Hi,
You can create a custom solution to track search statics.
For example, you can create a node in AEM to save all search related details. For each search you can create a node.
and at the EOD you can pull those details in JSON/XML/Database and delete those node to free up some space. But if you are planning to implement something like above for more than one instance architecture then you can directly save data on centralise location/database or sync between instance.
The idea is here to create custom solution if you don't have analytics implemented.
Thanks
Arun
Arun Patidar