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I'm not a statistician, so I would love an easy to understand answer. Is there a maximum likelihood estimator that can be stated as an explicit function of the observed data for the models enumerated below? If yes, what are the functions? I assume there's a different function for each parameter, right?

  1. Holt-Winter double exponential smoothing // for parameters a and b
  2. Brown's double exponential smoothing // for parameters a and b
  3. Peter Winters' triple exponential smoothing // for parameters a, b, and y.

I use the wikipedia definitions of these exponential smoothing models as defined in http://en.wikipedia.org/wiki/Exponential_smoothing#Double_exponential_smoothing

and

http://en.wikipedia.org/wiki/Exponential_smoothing#Triple_exponential_smoothing

If the answer is that there is no explicit function for the maximum likelihood estimators for these models, is there any other type of explicit function that can be used to estimate the parameters? I'm only looking for explicit functions of the observed data, not optimization programs.

Thanks so much to anyone who can help!

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"Traditional" exponential smoothing models are heuristic models and does not have an underlying statistical theory. So there is no maximum likelihood estimator. However, seminal work published in the Journal of the American Statistical Association by Ord, Koehler & Snyder in 1997 provided a statistical basis for exponential smoothing using a state space framework and provided a maximum likelihood estimation procedure. This book on Exponential smoothing extends the original work into all the forms of exponential smoothing including the ones that you mention. This book provides a clear example of how to implement maximum likelihood estimation. Forecast package in R uses the state space exponential smoothing framework. This package also uses Maximum Likelihood for estimation.

Here is the article that shows the log likelihood function for exponential smoothing also called as ETS models.

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  • $\begingroup$ thanks for the info. My follow up question is regarding the book you linked to in "Forecasting with Exponential Smoothing." You said it includes a clear example of how to implement maximum likelihood estimation. Is it something that I could then convert to an explicit equation that's just a simple function of the observed data? And by simple function, I mean it only uses the basic mathematical operators like +-x/^log, and ln. It's an $80 book, so just want to make sure that I don't buy it if it's not going to give me what I'm looking for. Thnx! $\endgroup$ – Tarak Nov 7 '14 at 18:19
  • $\begingroup$ the preview of the book has the maximum likelihood estimation part freely available to you to make a decision on whether to purchase or not. In addition, I have updated the post with the reference article that shows the log likelihood estimator for exponential smoothing with state space models. This is the same log likelihood in the book. The open source code for ETS is also available in R, you can find out the code implementation there as well. You can make your own call whether to purchase the book or not $\endgroup$ – forecaster Nov 8 '14 at 3:55
  • $\begingroup$ I read the article, and it states you have to minimize the L* equation (#9 in article) to obtain the parameter estimates. Then the procedure involves using an optimizer. So, I think the answer to my original question is that there is no explicit function of the observed data that can be used for parameter estimation. Please correct me if I'm wrong. Assuming I'm right, then is there any other less accurate methodology for parameter estimation that doesn't require an optimizer, and only uses as an explicit function of the observed data? Thnx! $\endgroup$ – Tarak Nov 8 '14 at 18:55

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