Wave-particle duality for a qubit system is discussed in the framework of quantum estimation theory. Two parameters are introduced for qubit states representing fringe visibility and which-way information. We analyze trade-off relations for mean-square errors upon estimating these two parameters. By going through our analysis, we clarify a role of nuisance parameter(s) for this model and conclude that a proper treatment needs to be involved in order to formulate this problem. (Nuisance parameters are those parametrizing a quantum statistical model, yet are of no interest.) We also address the problem of nuisance parameters in the general setting and propose a possible approach how to eliminate them.