Research article

On spectral numerical method for variable-order partial differential equations

  • Received: 29 January 2022 Revised: 28 February 2022 Accepted: 07 March 2022 Published: 28 March 2022
  • MSC : 26A33, 35A02, 35A25

  • In this research article, we develop a powerful algorithm for numerical solutions to variable-order partial differential equations (PDEs). For the said method, we utilize properties of shifted Legendre polynomials to establish some operational matrices of variable-order differentiation and integration. With the help of the aforementioned operational matrices, we reduce the considered problem to a matrix type equation (equations). The resultant matrix equation is then solved by using computational software like Matlab to get the required numerical solution. Here it should be kept in mind that the proposed algorithm omits discretization and collocation which save much of time and memory. Further the numerical scheme based on operational matrices is one of the important procedure of spectral methods. The mentioned scheme is increasingly used for numerical analysis of various problems of differential as well as integral equations in previous many years. Pertinent examples are given to demonstrate the validity and efficiency of the method. Also some error analysis and comparison with traditional Haar wavelet collocations (HWCs) method is also provided to check the accuracy of the proposed scheme.

    Citation: Kamal Shah, Hafsa Naz, Muhammad Sarwar, Thabet Abdeljawad. On spectral numerical method for variable-order partial differential equations[J]. AIMS Mathematics, 2022, 7(6): 10422-10438. doi: 10.3934/math.2022581

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  • In this research article, we develop a powerful algorithm for numerical solutions to variable-order partial differential equations (PDEs). For the said method, we utilize properties of shifted Legendre polynomials to establish some operational matrices of variable-order differentiation and integration. With the help of the aforementioned operational matrices, we reduce the considered problem to a matrix type equation (equations). The resultant matrix equation is then solved by using computational software like Matlab to get the required numerical solution. Here it should be kept in mind that the proposed algorithm omits discretization and collocation which save much of time and memory. Further the numerical scheme based on operational matrices is one of the important procedure of spectral methods. The mentioned scheme is increasingly used for numerical analysis of various problems of differential as well as integral equations in previous many years. Pertinent examples are given to demonstrate the validity and efficiency of the method. Also some error analysis and comparison with traditional Haar wavelet collocations (HWCs) method is also provided to check the accuracy of the proposed scheme.



    In survey sampling, it is well known fact that suitable use of the auxiliary information may improves the precision of an estimator for the unknown population parameters. The auxiliary information can be used either at the design stage or at estimation stage to increase the accuracy of the population parameter estimators. Several authors presented modified different type of estimators for estimating the finite population mean including [4,9,21,22,23,24,25,26,27].

    The problem of estimation of finite population mean or total in two-stage sampling scheme using the auxiliary information has been well established. The two stage sampling scheme is an improvement over the cluster sampling, when it is not possible or easy to calculate all the units from the selected clusters. One of the main characteristic could be the budget, and it becomes too difficult to collect information from all the units within the selected clusters. To overcome this, one way is to select clusters, called first stage unit (fsus) and from the given population of interest, select a subsample from the selected clusters called the second stage units (ssu). This also benefits to increase the size of the first stage samples which consist of clusters, and assume to be heterogeneous groups. If there is no variation within clusters then might not be possible to collect information from all the units within selected clusters. In many situations, it is not possible to obtain the complete list of ultimate sampling units in large scale sample surveys, while a list of primary units of clusters may be available. In such situations, we select a random sample of first stage units or primary units using certain probability sampling schemes i.e simple random sampling (with or without replacement), systematic sampling and probability proportional to size (PPS), and then we can perform sub-sampling in selected clusters (first stage units). This approach is called two-stage sampling scheme.

    Two-stage has a great varaity of applications, which go far beyond the immediate scop of sample survey. Whenever any process involves in chemical, physical, or biological tests that can be performed on a small amount of materail, it is likely to be drawn as a subsample from a larger amount that is itself a sample.

    In large scale survey sampling, it is usual to adopt multistage sampling to estimate the population mean or total of the study variable y. [13] proposed a general class of estimators of a finite population mean using multi-auxiliary information under two stage sampling scheme. [1] proposed an alternative class of estimators in two stage sampling with two auxiliary variables. [10] proposed estimators for finite population mean under two-stage sampling using multivariate auxiliary information. [12] suggested a detailed note on ratio estimates in multi-stage sampling. [6] given some stratagies in two stage sampling using auxiliary information. [3] suggested a class of predictive estimaotrs in two stage sampling using auxiliary information. [8] gave a generalized method of estimation for two stage sampling using two auxiliary variables. [5] suggested chain ratio estimators in two stage sampling. For certain related work, we refer some latest articles, i.e., [14,15,16,17,18,19,20].

