Suppose, if I draw samples of different sizes at time t=1,2,..,T,T+1,..,2T where the sample sizes could be different and the observed variable is a ratio. I want to test whether there is a structural break at time t=T for the observed variable. What statistical test is recommended? I want to confirm whether Bai & Perron's test can handle this problem.

Alternatively, at an aggregate level, I did test the two samples (post & pre structural break) for normality using Shapiro-Wilk test and these two are found not to be drawn from Normal distributions.