Distribution-free estimation with interval-censored contingent valuation data: Trobles with Turnbull?

Contingent valuation (CV) surveys frequently employ elicitation procedures that return interval-censored data on respondents' willingness to pay (WTP). Almost exclusively, CV practitioners have applied Turnbull's self-consistent algorithm to such data in order to obtain nonparametric maximum likelihood (NPML) estimates of the WTP distribution. This paper documents two failings of Turnbull's algorithm; (1) that it may not converge to NPML estimates and (2) that it may be very slow to converge. With regards to (1) we propose new starting and stopping criteria for the algorithm that guarantee convergence to the NPML estimates. With regards to (2) we present a smorgasbord of alternative NPML estimators and demonstrate, through Monte Carlo simulations, their performance advantages over Turnbull's algorithm.