Robust estimation of the mean
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Problem:
Suppose we want to estimate the mean µ of a random variable $X$ from a
sample $X_1 , dots , X_N$ drawn independently from the distribution
of $X$. We want an $varepsilon$-accurate estimate, i.e. one that
falls in the interval $(mu − varepsilon, mu + varepsilon)$.
Show that a sample of size $N = O( log (delta^{−1} ), sigma^2 / varepsilon^2 )$ is sufficient to compute an $varepsilon$-accurate
estimate with probability at least $1 −delta$.
Hint: Use the median of $O(log(delta^{−1}))$ weak estimates.
It is easy to use Chebyshev's inequality to find a weak estimate of $N = O( sigma^2 / delta varepsilon^2 )$.
However, I do not how to find inequality about their median. The wikipedia of median (https://en.wikipedia.org/wiki/Median#The_sample_median) says sample median asymptotically normal but this does not give a bound for specific $N$. Any suggestion is welcome.
probability median
asked Nov 15 at 17:02
Rikeijin
878
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up vote
2
down vote
favorite
Problem:
Suppose we want to estimate the mean µ of a random variable $X$ from a
sample $X_1 , dots , X_N$ drawn independently from the distribution
of $X$. We want an $varepsilon$-accurate estimate, i.e. one that
falls in the interval $(mu − varepsilon, mu + varepsilon)$.
Show that a sample of size $N = O( log (delta^{−1} ), sigma^2 / varepsilon^2 )$ is sufficient to compute an $varepsilon$-accurate
estimate with probability at least $1 −delta$.
Hint: Use the median of $O(log(delta^{−1}))$ weak estimates.
It is easy to use Chebyshev's inequality to find a weak estimate of $N = O( sigma^2 / delta varepsilon^2 )$.
However, I do not how to find inequality about their median. The wikipedia of median (https://en.wikipedia.org/wiki/Median#The_sample_median) says sample median asymptotically normal but this does not give a bound for specific $N$. Any suggestion is welcome.
probability median
asked Nov 15 at 17:02
Rikeijin
878
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
Problem:
Suppose we want to estimate the mean µ of a random variable $X$ from a
sample $X_1 , dots , X_N$ drawn independently from the distribution
of $X$. We want an $varepsilon$-accurate estimate, i.e. one that
falls in the interval $(mu − varepsilon, mu + varepsilon)$.
Show that a sample of size $N = O( log (delta^{−1} ), sigma^2 / varepsilon^2 )$ is sufficient to compute an $varepsilon$-accurate
estimate with probability at least $1 −delta$.
Hint: Use the median of $O(log(delta^{−1}))$ weak estimates.
It is easy to use Chebyshev's inequality to find a weak estimate of $N = O( sigma^2 / delta varepsilon^2 )$.
However, I do not how to find inequality about their median. The wikipedia of median (https://en.wikipedia.org/wiki/Median#The_sample_median) says sample median asymptotically normal but this does not give a bound for specific $N$. Any suggestion is welcome.
probability median
asked Nov 15 at 17:02
Rikeijin
878
Problem:
Suppose we want to estimate the mean µ of a random variable $X$ from a
sample $X_1 , dots , X_N$ drawn independently from the distribution
of $X$. We want an $varepsilon$-accurate estimate, i.e. one that
falls in the interval $(mu − varepsilon, mu + varepsilon)$.
Show that a sample of size $N = O( log (delta^{−1} ), sigma^2 / varepsilon^2 )$ is sufficient to compute an $varepsilon$-accurate
estimate with probability at least $1 −delta$.
Hint: Use the median of $O(log(delta^{−1}))$ weak estimates.
It is easy to use Chebyshev's inequality to find a weak estimate of $N = O( sigma^2 / delta varepsilon^2 )$.
However, I do not how to find inequality about their median. The wikipedia of median (https://en.wikipedia.org/wiki/Median#The_sample_median) says sample median asymptotically normal but this does not give a bound for specific $N$. Any suggestion is welcome.
probability median
probability median
asked Nov 15 at 17:02
Rikeijin
878
asked Nov 15 at 17:02
Rikeijin
878
asked Nov 15 at 17:02
Rikeijin
878
asked Nov 15 at 17:02
Rikeijin
878
asked Nov 15 at 17:02
Rikeijin
878
878
add a comment |
add a comment |
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