Loading [MathJax]/jax/element/mml/optable/BasicLatin.js
Research article Topical Sections

Improved estimation of population parameter of in the existence of nonresponse using auxiliary information

  • The issue of survey nonresponse causes substantial effects on the reliability, together with validity of study findings. This research develops precise estimation methods for two distinct nonresponse situations: when nonresponse happens to primary survey variables like the number of students, and when nonresponse includes survey variables together with the accompanying auxiliary variables. The proposed estimators receive full a performance assessment through derived equations for bias and mean square error (MSE) based on first-order approximation. The accuracy and reliability assessment depends heavily on MSE calculations, since this method effectively merges systematic error measurements with random error measurements. The MSE values of different estimators receive numerical evaluation through a comparative analysis under equivalent operational conditions for assessing proposed estimator performance outcomes. The research seeks to find the estimator with the minimal MSE because this selection results in the most trustworthy estimates under nonresponse conditions. All the findings from this study create important guidelines for building educational survey designs.

    Citation: Badr Aloraini. Improved estimation of population parameter of in the existence of nonresponse using auxiliary information[J]. AIMS Mathematics, 2025, 10(5): 12312-12342. doi: 10.3934/math.2025558

    Related Papers:

    [1] Yue Feng, Yujie Liu, Ruishu Wang, Shangyou Zhang . A conforming discontinuous Galerkin finite element method on rectangular partitions. Electronic Research Archive, 2021, 29(3): 2375-2389. doi: 10.3934/era.2020120
    [2] Chunmei Wang . Simplified weak Galerkin finite element methods for biharmonic equations on non-convex polytopal meshes. Electronic Research Archive, 2025, 33(3): 1523-1540. doi: 10.3934/era.2025072
    [3] Leilei Wei, Xiaojing Wei, Bo Tang . Numerical analysis of variable-order fractional KdV-Burgers-Kuramoto equation. Electronic Research Archive, 2022, 30(4): 1263-1281. doi: 10.3934/era.2022066
    [4] Guanrong Li, Yanping Chen, Yunqing Huang . A hybridized weak Galerkin finite element scheme for general second-order elliptic problems. Electronic Research Archive, 2020, 28(2): 821-836. doi: 10.3934/era.2020042
    [5] Victor Ginting . An adjoint-based a posteriori analysis of numerical approximation of Richards equation. Electronic Research Archive, 2021, 29(5): 3405-3427. doi: 10.3934/era.2021045
    [6] Jun Pan, Yuelong Tang . Two-grid H1-Galerkin mixed finite elements combined with L1 scheme for nonlinear time fractional parabolic equations. Electronic Research Archive, 2023, 31(12): 7207-7223. doi: 10.3934/era.2023365
    [7] Hongze Zhu, Chenguang Zhou, Nana Sun . A weak Galerkin method for nonlinear stochastic parabolic partial differential equations with additive noise. Electronic Research Archive, 2022, 30(6): 2321-2334. doi: 10.3934/era.2022118
    [8] Zexuan Liu, Zhiyuan Sun, Jerry Zhijian Yang . A numerical study of superconvergence of the discontinuous Galerkin method by patch reconstruction. Electronic Research Archive, 2020, 28(4): 1487-1501. doi: 10.3934/era.2020078
    [9] Suayip Toprakseven, Seza Dinibutun . A weak Galerkin finite element method for parabolic singularly perturbed convection-diffusion equations on layer-adapted meshes. Electronic Research Archive, 2024, 32(8): 5033-5066. doi: 10.3934/era.2024232
    [10] Bin Wang, Lin Mu . Viscosity robust weak Galerkin finite element methods for Stokes problems. Electronic Research Archive, 2021, 29(1): 1881-1895. doi: 10.3934/era.2020096
  • The issue of survey nonresponse causes substantial effects on the reliability, together with validity of study findings. This research develops precise estimation methods for two distinct nonresponse situations: when nonresponse happens to primary survey variables like the number of students, and when nonresponse includes survey variables together with the accompanying auxiliary variables. The proposed estimators receive full a performance assessment through derived equations for bias and mean square error (MSE) based on first-order approximation. The accuracy and reliability assessment depends heavily on MSE calculations, since this method effectively merges systematic error measurements with random error measurements. The MSE values of different estimators receive numerical evaluation through a comparative analysis under equivalent operational conditions for assessing proposed estimator performance outcomes. The research seeks to find the estimator with the minimal MSE because this selection results in the most trustworthy estimates under nonresponse conditions. All the findings from this study create important guidelines for building educational survey designs.



    For simplicity, we consider Poisson equation with a Dirichlet boundary condition as our model problem.

    Δu=f,inΩ, (1)
    u=g,onΩ, (2)

    where Ω is a bounded polygonal domain in R2.

    Using integration by parts, we can get the variational form: find uH1(Ω) satisfying u=gonΩ and

    (u,v)=(f,v),vH10(Ω). (3)

    Various finite element methods have been introduced to solve the Poisson equations (1)-(2), such as the Galerkin finite element methods (FEMs)[2, 3], the mixed FEMs [15] and the finite volume methods (FVMs) [6], etc. The FVMs emphasis on the local conservation property and discretize equations by asking the solution satisfying the flux conservation on a dual mesh consisting of control volumes. The mixed FEMs is another category method that based on the variable u and a flux variable usually written as p.

    The classical conforming finite element method obtains numerical approximate results by constructing a finite-dimensional subspace of H10(Ω). The finite element scheme has the same form with the variational form (3): find uhVhH1(Ω) satisfying uh=IhgonΩ and

    (uh,vh)=(f,vh),vhV0h, (4)

    where V0h is a subspace of Vh that satisfying vh=0 on Ω and Ih is the kth order Lagrange interpolation operator. The FE method is a popular and easy-to-implement numerical scheme, however, it is less flexible in constructing elements and generating meshes. These limitations are mainly due to the strong continuity requirements of functions in Vh. One solution to circumvent these limitations is using discontinuous approximations. Since the 1970th, many new finite element methods with discontinuous approximations have been developed, including the early proposed DG methods [1], local discontinuous Galerkin (LDG) methods [8], interior penalty discontinuous Galerkin (IPDG) methods [9], and the recently developed hybridizable discontinuous Galerkin (HDG) methods [7], mimetic finite differences method [10], virtual element (VE) method [4], weak Galerkin (WG) method [19, 20] and references therein.

    One obvious disadvantage of discontinuous finite element methods is their rather complex formulations which are often necessary to ensure connections of discontinuous solutions across element boundaries. For example, the IPDG methods add parameter depending interior penalty terms. Besides additional programming complexity, one often has difficulties in finding optimal values for the penalty parameters and corresponding efficient solvers. Most recently, Zhang and Ye [21] developed a discontinuous finite element method that has an ultra simple weak formulation on triangular/tetrahedal meshes. The corresponding numerical scheme can be written as: find uh˜Vh satisfying uh=IhgonΩ and

    (wuh,wvh)=(f,vh),vhV0h, (5)

    where ˜Vh is the DG finite element space and w is the weak gradient operator. The notion of weak gradient was first introduced by Wang and Ye in the weak Galerkin (WG) methods [19, 20]. The WG methods allow the use of totally discontinuous functions and provides stable numerical schemes that are parameter-independent and free of locking [17] in some applications. Another key feature in the WG methods is it can be used for arbitrary polygonal meshes. The WG finite element method has been rapidly developed and applied to other problems, including the Stokes and Navier-Stokes equations [11, 18], the biharmonic [14, 13] and elasticity equations [12, 17], div-curl systems and the Maxwell's equations and parabolic problem [23], etc. The introduction of the weak gradient operator in the conforming DG methods makes the scheme (5) maintain the simple formulation of conforming finite element method while have the flexibility of using discontinuous approximations. Hence, the programming complexity of this conforming DG scheme is significantly reduced. Furthermore, the scheme results in a simple symmetric and positive definite system.

