^INW=1MnE∑i=n+1N∑j=m+1MmINWx^ij−∑i=n+1N∑j=m+1MYij.Rewriting the regression model given byYij=mXij+eij as(19)Yij=mxij+mXij−mxij+eij,and substituting it in equation (15) gives(20)mINWx^ij=1gx^ij1mnλijb∑i∑jKXij−xijλijbmxij+mXij−mxij+eij.Hence, the first term in equation (18) before taking expectation is given as(21)1Mn1/mnb∑i=n+1N∑j=m+1MKXij−xij/λijbYijgx^ij=1Mngx^ij1mnλijb∑i=n+1N∑j=m+1MKXij−xijλijbmxij+1mnλijb∑i=n+1N∑j=m+1MKXij−xijλijbmXij−mxij+1mnλijb∑i=n+1N∑j=m+1MKXij−xijλijbeij.Simplifying equation (21), the following is obtained:(22)1Mn1mnλijbgx^ij∑i=n+1N∑j=m+1MKXij−xijλijbYij=1Mn1mnλijbgx^ij∑i=n+1N∑j=m+1Mgx^ijmxij+m1x^ij+m2x^ij,where(23)m1x^ij=1mnλijb∑i=n+1N∑j=m+1MKXij−xijλijbmXij−mxij,(24)m2x^ij=1mnλijb∑i=n+1N∑j=m+1MKXij−xijλijbeij.Taking conditional expectation of equation (22) leads to(25)E∑i=n+1N∑j=m+1MmINWx^ijxij=1MnE1mnλijb∑i=n+1N∑j=m+1Mmx^ij+m1x^ijgx^ij+m2x^ijgx^ij.The following theorem due to [10] and applied by [11] was used in obtaining asymptotic bias and variance of the estimator using conditional expectations.Theorem 1.
LetKw be a symmetric density function with ∫wkwdw=0 and ∫w2kwdw=k2. Assume n and N increase together such that n/N⟶π with 0<π<1. Besides, assume the sampled and nonsampled values of x are in the interval c,d and are obtained by densities ds and dp−s, respectively, where both are bounded away from zero on c,d with continuous second derivatives. If for any variable Z,EZ/U=u=Au+OB and VarZ/U=u=OC, then Z=Au+OpB+C1/2.
Using this theorem, the asymptotic bias and variance are derived in the following sections. From the conditions of the error term stated in equation (9), it follows that Eeij/Xij=0. Therefore, Em2x^ij=0. Thus, Em1x^ij can be obtained as follows:(26)E∑i=n+1N∑j=m+1Mm1x^ij=1Mn1mnλijbE∑i=n+1N∑j=m+1MKXij−xijλijb×mXij−mxij.
Using substitution and change of variable technique given by(27)w=V−xijλijbV=xij+λijbwdV=λijbdw,equation (26) can be simplified to(28)E∑i=n+1N∑j=m+1Mm1x^ij=1MnMn−mnmn∫kwmxij+λijbw−mxij∫gxij+λijbwdw.
Using Taylor’s series expansion about the pointxij, the kth order kernel can be derived as follows:(29)gxij+λijbw=gxij+g′xijλijbw+12g″xijλijb2w2+⋯+1k!gkxijλijbkwk+oλijb2.
Similarly,(30)mxij+λijbw=mxij+m′xijλijbw+12m″xijλijb2w2+⋯+1k!mkxijλijbkwk+oλijb2.
Therefore, expanding equation (28) up to order oλijb2 and simplifying gives(31)E∑i=n+1N∑j=m+1Mm1x^ij=1MnMn−mnmngx^ijm′xijλijb∫wkwdw+Mn−mnmng′x^ijm′xijλijb2∫w2kwdw+Mn−mnmn12gx^ijm″xijλijb2×∫w2kwdw+oλijb2.
Using the conditions due to [10] given by ∫−∞∞kwdw=1, ∫−∞∞wkwdw=0, and ∫−∞∞w2kwdw=dk, the derivation in equation (31) can further be simplified to obtain(32)E∑i=n+1N∑j=m+1Mm1x^ij=1MnMn−mnmng′xijm′xij+12gxijm′′′xijλijb2dk+oλijb2.
Hence, the expected value of the second term in equation (25) then becomes(33)E∑i=n+1N∑j=m+1Mm1x^ij=1MnMn−mnmn12gx^ijm′′xijgxij+g′xijm′xijgx^ijλijb2dk+oλijb2.
Simplifying equation (33) gives(34)E∑i=n+1N∑j=m+1Mm1x^ij=1MnMn−mnmnλijb2dkCx+oλijb2,where Cx=gx^ij−11/2m″xijgxij+g′xijm′xij and dk=∫w2kwdw.
Using the equation of the bias given in (16) and the conditional expectation in equation (25), the following equation for the conditional bias of the estimator was obtained:(35)BiasY¯¯^INWxij=1MnMn−mnmnλijb2dkCx+oλijb2.
In the next subsection, the asymptotic variance of the estimator is also derived.


