Asymptotic Noramlity of Maximum Likelihood Estimator and the distribution of Lagrange Multiplier

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    Asymptotic Noramlity of Maximum Likelihood Estimator and the
    distribution of Lagrange Multiplier and Likelihood Ratio test statistic
    This article gives the proof of the asymptotic normality of maximum likelihood estimator and the distribution of LM
    and LR statistics, which are frequenltly used in application of econometrics.
    TheoremLet yi(i = 1;;n)be the independently and identically distributed with probability density f(yi;0)
    characterized by0.And definethe likelihood functionto be L(y;)=
    n
    i= 1 f(yi;0).Now letˆ bethe maximum
    likelihood estimator;which maximizes L(y;0),such thatplimˆ = 0,where0 is the true value of parameter.
    Then under the regurality conditions, ˆ has asymptotic normalityˆ
    a
    N(0;E[I (0)] 1);where E[I (0)] =
    E0[ 2logL= 0 0
    0].
    Regularity Conditions
    (i)The first,second and third derivatives oflog f(yi;0)with respect to0 exist,are continuous and finite for
    almost all yi and for all.
    (ii)The conditions to obtain the expectaions of the first and second derivatives oflog f(yi;0)are met.
    (iii)For all values of,j 3log f(yi;0)=q jq kq l j is less than a function that has a finite expectation.
    Proof
    Now under some assumptions, we can interchange the operations of integration and differentiation.
    0
    Z
    f(yi;0)dyi =
    Z
    0
    f(yi;0)dyi
    =
    Z
    f(yi;0)
    log f(yi;0)
    0
    dyi
    =
    Z
    log f(yi;0)
    0
    f(yi;0)dyi
    = E0
    h
    log f(yi;0)
    0
    i
    = 0:
    (1)
    In order to obtain the result above, we used the following.
    f(yi;0)
    0
    =
    exp(log f(yi;0))
    0
    =
    exp(log f(yi;0))
    log f(yi;0)
    log f(yi;0)
    0
    = exp(log f(yi;

    資料の原本内容( この資料を購入すると、テキストデータがみえます。 )

    Asymptotic Noramlity of Maximum Likelihood Estimator and the
    distribution of Lagrange Multiplier and Likelihood Ratio test statistic
    This article gives the proof of the asymptotic normality of maximum likelihood estimator and the distribution of LM
    and LR statistics, which are frequenltly used in application of econometrics.
    TheoremLet yi(i = 1;;n)be the independently and identically distributed with probability density f(yi;0)
    characterized by0.And definethe likelihood functionto be L(y;)=
    n
    i..

    コメント7件

    kodai0627 購入
    tc
    2006/11/28 1:22 (10年前)

    naotan 購入
    参考になりました
    2006/11/28 9:55 (10年前)

    8oto8 購入
    参考にさせて頂きます!
    2006/12/10 12:59 (10年前)

    blacksarena 購入
    参考にします。
    2006/12/10 18:00 (10年前)

    leehikaru 購入
    参考にします。
    2006/12/17 4:30 (10年前)

    kanotch 購入
    参考になりました
    2006/12/29 15:31 (10年前)

    cticdire 購入
    b
    2007/01/08 22:03 (9年11ヶ月前)

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