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Censored regression analysis of multiclass passenger demand data subject to joint capacity constraints. (English) Zbl 0842.90076
Summary: In most passenger transportation systems, demand for seats is not recorded after all spaces for a particular trip have been sold out or after a booking limit has been reached. Thus historical booking data is comprised of ticket sales not demand – a condition known as censorship of the data. Data censorship is particularly complex when there are multiple classes of demand since the demand in one class can influence the degree of censorship in another. This paper examines the problem of simultaneously estimating passenger of demand models for two or more correlated classes of demand that are subject to a common capacity constraint. It is shown that the EM method of A. P. Dempster, N. M. Laird and D. B. Rubin [J. R. Stat. Soc., Ser. B 39, 1-38 (1977; Zbl 0346.62022)] can be adapted to provide maximum likelihood estimates of the parameters of the demand model under these circumstances. The problem of modelling demand for airline flights is discussed as a typical example of this estimation problem. Numerical examples show that, with reasonable sample sizes, it is possible to obtain good estimates even when 75% or more of the data have been censored.

MSC:
90B90 Case-oriented studies in operations research
Software:
bootstrap
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[1] T. Amemiya, Multivariate regression and simultaneous equation models when the dependent variables are truncated normal, Econometrica 42(1974)999–1012. · Zbl 0294.62076 · doi:10.2307/1914214
[2] T. Amemiya, Tobit models: A survey, Journal of Econometrics 24(1984)3–61. · Zbl 0539.62121 · doi:10.1016/0304-4076(84)90074-5
[3] P.P. Belobaba, Air travel demand and airline seat inventory management, Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, Report R87-7, Flight Transportation Laboratory, MA (May 1987).
[4] P.P. Belobaba, Application of a probabilistic decision model to airline seat inventory control, Operations Research 37(1989)183–197. · doi:10.1287/opre.37.2.183
[5] A.P. Dempster, N.M. Laird and D.B. Rubin, Maximum likelihood from incomplete data via the EM algorithm, Journal of the Royal Statistical Society B39(1977)1–38. · Zbl 0364.62022
[6] B. Efron and R.J. Tibshirani,An Introduction to the Bootstrap (Chapman and Hall, New York, 1993). · Zbl 0835.62038
[7] D.L. Harmer, A description of the surplus seats analysis system, Technical Report, The Consulting Division, Boeing Computer Services, Seattle, WA (1976).
[8] H.O. Hartley, Maximum likelihood estimation from incomplete data, Biometrics 14(1958)174–194. · Zbl 0081.13904 · doi:10.2307/2527783
[9] J.J. Heckman, The common structure of statistical models of truncation, sample selection and limited dependent variables and a simple estimator for such models, Annals of Economic and Social Measurement 5(1976)475–492.
[10] N.L. Johnson and S. Kotz,Distributions in Statistics: Continuous Multivariate Distributions (Wiley, New York, 1972). · Zbl 0248.62021
[11] R.A. Johnson and D.W. Wichern,Applied Multivariate Statistical Analysis (Prentice-Hall, Englewood Cliffs, NJ, 1992). · Zbl 0745.62050
[12] J.F. Lawless,Statistical Models and Methods for Lifetime Data (Wiley, New York, 1982). · Zbl 0541.62081
[13] G.S. Maddala,Limited-Dependent and Qualitative Variables in Econometrics (Cambridge University Press, Cambridge, UK, 1983). · Zbl 0527.62098
[14] J.I. McCool, Censored data, in:Encyclopedia of Statistical Sciences, Vol. 9 (Wiley, New York, 1982) p. 389–396.
[15] J.I. McGill, The multivariate hazard gradient and moments of the truncated multinormal distribution, Communications in Statistics 21(1992)3053–3060. · Zbl 0800.62267 · doi:10.1080/03610929208830962
[16] G.D. Murray, Contribution to discussion of paper by A.P. Dempster, N.M. Laird and D.B. Rubin, Journal of the Royal Statistical Society, Series B, 39(1977)27–28.
[17] W. Nelson,Applied Life Data Analysis (Wiley, New York, 1982). · Zbl 0579.62089
[18] W. Nelson and J. Schmee, Inference for (log) normal life distributions from small singly censored samples and BLUE’s, Technometrics 21(1979)43–54. · Zbl 0399.62100 · doi:10.2307/1268579
[19] G.G. Roussas,A First Course in Mathematical Statistics (Addison-Wesley, Reading, MA, 1973). · Zbl 0271.62001
[20] A.E. Sarhan and B.G. Greenberg, Estimation of location and scale parameters by order statistics from singly and doubly censored samples, Part I. The normal distribution up to samples of size 10, Annals of Mathematical Statistics 27(1956)427–451. · Zbl 0071.13502 · doi:10.1214/aoms/1177728267
[21] R. Shlifer and Y. Vardi, An airline overbooking policy, Transportation Science 9(1975)101–114. · doi:10.1287/trsc.9.2.101
[22] B.C. Smith, J.F. Leimkuhler and R.M. Darrow, Yield management at American airlines, Interfaces 22(1992)8–31. · doi:10.1287/inte.22.1.8
[23] N.K. Taneja,Airline Traffic Forecasting (Lexington Books, Lexington, MA, 1978). · Zbl 0428.90015
[24] J. Tobin, Estimation of relationships for limited dependent variables, Econometrica 26(1958)24–36. · Zbl 0088.36607 · doi:10.2307/1907382
[25] I.R. Weatherford and S.E. Bodily, A taxonomy and research overview of perishable-asset revenue management: Yield management, pricing, and overbooking, Operations Research 40(1992)831–844. · doi:10.1287/opre.40.5.831
[26] C.F.J. Wu, On the convergence properties of the EM algorithm, Annals of Statistics 11(1983)95–103. · Zbl 0517.62035 · doi:10.1214/aos/1176346060
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