### 2.2. Asymptotic Variance of the Proposed Estimator

Using equation (7), the conditional variance of the estimator is given as(36)VarY¯¯^INWxij=Var1Mn∑i=1n∑j=1mYij+∑i=n+1N∑j=m+1MmINWx^ij=1Mn2Var∑i=1n∑j=1mYij+∑i=n+1N∑j=m+1MmINWx^ij,where mINWx^ij is given by(37)mINWx^ij=1gx^ij1mnλijb∑i∑jKXij−xijλijbYij,where gx^ij=1/mnλijb∑i∑jKx−Xij/λijb is the estimated marginal density of auxiliary variables Xij; for details see [6, 7]. Rewriting the regression model Yij=mXij+eij as Yij=mxij+mXij−mxij+eij and substituting in equation (37) leads to(38)VarY¯¯^INWxij=1Mn2Var∑i=1n∑j=1mYij+1mnλijbgx^ij∑i=n+1N∑j=m+1Mgx^ijmxij+m1x^ij+m2x^ij.From equation (24),(39)m2x^ij=1mnb∑i=n+1N∑j=m+1MKXij−xijλijbeij.Hence,(40)Var∑i=n+1N∑j=m+1Mm2x^ij=1Mn2Mn−mnmnλijb2∑i=1n∑j=1mVarDx,where Dx=KXij−xij/λijbeij. Expressing equation (40) in terms of expectation, the following equation is obtained(41)Var∑i=n+1N∑j=m+1Mm2x^ij=1Mn2Mn−mn2mnλijb2EDx2−EDx2.Using the fact that the conditional expectationEeij/Xij=0, the second term in equation (41) reduces to zero. Therefore,(42)Var∑i=n+1N∑j=m+1Mm2x^ij=1Mn2Mn−mn2mnλijb2σij2,where Eeij/Xij2=σij2.LetX=Xij and x=xij and make the following substitutions:(43)w=X−xλijbX−x=λijbwdX=λijbdw,so that(44)Var∑i=n+1N∑j=m+1Mm2x^ij=Mn−mn2mnλijb2Mn2∫KX−xλijb2σx2gXdX.Using the change of variables technique and simplifying, equation (44) reduces to(45)Var∑i=n+1N∑j=m+1Mm2x^ij=Mn−mn2mnλijbMn2∫Kw2σx2gxdw+o1mnλijb.Following the same procedure for getting the variance ofm2x^ij,Var
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j can similarly be obtained as follows:(46)Var∑i=n+1N∑j=m+1Mm1x^ij=1Mn2Var∑i=n+1N∑j=m+1M1mnλijb∑i=1n∑j=1mKXij−xijλijb×mXij−mxij.Equation (46) can be rewritten as(47)Var∑i=n+1N∑j=m+1Mm1x^ij=Mn−mn2mnb2Mn2VarKXij−xijλijb2mXij−mxij2gXdX,where X=λijbw+x so that dX=λijbdw. Changing variables and applying Taylor’s series expansion about the point xij leads to(48)Var∑i=n+1N∑j=m+1Mm1x^ij=Mn−mn2mnλijb2Mn2∫Kw2mx+λijbw−mx2×gx+λijbwdw,which gives(49)Var∑i=n+1N∑j=m+1Mm1x^ij=Mn−mn2mnλijb2Mn2∫Kw2mx+m′xλijbw+⋯−mx2gx+g′xλijbwdw.Following the procedure by [12] and simplifying, equation (49) reduces to(50)Var∑i=n+1N∑j=m+1Mm1x^ij=oMn−mn2b2mnλijb.For large samples, asn⟶N,m⟶M, and b⟶0, then mnλijb⟶∞. Hence, the variance in equation (49) asymptotically tends to zero, i.e., Var∑i=1N∑j=1Mm1x^ij⟶0, so that the variance of the estimator of the population mean reduces to(51)VarY¯¯^INWxij=Mn−mn2mnλijbMn2∑i=n+1N∑j=m+1MVarmxij+m1x^ij+m2x^ijgx^ij.Simplifying equation (51) leads to(52)VarY¯¯^INWxij=Mn−mn2mnλijbMn2gx^ij2Var∑i=n+1N∑j=m+1Mm2x^ij.Substituting equation (45) in equation (52) yields the following:(53)VarY¯¯^INWxij=1Mn2Mn−mn2∫Kw2σxij2dwmnλijbgx^ij+oMn−mn2mnλijb+1mnλijb.


