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

Contingent valuation (CV) surveys frequently employ elicitation procedures that return interval-censored data on respondents’ willingness to pay (WTP). Almost without exception, 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 starting and stopping criteria for the algorithm that guarantee convergence to the NPML estimates. With regards to (2) we present a variety of alternative estimators and demonstrate, through Monte Carlo simulations, their performance advantages over Turnbull’s algorithm

Nonparametric Bayesian Volatility Estimation

Given discrete time observations over a fixed time interval, we study a nonparametric Bayesian approach to estimation of the volatility coefficient of a stochastic differential equation. We postulate a histogram-type prior on the volatility with piecewise constant realisations on bins forming a partition of the time interval. The values on the bins are assigned an inverse Gamma Markov chain (IGMC) prior. Posterior inference is straightforward to implement via Gibbs sampling, as the full conditional distributions are available explicitly and turn out to be inverse Gamma. We also discuss in detail the hyperparameter selection for our method. Our nonparametric Bayesian approach leads to good practical results in representative simulation examples. Finally, we apply it on a classical data set in change-point analysis: weekly closings of the Dow-Jones industrial averages