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

Abstract

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. Copyright Springer Science+Business Media, Inc. 2007Interval-censored data, Turnbull’s self-consistent algorithm, Nonparametric maximum likelihood, Contingent valuation,

Potential for Tradable Water Allocation and Rights in Jordan

“…This estimation technique does not impose ad hoc assumptions on the distribution of WTA, F(WTA). Denoting the lower bounds of the intervals in (10) as and the upper bounds as the probability of observing an individual i from this population indicating that his WTA lays in the interval between and is given by (Day, 2007;Turnbull, 1976):…”

“…Since (11) is strictly concave, the F m estimates of are unique. Estimation can be then carried out using Turnbull's self-consistent algorithm (Day, 2007;Gomez, Calle and Oller, 2004;Turnbull, 1976).…”

“…In this section we outline a procedure which allows estimation of farmers' unobservable WTA amounts using the estimated parameters. First notice that an estimate of the probability that the WTA of farmer i is in the basic bid interval m conditional on belonging to the WTA interval is given by (Day, 2007;Fay and Shih, 1998):…”

“…The supply curve is estimated using an innovative procedure that combines discrete and open ended contingent valuation (CV) questions and nonparametric methods. Previous studies using discrete contingent valuation questions and nonparametric procedures have focused on the estimation of the population willingness to pay function (WTP) (e.g., Day, 2007;Haab and McConnell, 1997).…”

Modeling multiple risks in the presence of double censoring

“…In this case, a type of selection bias for masked observations is introduced, thereby skewing the results from the models to some degree. Secondly, as noted in Day (2005), there are convergence issues occasionally encountered when applying Turnbull's SC algorithms. In recent years, there have been some innovative methods proposed to deal with this contingency.…”