### 2.3. Mean Squared Error of the Proposed Estimator

The conditionalMSE of the estimator of the finite population mean combines the conditional squared bias and the conditional variance of the estimator, that is,(54)MSEY¯¯^INWxij=VarY¯¯^INWxij+Bias2Y¯¯^INWxij,which on simplification leads to(55)MSEY¯¯^Xij=xij=1Mn2Mn−mn2Hwσxij2mnλijbgx^ij+Mn−mn24mn2Mn2λijb2dk2m′′xijgxij+2g′xijm′xijgx^ij2+o1MnMn−mn2mnλijb+1mnλijb,where Hw=∫Kw2dw and dk=∫w2Kwdw.From equation (55), it is noted that if the sample size is large, that is, as n⟶N and m⟶M, the MSE of Y¯¯^INW due to the kernel tends to zero for a sufficiently small bandwidth. The estimator Y¯¯^ is therefore asymptotically consistent since its MSE converges to zero in probability.


## 2.1. The Asymptotic Bias of the Proposed Estimator

The expected value of the proposed estimator is given by(13)EY¯¯^INW=1Mn∑i=1n∑j=1mYij+∑i=n+1N∑j=m+1MmINWx^ij.Rewriting equation (5) using the property of symmetry associated with Nadaraya–Watson estimator,(14)mINWx^ij=∑i∈s∑j∈sKXij−xij/λijbYij∑i∈s∑j∈sKXij−xij/λijb,i=1,2,…,n;j=1,2,…,m.Following the procedure by [9], equation (14) can be rewritten as(15)mINWx^ij=1gx^ij1mnλijb∑i∑jKXij−xijλijbYij,where gx^ij is the estimated marginal density of auxiliary variables Xij. The bias of the estimator can be written as(16)BiasY¯¯^INW=EY¯¯^INW−Y¯¯,(17)BiasY¯¯^INW=E1Mn∑i=1n∑j=1mYij+∑i=n+1N∑j=m+1MmINWx^ij−1Mn∑i=1n∑j=1mYij+∑i=n+1N∑j=m+1MYij,which reduces to(18)BiasY¯¯^INW=1MnE∑i=n+1N∑j=m+1MmINWx^ij−∑i=n+1N∑j=m+1MYij.Rewriting the regression model given byYij=mXij+eij as(19)Yij=mxij+mXij−mxij+eij,and substituting it in equation (15) gives(20)mINWx^ij=1gx^ij1mnλijb∑i∑jKXij−xijλijbmxij+mXij−mxij+eij.Hence, the first term in equation (18) before taking expectation is given as(21)1Mn1/mnb∑i=n+1N∑j=m+1MKXij−xij/λijbYijgx^ij=1Mngx^ij1mnλijb∑i=n+1N∑j=m+1MKXij−xijλijbmxij+1mnλijb∑i=n+1N∑j=m+1MKXij−xijλijbmXij−mxij+1mnλijb∑i=n+1N∑j=m+1MKXij−xijλijbeij.Simplifying equation (21), the following is obtained:(22)1Mn1mnλijbgx^ij∑i=n+1N∑j=m+1MKXij−xijλijbYij=1Mn1mnλijbgx^ij∑i=n+1N∑j=m+1Mgx^ijmxij+m1x^ij+m2x^ij,where(23)m1x^ij=1mnλijb∑i=n+1N∑j=m+1MKXij−xijλijbmXij−mxij,(24)m2x^ij=1mnλijb∑i=n+1N∑j=m+1MKXij−xijλijbeij.Taking conditional expectation of equation (22) leads to(25)E∑i=n+1N∑j=m+1MmINWx^ijxij=1MnE1mnλijb∑i=n+1N∑j=m+1Mmx^ij+m1x^ijgx^ij+m2x^ijgx^ij.The following theorem due to [10] and applied by [11] was used in obtaining asymptotic bias and variance of the estimator using conditional expectations.Theorem 1.
