Integration by parts with empirical measure

The name of the pictureThe name of the pictureThe name of the pictureClash Royale CLAN TAG#URR8PPP











up vote
1
down vote

favorite












I'm currently reading through the paper Asymptotic normality of nearest neighbor regression function estimates and am struggling to understand the asymptotic equalities that were shown in the proof of Lemma 4 (p. 923) where the last one was proven by using integration by parts:



$beginalign*&qquad fracm(x_0)sqrta_n^3int left[alpha_n(x_0)-alpha_n(x)right]K'left(fracF(x_0)-F(x)a_nright)F(d x)\ &= -fracm(x_0)sqrta_n^3int alpha_n(x)K'left(fracF(x_0)-F(x)a_nright)F(d x)\
&= -fracm(x_0)sqrta_nint Kleft(fracF(x_0)-F(x)a_nright)alpha_n(dx). endalign*$



Here $m(x) = E[Y|X=x]$, $alpha_n(x) = sqrtn [F_n(x)-F(x)]$ denotes the empirical process of the random variable $X$ (having continuous distribution function $F$), $K$ being a twice continuously differentiable probability kernel with bounded support and $na_n^3 to infty$, $a_n to 0$ for $n to infty$.



I know that I could expand the second to last term like so:
$beginalign
&quad-fracm(x_0)sqrta_n^3int alpha_n(x)K'left(fracF(x_0)-F(x)a_nright)F(d x)\
&= -fracm(x_0)sqrtnsqrta_n^3int F_n(x)K'left(fracF(x_0)-F(x)a_nright)F(d x) + fracm(x_0)sqrtnsqrta_n^3int F(x)K'left(fracF(x_0)-F(x)a_nright)F(d x)\ endalign$



where the second integral contains transformations with $F$, so a transformation of variables and the integral a la $u = F(x)$ is possible and then application of the "classic" integration by parts yields the first of the following terms and a "constant" term.



$beginalign &= fracm(x_0)sqrtnsqrta_nint Kleft(fracF(x_0)-F(x)a_nright) F(d x) -fracm(x_0)sqrtnsqrta_nint Kleft(fracF(x_0)-F(x)a_nright) F_n(d x)\
&= -frac1sqrta_nm(x_0)int Kleft(fracF(x_0)-F(x)a_nright) alpha_n(d x)endalign$



It would be nice if someone could provide me an explanation on the upper equalities since I'm not even sure if I'm on the right track. I only consider the classic integration by parts, maybe the Lebesgue-Stieltjes variant is more helpful.




integration probability-theory measure-theory regression empirical-processes




share|cite|improve this question























    up vote
    1
    down vote

    favorite












    I'm currently reading through the paper Asymptotic normality of nearest neighbor regression function estimates and am struggling to understand the asymptotic equalities that were shown in the proof of Lemma 4 (p. 923) where the last one was proven by using integration by parts:



    $beginalign*&qquad fracm(x_0)sqrta_n^3int left[alpha_n(x_0)-alpha_n(x)right]K'left(fracF(x_0)-F(x)a_nright)F(d x)\ &= -fracm(x_0)sqrta_n^3int alpha_n(x)K'left(fracF(x_0)-F(x)a_nright)F(d x)\
    &= -fracm(x_0)sqrta_nint Kleft(fracF(x_0)-F(x)a_nright)alpha_n(dx). endalign*$



    Here $m(x) = E[Y|X=x]$, $alpha_n(x) = sqrtn [F_n(x)-F(x)]$ denotes the empirical process of the random variable $X$ (having continuous distribution function $F$), $K$ being a twice continuously differentiable probability kernel with bounded support and $na_n^3 to infty$, $a_n to 0$ for $n to infty$.



    I know that I could expand the second to last term like so:
    $beginalign
    &quad-fracm(x_0)sqrta_n^3int alpha_n(x)K'left(fracF(x_0)-F(x)a_nright)F(d x)\
    &= -fracm(x_0)sqrtnsqrta_n^3int F_n(x)K'left(fracF(x_0)-F(x)a_nright)F(d x) + fracm(x_0)sqrtnsqrta_n^3int F(x)K'left(fracF(x_0)-F(x)a_nright)F(d x)\ endalign$



    where the second integral contains transformations with $F$, so a transformation of variables and the integral a la $u = F(x)$ is possible and then application of the "classic" integration by parts yields the first of the following terms and a "constant" term.



