|
7 |
edited title; edited title
|
||
When is the function of a median is closer to the median of a the function than the mean of a the function is to a the function of the mean.? |
||||
|
|
6 | edited title; edited body | ||
Under what conditions will When is the function of a median be is closer to the median of a function than the mean of a function is to a function of the mean?.I previously asked this on here on stats.stackexchange.com, but after not receiving an answer, was advised to post here on MO. Background notation: RV= random variable, $\mu=$ mean $m=$ median Jensen's Inequality considers the relationship between the mean of a function of an RV and the function of the mean of an RV. If $f(x)$ strictly convex: $$\mu (f(x)) > f(\mu (x))\mathrm{\hspace{20mm}(1)}$$ Conversely if $-f(x)$ is strictly convex: $$\mu (f(x)) < f(\mu (x))$$ An analogous property of the median has been presented (Merkle et al 2005, pdf). Motivation I have a nonlinear function (pdf) of positive random variables, too complex to post here, not directly pertinent to this question; I am looking for a more general answer. It is worth noting that it is, however, neither strictly concave nor convex. In practice, I find that the function of the medians provides a much better estimate of the median of the function than does the estimate of the mean of the function from the function of the means. I am interested in learning the conditions for which this is true. Question Under what conditions will the function of a median be closer to the median of a function than the mean of a function is to a function of the mean? Specifically for what types of $f(x)$ and $x$ is $$|\mu (f(x)) - f(\mu (x))| > |m (f(x)) - f(m (x))|$$ I previously asked this on here on stats.stackexchange.com, but after not receiving an answer, was advised to post here on MO. References |
||||
|
|
5 | fixed eqn by adding absolute value parens | ||
|
I previously asked this on here on stats.stackexchange.com, but after not receiving an answer, was advised to post here on MO. Background notation: RV= random variable, $\mu=$ mean $m=$ median Jensen's Inequality considers the relationship between the mean of a function of an RV and the function of the mean of an RV. If $f(x)$ strictly convex: $$\mu (f(x)) > f(\mu (x))\mathrm{\hspace{20mm}(1)}$$ Conversely if $-f(x)$ is strictly convex: $$\mu (f(x)) < f(\mu (x))$$ An analogous property of the median has been presented (Merkle et al 2005, pdf). Motivation I have a nonlinear function (pdf) of positive random variables, too complex to post here, not directly pertinent to this question; I am looking for a more general answer. It is worth noting that it is, however, neither strictly concave nor convex. In practice, I find that the function of the medians provides a much better estimate of the median of the function than does the estimate of the mean of the function from the function of the means. I am interested in learning the conditions for which this is true. Question Under what conditions will the function of a median be closer to the median of a function than the mean of a function is to a function of the mean? Specifically for what types of $f(x)$ and $x$ is $$\mu $|\mu (f(x)) - f(\mu (x)) x))| > |m (f(x)) - f(m (x))$$x))|$$ References |
||||
|
4 | Rollback to Revision 2 | ||
|
|
3 | deleted 189 characters in body | ||
|
|
2 | added link to pdf | ||
|
|
1 |
|
||
