Are samples from sliding windows independent samples? E.g. if I have window size of 90 seconds counting the number of cars on a street and I output the average within the window every second for 30 seconds, do I have 30 independent samples or not?

I'd say yes, as it looks (to me) like sampling with (partial?) replacement. But I'm not sure. I'm asking, because I thought if the samples were independent and I collected n>30, the central limit theorem could be used for further calculations.


1 Answer 1


No. Your have overlapping observations. The first and the second observations have 89 seconds of overlap, they are clearly heavily correlated.

  • 1
    $\begingroup$ It might be of interest to note that when the window size is odd, it is possible (although, I admit, unusual) for two overlapping windows to be uncorrelated. For instance, with a window of size $89$ and an AR-1 series with serial correlation around $-0.985737,$ successive averages will be uncorrelated (but still highly dependent). This indicates that it would be better not to substitute lack of correlation for independence in reasoning about this question. $\endgroup$
    – whuber
    May 13, 2014 at 20:41
  • $\begingroup$ @whuber, I agree that lack of correlation doesn't imply lack of dependence, but not sure I understand the AR-1 example. $\endgroup$
    – Aksakal
    May 13, 2014 at 21:16
  • $\begingroup$ Thanks for the comments. What type of statistical analyses would you suggest for such kind of data? Is paired t-test ok, if I want to test the hypothesis if the means of the two samples are equal? Are there other methods that make sense in this scenario? $\endgroup$
    – Frank D
    May 14, 2014 at 15:56
  • $\begingroup$ One way to proceed is with HAC estimator, such as Newey-West. Basically, you run a usual regression, then adjust the coefficient variances with a special procedure. $\endgroup$
    – Aksakal
    May 14, 2014 at 15:59

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