In a statement, her family said it had been "emotional week" after a farmer found what he suspected to be a grave close to where she disappeared in 1994.
They thanked him for contacting police, even though her remains were not found.
Arlene, who was 15 and from Castlederg, County Tyrone, was last seen in a car with the child killer Robert Howard.
He was acquitted of her murder in 2005 by a jury not told about his conviction for killing another teenage girl in England.
At Belfast Coroners Court on Monday, a statement from a senior police officer investigating Arlene's murder was read to the inquest.
Det Ch Supt Raymond Murray said police first received a phone call last Tuesday from the current owner of an unoccupied property in Killen, County Tyrone.
The owner informed them that a farmer, Noel Doherty, had discovered what he believed to be a grave in what would have been the back garden of the property.
"The actual excavation of the site took place on Friday 16th September 2016 and the experts concluded no human remains were present", the statement read.
"However, when they were present at the scene they indicated another area close by where, at some point, soil may have been disturbed.
"The scene was then held for a further day and this, too, was excavated. It was discovered to be an old sceptic tank.
"Nothing suspicious was found at either of the excavation sites. There are no outstanding lines of inquiry generated as a result of the search operation."
The Arkinson family did not attend the hearing but in a statement read by their legal representative, they said it had been "a very emotional week" and they thanked police, the media and the community for their help and support.
"They particularly want to thank the farmer, Mr Doherty, for reporting his suspicions to the police," their solicitor said.
"The family have to remain hopeful and do so while being bitterly disappointed that Arlene's remains haven't been found yet."
The Police Service of Northern Ireland (PSNI) believe Arlene, who has been missing for 22 years, was murdered.
Howard, who died in prison last year, remains their only suspect.
Howard was originally from County Laois in the Republic of Ireland and he picked up Arlene on 14 August 1994, after she attended a school disco in Bundoran, County Donegal.
Belfast Coroners Court was told on Monday that new legislation has now been passed in the Republic of Ireland that will enable An Garda Síochána (Irish police) to cooperate fully with the inquest.
The inquest was told that the coroner's legal representatives are to meet the Irish state solicitor in the next few days to identify what, if any, evidence Gardaí can contribute to the inquest.
The coroner said: "The elephant in the room is none of us knows what the Gardaí have concerning this and that is going to be question number one".
The inquest was adjourned until 7 October.
The inquest into the death of Arlene Arkinson has heard more detail about last week's unsuccessful search for the remains of the missing schoolgirl.