    Following the work in [21, 22], we propose a new conforming DG finite element method on rectangular partitions in this work. It can be obtained from the conforming formulation simply by replacing by w and enforcing the boundary condition strongly. The simplicity of the conforming DG formulation will ease the complexity for implementation of DG methods. We note that the conforming DG method in [21] is based on triangular/tetrahedal meshes. Then in [22], the method is extended to work on general polytopal meshes by raising the degree of polynomials used to compute weak gradient.

    In this paper, we keep the same finite element space as DG method, replace the boundary function with the average of the inner function, and use the weak gradient arising from local Raviart-Thomas (RT) elements [5] to approximate the classic gradient. Moreover, the derivation process in this paper is based on rectangular RT elements [16]. Error estimates of optimal order are established for the corresponding conforming DG approximation in both a discrete H1 norm and the L2 norm. Numerical verifications have been performed on different kinds of quadrangle finite element space. In particular, super-convergence phenomenon have been observed for Q0 elements.

    The rest of this paper is organized as follows: In Section 2, we shall present the conforming DG finite element scheme for the Poisson equation on rectangular partitions. Section 3 is devoted to a discussion of the stability and solvability of the new method. In Section 4, we shall prepare ourselves for error estimates by deriving some identities. Error estimates of optimal order in H1 and L2 norm are established in Section 5. In Section 6, we present some numerical results to illustrate the theory derived in earlier sections. Finally in section 7, we conclude our major contributions in this article.

    Throughout this paper, we adopt the standard definition of Sobolev space Hs(Ω). For any given open bounded domain KΩ, (,)s,K,s,K, and ||s,K are used to denote the inner product, norm and semi-norm, respectively. The space H0(K) coincides with L2(K), and the subscripts K in the inner product, norm, and semi-norm can be dropped in the case of K=Ω. In particular, the function space H10(Ω) is defined as

    H10(Ω)={vH1(Ω):v|Ω=0},

    and the space H(div,Ω) is defined as the set of vector-valued functions q, which together with their divergence are square integrable, i.e.

    H(div,Ω)={q[L2(Ω)]d:qL2(Ω)}.

    Assume that the domain Ω is of polygonal type and is partitioned into non-overlapping rectangles Th={T}. For each TTh, denote by T0 its interior and T its boundary. Denote by Eh={e} the set of all edges in Th, and E0h=EhΩ the set of all interior edges in Th. For each TTh and eEh, denote by hT and he the diameter of T and e, respectively. h=max is the meshsize of \mathcal{T}_h .

    For any interior edge e\in\mathcal{E}^0_h , let T_1 and T_2 be two rectangles sharing e , we define the average \{\cdot\} and the jump [\![ \cdot]\! ] on e for a scalar-valued function v by

    \begin{eqnarray} \{v\}& = &\frac{1}{2}(v|_{\partial T_1}+v|_{\partial T_2}),\; \; [\![ v]\! ] = v|_{\partial T_1}\boldsymbol{n}_1+v|_{\partial T_2}\boldsymbol{n}_2, \end{eqnarray} (6)

    where v|_{\partial T_{i}},i = 1,2 is the trace of v on \partial T_i , \boldsymbol{n</italic>}_1 and \boldsymbol{n</italic>}_2 are the two unit outward normal vectors on e , associated with T_1 and T_2 , respectively. If e is a boundary edge, we define

    \begin{eqnarray} \{v\}& = &v|_e \; \; \text{and}\; \; [\![ v]\! ] = v|_e\boldsymbol{n}. \end{eqnarray} (7)

    We define a discontinuous finite element space

    \begin{eqnarray} V_h = \{v\in L^2(\Omega): v|_T \in Q_k(T), \; \forall \; T\in \mathcal{T}_h\}, \end{eqnarray} (8)

    and its subspace

    \begin{eqnarray} V^0_h = \{v \in V_h:v = 0 \; \text{on} \; \partial \Omega\}, \end{eqnarray} (9)

    where Q_k(T), \;k \geq 1 denotes the set of polynomials with regard to quadrilateral elements. The weak gradient for a scalar-valued function v\in V_h is defined by the following definition

    Definition 2.1. For a given T \in \mathcal{T}_h and a function v\in V_h , the discrete weak gradient \nabla_d v \in RT_k(T) on T is defined as the unique polynomial such that

    \begin{eqnarray} (\nabla_{d}v,\boldsymbol{q})_T: = -(v,\nabla\cdot\boldsymbol{q})_T +\langle \{v\},\boldsymbol{q}\cdot \boldsymbol{n}\rangle_{\partial T},\; \; \; \; \forall\; \boldsymbol{q}\in RT_k(T), \end{eqnarray} (10)

    where \boldsymbol{n} is the unit outward normal on \partial T , RT_k(T) = [Q_k(T)]^2+\bf{x}Q_k(T) , and \{v\} is defined in (6) and (7).

    The weak gradient operator \nabla_{d} as defined in (10) is a local operator computed at each element. It can be extended to any function v \in V_h by taking weak gradient locally on each element T. More precisely, the weak gradient of any v \in V_h is defined element-by-element as follows:

    \begin{eqnarray*} (\nabla_d v)|_T = \nabla_{d}(v|_T). \end{eqnarray*}

    We introduce the following bilinear form:

    \begin{eqnarray*} a(v,w) = (\nabla_d v, \nabla_d w), \end{eqnarray*}

    the conforming DG algorithm to solve the problems (1) - (2) is given by

    Conforming DG algorithm 1. Find u_h\in V_h satisfying u_h = I_hg \; \mathit{\text{on}} \; \partial \Omega and

    \begin{eqnarray} a(u_h, v_h) = (f, v_h), \; \; \forall \; v_h\in V^0_h, \end{eqnarray} (11)

    where I_h is the kth order Lagrange interpolation.

    We will prove the existence and uniqueness of the solution of equation (11). Firstly, we present the following two useful inequalities to derive the forthcoming analysis.

    Lemma 3.1 (trace inequality). Let T be an element of the finite element partition \mathcal{T}_h , and e is an edge or face which is part of \partial T . For any function \varphi\in H^1(T) , the following trace inequality holds true (see [20] for details):

    \begin{eqnarray} \|\varphi\|_e^2\leq C(h_T^{-1}\|\varphi\|_T^2+h_T\|\nabla\varphi\|_T^2), \end{eqnarray} (12)

    where C is a constant independent of h .

    Lemma 3.2 (inverse inequality). Let \mathcal{T}_h be a finite element partition of \Omega that is shape regualr. Assume that \mathcal{T}_h satisfies all the assumptions A1-A4 in [20]. Then, for any piecewise polynomial function \varphi of degree n on \mathcal{T}_h , there exists a constant C = C(n) such that

    \begin{eqnarray} \|\nabla\varphi\|_T\leq C(n)h_T^{-1}\|\varphi\|_T,\; \; \forall T\in \mathcal{T}_h. \end{eqnarray} (13)

    Then, we define the following semi-norms in the discontinuous finite element space V_h

    \begin{eqnarray} |\!|\!| v |\!|\!|^2& = &a(v, v) = \sum\limits_{T\in \mathcal{T}_h}\| \nabla_d v\|_T^2, \end{eqnarray} (14)
    \begin{eqnarray} \| v \|_{1,h}^2& = &\sum\limits_{T\in \mathcal{T}_h}\|\nabla v\|_T^2+\sum\limits_{e\in\mathcal{E}_h^0}h_e^{-1}\|[\![ v]\! ]\|_e^2. \end{eqnarray} (15)

    We have the equivalence between the semi-norms |\!|\!| v |\!|\!| and \|v\|_{1,h} , and it is proved in the following lemma.