    In this article, we propose an improved generalized class of estimators using two auxiliary variables under two-stage sampling scheme. The biases and mean sqaure errors of the proposed generalized class of estimators are derived up to first order of approximation. Based on the numerical results, the proposed class of estimators are more efficient than their existing counterparts.

    Consider a finite population U = {U1,U2,...,UN} is divided into N first-stage units (fsus) clusters in the population. Let N be the total number of first stage unit in population, n be the number of first stage units selected in the sample, Mi be the number of second stage units (ssus) belongs to the ith first stage units (fsus), (i = 1, 2, …, N), and mi be the number of fsus selected from the ith fsu in the sample of n fsus, (i = 1, 2, …, n).

    Let yij, xij and zij be values of the study variable y and the auxiliary variables (xandz) respectively, for the jth ssus Ui=(j=1,2,...,Mi), in the ith fsus. The population mean of the study variable y and the auxiliary variables (x,z) are given by:

    ¯Y=1NNi=1ui¯Yi,¯X=1NNi=1ui¯Xi,¯Z=1NNi=1ui¯Zi,

    where

    ¯Yi=1MiMij=1yij,¯Xi=1MiMij=1xij,¯Zi=1MiMij=1zij,(i=1,2,...,N).
    ui=Mi¯M,and¯M=MN,M=Ni=1Mi,
    R=¯Y¯X,andRi=¯Yi¯Xi,
    S2by=1N1Ni=1(ui¯Yi¯Y)2,
    S2bx=1N1Ni=1(ui¯Xi¯X)2,
    S2bz=1N1Ni=1(ui¯Zi¯Z)2,
    Sbyx=1N1Ni=1(ui¯Yi¯Y)(ui¯Xi¯X),Sbyz=1N1Ni=1(ui¯Zi¯Z)(ui¯Yi¯Y),
    Sbxz=1N1Ni=1(ui¯Zi¯Z)(ui¯Xi¯X),S2iy=1Mi1Mij=1(yij¯Yi)2,
    S2ix=1Mi1Mij=1(xij¯Xi)2,S2iz=1Mi1Mij=1(zij¯Zi)2,
    Siyx=1Mi1(yij¯Yi)(xij¯Xi),Siyz=1Mi1(yij¯Yi)(zij¯Zi),
    Sixz=1Mi1Mij=1(xij¯Xi)(zij¯Zi),(i=1,2,...N).

    Similarly for sample data:

    ¯y=1nni=1ui¯yi=¯y(say),¯x=1nni=1ui¯xi=¯x(say),¯z=1nni=1ui¯zi=¯z(say),

    where

    ¯yi=1mimij=1yij,¯xi=1mimij=1xij,¯zi=1mimij=1zij,
    s2by=1n1ni=1(ui¯yi¯y)2,s2bx=1n1ni=1(ui¯xi¯x)2,
    s2bz=1n1ni=1(ui¯zi¯z)2,sbyx=1n1ni=1(ui¯yi¯y)(ui¯xi¯x),
    sbyz=1n1ni=1(ui¯yi¯y)(ui¯zi¯z),sbxz=1n1ni=1(ui¯xi¯x)(ui¯zi¯z),
    s2iy=1mi1(yij¯yi)2,s2ix=1mi1(xij¯xi)2,
    s2iz=1mi1(zij¯zi)2,siyx=1mi1mij=1(yij¯yi)(xij¯xi),
    siyz=1mi1mij=1(yij¯yi)(zij¯zi),sixz=1mi1mij=1(xij¯xi)(zij¯zi).

    In order to obtain the biases and mean sqaured errors, we consider the following relative error terms:

    e0=¯y¯Y¯Y,e1=¯x¯X¯X,e2=¯z¯Z¯Z,
    E(e20)=λC2by+1nNni=1u2iθiC2iy=Vy,
    E(e21)=λC2bx+1nNni=1u2iθiC2ix=Vx,
    E(e22)=λC2bz+1nNni=1u2iθiC2iz=Vz,
    E(e0e1)=λCbyx+1nNni=1u2iθiCiyx=Vyx,
    E(e0e2)=λCbyz+1nNni=1u2iθiCiyz=Vyz,
    E(e1e2)=λCbxz+1nNni=1u2iθiCixz=Vxz,
    Cby=Sby¯Y,Cbx=Sbx¯X,Cbz=Sbz¯Z,
    Cbyx=Sbyx¯Y¯X,Cbyz=Sbyz¯Y¯Z,Cbxz=Sbxz¯X¯Z,
    Ciyx=Siyx¯Y¯X,Ciyz=Sbyz¯Y¯Z,Cixz=Sixz¯X¯Z,
    Ciy=Siy¯Y,Cix=Six¯X,Ciz=Siz¯Z,

    where,

    θi=(1mi1Mi),λ=(1n1N).

