Research article

Empirical likelihood based heteroscedasticity diagnostics for varying coefficient partially nonlinear models

  • Received: 26 September 2024 Revised: 04 November 2024 Accepted: 06 December 2024 Published: 12 December 2024
  • MSC : 62G05, 62G20, 62H15

  • Heteroscedasticity diagnostics of error variance is essential before performing some statistical inference work. This paper is concerned with the statistical diagnostics for the varying coefficient partially nonlinear model. We propose a novel diagnostic approach for heteroscedasticity of error variance in the model by combining it with the empirical likelihood method. Under some mild conditions, the nonparametric version of the Wilks theorem is obtained. Furthermore, simulation studies and a real data analysis are implemented to evaluate the performances of our proposed approaches.

    Citation: Cuiping Wang, Xiaoshuang Zhou, Peixin Zhao. Empirical likelihood based heteroscedasticity diagnostics for varying coefficient partially nonlinear models[J]. AIMS Mathematics, 2024, 9(12): 34705-34719. doi: 10.3934/math.20241652

    Related Papers:

    [1] Yayun Fu, Mengyue Shi . A conservative exponential integrators method for fractional conservative differential equations. AIMS Mathematics, 2023, 8(8): 19067-19082. doi: 10.3934/math.2023973
    [2] Yong-Chao Zhang . Least energy solutions to a class of nonlocal Schrödinger equations. AIMS Mathematics, 2024, 9(8): 20763-20772. doi: 10.3934/math.20241009
    [3] Tingting Ma, Yuehua He . An efficient linearly-implicit energy-preserving scheme with fast solver for the fractional nonlinear wave equation. AIMS Mathematics, 2023, 8(11): 26574-26589. doi: 10.3934/math.20231358
    [4] Karmina K. Ali, Resat Yilmazer . Discrete fractional solutions to the effective mass Schrödinger equation by mean of nabla operator. AIMS Mathematics, 2020, 5(2): 894-903. doi: 10.3934/math.2020061
    [5] Erdal Bas, Ramazan Ozarslan . Theory of discrete fractional Sturm–Liouville equations and visual results. AIMS Mathematics, 2019, 4(3): 593-612. doi: 10.3934/math.2019.3.593
    [6] Dengfeng Lu, Shuwei Dai . On a class of three coupled fractional Schrödinger systems with general nonlinearities. AIMS Mathematics, 2023, 8(7): 17142-17153. doi: 10.3934/math.2023875
    [7] Mubashir Qayyum, Efaza Ahmad, Hijaz Ahmad, Bandar Almohsen . New solutions of time-space fractional coupled Schrödinger systems. AIMS Mathematics, 2023, 8(11): 27033-27051. doi: 10.3934/math.20231383
    [8] Xiaojun Zhou, Yue Dai . A spectral collocation method for the coupled system of nonlinear fractional differential equations. AIMS Mathematics, 2022, 7(4): 5670-5689. doi: 10.3934/math.2022314
    [9] Zunyuan Hu, Can Li, Shimin Guo . Fast finite difference/Legendre spectral collocation approximations for a tempered time-fractional diffusion equation. AIMS Mathematics, 2024, 9(12): 34647-34673. doi: 10.3934/math.20241650
    [10] Xiao-Yu Li, Yu-Lan Wang, Zhi-Yuan Li . Numerical simulation for the fractional-in-space Ginzburg-Landau equation using Fourier spectral method. AIMS Mathematics, 2023, 8(1): 2407-2418. doi: 10.3934/math.2023124
  • Heteroscedasticity diagnostics of error variance is essential before performing some statistical inference work. This paper is concerned with the statistical diagnostics for the varying coefficient partially nonlinear model. We propose a novel diagnostic approach for heteroscedasticity of error variance in the model by combining it with the empirical likelihood method. Under some mild conditions, the nonparametric version of the Wilks theorem is obtained. Furthermore, simulation studies and a real data analysis are implemented to evaluate the performances of our proposed approaches.