    Lemma 3.3. For any v\in V_h , the following equivalence holds true

    \begin{eqnarray} C_1\|v\|_{1,h}\leq|\!|\!| v |\!|\!|\leq C_2\|v\|_{1,h}, \end{eqnarray} (16)

    where C_1 and C_2 are two constants independent of h .

    Proof. It follows from the definition of \nabla_d v , integration by parts, the trace inequality, and the inverse inequality that

    \begin{eqnarray} \|\nabla_d v\|_{T_1}^2 & = & (\nabla_d v, \nabla_d v)_{T_1} = -(v, \nabla\cdot\nabla_d v)_{T_1} + \langle\{v\}\boldsymbol{n}, \nabla_d v\rangle_{\partial {T_1}}\\ & = & (\nabla v, \nabla_d v)_{T_1} - \langle (v-\{v\})\boldsymbol{n}, \nabla_d v\rangle_{\partial {T_1}}\\ &\leq& \|\nabla v\|_{T_1}\|\nabla_d v\|_{T_1}+\|(v-\{v\})\boldsymbol{n}\|_{\partial {T_1}}\|\nabla_d v\|_{\partial {T_1}}\\ &\leq&\|\nabla_d v\|_{T_1}(\|\nabla v\|_{T_1} + h_{T_1}^{-\frac{1}{2}}\|(v-\{v\})\boldsymbol{n}\|_{\partial {T_1}}). \end{eqnarray} (17)

    For any e\subset\partial T_1 , e = \partial T_1\cap\partial T_2 , we have

    \begin{eqnarray*} (v-\{v\})|_e\boldsymbol{n}_1 & = & v|_{\partial T_1}\boldsymbol{n}_1-\frac{1}{2}(v|_{\partial T_1}+v|_{\partial T_2})\boldsymbol{n}_1\\ & = & \frac{1}{2}(v|_{\partial T_1}\boldsymbol{n}_1+v|_{\partial T_2}\boldsymbol{n}_2)\\ & = & \frac{1}{2}[\![ v]\! ]_e. \end{eqnarray*}

    Then we can get

    \begin{eqnarray} \|(v-\{v\})\boldsymbol{n}\|_{\partial T_1}^2\leq\frac{1}{2}\sum\limits_{e\in\partial T_1}\|[\![ v]\! ]\|_e^2. \end{eqnarray} (18)

    Substituting (18) into (17) gives

    \begin{eqnarray*} \|\nabla_d v\|_{T_1}^2 &\leq& C_2\|\nabla_d v\|_{T_1}(\|\nabla v\|_{T_1}+\sum\limits_{e\in\partial T_1}h_e^{-\frac{1}{2}}\|[\![ v]\! ]\|_e), \end{eqnarray*}

    this completes the proof of the right-hand of (16).

    To prove the left-hand of (16), we consider the subspace of RT_k(T) for any T\in\mathcal{T}_h

    \begin{eqnarray*} D(k,T): = \{\bf{q}\in RT_k(T):\; \boldsymbol{q}\cdot\boldsymbol{n} = 0\; \text{on}\; \partial T\}. \end{eqnarray*}

    Note that D(k,T) is a dual space of [Q_{k-1}(T)]^2 [13]. Thus, for any \nabla v\in[Q_{k-1}(T)]^2 , we have

    \begin{eqnarray} \|\nabla v\|_T = \sup\limits_{\boldsymbol{q}\in D(k,T)}\frac{(\nabla v,\boldsymbol{q})_T}{\|\boldsymbol{q}\|_T}. \end{eqnarray} (19)

    Using the integration by parts, Cauchy-Schwarz inequality, the definition of D(k,T) and \nabla_d v , we get

    \begin{eqnarray*} (\nabla v,\boldsymbol{q})_T & = & -(v,\nabla\cdot\boldsymbol{q})_T + \langle v, \boldsymbol{q}\cdot\boldsymbol{n}\rangle_{\partial T}\\ & = & (\nabla_d v, \boldsymbol{q})_T-\langle \{v\},\boldsymbol{q}\cdot \boldsymbol{n}\rangle_{\partial T}\\ & = & (\nabla_d v, \boldsymbol{q})_T\\ &\leq& \|\nabla_d v\|_T\cdot\|\boldsymbol{q}\|_T, \end{eqnarray*}

    where we have used the fact that \boldsymbol{q</italic>}\cdot \boldsymbol{n}|_{\partial T} = 0 in the definition of D(k,T) . Combining the above result with (19), one has

    \begin{eqnarray} \|\nabla v\|_T \leq \|\nabla_d v\|_T. \end{eqnarray} (20)

    We define the space D_e(k,T) as the set of all \boldsymbol{q</italic>} \in RT_k(T) such that all degrees of freedom, except those for \boldsymbol{q</italic>} \cdot\boldsymbol{n}|_e , vanish. Note that D_e(k,T) is a dual space of [Q_k(e)]^2 [13]. Thus, we know

    \begin{eqnarray} \|[\![ v]\! ]\|_e = \sup\limits_{\boldsymbol{q}\in D_e(k,T)}\frac{\langle [\![ v]\! ], \boldsymbol{q}\cdot\boldsymbol{n}\rangle_e}{\|\boldsymbol{q}\cdot\boldsymbol{n}\|_e}. \end{eqnarray} (21)

    Following the integration by parts and the definition of \nabla_d , we can derive that

    \begin{eqnarray*} (\nabla_d v,\boldsymbol{q})_T = ( \nabla v, \boldsymbol{q} )_T - \langle v,\boldsymbol{q}\cdot\boldsymbol{n}\rangle_e + \langle \{v\}, \boldsymbol{q}\cdot\boldsymbol{n}\rangle_e. \end{eqnarray*}

    Together with (20), we obtain

    \begin{eqnarray*} |\langle [\![ v]\! ], \boldsymbol{q}\cdot\boldsymbol{n}\rangle_e| & = &2|(\nabla_d v,\boldsymbol{q})_T - (\nabla v, \boldsymbol{q})_T|\\ &\leq& 2|(\nabla_d v,\boldsymbol{q})_T|+2|(\nabla v, \boldsymbol{q})_T|\\ &\leq& C(\|\nabla_d v\|_T\|\boldsymbol{q}\|_T+\|\nabla v\|_T\|\boldsymbol{q}\|_T)\\ &\leq& C\|\nabla_d v\|_T\|\boldsymbol{q}\|_T. \end{eqnarray*}

    Substituting the above inequality into (21), by the scaling argument [13], for such \boldsymbol{q} \in D_e(k,T) , we have \|\boldsymbol{q}\|_T \leq h^{\frac{1}{2}}\|\boldsymbol{q}\cdot\boldsymbol{n}\|_e , then

    \begin{eqnarray} \|[\![ v]\! ]\|_e \leq C\frac{\|\nabla_d v\|_T\|\boldsymbol{q}\|_T}{\|\boldsymbol{q}\cdot\boldsymbol{n}\|_e} \leq Ch^{\frac{1}{2}}\|\nabla_d v\|_T. \end{eqnarray} (22)

    Combining (20) and (22) gives a proof of the left-hand of (16).