    In this section, we consider several estimators of the finite population mean under two-stage sampling that are available in the sampling literature, the properties of all estimators considered here are obtained up-to the first order of approximation.

    (ⅰ) The usual mean estimator ¯y=¯y0 and its variance under two-stage sampling are given by:

    ¯y0=1nni=1ui¯yi, (1)

    and

    V(¯y0)=¯Y2Vy=MSE(¯y0). (2)

    (ⅱ) The usual ratio estimator under two-stage sampling, is given by:

    ¯yR=¯y(¯X¯x), (3)

    where ¯X is the known population mean of x.

    The bias and MSE of ¯yR to first order of approximation, are given by:

    Bias(¯yR)=¯Y[VxVyx], (4)

    and

    MSE(¯yR)=¯Y2[Vy+Vx2Vyx]. (5)

    (ⅲ) [2] Exponential ratio type estimator under two-stage sampling, is given by:

    ¯yE=¯yexp(¯X¯x¯X+¯x). (6)

    The bias and MSE of ¯yE to first order of approximation, are given by:

    Bias(¯yE)=¯Y[38Vx12Vyx], (7)

    and

    MSE(¯yE)=¯Y2[Vy+14VxVyx]. (8)

    (ⅳ) The traditional difference estimator under two-stage sampling is given by:

    ¯yD=¯y+d(¯X¯x), (9)

    where d is the constant.

    The minimum variance of ¯yD, is given by:

    V(¯yDmin)=¯Y2Vy(1ρ2)=MSE(¯yD), (10)

    where ρ=VyxVyVx.

    The optimum value of d is dopt=¯YVyx¯XVx.

    (ⅴ) [7] Difference type estimator under two-stage sampling, is given by:

    ¯yRao=d0¯y+d1(¯X¯x), (11)

    where d0 and d1 are constants.

    The bias and minimum MSE of ¯yRao to first order of approximation, is given by:

    Bias(¯yRao)=(d01)¯Y, (12)

    and

    MSE(¯yRao)min¯Y2(VxVyV2yx)VxVyV2yx+Vx=¯Y2Vy(1ρ2)1+Vy(1ρ2). (13)

    The optimum values of d0 and d1 are:

    d0opt=VxVxVyV2yx+Vx and d1opt=¯YVyx¯X(VxVyV2yx+Vx).

    (ⅵ) The difference-in-ratio type estimator under two-stage sampling, is given by:

    ¯yDR=[d2¯y+d3(¯X¯x)](¯X¯x), (14)

    where d2 and d3 are constants.

    The bias and minimum MSE of ¯yDR to first order of approximation, are given by:

    Bias(¯yDR)¯Y(d21)d2¯YVyx+d3¯XVx+d2Vx¯Y, (15)

    and

    MSE(¯yDR)min¯Y2(V2xVyVxV2yxVxVy+V2yx)(V2xVxVy+V2yxVx). (16)

    The optimum values of d2 and d3 are:

    d2opt=Vx(Vx1)V2xVxVy+V2yxVx,
    d3opt=¯Y2(V2x+VxVyVxVyxV2yxVx+Vyx)¯X(V2xVxVy+V2yxVx).

    (ⅶ) The difference-in-exponential ratio type estimator under two-stage sampling, is given by:

    ¯yDE=[d4¯y+d5(¯X¯x)]exp(¯X¯x¯X¯x), (17)

    where d4 and d5 are constants.

    The bias and minimum MSE of ¯yDE to first order of approximation, are given by:

    Bias(¯yDE)=(d41)¯Y12d4¯YVyx+38d4¯YVx+12d5Vx, (18)

    and

    MSE(¯yDE)min¯Y2(V3x+16V2xVy16VxV2yx64VxVy+64V2yx)64VxVy64V2yx+64Vx. (19)

    The optimum values of d4 and d5 are:

    d4opt=18Vx(Vx8)(VxVyV2yx+Vx),
    d5opt=¯Y(V2x+4VxVyVxVyx4V2yx4Vx+8Vyx)18¯X(VxVyV2yx+Vx).

