For the South American (SA) monsoon, Silva et al. (2007) and Carvalho et al. (2012) proposed the Long-scale South American Monsoon Index (LISAM), based on the annual cycle of some fields.
For the 2012 paper, where several precipitation datasets are used to calculate what you want, the methodology is as follows:
Compute the empirical orthogonal function (EOF) of a precipitation gridded data on land over SA. (There are some details as weigth by latitude, you may want to do combined EOF etc etc... but the essence of the idea is the same). Note that EOF is computed on raw precipitation, not anomalies.
The first EOF represents the annual (seasonal) cycle of the monsoon. The First PC (PC1) shows it clearly (Figure 2 in Silva et al. 2007)
At last, having the time series of PC1, you can determine:
Start (onset): the day in which PC1 turns positive. End (demise): the day in which PC1 turns negative.
Further detailes are given in the cited papers. Hope it helps.