Re: 2 peculiar problems of data analysis

Formulating the response to Ray Kooperman above has helped me to solve my first problem. In case anyone is interested, I'll share my solution here.

n independant samples X_i are generated from a Gaussian pdf with mean mu and deviation sigma. What is actually measured is

x_i = X_i + eps_i,

where eps_i is Gaussian with mean zero and deviation e_i. The e_i are known, positive numbers. This means that the x_i are Gaussian with mean mu and deviation sqrt[sigma^2 + e_i^2]. Given the x_i and e_i, we can therefore use maximum likelihood methods to get estimates for mu and sigma. The log of the likelihood function is:

log(L) = -(1/2) sum_i[ (y_i - mu)^2 / (sigma^2 + e_i^2) ] - (1/2) sum_i[ log(sigma^2 + e_i^2) ] + constants

Setting the derivatives of this with respect to mu and sigma equal to zero yields 2 coupled equations for mu and sigma:

mu = sum_i[ y_i/(sigma^2 + e_i^2) ]/sum_i[ 1/(sigma^2 + e_i^2) ]

sum_i[ (y_i - mu)^2/(sigma^2 + e_i^2)^2 ] = sum_i[ 1/(sigma^2 + e_i^2) ]

These two equation have no analytical solution for mu and sigma (I think), but can easily be solved numerically. Note that when all of the e_i -> 0, we recover the standard formulae.

Are these expressions unbiased? I doubt it, since the limiting standard formulae are biased (you need to replace a 1/n with 1/(n-1) to get unbiased-ness). It is not obvious how to fix these to make them unbiased. Hopefully for large n it won't matter.

This only leaves my second, more difficult, problem to solve. Some kind of functional maximum likelihood approach seems promising, with g(k) and sigma(k) as the minimizing functions, although it is unclear how to enfore the "entropy condition" of minimizing "wiggliness".
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