Tag: asymptotics

In Deville’s approach, a population parameter of interest $Phi$ can be written as a functional $T$ with respect to a finite and discrete measure $M$, namely $Phi=T(M)$. The substitution estimator $hat{Phi}=T(hat{M})$ is the functional $T$ of a random measure $hat{M}$ that is associated with sampling weights $w_{k}, k in U$, and is ‘close’ to $M$. Suppose that $T$ is homogeneous of degree $alpha$, so that $T(r M)=r^{alpha} T(M)$, and $lim _{N rightarrow infty} N^{-alpha} T(M)<infty$. Under broad assumptions, Deville shows that
$$
begin{aligned}
sqrt{n} N^{-alpha}{T(hat{M})-T(M)} &=sqrt{n} N^{-alpha} int I_{T}(M, z) d(hat{M}-M)(z)+o_{p}(1) \
&=sqrt{n} N^{-alpha} sum_{k=1}^{N} u_{k}left(w_{k}-1right)+o_{p}(1)
end{aligned}
$$
The linearized variables $u_{k}$ are the influence functions $I_{T}left(M, z_{k}right)$, where $z_{k}$ is the value of the variable of interest for the $k$ th unit.

Result: Under broad assumptions, the substitution estimation of a functional $T(M)$ is linearizable. A linearized variable is $z_{k}=I Tleft(M ; x_{k}right)$ where $I T$ is the influence function of $T$ in $M$.

Proof of the result: Let us provide the space of measurements on $boldsymbol{R}^{q}$ with a metric $d$ accounting for the convergence: $dleft(M_{1}, M_{2}right) rightarrow 0$ if and only if $N^{-1}left(int y d M_{1}-int y d M_{2}right) rightarrow 0$ for any variable of interest $y$. The asymptotic postulates mean that $d(hat{M} / N, M / N)$ tends towards zero. We can visibly ensure that $d(hat{M} / N, M / N)$ is $O_{p}(1 / sqrt{n})$ according to the third postulate. Now, let us assume that $T$ can be derived in accordance with Fréchet, i.e., for any direction of the increase, in the space of "useful" measures provided with the abovementioned metric. Thus we have:
$$
N^{-alpha}(T(hat{M})-T(M))=frac{1}{N} sum_{U} z_{k}left(w_{k}-1right)+oleft(dleft(frac{hat{M}}{M}, frac{M}{N}right)right)
$$
The result is that:
$$
sqrt{n} N^{-alpha}(T(hat{M})-T(M))=frac{sqrt{n}}{N} sum_{U} z_{k}left(w_{k}-1right)+o_{p}(1) .
$$

In Deville’s approach, a population parameter of interest $Phi$ can be written as a functional $T$ with respect to a finite and discrete measure $M$, namely $Phi=T(M)$. The substitution estimator $hat{Phi}=T(hat{M})$ is the functional $T$ of a random measure $hat{M}$ that is associated with sampling weights $w_{k}, k in U$, and is ‘close’ to $M$. Suppose that $T$ is homogeneous of degree $alpha$, so that $T(r M)=r^{alpha} T(M)$, and $lim _{N rightarrow infty} N^{-alpha} T(M)<infty$. Under broad assumptions, Deville shows that
$$
begin{aligned}
sqrt{n} N^{-alpha}{T(hat{M})-T(M)} &=sqrt{n} N^{-alpha} int I_{T}(M, z) d(hat{M}-M)(z)+o_{p}(1) \
&=sqrt{n} N^{-alpha} sum_{k=1}^{N} u_{k}left(w_{k}-1right)+o_{p}(1)
end{aligned}
$$
The linearized variables $u_{k}$ are the influence functions $I_{T}left(M, z_{k}right)$, where $z_{k}$ is the value of the variable of interest for the $k$ th unit.

