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Problem

I have a time series data set with about 50 observations. I'd like to compute an interval that may contain the next/future value in the time series (the 51st data point). I tried using a 90% t-confidence interval (data isn't that normal) for this, so I calculated the mean, standard deviation, etc. However, the resulting interval captured less than 50% of the sample. That's not a very reassuring result given that it is a 90% CI and it doesn't give me much confidence on the ability of the interval to contain the next value observed in the time series.

After reading more about CI...

I started realizing that expecting a 90% confidence interval to contain about 90% of the sample is a popular misunderstanding because the confidence interval is a statement about the population statistic. Also, the "statistic" that my interval is discussing is the mean. However, this got me wondering if using a confidence interval to solve my problem even makes sense. I computed a 90% confidence interval around the mean of my data set but what I need is an interval that captures the next value in my data set. I believe those are two different things.

Questions

  1. Is there another method that's more appropriate? I saw something about using the RMSE instead of standard deviation in a confidence interval and adding the 90% t-value and RMSE based margin of error to the mean. I also saw the "prediction interval" method. Would bootstrapping be helpful? What is best? What sort of assumptions would be made about the data?

  2. Why doesn't a 90% CI capture at least 90% of the sample, mathematically speaking?

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Although I don’t have the perfect answer to your question, but apropos of the popular misconception - a 90% CI for the mean simply implies that “the population mean is likely to lie in the said interval in 90% of the samples”. Expanded - a X% Confidence Interval (where X could be 90%, 95% or 99% etc.) means that in X% of random samples drawn from the distribution, the estimated mean will lie in the stated interval. It does not mean that X% of the population lies in the CI.

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I don't think you are computing the interval you say you want to compute. If you take the mean and standard deviation of the entire data set and make a prediction interval, you are attempting some sort of unconditional prediction interval. But you said you want to predict the 51st value conditional on the previous 50 values. This is a one-step-ahead forecast, and is not expected to contain 90 percent of the dataset, it is expected to do what is advertised, that, given 50 values, the interval will contain the next value 90 percent of the time.

The whole idea of forecasting is to get a narrower interval, based on the fact that you have observed first 50 values.

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