    Lemma 3.4. The semi-norm |\!|\!| \cdot |\!|\!| defined in (14) is a norm in V_h^0 .

    Proof. We shall only verify the positivity property for |\!|\!| \cdot |\!|\!| . To this end, assume |\!|\!| v |\!|\!| = 0 for some v\in V_h^0 . By Lemma 3.3, it follows that \| v \|_{1,h} = 0 for all T\in\mathcal{T}_h , which means that \nabla v = {\mathit{\boldsymbol{0}}} for all elements T\in\mathcal{T}_h and [\![ v]\! ] = 0 for all edges e\in\mathcal{E}_h^0 . We can derive from \nabla v = {\mathit{\boldsymbol{0}}} for all T\in\mathcal{T}_h that v is a constant in each T . [\![ v]\! ] = 0 on each e\in\mathcal{E}_h^0 implies v is a continuous function. This two conclusions and v = 0 on \partial \Omega show that v = 0 , which completes the proof of the lemma.

    The above two lemmas imply the well posedness of the scheme (11). We prove the existence and uniqueness of solution of the conforming DG method in Theorem 3.1.

    Theorem 3.1. The conforming DG scheme (11) has and only has one solution.

    Proof. To prove the scheme (11) is uniquely solvable, it suffices to verify that the homogeneous equation has zero as its unique solution. To this end, let u_h\in V_h be the solution of the numerical scheme 11 with homogeneous data f = {0} , {g} = 0 . Letting v_h = u_h , we obtain

    \begin{eqnarray*} a(u_h, u_h) = 0, \end{eqnarray*}

    which leads to u_h = 0 by using Lemma 3.4. This completes the proof of the theorem.

    In this section, we will derive an error equation which will be used for the error estimates. For any \boldsymbol{q</italic>}\in H(div, \Omega) , we assume that there exist an interpolation operator \Pi_h satisfying \Pi_h\boldsymbol{q}\in H(div, \Omega)\cap RT_k(T) on each element T\in\mathcal{T}_h and

    \begin{eqnarray} (\nabla\cdot\boldsymbol{q},v)_T = (\nabla\cdot\Pi_h\boldsymbol{q}, v)_T, \; \; \forall v \in Q_k(T). \end{eqnarray} (23)

    For any w\in H^{1+k}(\Omega) with k\geq 1 , from Lemma 7.3 in [20], we have the estimate of \Pi_h as follows.

    \begin{eqnarray} \|\Pi_h(\nabla w)-\nabla w\|\leq Ch^k\|w\|_{1+k}. \end{eqnarray} (24)

    Moreover, it is easy to verify the following property holds true.

    Lemma 4.1. For any \boldsymbol{q}\in H(div, \Omega) ,

    \begin{eqnarray} \sum\limits_{T\in\mathcal{T}_h}(-\nabla\cdot\boldsymbol{q}, v)_T = \sum\limits_{T\in\mathcal{T}_h}(\Pi_h\boldsymbol{q}, \nabla_d v)_T, \; \; \forall v\in V_h^0. \end{eqnarray} (25)

    Proof. \Pi_h\boldsymbol{q}\in H(div, \Omega) implies that \Pi_h\boldsymbol{q} is continuous across each interior edge. Since v\in V_h^0 , we know that \{v\} = v = 0 on \partial \Omega . Then

    \begin{eqnarray} \sum\limits_{T\in\mathcal{T}_h}\langle\{v\}, \Pi_h\boldsymbol{q}\cdot\boldsymbol{n}\rangle_{\partial T} = 0. \end{eqnarray} (26)

    By the definition of \Pi_h and \nabla_d and the equation (26), we have

    \begin{eqnarray*} \sum\limits_{T\in\mathcal{T}_h}(-\nabla\cdot\boldsymbol{q}, v)_T & = & \sum\limits_{T\in\mathcal{T}_h}(-\nabla\cdot\Pi_h\boldsymbol{q}, v)_T\\ & = & \sum\limits_{T\in\mathcal{T}_h}(-\nabla\cdot\Pi_h\boldsymbol{q}, v)_T+ \sum\limits_{T\in\mathcal{T}_h}\langle\{v\}, \Pi_h\boldsymbol{q}\cdot\boldsymbol{n}\rangle_{\partial T}\\ & = & \sum\limits_{T\in\mathcal{T}_h}(\Pi_h\boldsymbol{q}, \nabla_d v)_T. \end{eqnarray*}

    This completes the proof of the lemma.

    Before establishing the error equation, we define a continuous finite element subspace of V_h as follows

    \begin{eqnarray} \tilde{V}_{h} = \{v \in H^{1}(\Omega) :v|_T \in Q_k(T), \; \forall T \in \mathcal{T}_{h}\}. \end{eqnarray} (27)

    so as a subspace of \tilde{V}_h

    \begin{eqnarray} \tilde{V}_h^0: = \{v\in\tilde{V}_h:v|_{\partial\Omega} = 0\}. \end{eqnarray} (28)

    Lemma 4.2. For any v\in\tilde{V}_h , we have

    \begin{eqnarray*} \nabla_d v = \nabla v. \end{eqnarray*}

    Proof. By the definition of \nabla_d and integration by parts, for any \boldsymbol{q</italic>}\in RT_k(T) , we have

    \begin{eqnarray*} (\nabla_d v, \boldsymbol{q})_T & = & -(v, \nabla\cdot\boldsymbol{q})_T + \langle\{v\}, \boldsymbol{q}\cdot\boldsymbol{n}\rangle_{\partial T}\\ & = & -(v,\nabla\cdot\boldsymbol{q})_T + \langle v, \boldsymbol{q}\cdot\boldsymbol{n}\rangle_{\partial T}\\ & = & (\nabla v, \boldsymbol{q})_T, \end{eqnarray*}

    which gives

    \begin{eqnarray*} (\nabla_d v-\nabla v, \boldsymbol{q})_T = 0, \; \; \forall \boldsymbol{q}\in RT_k(T). \end{eqnarray*}

    Letting \boldsymbol{q} be \nabla_d v-\nabla v in the above equation yields \|\nabla_d v-\nabla v\| = 0 , which completes the proof of the lemma.

    Let e_h = I_h u - u_h , where I_h is the kth order Lagrange interpolation, u\in H^{k+1}(\Omega) with k \geq 1 is the exact solution of the Poisson equations (1) - (2), and u_h\in V_h is the numerical solution of the scheme (11). The following estimate of the Lagrange interpolation operator I_h holds true.

    \begin{eqnarray} \|I_hu-u\|\leq Ch^{k+1}\|u\|_{k+1}, \end{eqnarray} (29)
    \begin{eqnarray} \|\nabla I_hu-\nabla u\|\leq Ch^k\|u\|_{k+1}. \end{eqnarray} (30)

    It is obvious that e_h\in V_h^0 and I_h u\in\tilde{V}_h . We have the following lemma:

    Lemma 4.3. Denote e_h = I_h u - u_h the error of conforming DG method arising from (11). For any v_h\in V_h^0 , we have

    \begin{eqnarray} a(e_h, v_h) = l_u( v_h), \end{eqnarray} (31)

    where

    \begin{eqnarray} l_u(v_h) = \sum\limits_{T\in{\mathcal{T}_h}}(\nabla I_{h} u-\Pi_h\nabla u, \nabla_d v_h). \end{eqnarray} (32)