    (ⅷ) The difference-difference type estimator under two stage sampling, is given by:

    ¯yDD=¯y+d6(¯X¯x)+d7(¯Z¯z), (20)

    where d6 and d7 are constants.

    The minimum variance or MSE of ¯yDD to first order of approximation, is given by:

    MSE(¯yDD)min¯Y2(VxVyVzVxV2yzV2xzVy+2VxzVyxVyzV2yxVz)VxVzV2xz. (21)

    The optimum values of d6 and d7 are:

    d6=¯Y(VxzVyzVyxVz)¯X(VxVzV2xz),
    d7=¯Y(VxVyzVxzVyx)¯Z(VxVzV2xz).

    (ix) The difference-difference type estimator under two stage sampling, is given by:

    ¯yDD(R)=d8¯y+d9(¯X¯x)+d10(¯Z¯z), (22)

    where d8, d9 and d10 are constants.

    The bias and MSE of ¯yDD(R) to first order of approximation is given by:

    Bias(¯yDD(R))=¯Y(d81), (23)

    and

    MSE(¯yDD(R))¯Y2(VxVyVzVxV2yzV2xzVy+2VxzVyxVyzV2yxVz)VxVyVzVxV2yzV2xzVy+2VxzVyxVyzV2yxVz+VxVzV2xz. (24)

    The optimum values of d8, d9 and d10 are given by:

    d8=VxVzV2xzVxVyVzVxV2yzV2xzVy+2VxzVyxVyzV2yxVz+VxVzV2xz,
    d9=¯Y(VxzVyzVyxVz)¯X(VxVyVzVxV2yzV2xzVy+2VxzVyxVyzV2yxVz+VxVzV2xz),
    d10=¯Y(VxVyzVxzVyx)¯Z(VxVyVzVxV2yzV2xzVy+2VxzVyxVyzV2yxVz+VxVzV2xz).

    The principal advantage of our proposed improved generalized class of estimators under two-stage sampling is that it is more flexible, efficient, than the existing estimators. The mean square errors based on two data sets are minimum and percentage relative efficiency is more than hundred as compared to the existing estimators considered here. We identified 11 estimators as members of the proposed class of estimators by substituting the different values of wi(i=1,2,3), δ and γ. On the lines of [2,7], we propose the following generalized improved class of estimators under two stage sampling for estimation of finite population mean using two auxiliary varaible as given by:

    ¯yG=[w1¯y+w2(¯X¯x)+w3(¯Z¯z)][{expδ(¯X¯x¯X+¯x)}(¯X¯x)γ], (25)

    where wi(i=1,2,3) are constants, whose values are to be determined; δ and γ are constants i.e., (0δ, γ1) and can be used to construct the different estimators.

    Using (25), solving ¯yG in terms of errors, we have

    ¯yG¯Y=(w11)¯Y+w1¯Y{e012α1e1+18α2e2112α1e0e1}
    w2¯X{e112α1e21}w3¯Z{e212α1e1e2},

    where

    α1=δ+2γ and α2=δ(δ+2)+4γ(δ+γ+1).

    The bias and MSE of ¯yG are given by:

    Bias(¯yG)(w11)¯Y+w1¯Y{18α2Vx12α1Vyx}+w2¯Xα1Vx2+w3¯Zα1Vxz2, (26)

    and

    MSE(¯yG)(w11)2+w21¯Y2A+w22¯X2B+w23¯Z2Cw1¯Y2Dw2¯Y¯XE
    w3¯Y¯ZF+2w1w2¯Y¯XG+2w1w3¯Y¯ZH+2w2w3¯X¯ZI, (27)

    where

    A=Vy+14Vx(α21+α2)2α1Vyx,B=Vx,C=Vz,
    D=14α2Vxα1Vyx,E=α1Vx,F=α1Vxz,
    G=α1VxVyx,H=α1VxzVxz,I=Vxz.

    Solving (27), the minimum MSE of ¯yG to first order of approximation are given by:

    MSE(¯yG)min=¯Y2[1Ω24Ω1], (28)

    where

    Ω1=ABCAI2BH2CG2+2GHI+BCI2,

    and

    Ω2=ABF2+ACE22AEFI+BCD22BDFH2CDEGD2I2+2DEHI
    +2DFGIE2H2+2EFGHF2G2+4BCD+BF24BFH+CE2
    4CEG4DI22EFI+4EHI+4FGI+4BC+4I2.