LetKw be a symmetric density function with ∫wkwdw=0 and ∫w2kwdw=k2. Assume n and N increase together such that n/N⟶π with 0<π<1. Besides, assume the sampled and nonsampled values of x are in the interval c,d and are obtained by densities ds and dp−s, respectively, where both are bounded away from zero on c,d with continuous second derivatives. If for any variable Z,EZ/U=u=Au+OB and VarZ/U=u=OC, then Z=Au+OpB+C1/2.
Using this theorem, the asymptotic bias and variance are derived in the following sections. From the conditions of the error term stated in equation (9), it follows that Eeij/Xij=0. Therefore, Em2x^ij=0. Thus, Em1x^ij can be obtained as follows:(26)E∑i=n+1N∑j=m+1Mm1x^ij=1Mn1mnλijbE∑i=n+1N∑j=m+1MKXij−xijλijb×mXij−mxij.
Using substitution and change of variable technique given by(27)w=V−xijλijbV=xij+λijbwdV=λijbdw,equation (26) can be simplified to(28)E∑i=n+1N∑j=m+1Mm1x^ij=1MnMn−mnmn∫kwmxij+λijbw−mxij∫gxij+λijbwdw.
Using Taylor’s series expansion about the pointxij, the kth order kernel can be derived as follows:(29)gxij+λijbw=gxij+g′xijλijbw+12g″xijλijb2w2+⋯+1k!gkxijλijbkwk+oλijb2.
Similarly,(30)mxij+λijbw=mxij+m′xijλijbw+12m″xijλijb2w2+⋯+1k!mkxijλijbkwk+oλijb2.
Therefore, expanding equation (28) up to order oλijb2 and simplifying gives(31)E∑i=n+1N∑j=m+1Mm1x^ij=1MnMn−mnmngx^ijm′xijλijb∫wkwdw+Mn−mnmng′x^ijm′xijλijb2∫w2kwdw+Mn−mnmn12gx^ijm″xijλijb2×∫w2kwdw+oλijb2.
Using the conditions due to [10] given by ∫−∞∞kwdw=1, ∫−∞∞wkwdw=0, and ∫−∞∞w2kwdw=dk, the derivation in equation (31) can further be simplified to obtain(32)E∑i=n+1N∑j=m+1Mm1x^ij=1MnMn−mnmng′xijm′xij+12gxijm′′′xijλijb2dk+oλijb2.
Hence, the expected value of the second term in equation (25) then becomes(33)E∑i=n+1N∑j=m+1Mm1x^ij=1MnMn−mnmn12gx^ijm′′xijgxij+g′xijm′xijgx^ijλijb2dk+oλijb2.
Simplifying equation (33) gives(34)E∑i=n+1N∑j=m+1Mm1x^ij=1MnMn−mnmnλijb2dkCx+oλijb2,where Cx=gx^ij−11/2m″xijgxij+g′xijm′xij and dk=∫w2kwdw.
Using the equation of the bias given in (16) and the conditional expectation in equation (25), the following equation for the conditional bias of the estimator was obtained:(35)BiasY¯¯^INWxij=1MnMn−mnmnλijb2dkCx+oλijb2.
In the next subsection, the asymptotic variance of the estimator is also derived.


