Remove the duck from the refrigerator. Let sit at room temperature for the 20 minutes that are needed for the next step.
Pour stock into a tall narrow stockpot. Be sure there is enough room left in the pot for the duck. By using a narrow pot, less stock is needed to cover the duck than in a wider pot. Add the wing tips, neck, giblets, and any blood from the duck. Cover the pot and bring to a boil over high heat.
Meanwhile, using the tines of a fork, thoroughly prick the duck all over, paying special attention to the fattiest areas. Insert the tines at an angle so there is a minimum risk of pricking the meat beneath. Carefully lower the duck into the boiling stock, neck end first, allowing the cavity to fill with stock so the duck sinks to the bottom of the pot. To keep the duck submerged, place a plate or pot cover over the duck to weight it down.
When the stock returns to a boil, reduce the heat and simmer 45 minutes. Even with a plate as weight, the duck will tend to float to the surface, so check about every 10 to 15 minutes to see that the duck remains submerged. Keep the stock at a gentle simmer; if it boils, the duck will rise to the surface.
When the duck has finished simmering, spoon 1 tablespoon of the duck fat off the top of the stock and spread it in the bottom of a shallow 12 x 8 x 1 1/2-inch roasting pan. Remove the plate and carefully lift out the duck, holding it over the pot to drain any liquid from the cavity. Place duck in roasting pan. Do not tuck the neck flap under the duck. Spread it out in the pan.
Pat the duck thoroughly dry and lightly coat the skin with the salt and pepper or the star anise rub, gently pressing against the skin. The duck is hot and the skin is tender, so work carefully.
The duck may be prepared ahead up to this point and refrigerated for a day. If made ahead, return duck to room temperature. If proceeding with roasting right away, for optimum results, leave the duck sitting out at room temperature for 30 minutes to permit the skin to dry and heat the oven to 500 degrees with oven rack on the second level from the bottom.
Place duck in oven legs first. Roast 30 minutes. After 10 minutes, spoon out the fat that accumulates in the roasting pan. Move the duck around in the pan with a wooden spatula to prevent the skin from sticking to the bottom of the pan. If it is easier, remove the pan from the oven being careful of the hot fat and spoon off fat. This will avoid getting fat on the inside of the oven, which would smoke. Make sure the oven door is closed, so that the temperature doesn't go down.
After the full 30 minutes, remove the duck from the pan. Pour or spoon off the fat, and deglaze pan with stock or water.
For the star anise rub: Place all ingredients for the rub in a spice mill. Process until a fine powder, stopping to shake several times for evenness.