    $beginalign &= fracm(x_0)sqrtnsqrta_nint Kleft(fracF(x_0)-F(x)a_nright) F(d x) -fracm(x_0)sqrtnsqrta_nint Kleft(fracF(x_0)-F(x)a_nright) F_n(d x)\
    &= -frac1sqrta_nm(x_0)int Kleft(fracF(x_0)-F(x)a_nright) alpha_n(d x)endalign$



    It would be nice if someone could provide me an explanation on the upper equalities since I'm not even sure if I'm on the right track. I only consider the classic integration by parts, maybe the Lebesgue-Stieltjes variant is more helpful.




    integration probability-theory measure-theory regression empirical-processes




    share|cite|improve this question





















      up vote
      1
      down vote

      favorite









      up vote
      1
      down vote

      favorite











      I'm currently reading through the paper Asymptotic normality of nearest neighbor regression function estimates and am struggling to understand the asymptotic equalities that were shown in the proof of Lemma 4 (p. 923) where the last one was proven by using integration by parts:



      $beginalign*&qquad fracm(x_0)sqrta_n^3int left[alpha_n(x_0)-alpha_n(x)right]K'left(fracF(x_0)-F(x)a_nright)F(d x)\ &= -fracm(x_0)sqrta_n^3int alpha_n(x)K'left(fracF(x_0)-F(x)a_nright)F(d x)\
      &= -fracm(x_0)sqrta_nint Kleft(fracF(x_0)-F(x)a_nright)alpha_n(dx). endalign*$



      Here $m(x) = E[Y|X=x]$, $alpha_n(x) = sqrtn [F_n(x)-F(x)]$ denotes the empirical process of the random variable $X$ (having continuous distribution function $F$), $K$ being a twice continuously differentiable probability kernel with bounded support and $na_n^3 to infty$, $a_n to 0$ for $n to infty$.



      I know that I could expand the second to last term like so:
      $beginalign
      &quad-fracm(x_0)sqrta_n^3int alpha_n(x)K'left(fracF(x_0)-F(x)a_nright)F(d x)\
      &= -fracm(x_0)sqrtnsqrta_n^3int F_n(x)K'left(fracF(x_0)-F(x)a_nright)F(d x) + fracm(x_0)sqrtnsqrta_n^3int F(x)K'left(fracF(x_0)-F(x)a_nright)F(d x)\ endalign$



      where the second integral contains transformations with $F$, so a transformation of variables and the integral a la $u = F(x)$ is possible and then application of the "classic" integration by parts yields the first of the following terms and a "constant" term.



      $beginalign &= fracm(x_0)sqrtnsqrta_nint Kleft(fracF(x_0)-F(x)a_nright) F(d x) -fracm(x_0)sqrtnsqrta_nint Kleft(fracF(x_0)-F(x)a_nright) F_n(d x)\
      &= -frac1sqrta_nm(x_0)int Kleft(fracF(x_0)-F(x)a_nright) alpha_n(d x)endalign$



      It would be nice if someone could provide me an explanation on the upper equalities since I'm not even sure if I'm on the right track. I only consider the classic integration by parts, maybe the Lebesgue-Stieltjes variant is more helpful.




      integration probability-theory measure-theory regression empirical-processes




      share|cite|improve this question











      I'm currently reading through the paper Asymptotic normality of nearest neighbor regression function estimates and am struggling to understand the asymptotic equalities that were shown in the proof of Lemma 4 (p. 923) where the last one was proven by using integration by parts:



      $beginalign*&qquad fracm(x_0)sqrta_n^3int left[alpha_n(x_0)-alpha_n(x)right]K'left(fracF(x_0)-F(x)a_nright)F(d x)\ &= -fracm(x_0)sqrta_n^3int alpha_n(x)K'left(fracF(x_0)-F(x)a_nright)F(d x)\
      &= -fracm(x_0)sqrta_nint Kleft(fracF(x_0)-F(x)a_nright)alpha_n(dx). endalign*$



      Here $m(x) = E[Y|X=x]$, $alpha_n(x) = sqrtn [F_n(x)-F(x)]$ denotes the empirical process of the random variable $X$ (having continuous distribution function $F$), $K$ being a twice continuously differentiable probability kernel with bounded support and $na_n^3 to infty$, $a_n to 0$ for $n to infty$.



      I know that I could expand the second to last term like so:
      $beginalign
      &quad-fracm(x_0)sqrta_n^3int alpha_n(x)K'left(fracF(x_0)-F(x)a_nright)F(d x)\
      &= -fracm(x_0)sqrtnsqrta_n^3int F_n(x)K'left(fracF(x_0)-F(x)a_nright)F(d x) + fracm(x_0)sqrtnsqrta_n^3int F(x)K'left(fracF(x_0)-F(x)a_nright)F(d x)\ endalign$



      where the second integral contains transformations with $F$, so a transformation of variables and the integral a la $u = F(x)$ is possible and then application of the "classic" integration by parts yields the first of the following terms and a "constant" term.