    The optimum values of wi(i=1,2,3) are given by:

    w1opt=Ω32Ω1,w2opt=¯YΩ42¯XΩ1, and w3opt=¯YΩ52¯ZΩ1,

    where

    Ω3=BCDBFHCEGDI2+EHI+FGI+2GI+2BC2I2,
    Ω4=ACEAFICDG+DHIEH2+FGH+CE2CGFI+2HI,
    Ω5=ABFAEIBDH+DGI+EGHFG2+BF2BHEI+2GI.

    From (28), we produce the following two estimators called ¯yG1 and ¯yG2. Put (δ=0,γ=1) and (δ=1,γ=0) in (25), we get the following two estimators respectively:

    (i)¯yG1=[w4¯y+w5(¯X¯x)+w6(¯Z¯z)](¯X¯x),
    (ii)¯yG2=[w7¯y+w8(¯X¯x)+w9(¯Z¯z)]exp(¯X¯x¯X+¯x),

    where wi(i=4,5,6,7,8,9) are constants. Solving ¯yG1, in terms of errors, we have:

    (¯yG1¯Y)=[¯Y+w4¯Y+w4¯Yeow5¯Xe1w6¯Ze2].[112e1+38e21],
    (¯yG1¯Y)=[w4¯Yw4¯Ye1+38w4¯Ye21+w4¯Ye012w4¯Ye0e1¯Y+12¯Ye138¯Ye21w5¯Xe1+12w5¯Xe21w6¯Ze2+12w6¯Ze1e2]. (29)

    The bias and MSE of ¯yG1, to first order of approximation is given by:

    Bias(¯yG1)=38w4¯YV2x12w4¯YVyx38¯YV2x+12w5¯XV2x+12w6¯ZVxz,

    By squaring and taking expectation of (29), we get the mean square error:

    MSE(¯yG1)=[w26V2z+w24V2y+w25V2x2w24RVyx+2w4RVyx2Rw5V2x2w4R2V2x+w24RV2x+2w4Rw5V2x2Rw6Vxz+w24¯Y22w4¯Y2+¯Y2+R2V2x2w4w5Vyx2w4w6Vyz+2w5w6Vxz+2w4Rw6Vxz]. (30)

    Differentiate (30) with respect to w4, w5 and w6, we get the optimum values of w4, w5 and w6 i.e.,

    w4opt=¯Y2(V2xV2zV2xz)[R2V4xV2z+RV4xV2z+¯Y2V2xV2z+R2V2xV2xz+V2xV2yV2zRV2xV2xz¯Y2V2xzV2xV2yzV2xzV2yV2yxV2z+2VxzVyxVyz],
    w5opt=[R3V4xV2zR2V4xV2zR3V2xV2xzRV2xV2yV2z+R2V2xV2xz¯Y2VyxV2z+RV2xV2yz+RV2xzV2y+RV2yxV2z+¯Y2VxzVyz2RVxzVyxVyz][R2V4xV2z+RV2xV2z+¯Y2V2xV2z+R2V2xV2xz+V2xV2yV2zRV2xV2xz¯Y2V2xzV2xV2yzV2xzV2yV2yxVz+2VxzVyxVyz],
    w6opt=¯Y2(V2xVyzVxzVyx)[R2V4xV2z+RV4xV2z+¯Y2V2xV2z+R2V2xV2xz+V2xV2yV2zRV2xV2xz¯Y2V2xzV2xV2yzV2xzV2yV2yxV2z+2VxzVyxVyz].

    Substituting the optimum values of w4, w5 and w6 in (30), we get the minimum mean square error of ¯yG1, given by:

    MSE(¯yG1)min=¯Y2[R2V4xV2zRV4xV2zR2V2xV2xzV2xV2yV2z+RV2xV2xz+V2xV2yz+V2xzV2y+V2yxV2z2VxzVyxVyz][R2V4xV2zRV4xV2z¯Y2V2xV2zR2V2xV2xzV2xV2yV2z+RV2xV2xz+¯Y2V2xz+V2xV2yz+V2xzV2y+V2yxV2z2VxzVyxVyz]. (31)

    Solving ¯yG2, in terms of errors, we have

    (¯yG2¯Y)=[w7¯Y+w7¯Ye0¯Yw8¯Xe1w9¯Ze2](1e1+e21),

    or

    (¯yG2¯Y)=[w7¯Y+w7¯Ye0¯Yw8¯Xe1w9¯ze2w7¯Ye1w7¯Ye1+¯Ye1+w8¯Xe21w9¯Ze1e2+w7¯Ye21¯Ye21]. (32)

    The Bias and MSE ¯yG2, to first order of approximation, is given by:

    Bias(¯yG2) = w8¯XV2xw9¯ZVxz+w7¯YV2x¯YV2x.

