Using Tocher's curve to convert subjective quantile-estimates into a probability distribution function

IIE Transactions - 1999
HON-SHIANG LAU1, AMY HING-LING LAU1, JOHN F. KOTTAS2
1College of Business Administration, Oklahoma State University, Stillwater, USA
2School of Business, College of William and Mary, Williamsburg, USA

Tóm tắt

One standard approach for estimating a subjective distribution is to elicit subjective quantiles from a human expert. However, most decision-making models require a random variable's moments and/or distribution function instead of its quantiles. In the literature little attention has been given to the problem of converting a given set of subjective quantiles into moments and/or a distribution function. We show that this conversion problem is far from trivial, and that the most commonly used conversion procedure often produces large errors. An alternative procedure using “Tocher's curve” is proposed, and its performance is evaluated with a wide variety of test distributions. The method is shown to be more accurate than a commonly used procedure.

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Tài liệu tham khảo

Hertz, D. (1964) Risk analysis in capital investment. Harvard Business Review, 42, 95–106.

Silverman, B. (1986) Density Estimation for Statistics and Data Analysis, Chapman and Hall, London.

Avramidis, A. and Wilson, J. (1994) A flexible method for estimating inverse distribution functions in simulation experiments. ORSA Journal on Computing, 6(4), 342–355.

Malcolm, D., Roseboom, J., Clark, C. and Fazar, W. (1959) Application of a technique for research and development program evaluation. Operation Research, 7, 646–669.

DeBrota, D., Dittus, R., Roberts, S. and Wilson, J. (1989) Visual interactive fitting of bounded Johnson distributions. Simulation, 52, 199–205.

AbouRizk, S., Halpin, D. and Wilson, J. (1991) Visual interactive fitting of beta distributions. Journal of Construction Engineering and Management, 117(4), 589–605.

Wagner, M. and Wilson, J. (1996) Using univariate Bezier distributions to model simulation input processes. IIE Transactions, 28, 699–711.

Hogarth, R. (1975) Cognitive processes and the assessment of subjective probability distributions. Journal of the American Statistical Association, 70, 271–294.

van Lenthe, J. (1994) Scoring-rule feedforward and the elicitation of subjective probability distributions. Organizational Behavior and Human Decision Processes, 59, 188–209.

Alpert, M. and Raiffa, H. (1982) A progress report on the training of probability assessors, in Judgement under Uncertainty: Heuristics and Biases, Kahneman, D., Slovic, P. and Tversky, A. (eds.), Cambridge University Press, New York, NY.

Holloway, C. (1979) Decision Making under Uncertainty, Prentice-Hall, Englewood Cliffs, NJ.

Samson, D. (1988) Managerial Decision Analysis, Irwin, IL.

Shields, M., Solomon, I. and Waller, W. (1987) Effects of alter-native sample space representations on the accuracy of auditors’ uncertainty judgments. Accounting, Organization and Society, 12(4), 375–385.

Keefer, D. and Verdini, W. (1993) Better estimation of PERT activity time parameters. Management Science, 39(9), 1086–1091.

Clemen, R. (1991) Making Hard Decisions, Duxbury Press, Belmont, CA.

Palisade Corporation, (1996) BESTFIT User' Guide, @RISK User' Guide, Newfield, NY.

Hertz, D. and Thomas, H. (1983) Risk Analysis and Its Applications, John Wiley, New York, NY. p. 161.

Law, A. and Kelton, W. (1982) Simulation Modeling and Analysis, McGraw-Hill, New York, NY.

Thesen, A. and Travis, L. (1992) Simulation for Decision Making, West Publishing, St. Paul, MN.

Banks, J., Carson, J. and Nelson, B. (1996) Discrete-Event System Simulation, Prentice-Hall, Englewood Cliffs, NJ.

Schriber, T. (1974) Simulation Using GPSS, John Wiley, New York, NY.

Ramberg, J. and Schmeiser, B. (1974) An Approximate method for generating asymmetric random variables. Communications of the ACM, 17(2), 78–82.

Kendall, M., Stuart, A. and Ord, J. (1987) Kendall' Advanced Theory of Statistics, Vol. 1, Oxford University Press, New York, NY. ch 10.

Tocher, K. (1963) The Art of Simulation, English Universities Press, London.

Anon, (1994) IMSL (1994), in IMSL Math/Library and IMSL Stat/Library, Visual Numerics Inc., Houston, TX.

Johnson, N., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions, Vol. 2, John Wiley, New York, NY.

Selvidge, J. (1980) Assessing the extremes of probability distributions by the quantile method. Decision Sciences, 11, 493–502.

Solomon, I. (1982) Probability assessment by individual auditors and audit teams: an empirical investigation. Journal of Accounting Research, 20, 689–710.

Lau, A., Lau, H. and Zhang, Y. (1996) A simple and logical alternative for making PERT time estimates. IIE Transactions, 28, 183–192.

Perry, C. and Greig, I. (1975) Estimating the mean and variance of subjective distributions in PERT and decision analysis. Management Science, 21, 1477–1480.

Hahn, G. and Shapiro, S. (1967) Statistical Models in Engineering, John Wiley, New York, NY.

Fama, E. (1965) Portfolio analysis in a stable Paretian market. Management Science, 11, 404–419.

Bowman, K. and Shenton, L. (1979) Approximate percentage points for Pearson distributions. Biometrika, 66, 127–132.

Bowman, K. and Shenton, L. (1979) Further approximate Pearson percentage points and Cornish–Fisher. Communications in Statistics, B8(3), 231–244.

Bowman, K. and Serbin, C. (1981) Explicit approximate solutions for SB. Communications in Statistics, B10(1), 1–15.

Bowman, K. and Shenton, L. (1980) Evaluation of the parameters of U by rational fractions. Communications in Statistics, B9(2), 127–132.

Ramberg, J., Dudewicz, E., Tadikamalla, P. and Mykytka, E. (1979) A probability distribution and its uses in fitting data. Technometrics, 21(2), 201–214.

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