Defining confidence intervalsI have a set of n observations O. I want to calculate the mean of these observations and their confidence interval. I can do this when the observations are iid, by looking up the t-table for the required confidence value and then calculate the interval as -- 2 t σ n However, if the i t h observation of this sequence is related to ( i − 1 ) t h observation, how do I now define the confidence interval. I believe the iid assumptions are no longer valid.

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Defining confidence intervalsI have a set of n observations O. I want to calculate the mean of these observations and their confidence interval. I can do this when the observations are iid, by looking up the t-table for the required confidence value and then calculate the interval as --  2  t      σ          n      However, if the       i          t      h       observation of this sequence is related to   (  i  −  1      )          t      h       observation, how do I now define the confidence interval. I believe the iid assumptions are no longer valid.

Kaitlyn Levine · Accepted Answer

Step 1This is very similar to a time-series problem, where you have Lag-1 correlation among data points. You first need to fit a model to this relationship and then remove it from each data point so you can analyze the residuals, which will hopefully be iid. For example, if       x    2    =      x    1    +      ϵ    1   where       ϵ    1    ∼  F  (  ϵ  )  ,  (  F  (  ϵ  ) is arbitrary) then you need to model this as a random walk and then you can calculate the mean value of   ϵ, since the residuals will be iid.Step 2In other words, correlations between successive points make your data non-stationary, and hence not iid. You need to compare apples to apples by adjuting for the correlation via some model and examining the residuals of the model using iid-based statistics.

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