To make the basic sauce, heat the duck fat and olive oil in a saucepan and fry the duck giblets, neck, gizzard and heart, until golden brow all over. Add the chopped onion and sauter gently until tender and transparent, then add the white wine. Boil for 5 minutes and add the stock and herbs. Season with salt and pepper and simmer for 1 hour. Strain and reserve. This stage can be done in advance.
To cook the duck, preheat the oven to 200°C. Prick the fatty parts of the bird with a skewer to let the fat run out. Put it in a roasting pan and roast it for 50 minutes. Spoon off the fat and add the basic sauce, the anchovies and tomato paste. Reduce the heat of the oven to 160°C. Cover the roasting pan with foil and cook for a further 30 minutes, then add the olives and lemon rind and cook for a further 20 minutes.
Now you should finish the roasting by crisping the skin. Turn the oven up to 220°C and cook uncovered for 20 minutes or so. Remove the duck to a warm dish and leave it to rest while you skim as much fat as possible from the sauce in the tin.
Slake the cornflour with 2 tablespoons of water in a cup, and stir it into the sauce; let it simmer, stirring, over a low heat. When it has thickened and turned transparent, taste for seasoning and keep hot.
Carve the duck, put the pieces onto a deep dish, pour over the olive sauce and serve hot.