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Nonparametric Estimation of a Survivor Function with Across- Interval-Censored Data


  • Mark Yuying An

    (Duke Uniersity)

  • Roberto Ayala

    (Central Bank, Ecuador)


In this paper we present a a generalized self-consistent algorithm that estimates a survivor function with across-interval- censored data. This algorithm is an iterative procedure based on Turnbull's (1974) reallocation idea. At each step of the iteration, the procedure first reduces the across-interval-censored problem to a singly-censored one, and then it applies the Kaplan-Meier estimation method. The main result of this paper is that our algorithm produces the maximum likelihood estimate. Unlike Turnbull (1974, 1976), we explicitly discuss situations in which corner solutions are encountered. The investigation is motivated from environmental economics where data from contingent valuation surveys are often used to nonparametrically estimate the willingness to pay distribution. In this estimation, the algorithm of Turnbull (1974, 1976) plays an instrumental role. However, there is a data grouping mechanism found in some contingent valuation surveys to which Turnbull's method does not apply. We refer to these cases as distinct bids and mixed bids, where across-interval-censored observations are common.
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Suggested Citation

  • Mark Yuying An & Roberto Ayala, 1996. "Nonparametric Estimation of a Survivor Function with Across- Interval-Censored Data," Econometrics 9611003, EconWPA.
  • Handle: RePEc:wpa:wuwpem:9611003
    Note: Type of Document - ; prepared on UNIX Sparc TeX; to print on PostScript; pages: 30 ; figures: request from author. We never published this piece and now we would like to reduce our mailing and xerox cost by posting it.

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    References listed on IDEAS

    1. Mark Yuying An, 1996. "Semiparametric Estimation of Willingness to Pay Distributions," Econometrics 9611001, EconWPA.
    2. Richard Carson & Robert Mitchell & Michael Hanemann & Raymond Kopp & Stanley Presser & Paul Ruud, 2003. "Contingent Valuation and Lost Passive Use: Damages from the Exxon Valdez Oil Spill," Environmental & Resource Economics, Springer;European Association of Environmental and Resource Economists, vol. 25(3), pages 257-286, July.
    3. An, Mark Y. & Roberto Ayala, 1995. "A Mixture Model of Willingness to Pay Distributions," Working Papers 95-21, Duke University, Department of Economics.
    4. Carson, Richard T & Wilks, Leanne & Imber, David, 1994. "Valuing the Preservation of Australia's Kakadu Conservation Zone," Oxford Economic Papers, Oxford University Press, vol. 46(0), pages 727-749, Supplemen.
    5. Cameron, Trudy Ann & James, Michelle D, 1987. "Efficient Estimation Methods for "Closed-ended' Contingent Valuation Surveys," The Review of Economics and Statistics, MIT Press, vol. 69(2), pages 269-276, May.
    6. Richard T. Carson, 2011. "Contingent Valuation," Books, Edward Elgar Publishing, number 2489.
    7. Daniel McFadden, 1994. "Contingent Valuation and Social Choice," American Journal of Agricultural Economics, Agricultural and Applied Economics Association, vol. 76(4), pages 689-708.
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    Cited by:

    1. An, Mark Yuying, 1996. "Semiparametric Estimation of Willingness to Pay Distributions," Working Papers 96-20, Duke University, Department of Economics.
    2. Riccardo Scarpa & Kenneth Willis & Guy Garrod, 2001. "Estimating Benefits for Effective Enforcement of Speed Reduction from Dichotomous-Choice CV," Environmental & Resource Economics, Springer;European Association of Environmental and Resource Economists, vol. 20(4), pages 281-304, December.
    3. Nick Hanley & Bengt Kristrom, "undated". "What’s it worth? Exploring value uncertainty using interval questions in Contingent Valuation," Working Papers 2002_10, Business School - Economics, University of Glasgow.
    4. An, Mark Y. & Roberto Ayala, 1995. "A Mixture Model of Willingness to Pay Distributions," Working Papers 95-21, Duke University, Department of Economics.
    5. An, Mark Yuying, 1996. "Statistical Inference of a Bivariate Proportional Hazard Model with Grouped Data," Working Papers 96-06, Duke University, Department of Economics.
    6. Ana Bedate & Luis Herrero & José Sanz, 2009. "Economic valuation of a contemporary art museum: correction of hypothetical bias using a certainty question," Journal of Cultural Economics, Springer;The Association for Cultural Economics International, vol. 33(3), pages 185-199, August.
    7. W. George Hutchinson & Riccardo Scarpa & Susan M. Chilton & T. McCallion, 2001. "Parametric and Non-Parametric Estimates of Willingness to Pay for Forest Recreation in Northern Ireland: A Discrete Choice Contingent Valuation Study with Follow-Ups," Journal of Agricultural Economics, Wiley Blackwell, vol. 52(1), pages 104-122.
    8. Brett Day, 2007. "Distribution-free estimation with interval-censored contingent valuation data: troubles with Turnbull?," Environmental & Resource Economics, Springer;European Association of Environmental and Resource Economists, vol. 37(4), pages 777-795, August.

    More about this item


    Across-Interval Censoring; Contingent valuation; Nonparametric Estimation; Self-Consistent Algorithm;

    JEL classification:

    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General
    • C24 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Truncated and Censored Models; Switching Regression Models; Threshold Regression Models
    • Q20 - Agricultural and Natural Resource Economics; Environmental and Ecological Economics - - Renewable Resources and Conservation - - - General


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