    Proof. Since I_h u\in\tilde{V}_h , we have \nabla_d I_h u = \nabla I_h u . Using the property (25), we can derive

    \begin{eqnarray*} \sum\limits_{T\in{\mathcal{T}_h}}(\nabla_d I_h u, \nabla_d v_h)_T & = & \sum\limits_{T\in{\mathcal{T}_h}}(\nabla I_h u, \nabla_d v_h)_T\\ & = & \sum\limits_{T\in{\mathcal{T}_h}}(\nabla I_h u -\Pi_h\nabla u + \Pi_h\nabla u, \nabla_d v_h)_T\\ & = & \sum\limits_{T\in{\mathcal{T}_h}}(\nabla I_h u -\Pi_h\nabla u, \nabla_d v_h)_T +\sum\limits_{T\in{\mathcal{T}_h}}(\Pi_h\nabla u, \nabla_d v_h)_T\\ & = & l_u(v_h) - \sum\limits_{T\in{\mathcal{T}_h}}(\nabla\cdot\nabla u, v_h)_T\\ & = & l_u(v_h) + (f,v_h). \end{eqnarray*}

    By the definition of the scheme (11), we have

    \begin{eqnarray*} \sum\limits_{T\in{\mathcal{T}_h}}(\nabla_d I_h u - \nabla_d u_h, \nabla_d v_h)_T = l_u(v_h). \end{eqnarray*}

    This completes the proof of the lemma.

    The goal of this section is to derive the error estimates in H^1 and L^2 norms for the conforming DG solution u_h .

    Theorem 5.1. Let u\in H^{k+1}(\Omega) with k \geq 1 be the exact solution of the Poisson equation (1) - (2), and u_h\in V_h be the numerical solution of the scheme (11). Let e_h = I_h u - u_h , there exists a constant C independent of h such that

    \begin{eqnarray} |\!|\!|e_h|\!|\!| \leq Ch^k|u|_{k+1}. \end{eqnarray} (33)

    Proof. Letting v_h = e_h in (31), and by the definition of |\!|\!|\cdot|\!|\!| , we have

    \begin{eqnarray} |\!|\!|e_h|\!|\!|^2 = l_u(e_h). \end{eqnarray} (34)

    From the Cauchy-Schwarz inequality, the triangle inequality, the definition of |\!|\!|\cdot|\!|\!| , (24), and (30), we arrive at

    \begin{eqnarray*} l_u(v_h)& = & \sum\limits_{T\in\mathcal{T}_h}(\nabla I_hu-\Pi_h(\nabla u), \nabla_d v_h)_T\\ &\leq& \sum\limits_{T\in\mathcal{T}_h}\|\nabla I_hu-\Pi_h(\nabla u)\|_T\|\nabla_d v_h\|_T\\ &\leq& \left(\sum\limits_{T\in\mathcal{T}_h}\|\nabla I_hu-\Pi_h(\nabla u)\|_T^2\right)^{\frac{1}{2}}\left(\sum\limits_{T\in\mathcal{T}_h}\|\nabla_d v_h\|_T^2\right)^{\frac{1}{2}}\\ & = & \left(\sum\limits_{T\in\mathcal{T}_h}\|\nabla I_hu-\nabla u+\nabla u-\Pi_h(\nabla u)\|_T^2\right)^{\frac{1}{2}}|\!|\!|v_h|\!|\!|\\ &\leq& \left(\sum\limits_{T\in\mathcal{T}_h}\|\nabla I_hu-\nabla u\|_T^2+\|\nabla u-\Pi_h(\nabla u)\|_T^2\right)^{\frac{1}{2}}|\!|\!|v_h|\!|\!|\\ &\leq& Ch^k|u|_{k+1}|\!|\!|v_h|\!|\!|. \end{eqnarray*}

    Then, we have

    \begin{eqnarray} l_u(e_h) \leq Ch^k|u|_{k+1}|\!|\!|e_h|\!|\!|. \end{eqnarray} (35)

    Substituting (35) to (34), we obtain

    \begin{eqnarray*} |\!|\!|e_h|\!|\!|^2 \leq Ch^k|u|_{k+1}|\!|\!|e_h|\!|\!|, \end{eqnarray*}

    which completes the proof of the lemma.

    It is obvious that \tilde{V}_h^0\subset V_h^0 . Let \tilde{u}_h\in\tilde{V}_h be the finite element solution for the problem (1)-(2) which satisfies \tilde{u}_h = I_hg on \partial\Omega and

    \begin{eqnarray} (\nabla\tilde{u}_h, \nabla v) = (f, v), \; \; \forall v\in\tilde{V}_h^0. \end{eqnarray} (36)

    For any v\in\tilde{V}_h^0\subset\tilde{V}_h , we have \nabla_d v = \nabla v , i.e.

    \begin{eqnarray} (\nabla_d u_h - \nabla\tilde{u}_h, \nabla v) = 0, \; \; \forall v\in\tilde{V}_h^0. \end{eqnarray} (37)

    In the rest of this section, we derive an optimal order error estimate for the conforming DG approximation (11) in L^2 norm by adopting the duality argument. To this end, we consider the following dual problem that seeks \Phi\in H_0^1(\Omega) satisfying

    \begin{eqnarray} -\nabla\cdot(\nabla\Phi) = u_h-\tilde{u}_h, \; \; in\; \Omega. \end{eqnarray} (38)

    Assume that the dual problem satisfies H^2 -regularity, which means the following priori estimate holds true

    \begin{eqnarray} \|\Phi\|_2 \leq C\|u_h-\tilde{u}_h\|. \end{eqnarray} (39)

    In the following of this paper, we note \varepsilon_h = u_h-\tilde{u}_h for simplicity.

    Theorem 5.2. Assume u\in H^{k+1}(\Omega) with k \geq 1 is the exact solution of the Poisson equation (1) - (2), and u_h\in V_h is the numerical solution obtained with the scheme (11). Furthermore, assume that (39) holds true. Then, there exists a constant C independent of h such that

    \begin{eqnarray} \|u-u_h\|\leq C h^{k+1}|u|_{k+1}. \end{eqnarray} (40)

    Proof. First, we shall derive the optimal order for \varepsilon_h in L^2 norm. Consider the corresponding conforming DG scheme defined in (11) and let \Phi_h\in V_h^0 be the solution satisfying

    \begin{eqnarray} a(\Phi_h, v) = (\varepsilon_h, v), \; \; \forall v\in V_h^0. \end{eqnarray} (41)

    Since I_h\Phi\in\tilde{V}_h , it follows from (37) that

    \begin{eqnarray*} (\nabla_d u_h-\nabla\tilde{u}_h, \nabla I_h\Phi) & = & 0,\\ \nabla_d I_h\Phi & = & \nabla I_h\Phi, \end{eqnarray*}

    which gives

    \begin{eqnarray} (\nabla_d u_h-\nabla\tilde{u}_h, \nabla_d I_h\Phi) = 0. \end{eqnarray} (42)

    Setting v = \varepsilon_h in (41), then by the definition of \varepsilon_h and (42), we have

    \begin{eqnarray*} \|\varepsilon_h\|^2& = &a(\Phi_h, \varepsilon_h) = \sum\limits_{T\in\mathcal{T}_h}(\nabla_d \Phi_h, \nabla_d\varepsilon_h)_T\\ & = & \sum\limits_{T\in\mathcal{T}_h}(\nabla_d (\Phi_h-I_h\Phi), \nabla_d u_h -\nabla\tilde{u}_h)_T\\ &\leq&|\!|\!| \Phi_h-I_h\Phi |\!|\!|(|\!|\!|u_h-I_h u|\!|\!|+\|\nabla(I_h u-\tilde{u}_h)\|). \end{eqnarray*}

    Then, by the Cauchy-Schwarz inequality, (33) and (39), we obtain

    \begin{eqnarray*} \|\varepsilon_h\|^2&\leq& Ch|\Phi|_2h^k|u|_{k+1} \leq Ch^{k+1}|u|_{k+1}\|\varepsilon_h\|, \end{eqnarray*}

    which gives

    \begin{eqnarray} \|\varepsilon_h\| \leq Ch^{k+1}|u|_{k+1}. \end{eqnarray} (43)

    Combining the error estimate of finite element solution, the triangle inequality and (43) yields (40), which completes the proof of the theorem.