## 2.2. Asymptotic Variance of the Proposed Estimator

Using equation (7), the conditional variance of the estimator is given as(36)VarY¯¯^INWxij=Var1Mn∑i=1n∑j=1mYij+∑i=n+1N∑j=m+1MmINWx^ij=1Mn2Var∑i=1n∑j=1mYij+∑i=n+1N∑j=m+1MmINWx^ij,where mINWx^ij is given by(37)mINWx^ij=1gx^ij1mnλijb∑i∑jKXij−xijλijbYij,where gx^ij=1/mnλijb∑i∑jKx−Xij/λijb is the estimated marginal density of auxiliary variables Xij; for details see [6, 7]. Rewriting the regression model Yij=mXij+eij as Yij=mxij+mXij−mxij+eij and substituting in equation (37) leads to(38)VarY¯¯^INWxij=1Mn2Var∑i=1n∑j=1mYij+1mnλijbgx^ij∑i=n+1N∑j=m+1Mgx^ijmxij+m1x^ij+m2x^ij.From equation (24),(39)m2x^ij=1mnb∑i=n+1N∑j=m+1MKXij−xijλijbeij.Hence,(40)Var∑i=n+1N∑j=m+1Mm2x^ij=1Mn2Mn−mnmnλijb2∑i=1n∑j=1mVarDx,where Dx=KXij−xij/λijbeij. Expressing equation (40) in terms of expectation, the following equation is obtained(41)Var∑i=n+1N∑j=m+1Mm2x^ij=1Mn2Mn−mn2mnλijb2EDx2−EDx2.Using the fact that the conditional expectationEeij/Xij=0, the second term in equation (41) reduces to zero. Therefore,(42)Var∑i=n+1N∑j=m+1Mm2x^ij=1Mn2Mn−mn2mnλijb2σij2,where Eeij/Xij2=σij2.LetX=Xij and x=xij and make the following substitutions:(43)w=X−xλijbX−x=λijbwdX=λijbdw,so that(44)Var∑i=n+1N∑j=m+1Mm2x^ij=Mn−mn2mnλijb2Mn2∫KX−xλijb2σx2gXdX.Using the change of variables technique and simplifying, equation (44) reduces to(45)Var∑i=n+1N∑j=m+1Mm2x^ij=Mn−mn2mnλijbMn2∫Kw2σx2gxdw+o1mnλijb.Following the same procedure for getting the variance ofm2x^ij,Var
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^
i
j can similarly be obtained as follows:(46)Var∑i=n+1N∑j=m+1Mm1x^ij=1Mn2Var∑i=n+1N∑j=m+1M1mnλijb∑i=1n∑j=1mKXij−xijλijb×mXij−mxij.Equation (46) can be rewritten as(47)Var∑i=n+1N∑j=m+1Mm1x^ij=Mn−mn2mnb2Mn2VarKXij−xijλijb2mXij−mxij2gXdX,where X=λijbw+x so that dX=λijbdw. Changing variables and applying Taylor’s series expansion about the point xij leads to(48)Var∑i=n+1N∑j=m+1Mm1x^ij=Mn−mn2mnλijb2Mn2∫Kw2mx+λijbw−mx2×gx+λijbwdw,which gives(49)Var∑i=n+1N∑j=m+1Mm1x^ij=Mn−mn2mnλijb2Mn2∫Kw2mx+m′xλijbw+⋯−mx2gx+g′xλijbwdw.Following the procedure by [12] and simplifying, equation (49) reduces to(50)Var∑i=n+1N∑j=m+1Mm1x^ij=oMn−mn2b2mnλijb.For large samples, asn⟶N,m⟶M, and b⟶0, then mnλijb⟶∞. Hence, the variance in equation (49) asymptotically tends to zero, i.e., Var∑i=1N∑j=1Mm1x^ij⟶0, so that the variance of the estimator of the population mean reduces to(51)VarY¯¯^INWxij=Mn−mn2mnλijbMn2∑i=n+1N∑j=m+1MVarmxij+m1x^ij+m2x^ijgx^ij.Simplifying equation (51) leads to(52)VarY¯¯^INWxij=Mn−mn2mnλijbMn2gx^ij2Var∑i=n+1N∑j=m+1Mm2x^ij.Substituting equation (45) in equation (52) yields the following:(53)VarY¯¯^INWxij=1Mn2Mn−mn2∫Kw2σxij2dwmnλijbgx^ij+oMn−mn2mnλijb+1mnλijb.


