Comparison　is　made　between　the　MINQUE　and　simple　estimate　of　the　error　variance　in　the　normal　linear　model　under　the　mean　square　errors　criterion,　where　the　model　matrix　need　not　have　full　rank　and　the　dispersion　matrix　can　be　singular.　Our　results　show　that　any　one　of　both　estimates　cannot　be　always　superior　to　the　other.　Some　sufficient　criteria　for　any　one　of　them　to　be　better　than　the　other　are　established.　Some　interesting　relations　between　these　two　estimates　are　also　given.　©　Springer-Verlag　2003.