    Fractional calculus is a popular subject because of having a lot of application areas of theoretical and applied sciences, like engineering, physics, biology, etc. Discrete fractional calculus is more recent area than fractional calculus and it was first defined by Diaz–Osler [1], Miller–Ross [2] and Gray–Zhang [3]. More recently, the theory of discrete fractional calculus have begun to develop rapidly with Goodrich–Peterson [4], Baleanu et al. [5,6], Ahrendt et al. [7], Atici–Eloe [8,9], Anastassiou [10], Abdeljawad et al. [11,12,13,14,15,16], Hein et al. [17] and Cheng et al. [18], Mozyrska [19] and so forth [20,21,22,23,24,25].

    Fractional Sturm–Liouville differential operators have been studied by Bas et al. [26,27], Klimek et al.[28], Dehghan et al. [29]. Besides that, Sturm–Liouville differential and difference operators were studied by [30,31,32,33]. In this study, we define DFHA operators and prove the self–adjointness of DFHA operator, some spectral properties of the operator.

    More recently, Almeida et al. [34] have studied discrete and continuous fractional Sturm–Liouville operators, Bas–Ozarslan [35] have shown the self–adjointness of discrete fractional Sturm–Liouville operators and proved some spectral properties of the problem.

    Sturm–Liouville equation having hydrogen atom potential is defined as follows

    d2Rdr2+ardRdr(+1)r2R+(E+ar)R=0(0<r<).

    In quantum mechanics, the study of the energy levels of the hydrogen atom leads to this equation. Where R is the distance from the mass center to the origin, is a positive integer, a is real number E is energy constant and r is the distance between the nucleus and the electron.

    The hydrogen atom is a two–particle system and it composes of an electron and a proton. Interior motion of two particles around the center of mass corresponds to the movement of a single particle by a reduced mass. The distance between the proton and the electron is identified r and r is given by the orientation of the vector pointing from the proton to the electron. Hydrogen atom equation is defined as Schrödinger equation in spherical coordinates and in consequence of some transformations, this equation is defined as

    y+(λl(l+1)x2+2xq(x))y=0.

    Spectral theory of hydrogen atom equation is studied by [39,40,41]. Besides that, we can observe that hydrogen atom differential equation has series solution as follows ([39], p.268)

    y(x)=a0xl+1{1kl11!(2l+2).2xk+(kl1)(kl2)2!(2l+2)(2l+3)(2xk)2++(1)n(kl1)(kl2)3.2.1(k1)!(2l+2)(2l+3)(2l+n)(2xk)n},k=1,2, (1.1)

    Recently, Bohner and Cuchta [36,37] studied some special integer order discrete functions, like Laguerre, Hermite, Bessel and especially Cuchta mentioned the difficulty in obtaining series solution of discrete special functions in his dissertation ([38], p.100). In this regard, finding series solution of DFHA equations is an open problem and has some difficulties in the current situation. For this reason, we study to obtain solutions of DFHA eq.s in a different way with representation of solutions.

    In this study, we investigate DFHA equation in Riemann–Liouville and Grü nwald–Letnikov sense. The aim of this study is to contribute to the spectral theory of DFHA operator and behaviors of eigenfunctions and also to obtain the solution of DFHA equation.

    We investigate DFHA equation in three different ways;

    i) (nabla left and right) Riemann–Liouville (R–L)sense,

    L1x(t)=μa(bμx(t))+(l(l+1)t22t+q(t))x(t)=λx(t), 0<μ<1,

    ii) (delta left and right) Grünwald–Letnikov (G–L) sense,

    L2x(t)=Δμ(Δμ+x(t))+(l(l+1)t22t+q(t))x(t)=λx(t), 0<μ<1,

    iii) (nabla left) Riemann–Liouville (R–L)sense,

    L3x(t)=μa(μax(t))+(l(l+1)t22t+q(t))x(t)=λx(t), 0<μ<1.