    By squaring and taking expectation of (32), we get the mean square error:

    MSE(¯yG2)=4Rw7Vyx4RV2xw8+¯Y22w7¯Y2+3R2V2x6w7R2V2x+4w7w9RVxz
    2w7w8Vyx2w7¯Yw9Vyz+2w8w9Vxz4Rw9Vxz4w27RVyx+3w37R2V2x
    +w29V2z+w28V2x+4w7w8RV2x+w27¯Y2. (33)

    Differentiate (33) with respect to w7, w8 and w9, we get the optimum values of w7, w8 and w9 i.e.,

    w7opt=(V2xV2zV2xz)(R2V2x+¯Y2)R2V4xV2z+¯Y2V2xV2yz¯Y2V2xV2z+¯Y2V2xz2¯YVxzVyxVyz+V2yxV2z,
    w8opt=[2¯Y2RV2xV2yz¯Y2R2V2xVxzVyz+R2V2xVyxV2z+¯Y3VxzVyz¯Y2VyzV2z4¯YRVxzVyxVyz+2RV2yxVz][R2V4xV2z+¯Y2V2xV2yz¯Y2V2xV2zR2V2xV2xz+¯Y2V2xz2¯YVxzVyxVyz+V2yxV2z],
    w9opt=¯YR2V4xVyz+¯Y3V2xVyz+R2V2xVxzVyx¯Y2VxzVyxR2V4xV2z+¯Y2V2xV2yz¯Y2V2xV2zR2V2xV2xz+¯Y2V2xz2¯YVxzVyxVyz+V2yxV2z.

    Substituting the optimum values of w7, w8 and w9 in (33), we get the minimum mean square error of ¯yG2, given by:

    MSE(¯yG2)min=(R2V2x¯Y2)(¯Y2V2xV2yz+V2yxV2z2¯YVxzVyxVyz)[¯Y2(V2xV2yzV2xV2z+V2xz)+2¯Y2VxzVyxVyzR2V4xV2z+R2V2xV2xzV2yxV2z]. (34)

    We can generate the considered and many more estimators from (25), by substituting the different values of wi(i=1,2,3), δ and γ, given in Table 1.

    Table 1.  Members of the proposed generalized family of estimators.
    w1 w2 w3 σ γ Estimator
    1 0 0 0 0 ¯y
    1 0 0 0 1 ¯yR
    1 0 0 1 0 ¯yE
    1 d 0 0 0 ¯yD
    d0 d1 0 0 0 ¯yRao
    d2 d3 0 0 1 ¯yDR
    d4 d5 0 1 0 ¯yDE
    0 d6 d7 0 0 ¯yDD
    d8 d9 d10 0 0 ¯yDD(R)
    w4 w5 w6 0 1 ¯yG1
    w7 w8 w9 1 0 ¯yG2

     | Show Table
    DownLoad: CSV

    Population 1. [Source: [11], Model Assisted Survey Sampling]

    There are 124 countries (second stage units) divided into 7 continents (first stage units) according to locations. Continent 7th consists of only one country therefore, we placed 7th continent in 6th continent.

    We considered:

    y = 1983 import (in millions U.S dollars),

    x = 1983 export (in millions U.S dollars),

    z = 1982 gross national product (in tens of millions of U.S dollars).

    The data are divided into 6 clusters, having N=6, and n=3. Also Ni=1Mi=124, ¯M=20.67. In Table 2, we show cluster sizes, and population means of the study variable (y) and the auxiliary variables (x,z). Tables 3 and 4 give some results.

