# Deterministic trend and noise

I have a chicken-egg problem in time series:

How do we remove the deterministic trend in time series when you have a guess that the noise is non-normal? How do we check if the noise is non-normal without removing the deterministic trend?

For my time-series, when I plot the histogram of the raw values and fit a density a log-normal seems to fit very well. However, this is raw data, and it has a trend, and potential seasonality. If I remove the trend and seasonality, would the resulting density still remain a log-normal? How would I remove the trend and seasonality if I don't know the noise distribution? If my noise is indeed log-normal, then I don't think I can use an OLS regression, and have to use a GLM (which seems not to handle log-normal, somehow, but that's a separate question). Any thoughts on this, please?

I hope I have made my question clear, but please comment if you need more information.

If a series $\{ x_t\}_{t=1}^\infty$ has a linear deterministic trend, it can be written $x_t=y_t+k t$ . An estimator for $k$ is $k=\frac{X_T - X_t}{T-t}$. To remove the trend from data, you take $y_t=x_t-kt$.
• Thanks for your comments. In line with @Aksakal, even if you assume the trend is linear, you need an assumption on the noise to show that $k$ or an averaged or OLS version of $k$ is the right estimate. For instance, if your distribution looks lognormal, then we may prefer X_T/X_t as opposed to a subtraction (not claiming this is how it should be done, but just an example). – user1669710 Feb 10 '15 at 20:53