How to reduce the required frequency of ANALYSE on a postgres table? I have a Postgres database into which about 20 million rows are inserted every day.

The records are inserted via a powershell script that calculates the average time taken to insert each record. It typically takes about 1.5ms per record (including some record parsing and a trigger).

Every now and again (typically after about 3 million rows are added) the processing time jumps up to about 12ms/record (it's a very noticeable step change that happens very quickly)

I've already found the cure. I stop the process that adds the records, do an ANALYSE on the table, and restart the process. The insert time goes back to 1.5ms/record. Although this only take 5-10 minutes, I do end up loosing several thousand records during this downtime. 

The database is already shutdown for a brief period at night and all the tables get ANALYSEd regularly, but is there any way I avoid doing additional ANALYSEs during the day?

(note - I've tried simply stopping and restart the INSERT process - it makes no difference. It's only stopping, doing an ANALYSE and then restarting that makes a difference)
Try to use Analyze without stopping the process. I think you already tried this, wny is this not working ? 
One option is to Analyze only specific columns , this will take less than analyzing the whole table.
Usually autovacuum is active on a postgres database, having this enabled is allowing the db to to the Analyze in the background.

Check :
https://www.postgresql.org/docs/9.6/static/routine-vacuuming.html#AUTOVACUUM

This is the formula when an analyze is run : 

analyze threshold = analyze base threshold + analyze scale factor * number of tuples


The params are here : 

https://www.postgresql.org/docs/9.6/static/runtime-config-autovacuum.html#GUC-AUTOVACUUM-ANALYZE-THRESHOLD

https://www.postgresql.org/docs/9.6/static/runtime-config-autovacuum.html#GUC-AUTOVACUUM-ANALYZE-SCALE-FACTOR

The values can be set on the table level , no need to have the same rule for all tables. 

ALTER TABLE a_lot_of_inserts SET ( autovacuum_analyze_threshold = 50 , autautovacuum_analyze_scale_factor = 0.01 ) 


Try to play with them until you find the right values.