## 2.3. Mean Squared Error of the Proposed Estimator

The conditionalMSE of the estimator of the finite population mean combines the conditional squared bias and the conditional variance of the estimator, that is,(54)MSEY¯¯^INWxij=VarY¯¯^INWxij+Bias2Y¯¯^INWxij,which on simplification leads to(55)MSEY¯¯^Xij=xij=1Mn2Mn−mn2Hwσxij2mnλijbgx^ij+Mn−mn24mn2Mn2λijb2dk2m′′xijgxij+2g′xijm′xijgx^ij2+o1MnMn−mn2mnλijb+1mnλijb,where Hw=∫Kw2dw and dk=∫w2Kwdw.From equation (55), it is noted that if the sample size is large, that is, as n⟶N and m⟶M, the MSE of Y¯¯^INW due to the kernel tends to zero for a sufficiently small bandwidth. The estimator Y¯¯^ is therefore asymptotically consistent since its MSE converges to zero in probability.


## 3. Simulation Study

A simulation experiment was conducted using R code in order to compare the performance of the proposed estimator in two-stage cluster sampling with the transformed estimator due to [13] and the nonparametric regression estimator due to [14]. An asymptotic framework is used where both the population number of clusters and the sample number of clusters are large. The number of clusters within each cluster Ni is held constant so that no cluster dominates the population.Both linear and nonlinear mean functions of auxiliary random variables due to [14] were considered in generating data, where x∈0,1. The equations of the mean functions used in simulating the data are given in Table 1.Table 1
Equations of mean functions simulated.
Mean function
Equation
Linear
m
1
x
^
=
1
+
2
x
−
0.5
Quadratic
m
2
x
^
=
1
+
2
x
−
0.5
2
Sine
m
3
x
^
=
2
+
s
i
n
2
π
x
Exponential
m
4
x
^
=
exp
−
8
x
Bump
m
5
x
^
=
1
+
2
x
−
0.5
+
exp
−
200
x
−
0.5
2
Jump
m
6
x
^
=
1
+
2
x
−
0.5
I
x
≤
0.65
+
0.65
I
x
≥
0.65The population auxiliary valuesxij of size M=2000 are generated as identical and independently distributed uniform 0,1 random variables. The survey values are only known for the respondents in the selected sample. Using the auxiliary values, the nonresponse values are generated, that is, for every generated value xij,i=1,2,…,M;j=1,2,…,Ni, the mean survey nonresponse values are generated as(56)y^ij=1Mmx^ijNi+e^ijNi,where e^ij are identically and independently distributed normal random variables with mean zero and variance one. Besides, a Gaussian kernel with mean zero and variance one was used. A Gaussian kernel was used since it has smooth and continuous derivatives at every data point. Besides, an optimal bandwidth generated using cross-validation technique due to [15] was used. It has been noted by [15] that this bandwidth would lead to more informative estimates compared to other choices. The local bandwidth factor λij given in equation (2) was generated using the algorithm due to [8].At stage one, a sample of clusters is generated first by simple random sampling using a sample of sizem=200. At stage two, subsamples of elements within every selected cluster are generated by simple random sampling with replacement using a random sample of size ni. The nonresponse mean survey values were then generated using equation (56). The estimates of the finite population mean were then computed using the estimator in equation (7). The values of bias and mean squared error values were also computed. The 95% confidence intervals were then constructed for the estimators of the finite population mean for comparative purposes.