    Definition 2.1. [42] Falling and rising factorial functions are defined as follows respectively

    tα_=Γ(t+1)Γ(tα+1), (2.1)
    t¯α=Γ(t+α)Γ(t), (2.2)

    where Γ is the gamma function, αR.

    Remark 2.1. Delta and nabla operators hold the following properties

    Δtα_=αtα1_,t¯α=αt¯α1. (2.3)

    Definition 2.2. [2,8,11] Nabla fractional sum operators are given as below,

    (i) The left fractional sum of order μ>0 is defined by

    μax(t)=1Γ(μ)ts=a+1(tρ(s))¯μ1x(s), tNa+1, (2.4)

    (ii) The right fractional sum of order μ>0 is defined by

    bμx(t)=1Γ(μ)b1s=t(sρ(t))¯μ1x(s), t b1N, (2.5)

    where ρ(t)=t1 is called backward jump operators, Na={a,a+1,...}, bN={b,b1,...}.

    Definition 2.3. [12,14] Nabla fractional difference operators are as follows,

    (i) The left fractional difference of order μ>0 is defined by

    μax(t)=n(nμ)ax(t)=nΓ(nμ)ts=a+1(tρ(s))¯nμ1x(s), tNa+1, (2.6)

    (ii) The right fractional difference of order μ>0 is defined by

    bμx(t)=(1)nn(nμ)ax(t)=(1)nΔnΓ(nμ)b1s=t(sρ(t))¯nμ1x(s), t b1N. (2.7)

    Fractional differences in (2.62.7) are called the Riemann–Liouville (R–L) definition of the μ-th order nabla fractional difference.

    Definition 2.4. [1,18] Fractional difference operators are given as follows

    (i) The delta left fractional difference of order μ, 0<μ1, is defined by

    Δμx(t)=1hμts=0(1)sμ(μ1)...(μs+1)s!x(ts), t=1,...,N. (2.8)

    (ii) The delta right fractional difference of order μ, 0<μ1, is defined by

    Δμ+x(t)=1hμNts=0(1)sμ(μ1)...(μs+1)s!x(t+s), t=0,..,N1, (2.9)

    fractional differences in (2.82.9) are called the Grünwald–Letnikov (G–L) definition of the μ-th order delta fractional difference.

    Definition 2.5 [14] Integration by parts formula for R–L nabla fractional difference operator is defined by, u is defined on bN and v is defined on Na,

    b1s=a+1u(s)μav(s)=b1s=a+1v(s)bμu(s). (2.10)

    Definition 2.6. [34] Integration by parts formula for G–L delta fractional difference operator is defined by, u, v is defined on {0,1,...,n}, then

    ns=0u(s)Δμv(s)=ns=0v(s)Δμ+u(s). (2.11)

    Definition 2.7. [17] f:NaR, s, Laplace transform is defined as follows,

    La{f}(s)=k=1(1s)k1f(a+k),

    where =C{1} and is called the set of regressive (complex) functions.

    Definition 2.8. [17] Let f,g:NaR, all tNa+1, convolution of f and g is defined as follows

    (fg)(t)=ts=a+1f(tρ(s)+a)g(s),

    where ρ(s) is the backward jump function defined in [42] as

    ρ(s)=s1.

    Theorem 2.1. [17] f,g:NaR, convolution theorem is expressed as follows,

    La{fg}(s)=La{f}La{g}(s).

    Lemma 2.1. [17] f:NaR, the following property is valid,

    La+1{f}(s)=11sLa{f}(s)11sf(a+1).

    Theorem 2.2. [17] f:NaR, 0<μ<1, Laplace transform of nabla fractional difference

    La+1{μaf}(s)=sμLa+1{f}(s)1sμ1sf(a+1),tNa+1.

    Definition 2.9. [17] For |p|<1, α>0, βR and tNa, Mittag–Leffler function is defined by

    Ep,α,β(t,a)=k=0pk(ta)¯αk+βΓ(αk+β+1).

