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Using the forecast function in R, I make a 1-step prediction for a log-transformed data set Y, ( Y = log(X) ). This prediction gives me a mean and a 95% prediction interval.

How valid is this prediction interval when considering the untransformed data? For example, the prediction may give a mean of 4 with prediction interval [3,5]. If we backtransform (with exp) this we get a mean of 54.6 with prediction interval [20.1, 148.4]. This seems to predict an exaggerated range of values over the predicted mean. Can someone explain this?

Further, if I wanted to draw from a probability distribution of my prediction, how should I do so?

kjetil b halvorsen
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user591497
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    Can you expand on why you find that surprising? The interval is width 2 on the log scale correspnding to a ratio of exp(2) on the original scale. – mdewey Sep 09 '18 at 15:30
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    Note that the mean doesn’t backtransform like that. E(log(x)) != log(E(x)). – The Laconic Sep 09 '18 at 22:27
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    @TheLaconic Nevertheless, it *is* valid to backtransform the prediction interval endpoints, which is what this question is about. – whuber Sep 10 '18 at 17:21
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    @whuber Agreed. But it is an important point the OP seemed to have missed, so I thought it worth a comment. – The Laconic Sep 10 '18 at 17:35

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