In the choice of bandwidth for kernel density estimator. Why usually minimize MISE instead of minimizing ISE?
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Before presenting my question (which I already formulate in the title of this post) is important to establish the context of my problem:
Let $xi$ be a random variable with density function $f$ unknow. Given a sample $widehatxi_1,ldots,widehatxi_N$ of $xi$ we have that the kernel density estimator (KDE) of $f$ is
$$widehatf_h(x):=frac1Nhsum_i=1^NmathcalKleft(fracx-widehatxi_ihright).$$
where $h> 0$ and $mathcalK$ is a probability density function such that $int x mathcalK(x)dx=0$ and $int x^2 mathcalK(x)dx=1$.
In the literature on this topic is customary to say tha $h$ must be chosen so as to minimize the MISE where
beginequation
mathrmMISE(h):=mathbbE_mathbbP^Nleft[ int left( widehatf_h(x)- f(x)right)^2dx right].
endequation
the randomness is in the vector $left(widehatxi_1,ldots,widehatxi_Nright)$, that has distribution $mathbbP^N=mathbbPtimescdots timesmathbbP$, where $mathbbP$ where
$mathbbP(A):=int_Af(x)dx.$
Therefore, $mathbbP$ is unknow. There are many techniques to estimate MISE.
The question: If $h_MISE$ minimizes $mathrmMISE$ then that $h_MISE$ is used to determine $widehatf_h_MISE$ for any sample of size $N$, the same is always used. Is not it better to find an $h$ for each sample?
Given a sample $widehatxi_1,ldots,widehatxi_N$, we consider the expression
$$mathrmISE(h):= int left( widehatf_h(x)- f(x)right)^2dx. $$
If $h_ISE$ minimizes $mathrmISE$. Is not $widehatf_h_ISE$ a better estimator than $widehatf_h_MISE$?
Why usually minimize MISE instead of minimizing ISE?
statistics probability-distributions density-function parameter-estimation
asked Apr 15 at 19:45
Diego Fonseca
1,442621
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up vote
0
down vote
favorite
Before presenting my question (which I already formulate in the title of this post) is important to establish the context of my problem:
Let $xi$ be a random variable with density function $f$ unknow. Given a sample $widehatxi_1,ldots,widehatxi_N$ of $xi$ we have that the kernel density estimator (KDE) of $f$ is
$$widehatf_h(x):=frac1Nhsum_i=1^NmathcalKleft(fracx-widehatxi_ihright).$$
where $h> 0$ and $mathcalK$ is a probability density function such that $int x mathcalK(x)dx=0$ and $int x^2 mathcalK(x)dx=1$.
In the literature on this topic is customary to say tha $h$ must be chosen so as to minimize the MISE where
beginequation
mathrmMISE(h):=mathbbE_mathbbP^Nleft[ int left( widehatf_h(x)- f(x)right)^2dx right].
endequation
the randomness is in the vector $left(widehatxi_1,ldots,widehatxi_Nright)$, that has distribution $mathbbP^N=mathbbPtimescdots timesmathbbP$, where $mathbbP$ where
$mathbbP(A):=int_Af(x)dx.$
Therefore, $mathbbP$ is unknow. There are many techniques to estimate MISE.
The question: If $h_MISE$ minimizes $mathrmMISE$ then that $h_MISE$ is used to determine $widehatf_h_MISE$ for any sample of size $N$, the same is always used. Is not it better to find an $h$ for each sample?
Given a sample $widehatxi_1,ldots,widehatxi_N$, we consider the expression
$$mathrmISE(h):= int left( widehatf_h(x)- f(x)right)^2dx. $$
If $h_ISE$ minimizes $mathrmISE$. Is not $widehatf_h_ISE$ a better estimator than $widehatf_h_MISE$?
Why usually minimize MISE instead of minimizing ISE?
statistics probability-distributions density-function parameter-estimation
asked Apr 15 at 19:45
Diego Fonseca
1,442621
Check this "sciencedirect.com/science/article/pii/016771529190163L" it will help you a lot.
– Angel
Sep 1 at 6:12
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Before presenting my question (which I already formulate in the title of this post) is important to establish the context of my problem:
Let $xi$ be a random variable with density function $f$ unknow. Given a sample $widehatxi_1,ldots,widehatxi_N$ of $xi$ we have that the kernel density estimator (KDE) of $f$ is
$$widehatf_h(x):=frac1Nhsum_i=1^NmathcalKleft(fracx-widehatxi_ihright).$$
where $h> 0$ and $mathcalK$ is a probability density function such that $int x mathcalK(x)dx=0$ and $int x^2 mathcalK(x)dx=1$.