    In this section, we shall present some numerical results for the conforming discontinuous Galerkin method analyzed in the previous sections.

    We solve the following Poisson equation on the unit square domain \Omega = (0,1)\times (0,1) ,

    \begin{align} -\Delta u & = 2 \pi^2 \sin(\pi x)\sin(\pi y) &&\hbox{in }\Omega && \end{align} (44)
    \begin{align} u& = 0 &&\hbox{on }\partial\Omega. \end{align} (45)

    The exact solution of the above problem is u = \sin(\pi x)\sin(\pi y) . Uniform square grids as shown in Figure 1 are used for computation.

    Figure 1.  The first three grids used in the computation.

    We first use the P_k conforming discontinuous Galerkin spaces (8) to compute the test case (44)-(45), where P_k denotes the set of polynomials of 2 variables of degree less than or equal to k. The weak gradient is computed locally using rectangular RT_k polynomials. The errors and the order of convergence of the conforming DG approximations are listed in Table 1. Optimal order of convergence is achieved in every case, which is consistent with our theory. In particular, a superconvergence of order \mathcal{O}(h^2) was observed in the discrete H^1 norm for P_0 elements. Furthermore, the results obtained with P_0 elements seems to be slightly better than that obtained with P_1 elements.

    Table 1.  Error profiles and convergence rates for test case (44)-(45) obtained with uniform grids and P_k conforming DG spaces.
    level \|u_h- Q_h u\|_0 rate {|\!|\!|} u_h- Q_h u{|\!|\!|} rate \#Dof
    by P_0 conforming discontinuous Galerkin elements
    6 0.1996E-02 1.97 0.8887E-02 1.98 1024
    7 0.5013E-03 1.99 0.2228E-02 2.00 4096
    8 0.1255E-03 2.00 0.5574E-03 2.00 16384
    by P_1 conforming discontinuous Galerkin elements
    6 0.2427E-02 1.97 0.1027E+00 1.02 3072
    7 0.6100E-03 1.99 0.5105E-01 1.01 12288
    8 0.1527E-03 2.00 0.2546E-01 1.00 49152
    by P_2 conforming discontinuous Galerkin elements
    5 0.1533E-03 3.00 0.2042E-01 2.03 1536
    6 0.1915E-04 3.00 0.5061E-02 2.01 6144
    7 0.2394E-05 3.00 0.1260E-02 2.01 24576
    by P_3 conforming discontinuous Galerkin elements
    5 0.7959E-05 4.00 0.1965E-02 3.00 2560
    6 0.4971E-06 4.00 0.2451E-03 3.00 10240
    7 0.3140E-07 3.98 0.3059E-04 3.00 40960
    by P_4 conforming discontinuous Galerkin elements
    4 0.1055E-04 4.97 0.1421E-02 4.05 960
    5 0.3314E-06 4.99 0.8735E-04 4.02 3840
    6 0.1057E-07 4.97 0.5417E-05 4.01 15360
    by P_5 conforming discontinuous Galerkin elements
    2 0.2835E-02 6.24 0.1450E+00 5.49 84
    3 0.4532E-04 5.97 0.4718E-02 4.94 336
    4 0.7115E-06 5.99 0.1478E-03 5.00 1344

     | Show Table
    DownLoad: CSV

    The same test case is also computed using the Q_k conforming DG finite element space, where Q_k denotes the set of polynomials of 2 variables defined on \Omega , and for each variable, the degree of the variable is at most k. Table 2 illustrates the numerical performance of the corresponding conforming DG scheme. It can be seen from numerical computing that, in this case, the results obtained with the Q_1 element are more accurate than those obtained with Q_0( = P_0) elements (see Table 1). All numerical results converge at the corresponding optimal order, which is consistent with the theory.

    Table 2.  Error profiles and convergence rates for test case (44)-(45) obtained with uniform grids and Q_k conforming DG spaces.
    level \|u_h- Q_hu\|_0 rate {|\!|\!|} u_h- Q_h u{|\!|\!|} rate \#Dof
    by Q_1 conforming discontinuous Galerkin elements
    6 0.4006E-03 1.99 0.2389E-02 1.99 4096
    7 0.1003E-03 2.00 0.5982E-03 2.00 16384
    8 0.2510E-04 2.00 0.1496E-03 2.00 65536
    by Q_2 conforming discontinuous Galerkin elements
    6 0.2360E-04 2.99 0.3186E-02 1.99 9216
    7 0.2953E-05 3.00 0.7976E-03 2.00 36864
    8 0.3692E-06 3.00 0.1995E-03 2.00 147456
    by Q_3 conforming discontinuous Galerkin elements
    5 0.1413E-04 4.08 0.1650E-02 2.97 4096
    6 0.8676E-06 4.03 0.2072E-03 2.99 16384
    7 0.5398E-07 4.01 0.2593E-04 3.00 65536
    by Q_4 conforming discontinuous Galerkin elements
    3 0.2226E-02 4.59 0.5414E-01 3.52 400
    4 0.9610E-04 4.53 0.3723E-02 3.86 1600
    5 0.3279E-05 4.87 0.2392E-03 3.96 6400

     | Show Table
    DownLoad: CSV

    To test the superconvergence of P_0 DG element, we solve the following 2nd order elliptic equation on the unit square domain \Omega = (0,1)\times (0,1) ,

    \begin{align*} -\Delta u +u & = f &&\hbox{in }\Omega &&\\ u& = 0 &&\hbox{on }\partial\Omega, \end{align*}

    where f is chosen so that the exact solution is not symmetric,

    \begin{align} u = (x-x^2)(y-y^3). \end{align} (46)

    Uniform square grids as shown in Figure 1 are used for numerical computation. The numerical results are listed in Table 3. Surprising, for this problem, the H^1 -like norm of error superconverges at 1.5 order, and the L^2 error has one order of superconvergence. But we do not yet know if such a superconvergence exists in general.

