Nonparametric Demand Forecasting with Right Censored Observations

Bin ZHANG, Zhongsheng HUA

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DOI: 10.4236/jsea.2009.24033                

Abstract

In a newsvendor inventory system, demand observations often get right censored when there are lost sales and no backordering. Demands for newsvendor-type products are often forecasted from censored observations. The Kap-lan-Meier product limit estimator is the well-known nonparametric method to deal with censored data, but it is unde-fined beyond the largest observation if it is censored. To address this shortfall, some completion methods are suggested in the literature. In this paper, we propose two hypotheses to investigate estimation bias of the product limit estimator, and provide three modified completion methods based on the proposed hypotheses. The proposed hypotheses are veri-fied and the proposed completion methods are compared with current nonparametric completion methods by simulation studies. Simulation results show that biases of the proposed completion methods are significantly smaller than that of those in the literature.

Keywords

Forecasting, Demand, Censored, Nonparametric, Product Limit Estimator

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	B. Zhang, X. Xu, and Z. Hua, “A binary solution method for the multi-product newsboy problem with budget con-straint,” International Journal of Production Economics, Vol. 117, No. 1, pp. 136–141, January 2009.
[2]	G. Hadley and T. M. Whitin, “Analysis of inventory sys-tems,” Prentice-Hall, Englewood Cliffs, 1963.
[3]	H. S. Lau and A. H. L. Lau, “Estimating the demand dis-tributions of single-period items having frequent stock-outs,” European Journal of Operational Research, Vol. 92, No. 2, pp. 254–265, July 1996.
[4]	Z. Hua and B. Zhang, “Improving density forecast by modeling asymmetric features: An application to S&P500 returns,” European Journal of Operational Research, Vol. 185, No. 2, pp. 716–725, March 2008.
[5]	E. L. Kaplan and P. Meier, “Nonparametric estimation from incomplete observations,” Journal of the American Statistical Association, Vol. 53, No. 282, pp. 457–481, June 1958.
[6]	B. Gillespie, J. Gillespie, and B. Iglewicz, “A comparison of the bias in four versions of the product-limit estima-tor,” Biometrika, Vol. 79, No. 1, pp. 149–155, March 1992.
[7]	S. A. Conrad, “Sales data and the estimation of demand,” Operational Research Quarterly, Vol. 27, No. 1, pp. 123–127, 1976.
[8]	W. Wecker, “Predicting demand from sales data in the presence of stockouts,” Management Science, Vol. 24, No. 10, pp. 1043–1054, June 1978.
[9]	D. B. Braden and M. Freimer, “Informational dynamics of censored observations,” Management Science, Vol. 37, No. 11, pp. 1390–1404, November 1991.
[10]	N. S. Agrawal and A. Smith, “Estimating negative bino-mial demand for retail inventory management with unob-servable lost sales,” Naval Research Logistics, Vol. 43, No. 6, pp. 839–861, September 1996.
[11]	P. C. Bell, “Adaptive sales forecasting with many stock-outs,” Journal of the Operational Research Society, Vol. 32, No. 10, pp. 865–873, October 1981.
[12]	P. C. Bell, “A new procedure for the distribution of peri-odicals,” Journal of the Operational Research Society, Vol. 29, No. 5, pp. 427–434, May 1978.
[13]	P. C. Bell, “Forecasting demand variation when there are stockouts,” Journal of the Operational Research Society, Vol. 51, No. 3, pp. 358–363, March 2000.
[14]	X. Ding, “Estimation and optimization in discrete inven-tory models,” Ph.D. thesis, The University of British Co-lumbia, Vancouver, Canada, 2002.
[15]	W. Zhang, B. Zhang, and Z. Hua, “Quasi-bootstrap pro-cedure for forecasting demand from sales data with stockouts,” in Proceedings of the 38th International Con-ference on Computers and Industrial Engineering, Vol. 1–3, pp. 423–429, October 2008.
[16]	B. Efron, “The two sample problem with censored data,” in Proceedings of the 5th Berkeley Symposium on Mathematical Statistics and Probability, Vol. 4, pp. 831–852, 1967.
[17]	R. D. Gill, “Censoring and stochastic integrals,” Mathe-matical Centre Tract No. 124, Mathematisch Centrum, Amsterdam, 1980.
[18]	Z. Chen and E. Phadia, “An optimal completion of the product limit estimator,” Statistics & Probability Letters, Vol. 76, No. 9, pp. 913–919, May 2006.
[19]	B. W. Brown, M. Jr. Hollander, and R. M. Korwar, “Non-parametric tests of independence for censored data, with applications to heart transplant studies,” in: F. Proschan and R. J. Serfling, (Eds.), “Reliability and biometry: sta-tistical analysis of lifelength,” Philadelphia: Society for Industrial and Applied Mathematics, pp. 291–302, 1974.
[20]	M. L. Moeschberger and J. P. Klein, “A comparison of several methods of estimating the survival function when there is extreme right censoring,” Biometrics, Vol. 41, No. 1, pp. 253–259, March 1985.
[21]	P. Meier, “Estimation of a distribution function from incomplete observations,” in: J. Gani (Eds.), “Perspec-tives in probability and statistics,” Applied Probability Turst, Sheffield, England, pp. 67–87, 1975.
[22]	J. H. J. Geurts, “Some small-samplel nonproportional hazards results for the Kaplan-Meier estimator,” Statistica Neerlandica, Vol. 39, No. 1, pp. 1–13, March 1985.
[23]	J. H. J. Geurts, “On the small-sample performance of Efron’s and of Gill’s version of the product limit estima-tor under nonproportional hazards,” Biometrics, Vol. 43, No. 3, pp. 683–692, September 1987.