In the literature on this topic is customary to say tha $h$ must be chosen so as to minimize the MISE where
beginequation
mathrmMISE(h):=mathbbE_mathbbP^Nleft[ int left( widehatf_h(x)- f(x)right)^2dx right].
endequation
the randomness is in the vector $left(widehatxi_1,ldots,widehatxi_Nright)$, that has distribution $mathbbP^N=mathbbPtimescdots timesmathbbP$, where $mathbbP$ where
$mathbbP(A):=int_Af(x)dx.$
Therefore, $mathbbP$ is unknow. There are many techniques to estimate MISE.
The question: If $h_MISE$ minimizes $mathrmMISE$ then that $h_MISE$ is used to determine $widehatf_h_MISE$ for any sample of size $N$, the same is always used. Is not it better to find an $h$ for each sample?
Given a sample $widehatxi_1,ldots,widehatxi_N$, we consider the expression
$$mathrmISE(h):= int left( widehatf_h(x)- f(x)right)^2dx. $$
If $h_ISE$ minimizes $mathrmISE$. Is not $widehatf_h_ISE$ a better estimator than $widehatf_h_MISE$?
Why usually minimize MISE instead of minimizing ISE?
statistics probability-distributions density-function parameter-estimation
asked Apr 15 at 19:45
Diego Fonseca
1,442621
Before presenting my question (which I already formulate in the title of this post) is important to establish the context of my problem:
Let $xi$ be a random variable with density function $f$ unknow. Given a sample $widehatxi_1,ldots,widehatxi_N$ of $xi$ we have that the kernel density estimator (KDE) of $f$ is
$$widehatf_h(x):=frac1Nhsum_i=1^NmathcalKleft(fracx-widehatxi_ihright).$$
where $h> 0$ and $mathcalK$ is a probability density function such that $int x mathcalK(x)dx=0$ and $int x^2 mathcalK(x)dx=1$.
In the literature on this topic is customary to say tha $h$ must be chosen so as to minimize the MISE where
beginequation
mathrmMISE(h):=mathbbE_mathbbP^Nleft[ int left( widehatf_h(x)- f(x)right)^2dx right].
endequation
the randomness is in the vector $left(widehatxi_1,ldots,widehatxi_Nright)$, that has distribution $mathbbP^N=mathbbPtimescdots timesmathbbP$, where $mathbbP$ where
$mathbbP(A):=int_Af(x)dx.$
Therefore, $mathbbP$ is unknow. There are many techniques to estimate MISE.
The question: If $h_MISE$ minimizes $mathrmMISE$ then that $h_MISE$ is used to determine $widehatf_h_MISE$ for any sample of size $N$, the same is always used. Is not it better to find an $h$ for each sample?
Given a sample $widehatxi_1,ldots,widehatxi_N$, we consider the expression
$$mathrmISE(h):= int left( widehatf_h(x)- f(x)right)^2dx. $$
If $h_ISE$ minimizes $mathrmISE$. Is not $widehatf_h_ISE$ a better estimator than $widehatf_h_MISE$?
Why usually minimize MISE instead of minimizing ISE?
statistics probability-distributions density-function parameter-estimation
statistics probability-distributions density-function parameter-estimation
asked Apr 15 at 19:45
Diego Fonseca
1,442621
asked Apr 15 at 19:45
Diego Fonseca
1,442621
asked Apr 15 at 19:45
Diego Fonseca
1,442621
asked Apr 15 at 19:45
Diego Fonseca
1,442621
asked Apr 15 at 19:45
Diego Fonseca
1,442621
1,442621
Check this "sciencedirect.com/science/article/pii/016771529190163L" it will help you a lot.
– Angel
Sep 1 at 6:12
add a comment |Â
Check this "sciencedirect.com/science/article/pii/016771529190163L" it will help you a lot.
– Angel
Sep 1 at 6:12
Check this "sciencedirect.com/science/article/pii/016771529190163L" it will help you a lot.
– Angel
Sep 1 at 6:12
Check this "sciencedirect.com/science/article/pii/016771529190163L" it will help you a lot.
– Angel
Sep 1 at 6:12
add a comment |Â
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Check this "sciencedirect.com/science/article/pii/016771529190163L" it will help you a lot.
– Angel
Sep 1 at 6:12