## 4. Simulation Results

The values of the bias, mean squared error, and confidence interval lengths are given in the following tables. Note thatY¯¯^INW is the estimator of the finite population mean proposed in this study and Y¯¯^TDM is the transformation of data method estimator of the finite population mean due to [13] whereas Y¯¯^REG is the nonparametric regression estimator due to [14]. Both Y¯¯^TDM and Y¯¯^REG were used for comparative purposes with the proposed estimator.The biases of the estimators considered are presented in Table2. Negative values of the bias imply underestimation while positive values of the bias indicate overestimation of the finite population mean by the different estimators. The proposed estimator has relatively smaller values of the bias followed by transformation of data method estimator due to [13]. The nonparametric-based estimator due to [14] has larger values compared to the other two estimators. It is also observed that the three estimators have relatively closer values of the bias in the quadratic mean function though the transformation of data method has positive bias at this mean function. Generally, among the three estimators of the finite population mean, the proposed estimator using the improved Nadaraya–Watson kernel regression technique performs better than the other two estimators in terms of bias.Table 2
Summary results of bias.
Estimator
Y
¯
¯
^
INW
Y
¯
¯
^
TDM
Y
¯
¯
^
REG
Linear
−0.00213
−0.1667
−0.3312
Quadratic
−0.0132
0.04171
−0.0966
Sine
−0.0521
−0.6416
−1.2311
Exponential
−0.0041
0.3592
0.7225
Bump
−0.0032
−0.2358
−0.4685
Jump
−0.0188
−0.2466
−0.4743Mean squared error combines both the variance and the squared bias terms of an estimator. The mean squared error values presented in Table3 were simulated using the different mean functions indicated. The quadratic mean function gives the smallest value of the mean squared error of the proposed estimator followed by the linear function. The estimator due to [14] has the largest value of the mean squared error in the jump function. Generally, it is noted from Table 3 that the mean squared error values for the proposed estimator are relatively smaller than the rest of the estimators considered. The transformation of data method estimator due to [13] follows closely in the second place with smaller mean squared error values compared to nonparametric regression-based estimator due to [14]. From this comparison of the mean squared error values, it can be concluded that the proposed estimator is more efficient than the other two estimators considered. It has got smaller MSE values in all the mean functions and thus outperforms the others in terms of efficiency.Table 3
Summary results of MSE values.
Estimator
Y
¯
¯
^
INW
Y
¯
¯
^
TDM
Y
¯
¯
^
REG
Linear
0.0334
0.1097
0.1321
Quadratic
0.0093
0.1455
0.5835
Sine
0.4215
1.5157
1.555
Exponential
0.3430
0.5220
1.3780
Bump
0.0634
0.2195
0.2508
Jump
0.2250
0.2951
1.1611The95% upper and lower confidence intervals were constructed for the estimators of the finite population mean. Confidence interval lengths were then obtained. The results are given in Table 4. From the values obtained, it is noted that the confidence interval lengths for the proposed estimator are much tighter than those of the estimators due to [13, 14]. Hence, at 95% level of confidence, the estimator proposed in this study performs better than its rival estimators.Table 4
Summary results of95% CI lengths.
Estimator
Y
¯
¯
^
INW
Y
¯
¯
^
TDM
Y
¯
¯
^
REG
Linear
0.7164
1.2984
1.4249
Quadratic
0.8270
1.4951
2.994
Sine
1.5269
2.5451
4.888
Exponential
2.2958
2.8322
4.6016
Bump
0.9872
1.8365
1.9630
Jump
1.8594
2.1297
4.2239


## 5. Conclusion

This study has developed an estimator of the finite population mean in two-stage cluster sampling, assuming random nonresponse occurs in the survey variable in the second stage of cluster sampling. Complete auxiliary information is assumed to be available in both stage one and stage two of cluster sampling. Kernel weights developed using the improved Nadaraya–Watson regression technique were used in the estimation process. The theoretical properties of the proposed estimator such as asymptotic bias, variance, and mean squared error were derived. Simulation results show that the proposed estimator has smaller values of the bias, smaller mean squared error values, and tighter confidence interval lengths compared to the other estimators. Therefore, the estimator of the finite population mean proposed in this study dominates the estimators due to [13, 14], respectively.



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*Source: 8090381-2020-04-21.xml*