Using the delta method, is there any clear procedure?
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I am currently enrolled in a course in theoretical statistics, exams are comming up and I am trying to study. The course litterature we are using is Van der Vaart's Asymptotic Statistics. I am having trouble figuring out how to use and apply the delta method. I feel as if it is quite vague how one should go about the delta method. Is there any particular procedure one can follow(like step1:, step2:, ...) to say derive asymptotic distribution? I am at a loss here, please help.
If no such clear procedure can be provided then, just so I can derive something from example, Let $X_1, dots , X_n$ i.i.d. copies of $X$ with a distribution $F$ density $f$ and
Var$(X) = sigma^2 < ∞$, $text{E}X=muneq 0$, $text{E}X^4 < ∞$. Consider the variance coefficient $$gamma=frac{sigma}{mu}.$$ Let $hat{gamma}=frac{S}{overline{X}}$, derive the asymptotic distribution of $hat{gamma}.$
statistics statistical-inference delta-method
asked Jan 2 at 16:03
L200123L200123
855
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add a comment |
$begingroup$
I am currently enrolled in a course in theoretical statistics, exams are comming up and I am trying to study. The course litterature we are using is Van der Vaart's Asymptotic Statistics. I am having trouble figuring out how to use and apply the delta method. I feel as if it is quite vague how one should go about the delta method. Is there any particular procedure one can follow(like step1:, step2:, ...) to say derive asymptotic distribution? I am at a loss here, please help.
If no such clear procedure can be provided then, just so I can derive something from example, Let $X_1, dots , X_n$ i.i.d. copies of $X$ with a distribution $F$ density $f$ and
Var$(X) = sigma^2 < ∞$, $text{E}X=muneq 0$, $text{E}X^4 < ∞$. Consider the variance coefficient $$gamma=frac{sigma}{mu}.$$ Let $hat{gamma}=frac{S}{overline{X}}$, derive the asymptotic distribution of $hat{gamma}.$
statistics statistical-inference delta-method
asked Jan 2 at 16:03
L200123L200123
855
$endgroup$
$begingroup$
In this case, your variance ratio is a certain nonlinear function of the sample average of the vectors $(X_i,X_i^2)$. These vectors obey the 2-dimensional CLT, and the Delta method in this case involves replacing the nonlinear function with its linear Taylor approximation in the neighborhood of the expectation of $(X_i,X_i^2)$.
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– kimchi lover
Jan 2 at 17:26
add a comment |
$begingroup$
I am currently enrolled in a course in theoretical statistics, exams are comming up and I am trying to study. The course litterature we are using is Van der Vaart's Asymptotic Statistics. I am having trouble figuring out how to use and apply the delta method. I feel as if it is quite vague how one should go about the delta method. Is there any particular procedure one can follow(like step1:, step2:, ...) to say derive asymptotic distribution? I am at a loss here, please help.
If no such clear procedure can be provided then, just so I can derive something from example, Let $X_1, dots , X_n$ i.i.d. copies of $X$ with a distribution $F$ density $f$ and
Var$(X) = sigma^2 < ∞$, $text{E}X=muneq 0$, $text{E}X^4 < ∞$. Consider the variance coefficient $$gamma=frac{sigma}{mu}.$$ Let $hat{gamma}=frac{S}{overline{X}}$, derive the asymptotic distribution of $hat{gamma}.$
statistics statistical-inference delta-method
asked Jan 2 at 16:03
L200123L200123
855
$endgroup$
I am currently enrolled in a course in theoretical statistics, exams are comming up and I am trying to study. The course litterature we are using is Van der Vaart's Asymptotic Statistics. I am having trouble figuring out how to use and apply the delta method. I feel as if it is quite vague how one should go about the delta method. Is there any particular procedure one can follow(like step1:, step2:, ...) to say derive asymptotic distribution? I am at a loss here, please help.
If no such clear procedure can be provided then, just so I can derive something from example, Let $X_1, dots , X_n$ i.i.d. copies of $X$ with a distribution $F$ density $f$ and
Var$(X) = sigma^2 < ∞$, $text{E}X=muneq 0$, $text{E}X^4 < ∞$. Consider the variance coefficient $$gamma=frac{sigma}{mu}.$$ Let $hat{gamma}=frac{S}{overline{X}}$, derive the asymptotic distribution of $hat{gamma}.$
statistics statistical-inference delta-method
statistics statistical-inference delta-method
asked Jan 2 at 16:03
L200123L200123
855
asked Jan 2 at 16:03
L200123L200123
855
asked Jan 2 at 16:03
L200123L200123
855
asked Jan 2 at 16:03
L200123L200123
855
asked Jan 2 at 16:03
L200123L200123
855
855
$begingroup$
In this case, your variance ratio is a certain nonlinear function of the sample average of the vectors $(X_i,X_i^2)$. These vectors obey the 2-dimensional CLT, and the Delta method in this case involves replacing the nonlinear function with its linear Taylor approximation in the neighborhood of the expectation of $(X_i,X_i^2)$.
$endgroup$
– kimchi lover
Jan 2 at 17:26
add a comment |
$begingroup$
In this case, your variance ratio is a certain nonlinear function of the sample average of the vectors $(X_i,X_i^2)$. These vectors obey the 2-dimensional CLT, and the Delta method in this case involves replacing the nonlinear function with its linear Taylor approximation in the neighborhood of the expectation of $(X_i,X_i^2)$.
$endgroup$
– kimchi lover
Jan 2 at 17:26
$begingroup$
In this case, your variance ratio is a certain nonlinear function of the sample average of the vectors $(X_i,X_i^2)$. These vectors obey the 2-dimensional CLT, and the Delta method in this case involves replacing the nonlinear function with its linear Taylor approximation in the neighborhood of the expectation of $(X_i,X_i^2)$.
$endgroup$
– kimchi lover
Jan 2 at 17:26
$begingroup$
In this case, your variance ratio is a certain nonlinear function of the sample average of the vectors $(X_i,X_i^2)$. These vectors obey the 2-dimensional CLT, and the Delta method in this case involves replacing the nonlinear function with its linear Taylor approximation in the neighborhood of the expectation of $(X_i,X_i^2)$.
$endgroup$
– kimchi lover
Jan 2 at 17:26
add a comment |
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$begingroup$
In this case, your variance ratio is a certain nonlinear function of the sample average of the vectors $(X_i,X_i^2)$. These vectors obey the 2-dimensional CLT, and the Delta method in this case involves replacing the nonlinear function with its linear Taylor approximation in the neighborhood of the expectation of $(X_i,X_i^2)$.
$endgroup$
– kimchi lover
Jan 2 at 17:26