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.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Tax calculation will be finalised during checkout.
This assumption has no substantive impact on the exposition. A more detailed development which takes account of differences between basic intervals and equivalence classes is provided in Day (2005).
As a matter of fact Mykland and Ren (1996) consider data subject to a slightly different censoring scheme than that considered here, though their results will apply equally to Type 2 interval-censored data.
More formally, this property follows from Corollary 1 of Theorem 2 proved by Nettleton (1999) for the specific case in which the objective function is globally concave.
In essence, a comprehensive set of convergence criteria such as these is anticipated by Gentleman and Geyer (1994) and underpins Mykland and Ren’s (1996) proposal that the algorithm should be restarted from alternative initial values if non-negativity of the Lagrange multipliers cannot be confirmed for a candidate solution.
Zhang and Jamshidian (2004) exploit similar computational savings for doubly censored data.
The data reduction procedures, the SC algorithm and ICM algorithm are adapted from the “Nonparametric density estimator for CVM data” written by Olvar Bergland of the Agricultural University of Norway. The SQP algorithm contains a solver for the linear complementarity problem written by Rob Dittmar of the Federal Reserve Bank of St. Louis. The rest of the code was written by the author and is available on request.
The comparison is complicated by differences in the hardware and software used to carry out the Monte Carlo experiments. Jongbloed (1998) employs code written in the Matlab 4.2c environment running on a Sun Sparcstation 4, whilst Zhang and Jamshidian (2004) use Microsoft Visual C++ running on a Dell 600 MHz PC.
An MY, Ayala R (1996) Nonparametric estimation of a survivor function with across-interval-censored data. Working Paper 96-02, Duke University, Durham, NC
Ayer M, Brunk HD, Ewing GM, Reid WT, Silverman E (1955) An empirical distribution function for sampling with incomplete information. Ann Math Stat 26:641–647
Bateman I, Day BH, Dupont D, Georgiou S (2004) Ohh la la! Testing the one-and-one-half bound dichotomous choice elicitation method for robustness to anomalies. CSERGE Working Paper, EDM-2004-06, University of East Anglia, UK
Boman M, Bostedt G, Kristrom B (1999) Obtaining welfare bounds in discrete-response valuation studies: a nonparametric approach. Land Econ 75:284–294
Boyle KJ, Bishop RC (1988) Welfare measurements using contingent valuation: a comparison of techniques. Am J Agric Econ 70:20–28
Carson RT, Wilks L, Imber D (1994) Valuing the preservation of Australia’s Kakadu conservation zone. Oxford Econ Papers 46:727–749
Carson RT, Groves T, Machina M (1999) Incentive and informational properties of preference questions. Plenary Address, European Association of Environmental and Resource Economists, Compatibility Issues in Stated Preference Surveys. Oslo, Norway, June
Chen HZ, Randall A (1997) Semi-nonparametric estimation of binary response models with an application to natural resource valuation. J Economet 76:323–340
Cooper J, Hanemann WM, Signorello G (2002) One-and-one-half-bound dichotomous choice contingent valuation. Rev Econ Stat 84(4):742–750
Creel M, Loomis J (1997) Semi-nonparametric distribution-free dichotomous choice contingent valuation. J Environ Econ Manage 32:341–358
Day BH (2005) Distribution-free estimation with interval-censored contingent valuation data: Troubles with Turnbull. CSERGE Working Paper, EDM-2005-07, University of East Anglia, UK
Day BH, Mourato S (1998) Willingness to pay for water quality maintenance in Chinese rivers. CSERGE Working Paper, WM 98-02, University College London, UK
Dempster AP, Laird NM, Rubin DB (1977) Maximum likelihood from incomplete data via the EM algorithm. J R Stat Soc, Ser B 39:1–38
Gentleman R, Geyer CJ (1994) Maximum likelihood for interval censored data: consistency and computation. Biometrika 81:618–623
Goetghebeur E, Ryan L (2000) Semiparametric regression analysis of interval-censored data. Biometrics 56:1139–1144
Gu MG, Zhang CH (1993) Asymptotic convergence of the self-consistent estimator of the distribution function based on doubly censored data. Ann Stat 21(2):611–624
Hoehn JP, Randall A (1987) A satisfactory benefit cost indicator from contingent valuation. J Environ Econ Manage 14:226–247
Haab TC, McConnell KE (1997) Referendum models and negative willingness to pay: alternative solutions. J Environ Econ Manage 32:251–270
Hanemann WM, Kanninen B (1999) The statistical analysis of discrete response CV data. In: Bateman IJ, Willis KG (eds) Valuing environmental preferences: theory and practice of the contingent valuation method in the U.S., EU, and developing countries. Oxford University Press, Oxford
Hanemann WM, Loomis J, Kanninen B (1991) Statistical efficiency of double-bounded dichotomous choice contingent valuation. Am J Agric Econ 73(4):1255–1263
Huang J, Wellner JA (1996) Interval censored survival data: a review of recent progress. Proceedings of the first Seattle symposium in biostatistics: survival analysis. Lecture notes in Statistics, vol 123. Springer-Verlag, Berlin, pp 123–169
Huhtala AH (2000) Binary choice valuation studies with heterogeneous preferences regarding the program being valued. Environ Resour Econ 16:263–279
Hutchinson G, Scarpa R, Chilton S, McCallion T (2001) Parametric and non-parametric estimates of willingness to pay for forest recreation in Northern Ireland: a multi-site analysis using discrete choice contingent valuation with follow-ups. J Agric Econ 52(1):104–122
Jongbloed G (1998) The iterative convex minorant algorithm for nonparametric estimation. J Comput Graph Stat 7:310–321
Kerr GN (2000) Dichotomous choice contingent valuation probability distributions. Aust J Agric Resour Econ 44(2):233–252
Mykland DA, Ren J-J (1996) Algorithms for computing self-consistent and maximum likelihood estimators with doubly censored data. Ann Stat 24:1740–1764
Nettleton D (1999) Convergence properties of the EM algorithm in constrained parameter spaces. Can J Stat 27(2):639–648
NOAA (1993) Appendix 1: report of the NOAA panel on contingent valuation. Federal Register, 58, pp 4602–2614
Nunes P, Van den Bergh J (2004) Can people value protection against invasive marine species? Evidence from a joint TC-CV survey in the Netherlands. Environ Resour Econ 28(4):517–532
Ready RC, Hu D (1995) Statistical approaches to the fat tail problem for dichotomous choice contingent valuation. Land Econ 71(4):491–499
Turnbull B (1976) The empirical distribution function with arbitrary grouped, censored and truncated data. J R Stat Soc Ser B 38:290–295
Wellner JA, Zhan Y (1997) A hybrid algorithm for computation of the nonparametric maximum likelihood estimator from censored data. J Am Stat Assoc 92:945–959
Zhang Y, Jamshidian M (2004) On algorithms for the nonparametric maximum likelihood estimator of the failure function with censored data. J Comput Graph Stat 13(1):123–140
About this article
Cite this article
Day, B. Distribution-free estimation with interval-censored contingent valuation data: troubles with Turnbull?. Environ Resource Econ 37, 777–795 (2007). https://doi.org/10.1007/s10640-006-9061-8
- Interval-censored data
- Turnbull’s self-consistent algorithm
- Nonparametric maximum likelihood
- Contingent valuation