    Table 2.  Cluster sizes with population means.
    No. of clusters Mi mi ui ¯Yi ¯Xi ¯Zi
    1 38 15 1.8387 2254.6 1901.1 1029.158
    2 14 6 0.6774 25533.14 22083.21 25671.57
    3 11 4 0.5323 3602.82 5835.455 5028.818
    4 33 13 1.5968 12156.79 12438.85 7533.939
    5 24 10 1.1613 34226.79 33198 16314.42
    6 4 2 0.1936 26392.5 29360.5 43967.75

     | Show Table
    DownLoad: CSV
    Table 3.  Statistical computation.
    (ui¯Yi¯Y)2 (ui¯Xi¯X)2 (ui¯Zi¯Z)2 (ui¯Yi¯Y)
    (ui¯Xi¯X)
    (ui¯Xi¯X)
    (ui¯Zi¯Z)
    (ui¯Yi¯Y)
    (ui¯Zi¯Z)
    109395354.7 116233397.3 69704363.4 112762554.6 90010971.3 87323155.9
    7243544.5 465731.6 51103837.3 1836722 4878593.3 19239878.4
    160959577.1 124780258.7 57219949.1 141720068 84498047.6 95969259.7
    23109139.4 31199313.33 3200403.1 26851243.6 9992516.3 8599916.4
    632160672.8 589329821.2 75771962.7 610369671.8 211316533.3 218860811.8
    90158398.4 73831543.1 2989684.1 81587582.8 14857085.7 16417829.8

     | Show Table
    DownLoad: CSV
    Table 4.  Statistical computations of variances and covariances.
    S2iy S2ix S2iz Sixy Sixz Siyz
    14634002.89 13229390.42 3667896.461 12035361.66 5676138.848 7031654.09
    5199331742 3024354709 6568461403 3920918987 42963119811 5785526585
    17474303.56 67544530.07 63348742.76 3322301379 62246450.49 32019714.86
    510689624 689903319 440717912.5 586829812.3 522773788 447429378.1
    1530618991 1588803380 408376223 1544450491 757258674.4 7559765056
    1361248223 1782024492 5663081987 1557362451 3157897870 2755900798

     | Show Table
    DownLoad: CSV
    S2by=204605337.7,S2bx=187168013,S2bz=51998039.99,
    Sbyx=195025568.6,Sbyz=89282170.45,Sbxz=83110749.52
    Vy=0.27028,Vx=0.25137,Vz=0.30933,
    Vyx=0.25723,Vyz=0.24573,Vxz=0.22493.
    ¯Y=14604.76564,¯X=14276.72113,¯Z=10241.22672.

    Population 2. [Source: [11], Model Assisted Survey Sampling]

    Similarly we considered the data as mentioned in Population 1,

    y = 1983 import (in millions U.S dollars),

    x = 1981 military expenditure (in tens of millions U.S dollars),

    z = 1980 population (in millions).

    The data are divided into 6 clusters having N=6, n=3, Ni=1Mi=124,¯M=20.67.

    In Table 5, we show cluster sizes, and means of the study variable (y) and the auxiliary variables (x,z). Tables 6 and 7 give some computation results.

    Table 5.  Cluster sizes with population means.
    No. of clusters Mi mi ui ¯Yi ¯Xi ¯Zi
    1 38 15 1.8387 13.03684 418.3421 11.88421
    2 14 6 0.6774 27.35 10065.21 26.1857
    3 11 4 0.5323 23.13636 484.45 21.8818
    4 33 13 1.5968 79.65455 3377.75 75.2424
    5 24 10 1.1613 20.28333 4929.41 20.9583
    6 4 2 0.1936 74.15 30676.25 70.975

     | Show Table
    DownLoad: CSV
    Table 6.  Statistical computation.
    (ui¯Yi¯Y)2 (ui¯Xi¯X)2 (ui¯Zi¯Z)2 (ui¯Yi¯Y)(ui¯Xi¯X) (ui¯Xi¯X)(ui¯Zi¯Z) (ui¯Yi¯Y)(ui¯Zi¯Z)
    109395354.7 11504653.07 171.5489 35430628.04 44434.42674 136842.964
    7243544.5 7165707.84 296.6622 −461113.24893 7355674.750 −47336.072
    160959577.1 15247937.12 544.5471 49517362.59 91132.5364 295949.600
    23109139.4 1556060.53 7312.6397 6074534.22 106667.8685 416415.347
    632160672.8 2494694.19 112.4143 39914472.69 −16750.3943 −268005.273
    90158398.4 3214861.18 451.071 −16997583.03 −38085.13867 201365.558

     | Show Table
    DownLoad: CSV
    Table 7.  Statistical computations of variances and covariances.
    S2iy S2ix S2iz Siyx Sixz Siyz
    270.9083357 594166.8257 222.4889331 6380.484353 5806.286629 245.4143812
    3906.928847 1281691972 3683.070549 2135135.281 2082979.098 3792.853077
    1339.404545 461472.2727 1174.031636 13075.32182 12298.74909 1253.851727
    45082.17318 53848774.81 83850.37836 1082424.717 1476243.493 43109.42511
    368.9423188 52672480.78 364.7838949 117010.6551 116939.2322 366.7860145
    18401.07 3453923758 16855.5025 7970505.317 7628400.308 17611.33833