    Theorem 2.3. [17] For |p|<1, α>0, βR, |1s|<1 and |s|α>p, Laplace transform of Mittag–Leffler function is as follows,

    La+1{Ep,α,β(.,a)}(s)=sαβ1sαp.

    Let us consider equations in three different forms;

    i) L1 DFHA operator L1 is defined in (nabla left and right) R–L sense,

    L1x(t)=μa(p(t)bμx(t))+(l(l+1)t22t+q(t))x(t)=λx(t), 0<μ<1, (3.1)

    where l is a positive integer or zero, q(t)+2tl(l+1)t2 are named potential function., λ is the spectral parameter, t[a+1,b1], x(t)l2[a+1,b1], a>0.

    ii) L2 DFHA operator L2 is defined in (delta left and right) G–L sense,

    L2x(t)=Δμ(p(t)Δμ+x(t))+(l(l+1)t22t+q(t))x(t)=λx(t), 0<μ<1, (3.2)

    where p,q,l,λ is as defined above, t[1,n], x(t)l2[0,n].

    iii) L3 DFHA operator L3 is defined in (nabla left) R–L sense,

    L3x(t)=μa(μax(t))+(l(l+1)t22t+q(t))x(t)=λx(t), 0<μ<1, (3.3)

    p,q,l,λ is as defined above, t[a+1,b1], a>0.

    Theorem 3.1. DFHA operator L1 is self–adjoint.

    Proof.

    u(t)L1v(t)=u(t)μa(p(t)bμv(t))+u(t)(l(l+1)t22t+q(t))v(t), (3.4)
    v(t)L1u(t)=v(t)μa(p(t)bμu(t))+v(t)(l(l+1)t22t+q(t))u(t). (3.5)

    Subtracting (1617) from each other

    u(t)L1v(t)v(t)L1u(t)=u(t)μa(p(t)bμv(t))v(t)μa(p(t)bμu(t))

    and applying definite sum operator to both side of the last equality, we have

    b1s=a+1(u(s)L1v(s)v(s)L1u(s))=b1s=a+1u(s)μa(p(s)bμv(s))b1s=a+1v(s)μa(p(s)bμu(s)). (3.6)

    Applying the integration by parts formula (2.10) to right hand side of (18), we have

    b1s=a+1(u(s)L1v(s)v(s)L1u(s))=b1s=a+1p(s)bμv(s)bμu(s)b1s=a+1p(s)bμu(s)bμv(s)=0,
    L1u,v=u,L1v.

    The proof completes.

    Theorem 3.2. Eigenfunctions, corresponding to distinct eigenvalues, of the equation (3.2) are orthogonal.

    Proof. Assume that λα and λβ are two different eigenvalues corresponds to eigenfunctions u(n) and v(n) respectively for the equation (3.1),

    μa(p(t)bμu(t))+(l(l+1)t22t+q(t))u(t)λαu(t)=0,μa(p(t)bμv(t))+(l(l+1)t22t+q(t))v(t)λβv(t)=0,

    Multiplying last two equations to v(n) and u(n) respectively, subtracting from each other and applying sum operator, since the self–adjointness of the operator L1, we get

    (λαλβ)b1s=a+1r(s)u(s)v(s)=0,

    since λαλβ,

    b1s=a+1r(s)u(s)v(s)=0,u(t),v(t)=0,

    and the proof completes.

    Theorem 3.3. All eigenvalues of the equation (3.1) are real.

    Proof. Assume λ=α+iβ, since the self–adjointness of the operator L1, we have

    L1u,u=u,L1u,λu,u=u,λu,
    (λ¯λ)u,u=0

    Since u,ur0,

    λ=¯λ

    and hence β=0. So, the proof is completed.

    Self–adjointness of L2 DFHA operator G–L sense, reality of eigenvalues and orthogonality of eigenfunctions of the equation 3.2 can be proven in a similar way to the Theorem 3.1–3.2–3.3 by means of Definition 2.5.

