PLT OF A MOONEY M20K-231; OBSERVED ANOTHER UNKNOWN ACFT JUST ABOVE HIM DURING A FINAL ILS APCH TO A NON CTLED ARPT. HE WAS NOT ON FLT PLAN OR CTLED; BUT BELIEVED THAT IT WAS AN ACR ACFT HE HAD HEARD CALLING INTO THE LCL FSS RADIO FACILITY EARLIER. NMAC.

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CAME OUT OF PHOENIX LATE AFTERNOON. WX BRIEFING EARLIER IN DAY HAD PREDICTED CLR SKIES. FLT WATCH OFF PHOENIX AFTER DEP HAD PREDICTED BROKEN TO SCATTERED. WX WAS CLR UNTIL AFTER MEDICINE BOW. CLOUD DECK APPEARED BELOW. STARTED LISTENING TO CASPER ATIS; ON I THINK 126.15. ATIS SAID TWR/APCH CLOSED; CONTACT CASPER RADIO ON 118.3. CALLED MULTIPLE TIMES. NO REPLY. A CARRIER ALSO STARTED CALLING 'CASPER RADIO' ON 118.3 -- NO REPLY. THE ACR REPEATED HIS CALLS TO 'CASPER RADIO' MULTIPLE TIMES AS DID I. NEITHER OF US EVER RECEIVED ANY REPLIES UNTIL MUCH LATER. I REPEATED CALLS TO CASPER RADIO; AND STARTED BROADCASTING MY POS AND INTENTIONS; INCLUDING FINALLY THAT I WAS ON THE ILS. HAD EACH TIME XMITTED I WAS LNDG AT CASPER. THE CARRIER DID NOT EVER BROADCAST HIS POS OR INTENTIONS. I ASSUMED; IN ERROR; HE WAS ON THE GND IN CASPER; LOOKING FOR A CLRNC. FIRST TIME ACR ANNOUNCED WHERE HE WAS; HE WAS ABOVE ME; AND I WAS ON ILS. MY ERRORS: A) ASSUMING ACR WAS ON GND; B) SHOULD HAVE CALLED ZDV; BUT ATIS SAID TO CALL 118.3; C) COULD HAVE ASKED ACR WHERE THEY WERE. OTHER SUGGESTIONS; A) ATIS IN CASPER COULD INCLUDE PHRASE 'OR ZDV ON XXX;' B) BE NICE IF CASPER RADIO WOULD ANSWER CALLS.