Efficiency of a Class of Unbiased Estimators for the Invariant Distribution Function of a Diffusion Process

We consider the problem of the estimation of the invariant distribution function of an ergodic diffusion process when the drift coefficient is unknown. The empirical distribution function is a natural estimator which is unbiased, uniformly consistent and efficient in different metrics. Here we study the properties of optimality for another kind of estimator recently proposed. We consider a class of unbiased estimators and we show that they are also efficient in the sense that their asymptotic risk, defined as the integrated mean square error, attains the same asymptotic minimax lower bound of the empirical distribution function.     

Approval voting and positional voting methods: Inference,relationship, examples

Approval voting is the voting method recently adopted by the Society for Social Choice and Welfare. Positional voting methods include the famous plurality, antiplurality, and Borda methods. We extend the inference framework of Tsetlin and Regenwetter (2003) from majority rule to approval voting and all positional voting methods. We also establish a link between approval voting and positional voting methods whenever Falmagne et al.s (1996) size-independent model of approval voting holds: In all such cases, approval voting mimics some positional voting method. We illustrate our inference framework by analyzing approval voting and ranking data, with and without the assumption of the size-independent model. For many of the existing data, including the Society for Social Choice and Welfare election analyzed by Brams and Fishburn (2001) and Saari (2001), low turnout implies that inferences drawn from such elections carry low (statistical) confidence. Whenever solid inferences are possible, we find that, under certain statistical assumptions, approval voting tends to agree with positional voting methods, and with Borda, in particular.Michel Regenwetter thanks the National Science Foundation for funding this research through NSF grant SBR 97-30076. Both authors thank the Fuqua School of Business for financially supporting their collaboration. Most of this research was done while Regenwetter was a faculty member at Fuqua. We thank Prof. Steven Brams for his valuable comments as a discussant of a previous version of this paper, given at the 2002 Public Choice meeting, and Prof. Donald Saari for his helpful comments in conversations and on another draft. We also thank the editor in charge and a referee for their valuable comments. Tsetlin acknowledges the support of the Centre for Decision Making and Risk Analysis at INSEAD.