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In many applications such as estimation theory, when we need to estimate a parameter then we usually consider in presence of white gaussian noise of zero mean and some standard deviation. During Maximum likelihood estimation, we also use this assumption. So, my question is -

  1. Do we consider noise to be uncorrelated or correlated in estimation?

  2. What is the difference between correlated and uncorrelated noise and its significance

  3. Why do we consider such a property of correlated or uncorrelated in estimation and when we say that measurement noise is gaussian.

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  1. "Noise" implies portions of an estimate that are random (e.g. unknowable, except in terms of distributional behavior). We consider noise to be uncorrelated. I suppose that within the realm of time series analysis processes like pure random walks might be considered a category of "correlated noise" although I would say that's a bit of a misnomer: random walks have (nonlinear) deterministic and random (noise) components.

  2. "Correlated noise" reads to me like an oxymoron... a little like saying "all the things we know about things we do not know about."

  3. Estimation error is not always assumed to be Gaussian, but is very commonly presumed to be so. I would imagine that the significance of the central limit theorem, and it's deep relationship to so many distributions and processes makes it such a commonly assumed for of noise.

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    $\begingroup$ Random means unknowable, except in terms of distribution. Correlated means (by certain definitions) a quantified measure of association; and one way of looking at association is that knowing something about X will tell you something about Y. But is noise is synonymous with random, then knowing about some X will not tell you anything (no association). Colored noise (if you mean the stuff that psycho-acoutistics people and engineers are talking about—pink noise, etc.) is likewise uncorrelated. The colored noises would simply have different distributions than Gaussian noise. $\endgroup$ – Alexis Jun 5 '14 at 21:45
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    $\begingroup$ Ok... imagine that you are watching a signal come in, bit by bit (or whatever your smallest chunks of signal are :). If that signal is purely noise, then the only thing you can do is make inferences about (and base decisions off/expectations about) the distribution of that noise. You can infer the distribution of the noise based on all the chunks you have already observed. Or you might know (based on theory or Some Authority) what the distribution is a priori. But for pure noise, you can do no better than guess the next chunk out the pipe based on the expectation of the distribution. $\endgroup$ – Alexis Jun 5 '14 at 22:07
  • $\begingroup$ No. 2 random variables are correlated iff there exists a linear dependence between them, that is, they have a non-zero covariance. Colored is a property of stochastic processes and it just means that the process gives rise to non-uniform power spectral density. Neither of these concepts are related to distribution. $\endgroup$ – RBF06 Feb 6 '19 at 18:24
  • $\begingroup$ @RBF06 "Colored is a property of stochastic processes and it just means that the process gives rise to non-uniform power spectral density. Neither of these concepts are related to distribution." That is an oxymoron, since by definition different densities implies different probability density functions (either in parameters, or in form) if the things having different densities are realizations from some kind of noise process. $\endgroup$ – Alexis Feb 7 '19 at 1:32
  • $\begingroup$ the concepts of correlation, statistical dependence, and power spectral density (noise color) are independent of probability distribution. Correlation and noise color are related in the sense that noise color arises from elements of a random processes being correlated with themselves. This has nothing to do with how any one of the elements is distributed. $\endgroup$ – RBF06 Feb 7 '19 at 13:38

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