Mixing
Estimating a parameter from a sequence of signals with non i.i.d. noise
$begingroup$
I have the following statistics/probability question:
Let $A$ and $B$ be two finite sets of real numbers. Let $n geq 1$. Let $T_1,...,T_n$ be random variables with values in $A$ (not necessarily independent). Let $Q$ be a random variable with values in $B$.
Let $epsilon_1, epsilon_2,...,epsilon_n$ be a sequence of independent random variables which are identically normally distributed, with mean 0 and variance $sigma>0$. Assume that
for all $1 leq i leq n$, $epsilon_i$ is independent of $T_i$ and of $Q$ (note that $epsilon_i$ may be correlated with $T_{i-1}$ for instance). Let
begin{equation*}
X_i=T_i+Q+epsilon_i
end{equation*}
I would like to construct a random variable $Q_n$ that is $(X_1,X_2,...,X_n)$-measurable, and such that for some constant $C$ and $d>0$ that depend only on $A$, $B$ and $sigma$, we have
begin{equation*}
left|Q_n-Qright|_{L^2} leq C n^{-d}.
end{equation*}
In words, knowing $X_1,X_2,...,X_n$, I would like to approximate $Q$ in an efficient way.
Many thanks!
probability-theory statistics normal-distribution parameter-estimation
edited Jan 30 at 13:48
YuiTo Cheng
2,1963937
asked Jan 30 at 13:02
BrunoBruno
112
$endgroup$
add a comment |
$begingroup$
I have the following statistics/probability question:
Let $A$ and $B$ be two finite sets of real numbers. Let $n geq 1$. Let $T_1,...,T_n$ be random variables with values in $A$ (not necessarily independent). Let $Q$ be a random variable with values in $B$.
Let $epsilon_1, epsilon_2,...,epsilon_n$ be a sequence of independent random variables which are identically normally distributed, with mean 0 and variance $sigma>0$. Assume that
for all $1 leq i leq n$, $epsilon_i$ is independent of $T_i$ and of $Q$ (note that $epsilon_i$ may be correlated with $T_{i-1}$ for instance). Let
begin{equation*}
X_i=T_i+Q+epsilon_i
end{equation*}
I would like to construct a random variable $Q_n$ that is $(X_1,X_2,...,X_n)$-measurable, and such that for some constant $C$ and $d>0$ that depend only on $A$, $B$ and $sigma$, we have
begin{equation*}
left|Q_n-Qright|_{L^2} leq C n^{-d}.
end{equation*}
In words, knowing $X_1,X_2,...,X_n$, I would like to approximate $Q$ in an efficient way.
Many thanks!
probability-theory statistics normal-distribution parameter-estimation
edited Jan 30 at 13:48
YuiTo Cheng
2,1963937
asked Jan 30 at 13:02
BrunoBruno
112
$endgroup$
add a comment |
$begingroup$
I have the following statistics/probability question:
Let $A$ and $B$ be two finite sets of real numbers. Let $n geq 1$. Let $T_1,...,T_n$ be random variables with values in $A$ (not necessarily independent). Let $Q$ be a random variable with values in $B$.
Let $epsilon_1, epsilon_2,...,epsilon_n$ be a sequence of independent random variables which are identically normally distributed, with mean 0 and variance $sigma>0$. Assume that
for all $1 leq i leq n$, $epsilon_i$ is independent of $T_i$ and of $Q$ (note that $epsilon_i$ may be correlated with $T_{i-1}$ for instance). Let
begin{equation*}
X_i=T_i+Q+epsilon_i
end{equation*}
I would like to construct a random variable $Q_n$ that is $(X_1,X_2,...,X_n)$-measurable, and such that for some constant $C$ and $d>0$ that depend only on $A$, $B$ and $sigma$, we have
begin{equation*}
left|Q_n-Qright|_{L^2} leq C n^{-d}.
end{equation*}
In words, knowing $X_1,X_2,...,X_n$, I would like to approximate $Q$ in an efficient way.
Many thanks!
probability-theory statistics normal-distribution parameter-estimation
edited Jan 30 at 13:48
YuiTo Cheng
2,1963937
asked Jan 30 at 13:02
BrunoBruno
112
$endgroup$
I have the following statistics/probability question:
Let $A$ and $B$ be two finite sets of real numbers. Let $n geq 1$. Let $T_1,...,T_n$ be random variables with values in $A$ (not necessarily independent). Let $Q$ be a random variable with values in $B$.
Let $epsilon_1, epsilon_2,...,epsilon_n$ be a sequence of independent random variables which are identically normally distributed, with mean 0 and variance $sigma>0$. Assume that
for all $1 leq i leq n$, $epsilon_i$ is independent of $T_i$ and of $Q$ (note that $epsilon_i$ may be correlated with $T_{i-1}$ for instance). Let
begin{equation*}
X_i=T_i+Q+epsilon_i
end{equation*}
I would like to construct a random variable $Q_n$ that is $(X_1,X_2,...,X_n)$-measurable, and such that for some constant $C$ and $d>0$ that depend only on $A$, $B$ and $sigma$, we have
begin{equation*}
left|Q_n-Qright|_{L^2} leq C n^{-d}.
end{equation*}
In words, knowing $X_1,X_2,...,X_n$, I would like to approximate $Q$ in an efficient way.
Many thanks!
probability-theory statistics normal-distribution parameter-estimation
probability-theory statistics normal-distribution parameter-estimation
edited Jan 30 at 13:48
YuiTo Cheng
2,1963937
asked Jan 30 at 13:02
BrunoBruno
112
edited Jan 30 at 13:48
YuiTo Cheng
2,1963937
asked Jan 30 at 13:02
BrunoBruno
112
edited Jan 30 at 13:48
YuiTo Cheng
2,1963937
edited Jan 30 at 13:48
YuiTo Cheng
2,1963937
edited Jan 30 at 13:48
YuiTo Cheng
2,1963937
2,1963937
asked Jan 30 at 13:02
BrunoBruno
112
asked Jan 30 at 13:02
BrunoBruno
112
asked Jan 30 at 13:02
BrunoBruno
112
112
add a comment |
add a comment |
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