    Table 3.  Error profiles and convergence rates for test case (46) obtained with uniform grids and P_0 conforming DG spaces.
    level \|u_h- Q_h u\|_0 rate {|\!|\!|} u_h- Q_h u{|\!|\!|} rate \#Dof
    by P_0 conforming discontinuous Galerkin elements
    3 0.8265E-02 1.06 0.4577E-01 1.14 16
    4 0.2772E-02 1.58 0.1732E-01 1.40 64
    5 0.7965E-03 1.80 0.6331E-02 1.45 256
    6 0.2142E-03 1.90 0.2290E-02 1.47 1024
    7 0.5564E-04 1.94 0.8213E-03 1.48 4096
    8 0.1419E-04 1.97 0.2928E-03 1.49 16384

     | Show Table
    DownLoad: CSV

    To test further the superconvergence of P_0 DG element, we solve the following 2nd order elliptic equations on the unit square domain \Omega = (0,1)\times (0,1) ,

    \begin{align*} -\nabla(a \nabla u ) & = f &&\hbox{in }\Omega &&\\ u& = 0 &&\hbox{on }\partial\Omega, \end{align*}

    where a = 1+x+y and f is chosen so that the exact solution is not symmetric,

    \begin{align} u = (x-x^3)(y^2-y^3). \end{align} (47)

    Uniform square grids as shown in Figure 1 are used for computation. The numerical results are listed in Table 4. Surprising, again, the H^1 -like norm of error superconverges at 1.5 order, and the L^2 error has one order of superconvergence for this problem.

    Table 4.  Error profiles and convergence rates for test case (47) obtained with uniform grids and P_0 conforming DG spaces.
    level \|u_h- Q_h u\|_0 rate {|\!|\!|} u_h- Q_h u{|\!|\!|} rate \#Dof
    by P_0 conforming discontinuous Galerkin elements
    3 0.4929E-02 0.97 0.5371E-01 0.80 16
    4 0.1917E-02 1.36 0.2401E-01 1.16 64
    5 0.6004E-03 1.67 0.9407E-02 1.35 256
    6 0.1682E-03 1.84 0.3507E-02 1.42 1024
    7 0.4457E-04 1.92 0.1275E-02 1.46 4096
    8 0.1148E-04 1.96 0.4576E-03 1.48 16384

     | Show Table
    DownLoad: CSV

    In this paper, we establish a new numerical approximation scheme based on the rectangular partition to solve second order elliptic equation. We derived the numerical scheme and then proved the optimal order of convergence of the error estimates in L^2 and H^1 norms of the conforming DG method. Numerical experiments are then present to verify the theoretical analysis, and all numerical results converging at the corresponding optimal order. Comparing with existing numerical methods, the confoming DG method has the following two characteristics: 1. The formulation is relatively simple. The stabilizer s(\cdot\; , \; \cdot) is no longer needed, and the boundary function u_b is omitted, which is replaced by the average of internal function u_0 ; 2. The projection operator Q_h used in the traditional WG method is replaced by the Lagrange interpolation operator I_h , which makes the theoretical analysis much easier. As can be seen from the numerical examples in Section 6, this method reduces the programming complexity while ensuring the optimal order of convergence.



