The District of Columbia's public housing agency is losing millions of dollars a year by failing to collect back rent, evict nonpaying tenants, perform routine maintenance and restore nearly 1,400 vacant units, federal auditors have found.

City housing officials have so little control over public housing rents, the auditors said, that in one case they continued to bill a dead woman for 18 months.

In addition, according to the audit by the federal Department of Housing and Urban Development, the city lost $1.5 million in interest by failing to invest large amounts of cash advanced by the federal government.

The 74-page audit by HUD Inspector General Charles L. Dempsey said the D.C. Department of Housing and Community Development has suffered "a breakdown in the internal control" over its 11,600 public housing units.

The audit noted that most of the same problems were cited in federal reports in 1970, 1975 and 1980, but said that the District has taken little corrective action.

James Clay, director of the city's housing department, acknowledged most of the findings yesterday, but said the department has made substantial progress in recent months. Since 45 public housing tenants were evicted last spring, he said, the proportion of tenants paying rent has increased from 68 percent to about 88 percent.

"We're trying to provide decent housing for people," Clay said. "You don't resolve problems that are systemic in a year or two years. I'm comfortable that we have a plan, a strategy.

"If you're trying to operate manually 12,000 accounts, there are going to be some screw ups, and we've had our share," he added.

The report is one of a series of federal audits of financially troubled housing authorities across the country. Clay confirmed that D.C. public housing, which receives more than half its budget from federal subsidies, would be more than $9 million in the red if the District government had not poured in an additional $13 million from 1981 to 1983.

Clay estimated last January that public housing tenants owed more than $1 million in rent. But the HUD audit said the back rent owed by 5,500 families as of last September totaled $4.8 million, with nearly two-thirds owed by tenants still living in public housing.

Clay said yesterday he believes the figure is $3.1 million; the report said the District "could not accurately determine" the overdue rent and has submitted different figures on different occasions.

Whatever the precise figure, the report said the District has evicted few nonpaying tenants, does not charge delinquent tenants a late fee and stopped referring such cases to a private collection agency 18 months ago. In addition, the auditors said, the city's plan to begin calling delinquent tenants foundered for months because new telephones had not been installed.

As an example of the city's confused rent records, the auditors said that two different tenants have been billed for rent for the same apartment since November 1982. One tenant was a woman who died the previous August, but continued to be billed $205 a month for 18 months; the other was her daughter, who had taken over the apartment.

In other cases cited by the audit, one tenant was charged twice a month for the same unit, $207 and $227, and was listed as delinquent; and several tenants were cited as owing back rent on one city list and being owed money on a second list. The auditors selected five tenants at the Arthur Capper project who were assessed no rent at all, and found that four of them should have been charged some rent; records were incomplete for the fifth.

Clay blamed the record-keeping problems on a lack of computerized equipment, which he expects this winter. Clay said the city stopped using a collection agency because some former tenants could not be found and others could not afford to pay the back rent. He added that the housing agency has been "delinquent" in failing to write off some of the rents as bad debts.

Clay said the city has had continued difficulties in enlisting federal marshals to carry out evictions, but that he will continue to try to evict delinquent families "as a last resort."

Another major finding by the auditors is that 1,396 public housing units, or more than 1 in 10, remain vacant. They said the District's slowness in repairing and renting empty units -- 63 percent of which have been vacant more than six months, and some more than three years -- is costing the city $1.4 million a year in rental income.

The city's contention that many units remain vacant because of scheduled modernization work, the auditors said, is "somewhat misleading . . . We noted instances in which units were vacant for long periods of time after modernization work was completed or before it was begun."

Last October, for example, the city finished moving all residents out of the 214-unit Greenleaf Gardens on Delaware Avenue SW so it could be converted to a project for the elderly. But the auditors said the city only recently solicited bids for an architectural and engineering study of the building and is losing $19,000 a month while the project remains vacant.

Clay said the families had to be moved because the Greenleaf units had become unsafe and unsanitary. He said it "takes longer than I would like" to restore vacant units, but blamed this on vandalism, high repair costs and delays by HUD in approving plans to repair or demolish projects.

The auditors called the public housing maintenance program "inefficient, ineffective and uneconomical," and said the city performed no preventive maintenance to stop further deterioration of aging buildings. The auditors said that high-paid skilled workers such as carpenters and painters were being used to sweep hallways, mop floors, remove trash and clean parking lots.

The auditors also found that maintenance workers received no written instructions and that, in one review of 35 workers, there was no documentation for how the workers spent 60 percent of their time. Clay acknowledged these problems and said the city hopes to start a preventive maintenance program "in the next year or so."

The report criticized the housing department for failing to invest federal grants advanced for two scattered-site housing projects, causing the city to lose $1.5 million. Although this surplus federal cash totaled $4.7 million at one point, the housing department reported no investment income, not even from an interest-bearing account, from 1981 to 1983. A city housing official told HUD it was possible that the District government had invested the housing money and retained the interest, but that he wasn't sure, according to the report.

The audit also said that the city:

*May have to return $1.1 million to HUD because it has not performed required energy audits on most projects and does not monitor most fuel consumption. Clay said the energy audits should be completed this month.

*Lost up to $150,000 by undercharging some tenants and failing to recertify others as still eligible for public housing.

*Could not locate records for nine ranges and 60 refrigerators owned by the housing department.