Tuesday, June 10, 2014

mean, median and mode

this page explains a mean minimizes the $\ell^2$ norm of the residual:$\min_{m_2} \sum_i (m_2-d_i)^2$ 
a median minimizes its $\ell^1$ norm and a mode minimizes the zero norm of the residual, namely $\ell^0=\vert m_0-d_i\vert^0$.See the wikipedia page about median.

from here, it was further explained that
Inder Jeet Taneja’s book draft has a nice survey of the results: if you fix the upper and lower boundary, and maximize entropy, you’ll get the uniform distribution. If you fix the mean and the expected L2 norm (d^2) between the mean and the distribution, maximizing the entropy you’ll get the Gaussian. If you fix the expected L1 norm (|d|) between the mean and the distribution, maximizing the entropy you’ll get the Laplace (also referred to as Double Exponential). Moreover, log(1+d^2) norm will yield the Cauchy distribution – a special case of the standard heavy-tailed Student distribution.
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