      $beginalign &= fracm(x_0)sqrtnsqrta_nint Kleft(fracF(x_0)-F(x)a_nright) F(d x) -fracm(x_0)sqrtnsqrta_nint Kleft(fracF(x_0)-F(x)a_nright) F_n(d x)\
      &= -frac1sqrta_nm(x_0)int Kleft(fracF(x_0)-F(x)a_nright) alpha_n(d x)endalign$



      It would be nice if someone could provide me an explanation on the upper equalities since I'm not even sure if I'm on the right track. I only consider the classic integration by parts, maybe the Lebesgue-Stieltjes variant is more helpful.





      integration probability-theory measure-theory regression empirical-processes





      share|cite|improve this question










      share|cite|improve this question




      share|cite|improve this question









      asked Jul 26 at 11:43









      PeterMcCoy

      37139




      37139

























          active

          oldest

          votes











          Your Answer




          StackExchange.ifUsing("editor", function ()
          return StackExchange.using("mathjaxEditing", function ()
          StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
          StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
          );
          );
          , "mathjax-editing");

          StackExchange.ready(function()
          var channelOptions =
          tags: "".split(" "),
          id: "69"
          ;
          initTagRenderer("".split(" "), "".split(" "), channelOptions);

          StackExchange.using("externalEditor", function()
          // Have to fire editor after snippets, if snippets enabled
          if (StackExchange.settings.snippets.snippetsEnabled)
          StackExchange.using("snippets", function()
          createEditor();
          );

          else
          createEditor();

          );

          function createEditor()
          StackExchange.prepareEditor(
          heartbeatType: 'answer',
          convertImagesToLinks: true,
          noModals: false,
          showLowRepImageUploadWarning: true,
          reputationToPostImages: 10,
          bindNavPrevention: true,
          postfix: "",
          noCode: true, onDemand: true,
          discardSelector: ".discard-answer"
          ,immediatelyShowMarkdownHelp:true
          );



          );








           

          draft saved


          draft discarded


















          StackExchange.ready(
          function ()
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2863338%2fintegration-by-parts-with-empirical-measure%23new-answer', 'question_page');

          );

          Post as a guest






















          active

          oldest

          votes













          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes










           

          draft saved


          draft discarded


























           


          draft saved


          draft discarded














          StackExchange.ready(
          function ()
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2863338%2fintegration-by-parts-with-empirical-measure%23new-answer', 'question_page');

          );

          Post as a guest
































































          Comments

          Post a Comment

          Popular posts from this blog

          Color the edges and diagonals of a regular polygon

          Image
          Clash Royale CLAN TAG #URR8PPP up vote 5 down vote favorite 1 Here is the problem: For what $n$ is it possible to color the edges and diagonals of an $n$-side regular polygon with $dfracbinomn23$ colors, such that you use every color exactly three times and for every color the three segments (edges or diagonals) with that color form a triangle? Trivially $nequiv 0 text or 1 pmod3$. I can also prove that if the statement is true for $k$ than it is true for $3k$ as well. How to finish? Please help! Thanks combinatorial-geometry share | cite | improve this question edited Aug 6 at 21:57 alcana 118 4 asked Aug 6 at 21:15 Leo Gardner 356 11 Even assuming "polyhpn" in the title is a typo for polygon , it is unclear what you mean by "the edges and sides". Aren't the side of a regular polygon the same as the edges? A regular $n$-sided polygon has $n$ edges. – hardmath Aug 6 at 21:20 3 $n$ ne
          Read more

          Relationship between determinant of matrix and determinant of adjoint?

          Image
          Clash Royale CLAN TAG #URR8PPP up vote 2 down vote favorite We are studying adjoints in class, and I was curious if there is a relationship between the determinant of matrix A, and the determinant of the adjoint of matrix A? I assume there would be a relationship because finding the adjoint requires creating a cofactor matrix and then transposing it. linear-algebra determinant adjoint-operators share | cite | improve this question asked Nov 17 '17 at 17:32 CluelessCoder 32 5 For a square matrix $A$ of order $n$, we have $A(operatornameadj A)=(operatornameadj A)A=|A|I_n$ where $I_n$ is the identity matrix of order $n$. This should tell you about the determinant of the adjoint in terms of that of $A$ (use multiplicative property and other elementary properties of determinants). – Prasun Biswas Nov 17 '17 at 17:35 1 @CluelessCoder: see here: math.stackexchange.com/questions/516127/… – Hector Blandin Nov 17 &
          Read more

          What is the equation of a 3D cone with generalised tilt?

          Image
          Clash Royale CLAN TAG #URR8PPP up vote 2 down vote favorite What is the equation of a 3D cone with generalized tilt? I've noticed that in most equations given to represent a cone, there is no parameter which defines the tilt of the cone in 3D space and that most of them have their point at the origin $(0,0,0)$ - I was wondering if anyone could give me a more generalized cone equation for a cone in any position in 3D space 3d share | cite | improve this question edited Jul 25 '16 at 0:05 ervx 9,425 3 13 36 asked Jul 24 '16 at 15:38 Charlene 11 2 2 Welcome to Math.SE! Could you please explain a few things to help people give an answer useful to you: 1. What do you mean by "generalized tilt"? 2. Are you asking about a right circular cone? Does the angle at the vertex matter? 3. What form of answer (implicit, parametric...?) are you looking for? 4. If this is homework, what tools do you have available, an
          Read more