     | Show Table
    DownLoad: CSV
    S2by=2002.428957,S2bx=8236782.79,S2bz=1782.2076,
    Sbyx=28451.30273,Sbyz=1888.920758,Sxz=27939.79,
    Vy=0.48633,Vx=0.39654,Vz=0.72000,
    Vyx=0.14250,Vyz=0.48726,Vxz=0.16552,
    ¯Y=36.7702,¯X=4163.56,¯Z=34.8552.

    The results based on Tables 27 are given in Tables 8 and 9 having biasses, mean square errors, and percentage relative efficiencies of the poposed and exisitng estimators w.r.t ¯y0. Tables 8 and 9 show that the proposed estimators perform well as compared to the existing estimators considered here.

    Table 8.  Biases of different estimators in both data sets.
    Estimator Population 1 Population 2
    ¯y0, ¯yD, ¯yDD 0 0
    ¯yR −85.58393 9.341102
    ¯yE −501.692 2.847944
    ¯yRao −102.2916 −11.14854
    ¯yDR 62517.58 −5.729291
    ¯yDE −1040.227 −4.254352
    ¯yDD(R) −8687.674 0.8665627
    ¯yG1 −911.2082 42.56601
    ¯yG2 4097419 −660.0231

     | Show Table
    DownLoad: CSV
    Table 9.  MSE and PRE of different estimators w.r.t ¯y0.
    Population 1 Population 2
    Estimators MSE PRE MSE PRE
    ¯y0 57649726.19 100 657.541 100
    ¯yR 1532261.42 3762.39 808.353 81.3433
    ¯yE 16186983.44 356.149 598.912 109.789
    ¯yD 1503085.83 3835.42 588.307 111.768
    ¯yRao 1492567.93 3862.45 409.935 160.401
    ¯yDR 1489069.32 3871.53 341.831 192.358
    ¯yDE 1189664.93 4845.88 366.982 179.176
    ¯yDD 1025752.55 5620.24 208.189 315.839
    ¯yDD(R) 1020843.33 5647.26 180.409 364.472
    ¯yG1 1019205.50 5656.34 165.866 396.429
    ¯yG2 747118.42 7716.28 159.646 411.875

     | Show Table
    DownLoad: CSV

    The following expression is used to obtain the Percent Relative Efficiency (PRE), i.e.,

    PRE=MSE(¯y0)MSE(¯yi)×100,

    where i=0,R,E,D,Rao,DR,DE,DD,DD(R),G1,G2.

    As mentioned above, we used two real data sets to obtain the biases, MSEs or variances and PREs of all estimators under two-stage sampling scheme when using two auxiliary variables. In Tables 24 and Tables 57, we present the summary statistic of both population. From Tables 8 and 9, we observed that the proposed class of estimators ¯yG1 and ¯yG2 are more precise than the existing estimators ¯y0, ¯yR, ¯yE, ¯yD, ¯yRao, ¯yDR, ¯yDE, ¯yDD, ¯yDD(R) in terms of MSEs and PREs. It is clear that the proposed improved generalized class of estimators, i.e., performs better than the estimators. As we increase the sample size the mean square error values decreases, and percentage relative efficiency give best results, which are the expected results.

    In this manuscript, we proposed a generalized class of estimators using two auxiliary variables under two-stage sampling for estimating the finite population mean. In addition, some well-known estimators of population mean like traditional unbiased estimator, usual ratio, exponential ratio type, traditional difference type, Rao difference type, difference-in- ratio type, difference-in-exponential ratio type, difference-in-difference, difference-difference ratio type estimator are created to be members of our suggested improved generalized class of estimators. Expression for the biases and mean squared error have been generated up to the first order of approximation. We identified 11 estimators as members of the proposed class of estimators by substituting the different values of wi(i=1,2,3), δ and γ. Both generalized class of estimators ¯yG1 and ¯yG2 perform better as compared to all other considered estimators, although ¯yG2 is the best. In Population 2, the performance of ratio estimator (¯yR) is weak. The gain in Population 1 is more as compared to Population 2.

    The authors are thankful to the Editor-in-Chief and two anonymous referees for their careful reading of the paper and valuable comments which leads to a significant improvement in article.

    The authors declare no conflict of interest.



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