Heat oven to 350 degrees. Spray Bundt pan with non-stick cooking spray, making sure all pan crevices are equally coated. Spray it again for good measure.
Finely chop the pecans and place them in the bottom of the Bundt pan. Evenly drop the brown sugar on top of the pecans.
In a large bowl, mix together the yellow cake mix, instant pudding, eggs, water, rum and oil. Pour batter over the brown sugar/pecan mixture.
Bake the cake for 1 hour - do not over bake this cake! Your cooking time may need to be adjusted based on your pan; black pans need less cooking time.
While the cake bakes, prepare the glaze: melt butter in a medium saucepan and then stir in water & sugar. Bring the mixture to a boil for about 4 minutes, stirring constantly. Remove the glaze from the burner and pour in rum. Stir thoroughly. Return the saucepan to the burner and heat the glaze for a minute or two.
When the cake is finished baking, remove it from the oven and pour about 1/3 of the glaze on top of it while it is still warm and in the pan. Let it sit like this for a few minutes.
Turn the cake onto a serving platter. Prick top and sides of the cake with toothpicks or a fork. Repeat this multiple times! Pour remaining glaze evenly on the top and sides of the cake.
Let the cake return to room temperature. Cover the cake with a domed cake lid OR cover it entirely with plastic wrap. (The latter method is preferred.)
Let the cake rest, covered & coated in the glaze for a minimum of 2-4 hours before serving. The cake is best if allowed to rest covered overnight.