# Maximum Likelihood Estimator for Exponential Smoothing

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!

"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.