Update row with data from another row in the same table

I've got a table which looks something like this

ID   |   NAME    |  VALUE  |
----------------------------
 1   |   Test    |  VALUE1 |
 2   |   Test2   |  VALUE2 |
 1   |   Test2   |         |
 4   |   Test    |         |
 1   |   Test3   |  VALUE3 |

I'm looking for a way to update the values 'Test2' and 'Test' with the data from other rows in the 'VALUE' column with the same 'NAME' (The ID is not unique here, a composite key of the ID and NAME make a row unique). For example, the output I'm looking for is:

ID   |   NAME    |  VALUE  |
----------------------------
 1   |   Test    |  VALUE1 |
 2   |   Test2   |  VALUE2 |
 1   |   Test2   |  VALUE2 |
 4   |   Test    |  VALUE1 |
 1   |   Test3   |  VALUE3 |

If it was in another table I'd be fine, but I'm at a loss as to how I can reference a different row within the current table with the same NAME value.

Update

After modifying manji query, below is the query I used for a working solution. Thanks all!

UPDATE data_table dt1, data_table dt2 
SET dt1.VALUE = dt2.VALUE 
WHERE dt1.NAME = dt2.NAME AND dt1.VALUE = '' AND dt2.VALUE != '' 

Try this:

UPDATE data_table t, (SELECT DISTINCT ID, NAME, VALUE
                        FROM data_table
                       WHERE VALUE IS NOT NULL AND VALUE != '') t1
   SET t.VALUE = t1.VALUE
 WHERE t.ID = t1.ID
   AND t.NAME = t1.NAME

Here's my go:

UPDATE test as t1 
    INNER JOIN test as t2 ON 
        t1.NAME = t2.NAME AND 
        t2.value IS NOT NULL 
SET t1.VALUE = t2.VALUE;

EDIT: Removed superfluous t1.id != t2.id condition.

Update MyTable
Set Value =	(
				Select Min( T2.Value )
				From MyTable As T2
				Where T2.Id <> MyTable.Id
					And T2.Name = MyTable.Name
				)
Where ( Value Is Null Or Value = '' )
	And Exists	(
				Select 1
				From MyTable As T3
				Where T3.Id <> MyTable.Id
					And T3.Name = MyTable.Name
				)

UPDATE financialyear
   SET firstsemfrom = dt2.firstsemfrom,
       firstsemto = dt2.firstsemto,
       secondsemfrom = dt2.secondsemfrom,
       secondsemto = dt2.secondsemto
  from financialyear dt2
 WHERE financialyear.financialyearkey = 141
   AND dt2.financialyearkey = 140

UPDATE t SET t.VALUE = t1.VALUE

FROM data_table t,

(
  
SELECT DISTINCT ID, NAME, VALUE
	FROM data_table
	WHERE VALUE IS NOT NULL AND VALUE != ''

) t1

WHERE t.ID = t1.ID

AND t.NAME = t1.NAME

If you just need to insert a new row with a data from another row,

    insert into ORDER_ITEM select * from ORDER_ITEM where ITEM_NUMBER =123;