I am looking for a test, or set thereof, that allows me to quantify if a time series has changed from one point in time to another.
Problem Description: I have a time series, consisting of measurements at discrete points in time $(t_1, t_2, \ldots, t_n)$. At each $t_i$, a sample of size $N$ is acquired. From prior analysis I know that each sample at $t_i,\, 1 \leq i \leq n$, is normally distributed. Loosely speaking, I am looking to quantify if and to what degree the time series has changed.
My thoughts so far: My initial idea was to use a two-sample two-sided t-test for a given pair $(t_i,\,t_j),i \neq j$ where I choose the sample size $N$ to match my power pre-registration. For the pre-registration, I essentially ran pwr.t.test(n = NULL, d = x, sig.level = 0.05, power = 0.95, alternative = "two.sided", type = "two.sample") in R (using pwr package), where I chose the effect size $x$ to match values from a pilot study.
Update changepoint detection: After reading a bit through how changepoint detection is working, most of the approaches seem to be centered around fitting ARIMA models pre and post intervention and compare coefficients. Is there any special treatment available for when a distribution is available at each point in time, as opposed to only having a single measurement?
Update related questions: While I wasn't able to find an exact answer on StackExchange, I found two related questions. a) Building a time series that includes multiple observations for each date : Answers suggest to use panel data models, however this depends on how exactly the samples are drawn and in my case I do not necessarily measure the exact same subjects over time and b) Finding the change point in data from a piecewise linear function : where answers include R package suggestions for changepoint detection.
I would be happy for any pointers, relevant papers or general guidance for this particular problem.
