I need an estimate of the average value $\bar{X}$ of the data within the shaded area of this graph, and its corresponding confidence interval.
In the ideal case, the data would reflect some minor oscillations. I'm tempted to ignore this, and use the $CLT$ so the standard error would be $SE=\frac{S}{\sqrt{N}}$, where $S$ is the sample standard deviation, and $N$ is the total number of data points in the range. But I'm uncomfortable with this approach because I know this is not completely correct.
But the main problem is that sometimes the data is not stationary for the whole range, and can show multiple stationary segments, as shown below.
These issues invalidate any assumption of normality. I'm struggling to identify a proper method to calculate a single value for the mean and confidence interval for the whole data range. Any ideas?
Best Answer
You may consider a nonstationary time series as consisting of a number of time segments that are stationary. You can do it either manually or using methods related to the algorithm discussed here. Once segments are identified, you can estimate the mean, variance and the autocovariance structure of each stationary segment.
The fact that time series is non-stationary does not automatically make the assumption of normality inappropriate. Also note that for a non-stationary time serires the long-run mean is undefined. For example, how would you define the mean of a random walk? The local means of stationary segements we refer to as levels of the process. To estimate levels at any given point of (discrete) time, we use methods appropriate for non-stationary time series, such as those of exponential smoothing.