    Theorem 3.4.

    L3x(t)=μa(μax(t))+(l(l+1)t22t+q(t))x(t)=λx(t),0<μ<1, (3.7)
    x(a+1)=c1,μax(a+1)=c2, (3.8)

    where p(t)>0, r(t)>0, q(t) is defined and real valued, λ is the spectral parameter. The sum representation of solution of the problem (3.7)(3.8) is given as follows,

    x(t)=c1((1+l(l+1)(a+1)22a+1+q(a+1))Eλ,2μ,μ1(t,a)λEλ,2μ,2μ1(t,a))+c2(Eλ,2μ,2μ1(t,a)Eλ,2μ,μ1(t,a))ts=a+1Eλ,2μ,2μ1(tρ(s)+a)(l(l+1)s22s+q(s))x(s). (3.9)

    Proof. Taking Laplace transform of the equation (3.7) by Theorem 2.2 and take (l(l+1)t22t+q(t))x(t)=g(t),

    La+1{μa(μax)}(s)+La+1{g}(s)=λLa+1{x}(s),=sμLa+1{μax}(s)1sμ1sc2=λLa+1{x}(s)La+1{g}(s),=sμ(sμLa+1{x}(s)1sμ1sc1)1sμ1sc2=λLa+1{x}(s)La+1{g}(s),
    =La+1{x}(s)=1sμ1s1s2μλ(sμc1+c2)1s2μλLa+1{g}(s).

    Using Lemma 2.1, we have

    La{x}(s)=c1(sμλs2μλ)1ss2μλ(11sLa{g}(s)11sg(a+1))+c2(1sμs2μλ). (3.10)

    Now, taking inverse Laplace transform of the equation (3.10) and applying convolution theorem, then we have the representation of solution of the problem (3.7)(3.8), |λ|<1, |1s|<1 and |s|α>λ from Theorem 2.3., i.e.

    L1a{sμs2μλ}=Eλ,2μ,μ1(t,a),L1a{1s2μλ}=Eλ,2μ,2μ1(t,a),
    L1a{1s2μλLa{q(s)x(s)}}=ts=a+1Eλ,2μ,2μ1(tρ(s)+a)q(s)x(s).

    Consequently, we have sum representation of solution for DFHA problem 3.7–3.8

    x(t)=c1((1+l(l+1)(a+1)22a+1+q(a+1))Eλ,2μ,μ1(t,a)λEλ,2μ,2μ1(t,a))+c2(Eλ,2μ,2μ1(t,a)Eλ,2μ,μ1(t,a))ts=a+1Eλ,2μ,2μ1(tρ(s)+a)(l(l+1)s22s+q(s))x(s).

    Presume that c1=1,c2=0,a=0 in the representation of solution (3.9) and hence we may observe the behaviors of solutions in following figures (Figures 17) and tables (Tables 13);

    Figure 1.  q(t)=0,λ=0.01,l=1.
    Figure 2.  λ=0.01,μ=0.45,l=1.
    Figure 3.  q(t)=0,μ=0.4,l=1.
    Figure 4.  q(t)=0,l=1,λ=0.01,t=4.
    Figure 5.  q(t)=0,λ=0.01,l=0.01.
    Figure 6.  q(t)=0,λ=0.01,μ=0.9.
    Figure 7.  q(t)=0,μ=0.9,l=0.01.
    Table 1.  q(t)=0,λ=0.01,l=1.
    x(t) μ=0.3 μ=0.35 μ=0.4 μ=0.45 μ=0.5
    x(1) 1 1 1 1 1
    x(2) 0.612 0.714 1.123 0.918 1.020
    x(3) 0.700 0.900 1.515 1.370 1.641
    x(5) 0.881 1.336 2.402 2.747 3.773
    x(7) 1.009 1.740 3.352 4.566 7.031
    x(9) 1.099 2.100 4.332 6.749 11.461
    x(12) 1.190 2.570 5.745 10.623 20.450
    x(15) 1.249 2.975 6.739 15.149 32.472
    x(16) 1.264 3.098 7.235 16.793 37.198
    x(18) 1.289 3.330 8.233 20.279 47.789
    x(20) 1.309 3.544 9.229 24.021 59.967

     | Show Table
    DownLoad: CSV
    Table 2.  λ=0.01,μ=0.45,l=1.
    x(t) q(t)=1 q(t)=t q(t)=t
    x(1) 1 1 1
    x(2) 7.371017 4.411017 5.771017
    x(3) 0.131 0.057 0.088
    x(5) 0.123 0.018 0.049
    x(7) 0.080 0.006 0.021
    x(9) 0.050 0.003 0.011
    x(12) 0.028 0.001 0.005
    x(15) 0.017 0.0008 0.003
    x(16) 0.015 0.0006 0.0006
    x(18) 0.012 0.0005 0.002
    x(20) 0.010 0.0003 0.001

     | Show Table
    DownLoad: CSV
    Table 3.  q(t)=0,μ=0.4,l=1.
    x(t) λ=0.1 λ=0.11 λ=0.12
    x(1) 1 1 1
    x(2) 1 1.025 1.052
    x(3) 1.668 1.751 1.841
    x(5) 3.876 4.216 4.595
    x(7) 7.243 8.107 9.095
    x(9) 11.941 13.707 12.130
    x(12) 22.045 26.197 25.237
    x(15) 36.831 45.198 46.330
    x(16) 43.042 53.369 55.687
    x(18) 57.766 73.092 78.795
    x(20) 76.055 98.154 127.306

     | Show Table
    DownLoad: CSV

    We have analyzed DFHA equation in Riemann–Liouville and Grü nwald–Letnikov sense. Self–adjointness of the DFHA operator is presented and also, we have proved some significant spectral properties for instance, orthogonality of distinct eigenfunctions, reality of eigenvalues. Moreover, we give sum representation of the solutions for DFHA problem and find the solutions of the problem. We have carried out simulation analysis with graphics and tables. The aim of this paper is to contribute to the theory of hydrogen atom fractional difference operator.

    We observe the behaviors of solutions by changing the order of the derivative μ in Figure 1 and Figure 5, by changing the potential function q(t) in Figure 2, we compare solutions under different λ eigenvalues in Figure 3, and Figure 7, also we observe the solutions by changing μ with a specific eigenvalue in Figure 4 and by changing l values in Figure 6.

    We have shown the solutions by changing the order of the derivative μ in Table 1, by changing the potential function q(t) and λ eigenvalues in Table 2, Table 3.

    The authors would like to thank the editor and anonymous reviewers for their careful reading of our manuscript and their many insightful comments and suggestions.

    The authors declare no conflict of interest.



    [1] T. Li, C. Mei, Estimation and inference for varying coefficient partially nonlinear models, J. Stat. Plan. Infer., 143 (2013), 2023–2037. http://dx.doi.org/10.1016/j.jspi.2013.05.011 doi: 10.1016/j.jspi.2013.05.011
    [2] X. Zhou, P. Zhao, X. Wang, Empirical likelihood inferences for varying coefficient partially nonlinear models, J. Appl. Stat., 44 (2017), 474–492. http://dx.doi.org/10.1080/02664763.2016.1177496 doi: 10.1080/02664763.2016.1177496
    [3] Y. Jiang, Q. Ji, B. Xie, Robust estimation for the varying coefficient partially nonlinear models, J. Comput. Appl. Math., 326 (2017), 31–43. http://dx.doi.org/10.1016/j.cam.2017.04.028 doi: 10.1016/j.cam.2017.04.028
    [4] Y. Xiao, Z. Chen, Bias-corrected estimations in varying-coefficient partially nonlinear models with measurement error in the nonparametric part, J. Appl. Stat., 45 (2018), 586–603. http://dx.doi.org/10.1080/02664763.2017.1288201 doi: 10.1080/02664763.2017.1288201
    [5] S. Dai, Z. Huang, Estimation for varying coefficient partially nonlinear models with distorted measurement errors, J. Korean Stat. Soc., 48 (2019), 117–133. http://dx.doi.org/10.1016/j.jkss.2018.09.001 doi: 10.1016/j.jkss.2018.09.001
    [6] Y. Qian, Z. Huang, Statistical inference for a varying-coefficient partially nonlinear model with measurement errors, Stat. Methodol., 32 (2016), 122130. http://dx.doi.org/10.1016/j.stamet.2016.05.004 doi: 10.1016/j.stamet.2016.05.004
    [7] X. Wang, P. Zhao, H. Du, Statistical inferences for varying coefficient partially nonlinear model with missing covariates, Commun. Stat.-Theor. M., 50 (2021), 2599–2618. http://dx.doi.org/10.1080/03610926.2019.1674870 doi: 10.1080/03610926.2019.1674870
    [8] L. Xia, X. Wang, P. Zhao, Y. Song, Empirical likelihood for varying coefficient partially nonlinear model with missing responses, AIMS Mathematics, 6 (2021), 7125–7152. http://dx.doi.org/10.3934/math.2021418 doi: 10.3934/math.2021418
    [9] Y. Xiao, L. Liang, Robust estimation and variable selection for varying-coefficient partially nonlinear models based on modal regression, J. Korean Stat. Soc., 51 (2020), 692–715. http://dx.doi.org/10.1007/s42952-021-00158-w doi: 10.1007/s42952-021-00158-w
    [10] Y. Xiao, Y. Shi, Robust estimation for varying-coefficient partially nonlinear model with nonignorable missing response, AIMS Mathematics, 8 (2023), 29849–29871. http://dx.doi.org/10.3934/math.20231526 doi: 10.3934/math.20231526
    [11] X. Zhou, P. Zhao, Estimation and inferences for varying coefficient partially nonlinear quantile models with censoring indicators missing at random, Comput. Stat., 37 (2022), 1727–1750. http://dx.doi.org/10.1007/s00180-021-01192-2 doi: 10.1007/s00180-021-01192-2
    [12] H. Wong, F. Liu, M. Chen, W. Ip, Empirical likelihood based diagnostics for heteroscedasticity in partial linear models, Comput. Stat. Data Anal., 53 (2009), 3466–3477. http://dx.doi.org/10.1016/j.csda.2009.02.029 doi: 10.1016/j.csda.2009.02.029
    [13] H. Wong, F. Liu, M. Chen, W. Ip, Empirical likelihood based diagnostics for heteroscedasticity in partially linear errors-in-variables models, J. Stat. Plan. Infer., 139 (2009), 916–929. http://dx.doi.org/10.1016/j.jspi.2008.05.049 doi: 10.1016/j.jspi.2008.05.049
    [14] G. Fan, H. Liang, J. Wang, Empirical likelihood for a heteroscedastic partial linear errors-in-variables model, Commun. Stat.-Theor. M., 41 (2012), 108–127. http://dx.doi.org/10.1080/03610926.2010.517357 doi: 10.1080/03610926.2010.517357
    [15] J. Lin, Y. Zhao, H. Wang, Heteroscedasticity diagnostics in varying-coefficient partially linear regression models and applications in analyzing Boston housing data, J. Applied Statistics, 42 (2015), 2432–2448. http://dx.doi.org/10.1080/02664763.2015.1043623 doi: 10.1080/02664763.2015.1043623
    [16] F. Liu, C. Li, P. Wang, X. Kang, Empirical likelihood based diagnostics for heteroscedasticity in semiparametric varying-coefficient partially linear errors-in-variables models, Commun. Stat.-Theor. M., 47 (2018), 5485–5496. http://dx.doi.org/10.1080/03610926.2017.1395050 doi: 10.1080/03610926.2017.1395050
    [17] F. Liu, W. Gao, J. He, X. Fu, X. Kang, Empirical likelihood based diagnostics for heteroscedasticity in semiparametric varying-coefficient partially linear models with missing responses, J. Syst. Sci. Complex., 34 (2021), 1175–1188. http://dx.doi.org/10.1007/s11424-020-9240-7 doi: 10.1007/s11424-020-9240-7
    [18] A. Owen, Empirical likelihood ratio confidence intervals for single functional, Biometrika, 75 (1988), 237–249. http://dx.doi.org/10.1093/biomet/75.2.237 doi: 10.1093/biomet/75.2.237
    [19] A. Owen, Empirical likelihood ratio confidence regions, Ann. Stat., 18 (1990), 90–120. http://dx.doi.org/10.1214/aos/1176347494 doi: 10.1214/aos/1176347494
    [20] A. Owen, Empirical likelihood for linear models, Ann. Stat., 19 (1991), 1725–1747. http://dx.doi.org/10.1214/aos/1176348368 doi: 10.1214/aos/1176348368
    [21] J. Fan, T. Huang, Profile likelihood inferences on semiparametric varying-coefficient partially linear models, Bernoulli, 11 (2005), 1031–1057. http://dx.doi.org/10.3150/bj/1137421639 doi: 10.3150/bj/1137421639
    [22] Y. Sun, H. Yan, W. Zhang, Z. Lu, A semiparametric spatial dynamic model, Ann. Stat., 42 (2014), 700–727. http://dx.doi.org/10.1214/13-AOS1201 doi: 10.1214/13-AOS1201
    [23] J. Du, X. Sun, R. Cao, Z. Zhang, Statistical inference for partially linear additive spatial autoregressive models, Spat. Stat., 25 (2018), 52–67. http://dx.doi.org/10.1016/j.spasta.2018.04.008 doi: 10.1016/j.spasta.2018.04.008
    [24] S. Cheng, J. Chen, Estimation of partially linear single-index spatialautoregressive model, Stat. Papers, 62 (2021), 495–531. http://dx.doi.org/10.1007/s00362-019-01105-y doi: 10.1007/s00362-019-01105-y
    [25] H. Zhang, G. Cheng, Y. Liu, Linear or nonlinear? Automatic structure discovery for partially linear models, J. Am. Stat. Assoc., 106 (2011), 1099–1112. http://dx.doi.org/10.1198/jasa.2011.tm10281 doi: 10.1198/jasa.2011.tm10281
    [26] S. Wang, M. Wu, Z. Jia, Inequalities in matrix theory (Chinese), 2 Eds., Beijing: Science Press, 2006.
    [27] J. Gao, Asymptotic theory for partly linear models, Commun. Stat.-Theor. M., 24 (1995), 1985–2009. http://dx.doi.org/10.1080/03610929508831598 doi: 10.1080/03610929508831598
  • This article has been cited by:

    1. Erdal Bas, Funda Metin Turk, Ramazan Ozarslan, Ahu Ercan, Spectral data of conformable Sturm–Liouville direct problems, 2021, 11, 1664-2368, 10.1007/s13324-020-00428-6
    2. Tom Cuchta, Dallas Freeman, Discrete Polylogarithm Functions, 2023, 84, 1338-9750, 19, 10.2478/tmmp-2023-0012
    3. B. Shiri, Y. Guang, D. Baleanu, Inverse problems for discrete Hermite nabla difference equation, 2025, 33, 2769-0911, 10.1080/27690911.2024.2431000
    4. Muhammad Sulthan Zacky, Heru Sukamto, Lila Yuwana, Agus Purwanto, Eny Latifah, The performance of space-fractional quantum carnot engine, 2025, 100, 0031-8949, 025306, 10.1088/1402-4896/ada9de
  • Reader Comments
  • © 2024 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(478) PDF downloads(42) Cited by(0)

Figures and Tables

Figures(2)  /  Tables(1)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog