Defining confidence intervals. I 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 -- 2t sigma/(sqrtn)

Sonia Rowland 2022-10-01 Answered
Defining confidence intervals
I 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 I believe the iid assumptions are no longer valid.
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Answers (1)

Kaitlyn Levine
Answered 2022-10-02 Author has 12 answers
Step 1
This 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 2
In 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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