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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.

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  1. 1.

    Extending these ideas, semi-nonparametric distribution-free estimators have been developed to allow analysts to investigate how WTP is affected by covariates (e.g. Chen and Randall 1997; Creel and Loomis 1997).

  2. 2.

    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).

  3. 3.

    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.

  4. 4.

    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.

  5. 5.

    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.

  6. 6.

    Zhang and Jamshidian (2004) exploit similar computational savings for doubly censored data.

  7. 7.

    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.

  8. 8.

    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.


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Correspondence to Brett Day.

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Day, B. Distribution-free estimation with interval-censored contingent valuation data: troubles with Turnbull?. Environ Resource Econ 37, 777–795 (2007).

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  • Interval-censored data
  • Turnbull’s self-consistent algorithm
  • Nonparametric maximum likelihood
  • Contingent valuation