Evaluate statistically if time series is Jump or Trend? Q1.
Is there any statistical test to differentiate "Trend" and "Jumps". Below example.
I have tried Successive variation test (Neumann), all data sets yield same results.
It is clear that Example 1 has trend, Jumps in case of example 2 & 3.
What statistical will tell us that Example 1 has trend while other's do not.
Unfortunately will not be able to share the data, just looking for some clues.




Q2.
How do you statistically conclude that there is trend / No trend ? |or| How do you say data is non-stationary and has a trend ?
If slope is the measure of trend, is there a threshold for slope after which we can consider that the time series has a trend ?

You could use the $R^2$ (or mean squared error or mean absolute error) metric to see if the trend is really approximating the data appropriately. In the first case the performance metric should be higher (for $R^2$) or lower (for mean squared error or mean absolute error) indicating a trend in comparison to the last two cases.
Note, that trend could also imply that the series is stationary. This would be true if the slope of the linear approximation is vanishing.
