One option if this becomes a serious problem and your PD wants you to be the detective is to switch to an 800# and take ANI instead of CLID. People can block / alter CLID but since you are paying for the call with 800 service the ANI is never blocked and usually comes along on the 8K out of band supervision data. You either get it real time or at the end of the month with your bill.
Many service providers can switch you to 800 service on a PRI, including setting up a "number changed" recording on your old TN pointing to your new 800# then delivering real time ANI via the CLID mechanism. A harasser would love having toll free dialing. I've dealt with 2 systems that got realtime ANI as if it were CLID data for compatibility and simplicity. Prior to the rollout of CLID, ANI was delivered on DID trunks after the DN dialed as a string of digits that had to be logged into the rs-232 log output and processed by an external computer (usually in an answering service setup) that would notate the actual caller's number regardless of *69, etc. on the call ticket or computer display in front of the operator. CLID just made life alot easier since you no longer needed a DID trunk to get realtime ani data.
Small time 800 numbers (1 or 2 voice circuits) terminating onto ISDN often carry real time ani on the D channel of the ISDN adapter again making life easy.
Note however that with the advent of VOIP even the previously reliable and unblockable ANI is no longer totally secure. When I dial out on my VOIP service to my 800# w/ real time ani, I get a new york telephone number that is the outbound trunk of the VOIP provider - same for EVERY voip subscriber. Thus VOIP has become a truly real time anonymizer since if you have a warrant the best a VOIP provider could do is give you the credit card used to pay for a particular call made at a given time. Here again, some VOIP services don't charge anything for 800# calling thus no credit card, no direct dialing, but you fool every ANI and SS7 billing record along the way.
The local phone company should be able to put what is known as a "trap and trace" on your pilot number for the PRI. Your local PD will assign a detective who will get a printout of EVERY phone call that came in on that TN down to the second it rang and supervised and disconnected. The police will give you a daily logbook to enter each call you receive as normal or harassing. Be sure to use a watch synchronized to GPS or USNO. After you log 2 harassing calls they will compare against the trap and trace printout to find out where it is comming from. This is probable cause and a search warrant will be issued and the harasser gets a visit in person.
I would not change to 800 service if you do the trap and trace with a police detective since the caller will no doubt use coin service phones to call you. Keeping the number local will keep him on a unmeasured residential phone most likely or a cellphone.
This is unaffected by the call delivery to you, but is still vulnerable to VOIP hiding the billing number of the calling party unfortunately.
Good luck with your problem. I've had to prosecute a harassing caller on my business one time that went on for 6 months. It turned out to be a pizza delivery kid who didn't like his tip when he was late delivering pizza to the night crew who gave our inwats line as the phone number for pizza delivery along with our address. All the harassing calls happened in the daytime since the kid didn't have a CP and was busy at night. He was basically using the inwats to run up our bill and staying silent for as long as we would listen to the call.
Don't expect much information to be given to you from your detective - basically they'll ask you if you know anyone with the surname of the caller but they may not give you his TN and won't show you the log printouts from the phone company. Because that is all evidence they don't want to contaminate you as a witness for prosecution. Cases are handled based on the level of harassment (death threats get faster response than silent hangup calls or misprogrammed fax/alarm dialers) Good luck whichever way you go.