Result: Under broad assumptions, the substitution estimation of a functional $T(M)$ is linearizable. A linearized variable is $z_{k}=I Tleft(M ; x_{k}right)$ where $I T$ is the influence function of $T$ in $M$.

Proof of the result: Let us provide the space of measurements on $boldsymbol{R}^{q}$ with a metric $d$ accounting for the convergence: $dleft(M_{1}, M_{2}right) rightarrow 0$ if and only if $N^{-1}left(int y d M_{1}-int y d M_{2}right) rightarrow 0$ for any variable of interest $y$. The asymptotic postulates mean that $d(hat{M} / N, M / N)$ tends towards zero. We can visibly ensure that $d(hat{M} / N, M / N)$ is $O_{p}(1 / sqrt{n})$ according to the third postulate. Now, let us assume that $T$ can be derived in accordance with Fréchet, i.e., for any direction of the increase, in the space of "useful" measures provided with the abovementioned metric. Thus we have:
$$
N^{-alpha}(T(hat{M})-T(M))=frac{1}{N} sum_{U} z_{k}left(w_{k}-1right)+oleft(dleft(frac{hat{M}}{M}, frac{M}{N}right)right)
$$
The result is that:
$$
sqrt{n} N^{-alpha}(T(hat{M})-T(M))=frac{sqrt{n}}{N} sum_{U} z_{k}left(w_{k}-1right)+o_{p}(1) .
$$

In Deville’s approach, a population parameter of interest $Phi$ can be written as a functional $T$ with respect to a finite and discrete measure $M$, namely $Phi=T(M)$. The substitution estimator $hat{Phi}=T(hat{M})$ is the functional $T$ of a random measure $hat{M}$ that is associated with sampling weights $w_{k}, k in U$, and is ‘close’ to $M$. Suppose that $T$ is homogeneous of degree $alpha$, so that $T(r M)=r^{alpha} T(M)$, and $lim _{N rightarrow infty} N^{-alpha} T(M)<infty$. Under broad assumptions, Deville shows that
$$
begin{aligned}
sqrt{n} N^{-alpha}{T(hat{M})-T(M)} &=sqrt{n} N^{-alpha} int I_{T}(M, z) d(hat{M}-M)(z)+o_{p}(1) \
&=sqrt{n} N^{-alpha} sum_{k=1}^{N} u_{k}left(w_{k}-1right)+o_{p}(1)
end{aligned}
$$
The linearized variables $u_{k}$ are the influence functions $I_{T}left(M, z_{k}right)$, where $z_{k}$ is the value of the variable of interest for the $k$ th unit.

Result: Under broad assumptions, the substitution estimation of a functional $T(M)$ is linearizable. A linearized variable is $z_{k}=I Tleft(M ; x_{k}right)$ where $I T$ is the influence function of $T$ in $M$.

Proof of the result: Let us provide the space of measurements on $boldsymbol{R}^{q}$ with a metric $d$ accounting for the convergence: $dleft(M_{1}, M_{2}right) rightarrow 0$ if and only if $N^{-1}left(int y d M_{1}-int y d M_{2}right) rightarrow 0$ for any variable of interest $y$. The asymptotic postulates mean that $d(hat{M} / N, M / N)$ tends towards zero. We can visibly ensure that $d(hat{M} / N, M / N)$ is $O_{p}(1 / sqrt{n})$ according to the third postulate. Now, let us assume that $T$ can be derived in accordance with Fréchet, i.e., for any direction of the increase, in the space of "useful" measures provided with the abovementioned metric. Thus we have:
$$
N^{-alpha}(T(hat{M})-T(M))=frac{1}{N} sum_{U} z_{k}left(w_{k}-1right)+oleft(dleft(frac{hat{M}}{M}, frac{M}{N}right)right)
$$
The result is that:
$$
sqrt{n} N^{-alpha}(T(hat{M})-T(M))=frac{sqrt{n}}{N} sum_{U} z_{k}left(w_{k}-1right)+o_{p}(1) .
$$

In Deville’s approach, a population parameter of interest $Phi$ can be written as a functional $T$ with respect to a finite and discrete measure $M$, namely $Phi=T(M)$. The substitution estimator $hat{Phi}=T(hat{M})$ is the functional $T$ of a random measure $hat{M}$ that is associated with sampling weights $w_{k}, k in U$, and is ‘close’ to $M$. Suppose that $T$ is homogeneous of degree $alpha$, so that $T(r M)=r^{alpha} T(M)$, and $lim _{N rightarrow infty} N^{-alpha} T(M)<infty$. Under broad assumptions, Deville shows that
$$
begin{aligned}
sqrt{n} N^{-alpha}{T(hat{M})-T(M)} &=sqrt{n} N^{-alpha} int I_{T}(M, z) d(hat{M}-M)(z)+o_{p}(1) \
&=sqrt{n} N^{-alpha} sum_{k=1}^{N} u_{k}left(w_{k}-1right)+o_{p}(1)
end{aligned}
$$
The linearized variables $u_{k}$ are the influence functions $I_{T}left(M, z_{k}right)$, where $z_{k}$ is the value of the variable of interest for the $k$ th unit.

Result: Under broad assumptions, the substitution estimation of a functional $T(M)$ is linearizable. A linearized variable is $z_{k}=I Tleft(M ; x_{k}right)$ where $I T$ is the influence function of $T$ in $M$.

Proof of the result: Let us provide the space of measurements on $boldsymbol{R}^{q}$ with a metric $d$ accounting for the convergence: $dleft(M_{1}, M_{2}right) rightarrow 0$ if and only if $N^{-1}left(int y d M_{1}-int y d M_{2}right) rightarrow 0$ for any variable of interest $y$. The asymptotic postulates mean that $d(hat{M} / N, M / N)$ tends towards zero. We can visibly ensure that $d(hat{M} / N, M / N)$ is $O_{p}(1 / sqrt{n})$ according to the third postulate. Now, let us assume that $T$ can be derived in accordance with Fréchet, i.e., for any direction of the increase, in the space of "useful" measures provided with the abovementioned metric. Thus we have:
$$
N^{-alpha}(T(hat{M})-T(M))=frac{1}{N} sum_{U} z_{k}left(w_{k}-1right)+oleft(dleft(frac{hat{M}}{M}, frac{M}{N}right)right)
$$
The result is that:
$$
sqrt{n} N^{-alpha}(T(hat{M})-T(M))=frac{sqrt{n}}{N} sum_{U} z_{k}left(w_{k}-1right)+o_{p}(1) .
$$

In Deville’s approach, a population parameter of interest $Phi$ can be written as a functional $T$ with respect to a finite and discrete measure $M$, namely $Phi=T(M)$. The substitution estimator $hat{Phi}=T(hat{M})$ is the functional $T$ of a random measure $hat{M}$ that is associated with sampling weights $w_{k}, k in U$, and is ‘close’ to $M$. Suppose that $T$ is homogeneous of degree $alpha$, so that $T(r M)=r^{alpha} T(M)$, and $lim _{N rightarrow infty} N^{-alpha} T(M)<infty$. Under broad assumptions, Deville shows that
$$
begin{aligned}
sqrt{n} N^{-alpha}{T(hat{M})-T(M)} &=sqrt{n} N^{-alpha} int I_{T}(M, z) d(hat{M}-M)(z)+o_{p}(1) \
&=sqrt{n} N^{-alpha} sum_{k=1}^{N} u_{k}left(w_{k}-1right)+o_{p}(1)
end{aligned}
$$
The linearized variables $u_{k}$ are the influence functions $I_{T}left(M, z_{k}right)$, where $z_{k}$ is the value of the variable of interest for the $k$ th unit.

Result: Under broad assumptions, the substitution estimation of a functional $T(M)$ is linearizable. A linearized variable is $z_{k}=I Tleft(M ; x_{k}right)$ where $I T$ is the influence function of $T$ in $M$.

Proof of the result: Let us provide the space of measurements on $boldsymbol{R}^{q}$ with a metric $d$ accounting for the convergence: $dleft(M_{1}, M_{2}right) rightarrow 0$ if and only if $N^{-1}left(int y d M_{1}-int y d M_{2}right) rightarrow 0$ for any variable of interest $y$. The asymptotic postulates mean that $d(hat{M} / N, M / N)$ tends towards zero. We can visibly ensure that $d(hat{M} / N, M / N)$ is $O_{p}(1 / sqrt{n})$ according to the third postulate. Now, let us assume that $T$ can be derived in accordance with Fréchet, i.e., for any direction of the increase, in the space of "useful" measures provided with the abovementioned metric. Thus we have:
$$
N^{-alpha}(T(hat{M})-T(M))=frac{1}{N} sum_{U} z_{k}left(w_{k}-1right)+oleft(dleft(frac{hat{M}}{M}, frac{M}{N}right)right)
$$
The result is that:
$$
sqrt{n} N^{-alpha}(T(hat{M})-T(M))=frac{sqrt{n}}{N} sum_{U} z_{k}left(w_{k}-1right)+o_{p}(1) .
$$

In Deville’s approach, a population parameter of interest $Phi$ can be written as a functional $T$ with respect to a finite and discrete measure $M$, namely $Phi=T(M)$. The substitution estimator $hat{Phi}=T(hat{M})$ is the functional $T$ of a random measure $hat{M}$ that is associated with sampling weights $w_{k}, k in U$, and is ‘close’ to $M$. Suppose that $T$ is homogeneous of degree $alpha$, so that $T(r M)=r^{alpha} T(M)$, and $lim _{N rightarrow infty} N^{-alpha} T(M)<infty$. Under broad assumptions, Deville shows that
$$
begin{aligned}
sqrt{n} N^{-alpha}{T(hat{M})-T(M)} &=sqrt{n} N^{-alpha} int I_{T}(M, z) d(hat{M}-M)(z)+o_{p}(1) \
&=sqrt{n} N^{-alpha} sum_{k=1}^{N} u_{k}left(w_{k}-1right)+o_{p}(1)
end{aligned}
$$
The linearized variables $u_{k}$ are the influence functions $I_{T}left(M, z_{k}right)$, where $z_{k}$ is the value of the variable of interest for the $k$ th unit.

Result: Under broad assumptions, the substitution estimation of a functional $T(M)$ is linearizable. A linearized variable is $z_{k}=I Tleft(M ; x_{k}right)$ where $I T$ is the influence function of $T$ in $M$.

Proof of the result: Let us provide the space of measurements on $boldsymbol{R}^{q}$ with a metric $d$ accounting for the convergence: $dleft(M_{1}, M_{2}right) rightarrow 0$ if and only if $N^{-1}left(int y d M_{1}-int y d M_{2}right) rightarrow 0$ for any variable of interest $y$. The asymptotic postulates mean that $d(hat{M} / N, M / N)$ tends towards zero. We can visibly ensure that $d(hat{M} / N, M / N)$ is $O_{p}(1 / sqrt{n})$ according to the third postulate. Now, let us assume that $T$ can be derived in accordance with Fréchet, i.e., for any direction of the increase, in the space of "useful" measures provided with the abovementioned metric. Thus we have:
$$
N^{-alpha}(T(hat{M})-T(M))=frac{1}{N} sum_{U} z_{k}left(w_{k}-1right)+oleft(dleft(frac{hat{M}}{M}, frac{M}{N}right)right)
$$
The result is that:
$$
sqrt{n} N^{-alpha}(T(hat{M})-T(M))=frac{sqrt{n}}{N} sum_{U} z_{k}left(w_{k}-1right)+o_{p}(1) .
$$

In Deville’s approach, a population parameter of interest $Phi$ can be written as a functional $T$ with respect to a finite and discrete measure $M$, namely $Phi=T(M)$. The substitution estimator $hat{Phi}=T(hat{M})$ is the functional $T$ of a random measure $hat{M}$ that is associated with sampling weights $w_{k}, k in U$, and is ‘close’ to $M$. Suppose that $T$ is homogeneous of degree $alpha$, so that $T(r M)=r^{alpha} T(M)$, and $lim _{N rightarrow infty} N^{-alpha} T(M)<infty$. Under broad assumptions, Deville shows that
$$
begin{aligned}
sqrt{n} N^{-alpha}{T(hat{M})-T(M)} &=sqrt{n} N^{-alpha} int I_{T}(M, z) d(hat{M}-M)(z)+o_{p}(1) \
&=sqrt{n} N^{-alpha} sum_{k=1}^{N} u_{k}left(w_{k}-1right)+o_{p}(1)
end{aligned}
$$
The linearized variables $u_{k}$ are the influence functions $I_{T}left(M, z_{k}right)$, where $z_{k}$ is the value of the variable of interest for the $k$ th unit.

Result: Under broad assumptions, the substitution estimation of a functional $T(M)$ is linearizable. A linearized variable is $z_{k}=I Tleft(M ; x_{k}right)$ where $I T$ is the influence function of $T$ in $M$.

Proof of the result: Let us provide the space of measurements on $boldsymbol{R}^{q}$ with a metric $d$ accounting for the convergence: $dleft(M_{1}, M_{2}right) rightarrow 0$ if and only if $N^{-1}left(int y d M_{1}-int y d M_{2}right) rightarrow 0$ for any variable of interest $y$. The asymptotic postulates mean that $d(hat{M} / N, M / N)$ tends towards zero. We can visibly ensure that $d(hat{M} / N, M / N)$ is $O_{p}(1 / sqrt{n})$ according to the third postulate. Now, let us assume that $T$ can be derived in accordance with Fréchet, i.e., for any direction of the increase, in the space of "useful" measures provided with the abovementioned metric. Thus we have:
$$
N^{-alpha}(T(hat{M})-T(M))=frac{1}{N} sum_{U} z_{k}left(w_{k}-1right)+oleft(dleft(frac{hat{M}}{M}, frac{M}{N}right)right)
$$
The result is that:
$$
sqrt{n} N^{-alpha}(T(hat{M})-T(M))=frac{sqrt{n}}{N} sum_{U} z_{k}left(w_{k}-1right)+o_{p}(1) .
$$

In Deville’s approach, a population parameter of interest $Phi$ can be written as a functional $T$ with respect to a finite and discrete measure $M$, namely $Phi=T(M)$. The substitution estimator $hat{Phi}=T(hat{M})$ is the functional $T$ of a random measure $hat{M}$ that is associated with sampling weights $w_{k}, k in U$, and is ‘close’ to $M$. Suppose that $T$ is homogeneous of degree $alpha$, so that $T(r M)=r^{alpha} T(M)$, and $lim _{N rightarrow infty} N^{-alpha} T(M)<infty$. Under broad assumptions, Deville shows that
$$
begin{aligned}
sqrt{n} N^{-alpha}{T(hat{M})-T(M)} &=sqrt{n} N^{-alpha} int I_{T}(M, z) d(hat{M}-M)(z)+o_{p}(1) \
&=sqrt{n} N^{-alpha} sum_{k=1}^{N} u_{k}left(w_{k}-1right)+o_{p}(1)
end{aligned}
$$
The linearized variables $u_{k}$ are the influence functions $I_{T}left(M, z_{k}right)$, where $z_{k}$ is the value of the variable of interest for the $k$ th unit.

Result: Under broad assumptions, the substitution estimation of a functional $T(M)$ is linearizable. A linearized variable is $z_{k}=I Tleft(M ; x_{k}right)$ where $I T$ is the influence function of $T$ in $M$.

Proof of the result: Let us provide the space of measurements on $boldsymbol{R}^{q}$ with a metric $d$ accounting for the convergence: $dleft(M_{1}, M_{2}right) rightarrow 0$ if and only if $N^{-1}left(int y d M_{1}-int y d M_{2}right) rightarrow 0$ for any variable of interest $y$. The asymptotic postulates mean that $d(hat{M} / N, M / N)$ tends towards zero. We can visibly ensure that $d(hat{M} / N, M / N)$ is $O_{p}(1 / sqrt{n})$ according to the third postulate. Now, let us assume that $T$ can be derived in accordance with Fréchet, i.e., for any direction of the increase, in the space of "useful" measures provided with the abovementioned metric. Thus we have:
$$
N^{-alpha}(T(hat{M})-T(M))=frac{1}{N} sum_{U} z_{k}left(w_{k}-1right)+oleft(dleft(frac{hat{M}}{M}, frac{M}{N}right)right)
$$
The result is that:
$$
sqrt{n} N^{-alpha}(T(hat{M})-T(M))=frac{sqrt{n}}{N} sum_{U} z_{k}left(w_{k}-1right)+o_{p}(1) .
$$

In Deville’s approach, a population parameter of interest $Phi$ can be written as a functional $T$ with respect to a finite and discrete measure $M$, namely $Phi=T(M)$. The substitution estimator $hat{Phi}=T(hat{M})$ is the functional $T$ of a random measure $hat{M}$ that is associated with sampling weights $w_{k}, k in U$, and is ‘close’ to $M$. Suppose that $T$ is homogeneous of degree $alpha$, so that $T(r M)=r^{alpha} T(M)$, and $lim _{N rightarrow infty} N^{-alpha} T(M)<infty$. Under broad assumptions, Deville shows that
$$
begin{aligned}
sqrt{n} N^{-alpha}{T(hat{M})-T(M)} &=sqrt{n} N^{-alpha} int I_{T}(M, z) d(hat{M}-M)(z)+o_{p}(1) \
&=sqrt{n} N^{-alpha} sum_{k=1}^{N} u_{k}left(w_{k}-1right)+o_{p}(1)
end{aligned}
$$
The linearized variables $u_{k}$ are the influence functions $I_{T}left(M, z_{k}right)$, where $z_{k}$ is the value of the variable of interest for the $k$ th unit.

Result: Under broad assumptions, the substitution estimation of a functional $T(M)$ is linearizable. A linearized variable is $z_{k}=I Tleft(M ; x_{k}right)$ where $I T$ is the influence function of $T$ in $M$.

Proof of the result: Let us provide the space of measurements on $boldsymbol{R}^{q}$ with a metric $d$ accounting for the convergence: $dleft(M_{1}, M_{2}right) rightarrow 0$ if and only if $N^{-1}left(int y d M_{1}-int y d M_{2}right) rightarrow 0$ for any variable of interest $y$. The asymptotic postulates mean that $d(hat{M} / N, M / N)$ tends towards zero. We can visibly ensure that $d(hat{M} / N, M / N)$ is $O_{p}(1 / sqrt{n})$ according to the third postulate. Now, let us assume that $T$ can be derived in accordance with Fréchet, i.e., for any direction of the increase, in the space of "useful" measures provided with the abovementioned metric. Thus we have:
$$
N^{-alpha}(T(hat{M})-T(M))=frac{1}{N} sum_{U} z_{k}left(w_{k}-1right)+oleft(dleft(frac{hat{M}}{M}, frac{M}{N}right)right)
$$
The result is that:
$$
sqrt{n} N^{-alpha}(T(hat{M})-T(M))=frac{sqrt{n}}{N} sum_{U} z_{k}left(w_{k}-1right)+o_{p}(1) .
$$