, Volume 50, Issue 3, pp 301–321 | Cite as

A constrained spline estimator of a hazard function

  • Bruce Bloxom


A constrained quadratic spline is proposed as an estimator of the hazard function of a random variable. A maximum penalized likelihood procedure is used to fit the estimator to a sample of psychological response times. The results of a small simulation study suggest that, with a sample size of 500, the procedure may provide a reasonably precise estimate of the shape of a hazard function.

Key words

hazard function quadratic spline maximum penalized likelihood constrained estimation response time distribution 


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  1. Akaike, H. (1974). A new look at the statistical model identification.IEEE Transactions on Automatic Control, AC-19, 716–723.Google Scholar
  2. Anderson, J. A., & Senthilselvan, A. (1980). Smooth estimates for the hazard function.Journal of the Royal Statistical Society (Series B),42, 322–327.Google Scholar
  3. Ashby, F. G. (1982). Testing assumptions of exponential additive reaction time models.Memory and Cognition, 10, 125–134.Google Scholar
  4. Barlow, R. E., & Proschan, F. (1975).Statistical theory of reliability and life testing: Probability models. New York: Holt, Rinehart & Winston.Google Scholar
  5. Bartoszynski, R., Brown, B. W., McBride, C. M., & Thompson, J. R. (1981). Some nonparametric techniques for estimating the intensity function of a cancer related nonstationary Poisson process.Annals of Statistics, 9, 1050–1060.Google Scholar
  6. Bloxom, B. (1983). Some problems in estimating response time distributions. In H. Wainer & S. Messick (Eds.),Principals of modern psychological measurement: A festschrift in honor of Frederick M. Lord (pp. 303–328). Hillsdale, NJ: Lawrence Erlbaum.Google Scholar
  7. Bloxom, B. (1984). Estimating response time hazard functions: An exposition and extension.Journal of Mathematical Psychology, 28, 401–420.Google Scholar
  8. Burbeck, S. L., & Luce, R. D. (1982). Evidence from auditory simple reaction times for both change and level detectors.Perception and Psychophysics, 32, 117–133.Google Scholar
  9. de Boor, C. (1978).A practical guide to splines. New York: Springer-Verlag.Google Scholar
  10. Fiacco, A. V., & McCormick, G. P. (1968).Nonlinear programming: Sequential unconstrained minimization techniques. New York: John Wiley & Sons.Google Scholar
  11. Glaser, R. E. (1980). Bathtub and related failure rate characterizations.Journal of the American Statistical Association, 75, 667–672.Google Scholar
  12. Hall, P. (1982). Limit theorems for stochastic measures of the accuracy of density estimators.Stochastic Processes and Their Applications, 13, 11–25.Google Scholar
  13. Johnston, N. I., & Kotz, S. (1970).Continuous univariate distributions-1. Boston: Houghton Mifflin.Google Scholar
  14. Kohfeld, D. L. (1981, November). Stages of reaction time in children and adults. Paper presented at the meeting of the Psychonomic Society, Philadelphia, PA.Google Scholar
  15. Kronmal, R. A., & Tarter, M. E. (1968). The estimation of probability densities by Fourier series methods.Journal of the American Statistical Association, 63, 925–952.Google Scholar
  16. Lawless, J. F. (1983). Statistical methods in reliability.Technometrics, 25, 305–316.Google Scholar
  17. Lewis, P. A. W., & Uribe, L. (1981). The new Naval Postgraduate School random number package LLRANDOMII [Computer program]. Monterey, CA: Naval Postgraduate School. (Naval Postgraduate School Report NPS55-81-005)Google Scholar
  18. Lii, K. S., & Rosenblatt, M. (1975). Asymptotic behavior of a spline estimate of a density function.Computation and Mathematics with Applications, 1, 223–235.Google Scholar
  19. Mendelsohn, J., & Rice, J. (1982). Deconvolution of microfluorometric histograms with B-splines.Journal of the American Statistical Association, 77, 748–753.Google Scholar
  20. Miller, D. L., & Singpurwalla, N. D. (1980). Failure rate estimation using random smoothing.Sankhya (Series B),42, 217–228.Google Scholar
  21. Mylander, W. C., Holmes, R. L., & McCormick, G. P. (1973).A guide to SUMT-Version 4: The computer program implementing the sequential unconstrained minimization technique for nonlinear programming [Computer program manual]. McLean, VA: Research Analysis Corporation. (Research Report RAC-P-63)Google Scholar
  22. Neftci, S. N. (1982). Specification of economic time series models using Akaike's criterion.Journal of the American Statistical Association, 77, 537–540.Google Scholar
  23. Rice, J., & Rosenblatt, M. (1976). Estimation of the log survivor function and hazard function.Sankhya (Series A),38, 60–78.Google Scholar
  24. Rice, J., & Rosenblatt, M. (1983). Smoothing splines: Regression, derivatives and deconvolution.Annals of Statistics, 11, 141–156.Google Scholar
  25. Schwartz, G. (1978). Estimating the dimension of a model.Annals of Statistics, 6, 461–464.Google Scholar
  26. Shumaker, L. L. (1981).Spline functions: Basic theory. New York: Wiley.Google Scholar
  27. Singpurwalla, N. D., & Wong, M. Y. (1983). Estimation of the failure rate—A survey of nonparametric methods. Part I: Non-Bayesian methods.Communications in Statistics—Theory and Methods, A12, 559–588.Google Scholar
  28. Towsend, J. T., & Ashby, F. G. (1978). Methods of modelling capacity in simple processing systems. In N. J. Castellan & F. Restle (Eds.),Cognitive theory: Volume 3 (pp. 199–239). Hillsdale, NJ: Lawrence Erlbaum.Google Scholar
  29. Wahba, G. (1976). Histoplines with knots which are order statistics.Journal of the Royal Statistical Society (Series B),38, 140–151.Google Scholar
  30. Wold, S. (1974). Spline functions in data analysis.Technometrics, 16, 1–11.Google Scholar
  31. Wright, J. W., & Wegman, E. J. (1980). Isotonic, convex and related splines.Annals of Statistics, 8, 1023–1035.Google Scholar

Copyright information

© The Psychometric Society 1985

Authors and Affiliations

  • Bruce Bloxom
    • 1
  1. 1.Naval Postgraduate School, and Navy Personnel Research and Development CenterVanderbilt UniversityUSA

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