    [1] A. Ahmadini, T. Yadav, S. Yadav, A. Al Luhayb, Restructured searls family of estimators of population mean in the presence of nonresponse, Front. Appl. Math. Stat., 8 (2022), 969068. https://doi.org/10.3389/fams.2022.969068 doi: 10.3389/fams.2022.969068
    [2] S. Ahmad, S. Hussain, M. Aamir, F. Khan, M. Alshahrani, M. Alqawba, Estimation of finite population mean using dual auxiliary variable for nonresponse using simple random sampling, AIMS Mathematics, 7 (2022), 4592–4613. https://doi.org/10.3934/math.2022256 doi: 10.3934/math.2022256
    [3] F. Almulhim, H. Aljohani, R. Aldallal, M. Mustafa, M. Alsolmi, A. Elshenawy, et al., Estimation of finite population mean using dual auxiliary information under nonresponse with simple random sampling, Alex. Eng. J., 100 (2024), 286–299. https://doi.org/10.1016/j.aej.2024.04.058 doi: 10.1016/j.aej.2024.04.058
    [4] N. Adichwal, A. Ahmadini, Y. Raghav, R. Singh, I. Ali, Estimation of general parameters using auxiliary information in simple random sampling without replacement, J. King Saud Univ. Sci., 34 (2022), 101754. https://doi.org/10.1016/j.jksus.2021.101754 doi: 10.1016/j.jksus.2021.101754
    [5] V. Adamo, On the exchangeability property in causal models, Computational Journal of Mathematical and Statistical Sciences, 3 (2024), 228–239.
    [6] H. Alshanbari, A. Gemeay, A. El-Bagoury, S. Khosa, E. Hafez, A. Muse, A novel extension of Fréchet distribution: Application on real data and simulation, Alex. Eng. J., 61 (2022), 7917–7938. https://doi.org/10.1016/j.aej.2022.01.013 doi: 10.1016/j.aej.2022.01.013
    [7] M. Bakr, A. Al-Babtain, Z. Mahmood, R. Aldallal, S. Khosa, M. Abd El-Raouf, et al., Statistical modelling for a new family of generalized distributions with real data applications, Math. Biosci. Eng., 19 (2022), 8705–8740. https://doi.org/10.3934/mbe.2022404 doi: 10.3934/mbe.2022404
    [8] Z. Basit, S. Masood, I. Bhatti, A class of estimators for estimation of population mean under random nonresponse in two phase successive sampling, J. Stat. Theory Appl., 22 (2023), 309–338. https://doi.org/10.1007/s44199-023-00065-5 doi: 10.1007/s44199-023-00065-5
    [9] W. Cochran, The estimation of the yields of cereal experiments by sampling for the ratio of grain to total produce, J. Agr. Sci., 30 (1940), 262–275. https://doi.org/10.1017/S0021859600048012 doi: 10.1017/S0021859600048012
    [10] A. Dyab, S. El Sayed, A. Abdelfattah, Short dynamic panel data models in the presence of a higher frequency regressor(s), Computational Journal of Mathematical and Statistical Sciences, 2 (2023), 251–274.
    [11] D. Gujarati, D. Porter, Basic econometrics, New York: McGraw-Hill, 2009.
    [12] A. Gemeay, K. Karakaya, M. Bakr, O. Balogun, M. Atchadé, E. Hussam, Power Lambert uniform distribution: statistical properties, actuarial measures, regression analysis, and applications, AIP Adv., 13 (2023), 095319. https://doi.org/10.1063/5.0170964 doi: 10.1063/5.0170964
    [13] L. Grover, P. Kaur, A generalized class of ratio type exponential estimators of population mean under linear transformation of auxiliary variable, Commun. Stat. Simul. Comput., 43 (2014), 1552–1574. https://doi.org/10.1080/03610918.2012.736579 doi: 10.1080/03610918.2012.736579
    [14] S. Hussain, S. Ahmad, S. Akhtar, A. Javed, U. Yasmeen, Estimation of finite population distribution function with dual use of auxiliary information under nonresponse, PLoS One, 15 (2020), e0243584. https://doi.org/10.1371/journal.pone.0243584 doi: 10.1371/journal.pone.0243584
    [15] M. Hansen, W. Hurwitz, The problem of nonresponse in sample surveys, J. Am. Stat. Assoc., 41 (1946), 517–529. https://doi.org/10.1080/01621459.1946.10501894 doi: 10.1080/01621459.1946.10501894
    [16] S. Hussain, M. Zichuan, S. Hussain, A. Iftikhar, M. Asif, S. Akhtar, et al., On estimation of distribution function using dual auxiliary information under nonresponse using simple random sampling, J. Probab. Stat., 2020 (2020), 1693612. https://doi.org/10.1155/2020/1693612 doi: 10.1155/2020/1693612
    [17] M. Ismail, Combination of ratio and regression estimator of population mean in presence of nonresponse, Gazi Univ. J. Sci., 30 (2017), 634–642.
    [18] B. Khare, S. Kumar, Chain ratio-regression estimators in two phase sampling in the presence of nonresponse, ProbStat Forum, 8 (2015), 95–102.
    [19] B. Khare, R. Sinha, On class of estimators for population mean using multi-auxiliary characters in the presence of nonresponse, Statistics in Transition, 10 (2009), 3–14.
    [20] R. Kurbah, P. Khongji, A new exponential ratio-type estimator of population mean using mean and median with two auxiliary variables under double sampling, Thail. Statist., 22 (2024), 458–470.
    [21] N. Koyuncu, C. Kadilar, Ratio and product estimators in stratified random sampling, J. Stat. Plan. Infer., 139 (2009), 2552–2558. https://doi.org/10.1016/j.jspi.2008.11.009 doi: 10.1016/j.jspi.2008.11.009
    [22] S. Lone, M. Subzar, A. Sharma, Enhanced estimators of population variance with the use of supplementary information in survey sampling, Math. Probl. Eng., 2021 (2021), 9931217. https://doi.org/10.1155/2021/9931217 doi: 10.1155/2021/9931217
    [23] S. Muneer, J. Shabbir, A. Khalil, A generalized exponential type estimator of population mean in the presence of nonresponse, Statistics in Transition, 19 (2018), 259–276.
    [24] M. Qureshi, Y. Faizan, A. Shetty, M. Ahelali, M. Hanif, O. Alamri, Ln-type estimators for the estimation of the population mean of a sensitive study variable using auxiliary information, Heliyon, 10 (2024), e23066. https://doi.org/10.1016/j.heliyon.2023.e23066 doi: 10.1016/j.heliyon.2023.e23066
    [25] T. Raja, S. Maqbool, Ratio type estimators for estimating population mean by using some auxiliary information, International Journal of Agriculture Sciences, 16 (2024), 12799–12801.
    [26] M. Rueda, A. Arcos, M. Martínez-Miranda, Y. Román, Some improved estimators of finite population quantile using auxiliary information in sample surveys, Comput. Stat. Data Anal., 45 (2004), 825–848. https://doi.org/10.1016/S0167-9473(03)00097-5 doi: 10.1016/S0167-9473(03)00097-5
    [27] T. Rao, On certain methods of improving ratio and regression estimators, Commun. Stat. Theor. M., 20 (1991), 3325–3340.
    [28] F. Riad, E. Hussam, A. Gemeay, R. Aldallal, A. Afify, Classical and Bayesian inference of the weighted-exponential distribution with an application to insurance data, Math. Biosci. Eng., 19 (2022), 6551–6581. https://doi.org/10.3934/mbe.2022309 doi: 10.3934/mbe.2022309
    [29] R. Sinha, B. Khanna, Efficient estimation for ratio of means using dual auxiliary information under nonresponse, Thail. Statist., 22 (2025), 736–749.
    [30] S. Singh, Advanced sampling theory with applications: how Michael "selected" Amy, volume I, Dordrecht: Springer, 2003. https://doi.org/10.1007/978-94-007-0789-4
    [31] H. Singh, N. Garg, Modified correlated measurement errors model for estimation of population mean utilizing auxiliary information, Sci. Rep., 14 (2024), 11086. https://doi.org/10.1038/s41598-024-61609-y doi: 10.1038/s41598-024-61609-y
    [32] A. Salama, H. Elagamy, Neutrosophic fuzzy ideal open sets, ideal closed sets, and their applications in geographic information systems, Journal of Business and Environmental Science, 3 (2024), 302–321. https://doi.org/10.21608/jcese.2024.281435.1058 doi: 10.21608/jcese.2024.281435.1058
    [33] H. Semary, S. Ahmad, I. Elbatal, C. Chesneau, M. Elgarhy, E. Almetwally, New comprehensive class of estimators for population proportion using auxiliary attribute: simulation and an application, Alex. Eng. J., 99 (2024), 130–136. https://doi.org/10.1016/j.aej.2024.04.065 doi: 10.1016/j.aej.2024.04.065
    [34] A. Teamah, A. Elbanna, A. Gemeay, An alternative version of half-logistic distribution: properties, estimation and application, Thail. Statist., 23 (2024), 97–114.
    [35] K. Ullah, Z. Hussain, I. Hussain, S. Cheema, Z. Almaspoor, M. El-Morshedy, Estimation of finite population mean in simple and stratified random sampling by utilizing the auxiliary, ranks, and square of the auxiliary information, Math. Probl. Eng., 2022 (2022), 5263492. https://doi.org/10.1155/2022/5263492 doi: 10.1155/2022/5263492
    [36] D. Watson, The estimation of leaf area in field crops, J. Agr. Sci., 27 (1937), 474–483. https://doi.org/10.1017/S002185960005173X doi: 10.1017/S002185960005173X
    [37] M. Yaqub, F. Sohil, J. Shabbir, M. Sohail, Estimation of population distribution function in the presence of nonresponse using stratified random sampling, Commun. Stat.-Simul. Comput., 53 (2024), 2498–2526. https://doi.org/10.1080/03610918.2022.2078492 doi: 10.1080/03610918.2022.2078492
    [38] E. Yıldırım, E. Ilıkkan, A. Gemeay, N. Makumi, M. Bakr, O. Balogun, Power unit Burr-Ⅻ distribution: statistical inference with applications, AIP Adv., 13 (2023), 105107. https://doi.org/10.1063/5.0171077 doi: 10.1063/5.0171077
  • This article has been cited by:

    1. Xiu Ye, Shangyou Zhang, A weak divergence CDG method for the biharmonic equation on triangular and tetrahedral meshes, 2022, 178, 01689274, 155, 10.1016/j.apnum.2022.03.017
    2. Jun Zhou, Da Xu, Wenlin Qiu, Leijie Qiao, An accurate, robust, and efficient weak Galerkin finite element scheme with graded meshes for the time-fractional quasi-linear diffusion equation, 2022, 124, 08981221, 188, 10.1016/j.camwa.2022.08.022
    3. Xiu Ye, Shangyou Zhang, A conforming discontinuous Galerkin finite element method for the Stokes problem on polytopal meshes, 2021, 93, 0271-2091, 1913, 10.1002/fld.4959
    4. Xiu Ye, Shangyou Zhang, Constructing a CDG Finite Element with Order Two Superconvergence on Rectangular Meshes, 2023, 2096-6385, 10.1007/s42967-023-00330-5
    5. Yan Yang, Xiu Ye, Shangyou Zhang, A pressure-robust stabilizer-free WG finite element method for the Stokes equations on simplicial grids, 2024, 32, 2688-1594, 3413, 10.3934/era.2024158
    6. Xiu Ye, Shangyou Zhang, A superconvergent CDG finite element for the Poisson equation on polytopal meshes, 2023, 0044-2267, 10.1002/zamm.202300521
    7. Xiu Ye, Shangyou Zhang, Two-Order Superconvergent CDG Finite Element Method for the Heat Equation on Triangular and Tetrahedral Meshes, 2024, 2096-6385, 10.1007/s42967-024-00444-4
    8. Xiu Ye, Shangyou Zhang, Order two superconvergence of the CDG finite elements for non-self adjoint and indefinite elliptic equations, 2024, 50, 1019-7168, 10.1007/s10444-023-10100-9
    9. Fuchang Huo, Weilong Mo, Yulin Zhang, A locking-free conforming discontinuous Galerkin finite element method for linear elasticity problems, 2025, 465, 03770427, 116582, 10.1016/j.cam.2025.116582
  • Reader Comments
  • © 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(173) PDF downloads(11) Cited by(0)

Figures and Tables

Figures(4)  /  Tables(25)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog