Semiparametric model with finite-dimensional nuisance parameters












0












$begingroup$


The finite-dimensional parameter vector $boldsymbol{gamma} in mathbb{R}^q$ is partitioned in two sub-vectors, i.e. $boldsymbol{gamma} = (boldsymbol{phi}^T,boldsymbol{zeta}^T)^T in Psi times Xi subseteqmathbb{R}^n times mathbb{R}^k$ with $n+k=q$. In particular, $boldsymbol{phi}$ represents the sub-vector of the parameters of interest, while $boldsymbol{zeta}$ is the sub-vector of the finite-dimensional nuisance parameters. Let us define a semi-parametric model as follows:
begin{equation}
mathcal{P}_{boldsymbol{phi},boldsymbol{zeta},l} triangleq leftlbrace p_X | p_X(mathbf{x}|boldsymbol{phi},boldsymbol{zeta},l), boldsymbol{phi} in Psi, boldsymbol{zeta} in Xi, l in mathcal{L} rightrbrace ,
end{equation}
where $l in mathcal{L}$ is a nuisance function.
Let $p_0(mathbf{x})=p_X(mathbf{x}|boldsymbol{phi}_0,boldsymbol{zeta}_0,l_0)$ be the true pdf and




  • let
    begin{equation}
    mathbf{s}_{boldsymbol{phi}_0}(mathbf{x})=nabla_{boldsymbol{phi}}ln p_X(mathbf{x}|boldsymbol{phi}_0,boldsymbol{zeta}_0,l_0)
    end{equation}
    be the score vector of the vector of the parameters of interest. We indicate with $mathcal{T}_{boldsymbol{phi}_0}$ the relative tangent space.

  • let
    begin{equation}
    mathbf{s}_{boldsymbol{zeta}_0}(mathbf{x})=nabla_{boldsymbol{zeta}}ln p_X(mathbf{x}|boldsymbol{phi}_0,boldsymbol{zeta}_0,l_0)
    end{equation}
    be the score vector of the finite-dimensional nuisance parameters. We indicate with $mathcal{T}_{boldsymbol{zeta}_0}$ the relative tangent space.


  • let $mathcal{T}_{l_0}$ the semiparametric nuisance tangent space.



    I need to find the inverse of the efficient Fisher Information Matrix (FIM) for $boldsymbol{phi}_0$, i.e. $overline{mathrm{FIM}}(boldsymbol{phi}_0)^{-1}$.




I followed this procedure:



1) By indicating with $mathbf{s}_{boldsymbol{gamma}_0} = [mathbf{s}_{boldsymbol{phi}_0}^T, ; mathbf{s}_{boldsymbol{zeta}_0}^T]^T$,
evaluate the efficient score vector for $boldsymbol{gamma}_0$ as:
begin{equation}
bar{mathbf{s}}_{boldsymbol{gamma}_0}(mathbf{x}) = mathbf{s}_{boldsymbol{gamma}_0}(mathbf{x}) - Pi(mathbf{s}_{boldsymbol{gamma}_0}(mathbf{x})|mathcal{T}_{l_0}).
end{equation}
2) Then, obtain the efficient FIM for $boldsymbol{gamma}_0$ as:
begin{equation}
overline{mathrm{FIM}}(boldsymbol{gamma}_0) = E_0{bar{mathbf{s}}_{boldsymbol{gamma}_0}bar{mathbf{s}}_{boldsymbol{gamma}_0}^T} triangleq
left( begin{array}{cc}
F_{boldsymbol{phi}boldsymbol{phi}} & F_{boldsymbol{phi}boldsymbol{zeta}}\
F_{boldsymbol{zeta}boldsymbol{phi}} & F_{boldsymbol{zeta}boldsymbol{zeta}}
end{array}right) ,
end{equation}
3) Use the Woodbury matrix identity to evaluate the inverse of the $n times n$ top-left sub-matrix of $overline{mathrm{FIM}}(boldsymbol{gamma}_0)$ as:
begin{equation}
overline{mathrm{FIM}}(boldsymbol{phi}_0)^{-1} = (F_{boldsymbol{phi}boldsymbol{phi}} - F_{boldsymbol{phi}boldsymbol{zeta}}F_{boldsymbol{zeta}boldsymbol{zeta}}^{-1}F_{boldsymbol{zeta}boldsymbol{phi}}^T)^{-1}.
end{equation}



Is it correct?






probability probability-theory statistics probability-distributions statistical-inference







share|cite|improve this question











$endgroup$

















    0












    $begingroup$


    The finite-dimensional parameter vector $boldsymbol{gamma} in mathbb{R}^q$ is partitioned in two sub-vectors, i.e. $boldsymbol{gamma} = (boldsymbol{phi}^T,boldsymbol{zeta}^T)^T in Psi times Xi subseteqmathbb{R}^n times mathbb{R}^k$ with $n+k=q$. In particular, $boldsymbol{phi}$ represents the sub-vector of the parameters of interest, while $boldsymbol{zeta}$ is the sub-vector of the finite-dimensional nuisance parameters. Let us define a semi-parametric model as follows:
    begin{equation}
    mathcal{P}_{boldsymbol{phi},boldsymbol{zeta},l} triangleq leftlbrace p_X | p_X(mathbf{x}|boldsymbol{phi},boldsymbol{zeta},l), boldsymbol{phi} in Psi, boldsymbol{zeta} in Xi, l in mathcal{L} rightrbrace ,
    end{equation}
    where $l in mathcal{L}$ is a nuisance function.
    Let $p_0(mathbf{x})=p_X(mathbf{x}|boldsymbol{phi}_0,boldsymbol{zeta}_0,l_0)$ be the true pdf and




    • let
      begin{equation}
      mathbf{s}_{boldsymbol{phi}_0}(mathbf{x})=nabla_{boldsymbol{phi}}ln p_X(mathbf{x}|boldsymbol{phi}_0,boldsymbol{zeta}_0,l_0)
      end{equation}
      be the score vector of the vector of the parameters of interest. We indicate with $mathcal{T}_{boldsymbol{phi}_0}$ the relative tangent space.

    • let
      begin{equation}
      mathbf{s}_{boldsymbol{zeta}_0}(mathbf{x})=nabla_{boldsymbol{zeta}}ln p_X(mathbf{x}|boldsymbol{phi}_0,boldsymbol{zeta}_0,l_0)
      end{equation}
      be the score vector of the finite-dimensional nuisance parameters. We indicate with $mathcal{T}_{boldsymbol{zeta}_0}$ the relative tangent space.


    • let $mathcal{T}_{l_0}$ the semiparametric nuisance tangent space.



      I need to find the inverse of the efficient Fisher Information Matrix (FIM) for $boldsymbol{phi}_0$, i.e. $overline{mathrm{FIM}}(boldsymbol{phi}_0)^{-1}$.




    I followed this procedure:



    1) By indicating with $mathbf{s}_{boldsymbol{gamma}_0} = [mathbf{s}_{boldsymbol{phi}_0}^T, ; mathbf{s}_{boldsymbol{zeta}_0}^T]^T$,
    evaluate the efficient score vector for $boldsymbol{gamma}_0$ as:
    begin{equation}
    bar{mathbf{s}}_{boldsymbol{gamma}_0}(mathbf{x}) = mathbf{s}_{boldsymbol{gamma}_0}(mathbf{x}) - Pi(mathbf{s}_{boldsymbol{gamma}_0}(mathbf{x})|mathcal{T}_{l_0}).
    end{equation}
    2) Then, obtain the efficient FIM for $boldsymbol{gamma}_0$ as:
    begin{equation}
    overline{mathrm{FIM}}(boldsymbol{gamma}_0) = E_0{bar{mathbf{s}}_{boldsymbol{gamma}_0}bar{mathbf{s}}_{boldsymbol{gamma}_0}^T} triangleq
    left( begin{array}{cc}
    F_{boldsymbol{phi}boldsymbol{phi}} & F_{boldsymbol{phi}boldsymbol{zeta}}\
    F_{boldsymbol{zeta}boldsymbol{phi}} & F_{boldsymbol{zeta}boldsymbol{zeta}}
    end{array}right) ,
    end{equation}
    3) Use the Woodbury matrix identity to evaluate the inverse of the $n times n$ top-left sub-matrix of $overline{mathrm{FIM}}(boldsymbol{gamma}_0)$ as:
    begin{equation}
    overline{mathrm{FIM}}(boldsymbol{phi}_0)^{-1} = (F_{boldsymbol{phi}boldsymbol{phi}} - F_{boldsymbol{phi}boldsymbol{zeta}}F_{boldsymbol{zeta}boldsymbol{zeta}}^{-1}F_{boldsymbol{zeta}boldsymbol{phi}}^T)^{-1}.
    end{equation}



    Is it correct?






    probability probability-theory statistics probability-distributions statistical-inference







    share|cite|improve this question











    $endgroup$















      0












      0








      0





      $begingroup$


      The finite-dimensional parameter vector $boldsymbol{gamma} in mathbb{R}^q$ is partitioned in two sub-vectors, i.e. $boldsymbol{gamma} = (boldsymbol{phi}^T,boldsymbol{zeta}^T)^T in Psi times Xi subseteqmathbb{R}^n times mathbb{R}^k$ with $n+k=q$. In particular, $boldsymbol{phi}$ represents the sub-vector of the parameters of interest, while $boldsymbol{zeta}$ is the sub-vector of the finite-dimensional nuisance parameters. Let us define a semi-parametric model as follows:
      begin{equation}
      mathcal{P}_{boldsymbol{phi},boldsymbol{zeta},l} triangleq leftlbrace p_X | p_X(mathbf{x}|boldsymbol{phi},boldsymbol{zeta},l), boldsymbol{phi} in Psi, boldsymbol{zeta} in Xi, l in mathcal{L} rightrbrace ,
      end{equation}
      where $l in mathcal{L}$ is a nuisance function.
      Let $p_0(mathbf{x})=p_X(mathbf{x}|boldsymbol{phi}_0,boldsymbol{zeta}_0,l_0)$ be the true pdf and




      • let
        begin{equation}
        mathbf{s}_{boldsymbol{phi}_0}(mathbf{x})=nabla_{boldsymbol{phi}}ln p_X(mathbf{x}|boldsymbol{phi}_0,boldsymbol{zeta}_0,l_0)
        end{equation}
        be the score vector of the vector of the parameters of interest. We indicate with $mathcal{T}_{boldsymbol{phi}_0}$ the relative tangent space.

      • let
        begin{equation}
        mathbf{s}_{boldsymbol{zeta}_0}(mathbf{x})=nabla_{boldsymbol{zeta}}ln p_X(mathbf{x}|boldsymbol{phi}_0,boldsymbol{zeta}_0,l_0)
        end{equation}
        be the score vector of the finite-dimensional nuisance parameters. We indicate with $mathcal{T}_{boldsymbol{zeta}_0}$ the relative tangent space.


      • let $mathcal{T}_{l_0}$ the semiparametric nuisance tangent space.



        I need to find the inverse of the efficient Fisher Information Matrix (FIM) for $boldsymbol{phi}_0$, i.e. $overline{mathrm{FIM}}(boldsymbol{phi}_0)^{-1}$.




      I followed this procedure:



      1) By indicating with $mathbf{s}_{boldsymbol{gamma}_0} = [mathbf{s}_{boldsymbol{phi}_0}^T, ; mathbf{s}_{boldsymbol{zeta}_0}^T]^T$,
      evaluate the efficient score vector for $boldsymbol{gamma}_0$ as:
      begin{equation}
      bar{mathbf{s}}_{boldsymbol{gamma}_0}(mathbf{x}) = mathbf{s}_{boldsymbol{gamma}_0}(mathbf{x}) - Pi(mathbf{s}_{boldsymbol{gamma}_0}(mathbf{x})|mathcal{T}_{l_0}).
      end{equation}
      2) Then, obtain the efficient FIM for $boldsymbol{gamma}_0$ as:
      begin{equation}
      overline{mathrm{FIM}}(boldsymbol{gamma}_0) = E_0{bar{mathbf{s}}_{boldsymbol{gamma}_0}bar{mathbf{s}}_{boldsymbol{gamma}_0}^T} triangleq
      left( begin{array}{cc}
      F_{boldsymbol{phi}boldsymbol{phi}} & F_{boldsymbol{phi}boldsymbol{zeta}}\
      F_{boldsymbol{zeta}boldsymbol{phi}} & F_{boldsymbol{zeta}boldsymbol{zeta}}
      end{array}right) ,
      end{equation}
      3) Use the Woodbury matrix identity to evaluate the inverse of the $n times n$ top-left sub-matrix of $overline{mathrm{FIM}}(boldsymbol{gamma}_0)$ as:
      begin{equation}
      overline{mathrm{FIM}}(boldsymbol{phi}_0)^{-1} = (F_{boldsymbol{phi}boldsymbol{phi}} - F_{boldsymbol{phi}boldsymbol{zeta}}F_{boldsymbol{zeta}boldsymbol{zeta}}^{-1}F_{boldsymbol{zeta}boldsymbol{phi}}^T)^{-1}.
      end{equation}



      Is it correct?






      probability probability-theory statistics probability-distributions statistical-inference







      share|cite|improve this question











      $endgroup$




      The finite-dimensional parameter vector $boldsymbol{gamma} in mathbb{R}^q$ is partitioned in two sub-vectors, i.e. $boldsymbol{gamma} = (boldsymbol{phi}^T,boldsymbol{zeta}^T)^T in Psi times Xi subseteqmathbb{R}^n times mathbb{R}^k$ with $n+k=q$. In particular, $boldsymbol{phi}$ represents the sub-vector of the parameters of interest, while $boldsymbol{zeta}$ is the sub-vector of the finite-dimensional nuisance parameters. Let us define a semi-parametric model as follows:
      begin{equation}
      mathcal{P}_{boldsymbol{phi},boldsymbol{zeta},l} triangleq leftlbrace p_X | p_X(mathbf{x}|boldsymbol{phi},boldsymbol{zeta},l), boldsymbol{phi} in Psi, boldsymbol{zeta} in Xi, l in mathcal{L} rightrbrace ,
      end{equation}
      where $l in mathcal{L}$ is a nuisance function.
      Let $p_0(mathbf{x})=p_X(mathbf{x}|boldsymbol{phi}_0,boldsymbol{zeta}_0,l_0)$ be the true pdf and




      • let
        begin{equation}
        mathbf{s}_{boldsymbol{phi}_0}(mathbf{x})=nabla_{boldsymbol{phi}}ln p_X(mathbf{x}|boldsymbol{phi}_0,boldsymbol{zeta}_0,l_0)
        end{equation}
        be the score vector of the vector of the parameters of interest. We indicate with $mathcal{T}_{boldsymbol{phi}_0}$ the relative tangent space.

      • let
        begin{equation}
        mathbf{s}_{boldsymbol{zeta}_0}(mathbf{x})=nabla_{boldsymbol{zeta}}ln p_X(mathbf{x}|boldsymbol{phi}_0,boldsymbol{zeta}_0,l_0)
        end{equation}
        be the score vector of the finite-dimensional nuisance parameters. We indicate with $mathcal{T}_{boldsymbol{zeta}_0}$ the relative tangent space.


      • let $mathcal{T}_{l_0}$ the semiparametric nuisance tangent space.



        I need to find the inverse of the efficient Fisher Information Matrix (FIM) for $boldsymbol{phi}_0$, i.e. $overline{mathrm{FIM}}(boldsymbol{phi}_0)^{-1}$.




      I followed this procedure:



      1) By indicating with $mathbf{s}_{boldsymbol{gamma}_0} = [mathbf{s}_{boldsymbol{phi}_0}^T, ; mathbf{s}_{boldsymbol{zeta}_0}^T]^T$,
      evaluate the efficient score vector for $boldsymbol{gamma}_0$ as:
      begin{equation}
      bar{mathbf{s}}_{boldsymbol{gamma}_0}(mathbf{x}) = mathbf{s}_{boldsymbol{gamma}_0}(mathbf{x}) - Pi(mathbf{s}_{boldsymbol{gamma}_0}(mathbf{x})|mathcal{T}_{l_0}).
      end{equation}
      2) Then, obtain the efficient FIM for $boldsymbol{gamma}_0$ as:
      begin{equation}
      overline{mathrm{FIM}}(boldsymbol{gamma}_0) = E_0{bar{mathbf{s}}_{boldsymbol{gamma}_0}bar{mathbf{s}}_{boldsymbol{gamma}_0}^T} triangleq
      left( begin{array}{cc}
      F_{boldsymbol{phi}boldsymbol{phi}} & F_{boldsymbol{phi}boldsymbol{zeta}}\
      F_{boldsymbol{zeta}boldsymbol{phi}} & F_{boldsymbol{zeta}boldsymbol{zeta}}
      end{array}right) ,
      end{equation}
      3) Use the Woodbury matrix identity to evaluate the inverse of the $n times n$ top-left sub-matrix of $overline{mathrm{FIM}}(boldsymbol{gamma}_0)$ as:
      begin{equation}
      overline{mathrm{FIM}}(boldsymbol{phi}_0)^{-1} = (F_{boldsymbol{phi}boldsymbol{phi}} - F_{boldsymbol{phi}boldsymbol{zeta}}F_{boldsymbol{zeta}boldsymbol{zeta}}^{-1}F_{boldsymbol{zeta}boldsymbol{phi}}^T)^{-1}.
      end{equation}



      Is it correct?






      probability probability-theory statistics probability-distributions statistical-inference




      probability probability-theory statistics probability-distributions statistical-inference






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Jan 28 at 13:40







      Vuk

















      asked Jan 28 at 13:17









      VukVuk

      768




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