On Thu, 10 Oct 2013 22:53:46 +0200, Cristiano wrote: > On 10/10/2013 22:35, James Waldby wrote: >> On Thu, 10 Oct 2013 22:15:46 +0200, Cristiano wrote:
>>> In this link: >>> http://en.wikipedia.org/wiki/Weighted_arithmetic_mean#Statistical_properties >>> >>> there is the formula sigma(x_bar) = sqrt(...). >>> I need to find the n w's which give the smallest possible sigma(x_bar). >>> Please, could somebody help me? >> >> Your question might not be well-posed. If you are given a set of >> sigma_i values and want to minimize their weighted sum of squared >> values, and weights add up to 1, find the minimum sigma_j, >> set w_j = 1, and set w_i = 0 for i other than j. If weights don't >> have to add up to 1, set them all to zero, which gives sigma(x_bar)=0. > > The additional constraint is that all the w's must be greater than 0.
In that case, choose some small epsilon, eg epsilon = 10^(-99), and set w_j = 1 - (n-1)*epsilon and all other w_i = epsilon. Then all the w's are greater than 0. Note, in this formulation, sigma(x_bar) doesn't actually achieve sigma_j as its minimal value, but approaches that value arbitrarily closely as epsilon approaches zero.
In saying the question might not be well-posed, I meant that as stated it is underconstrained, so trivial solutions exist that probably don't match up with your expectations. Not knowing what you are aiming at or what your expectations are or what sort of data is involved makes it difficult to give a better answer.