Kybernetika 54 no. 6, 1156-1166, 2018

On quantile optimization problem based on information from censored data

Petr VolfDOI: 10.14736/kyb-2018-6-1156


Stochastic optimization problem is, as a rule, formulated in terms of expected cost function. However, the criterion based on averaging does not take in account possible variability of involved random variables. That is why the criterion considered in the present contribution uses selected quantiles. Moreover, it is assumed that the stochastic characteristics of optimized system are estimated from the data, in a non-parametric setting, and that the data may be randomly right-censored. Therefore, certain theoretical results concerning estimators of distribution function and quantiles under censoring are recalled and then utilized to prove consistency of solution based on estimates. Behavior of solutions for finite data sizes is studied with the aid of randomly generated example of a newsvendor problem.


optimization, censored data, product-limit estimator, empirical quantile, newsvendor problem


62N02, 62P25


  1. P. Andersen, O. Borgan, R. Gill and N. Keiding: Models Based on Counting Processes. Springer, New York 1993.   DOI:10.1007/978-1-4612-4348-9
  2. N. Breslow and J. E. Crowley: A large sample study of the life table and product limit estimates under random censorship. Ann. Statist. 2 (1974), 437-453.   DOI:10.1214/aos/1176342705
  3. J. D. Kalbfleisch and R. L. Prentice: The Statistical Analysis of Failure Time Data (Second edition). Wiley, New York 2002.   DOI:10.1002/9781118032985
  4. V. Kaňková: Empirical estimates in stochastic optimization via distribution tails. Kybernetika 46 (2010), 459-471.   CrossRef
  5. A. I. Kibzun and Yu. S. Kan: Stochastic Programming Problem with Probability and Quantile Functions. Wiley, Chichester 1996.   DOI:10.1016/s0166-218x(97)81420-5
  6. J. H. Kim and W. Powell: Quantile optimization for heavy-tailed distributions using asymmetric signum functions. Working Paper, Princeton University, 2011. Retrieved 12.01.2016 from   CrossRef
  7. A. V. Peterson: Expressing the Kaplan-Meier estimator as a function of empirical subsurvival functions. J. Amer. Stat. Assoc. 72 (1977), 360, 854-858.   DOI:10.1080/01621459.1977.10479970
  8. N. C. Petruzzi and M. Dada: Pricing and the newsvendor problem: A review with extensions. Oper. Res. 47 (1999), 2, 183-194.   DOI:10.1287/opre.47.2.183
  9. L. Rejto: On fixed censoring model and consequences for the stochastic case. In: Trans. 9th Prague Conference on Stochastic Decision Functions 1982, Academia, Prague 1983, pp. 141-147.   CrossRef
  10. G. A. Timofeeva: Optimal and suboptimal solutions to stochastically uncertain problem of quantile optimisation. Automat. Remote Control 68 (2007), 3, 1145-1157.   DOI:10.1134/s000511790707003x
  11. P. Volf: On precision of optimization in the case of incomplete information. Bull. Czech Econometr. Soc. 19 (2012), 170-184.   CrossRef