Estimation of the inverse function for random variate generation

ABSTRACT

A regression method for estimating the inverse of a continuous cumulative probability function F(x) is presented. It is assumed that an ordered sample, X1, …, Xn, of identically and independently distributed random variables is available. A reference distribution F0(x) with known inverse F0-1(p) is used to calculate the quantities Wi = i ln[F0(Xi)/F0(Xi+1)]. These quantities are used to estimate the function &ggr;(p) = pd ln≥F0[F-1(p)]⋦/dp from which an estimate of F-1(p) is derived. The method produces an estimate in a form that is convenient for random variate generation. The procedure is illustrated using data from a study of oil and gas lease bidding.

REFERENCES

• 1: Abramowitz, M., and Stegun, I. Handbook of Mathematical Functions. National Bureau of Standards, Washington, D.C., 1964.
• 2: Dougherty, E.L., and Lohrenz, J. Statistical analyses of bids for Federal offshore leases. J. Petroleum Tech. 28 (Nov. 1976), 1377-1390.
• 3: Draper, N., and Smith, H. Applied Regression Analysis, 2nd ed. Wiley, New York, 1981.
• 4: Genter, F.C., and Bruckner, L.A. Investigation of the lognormality of offshore oil and gas lease bidding data: Cramer yon Mises one-sample test. Informal Rept. LA-7339-MS, Los Alamos Scientific Lab., Los Alamos, N. Mex., June 1978.
• 5: Stephen C. Hora, A comparison of three methods of modeling input distributions, Proceedings of the 13th conference on Winter simulation, p.309-312, December 09-11, 1981, Atlanta, Georgia
• 6: Kennedy, W.J., Jr., and Gentle, J.E. Statistical Computing. Marcel Dekker, New York, 1980.
• 7: Malmquist, S. On a property of order statistics from a rectangular distribution. Skandioovisk Aktoarietidskrifl 33 (1950), 214-222.
• 8: Mitchell, R.L., and Stone, C.R. Table-lookup methods for generating arbitrary random numbers. IEEE Transactions on Computers C-26, 10 (Oct. 1975), 1006-1008.
• 9: Neill, J.W., and Johnson, D.E. Testing for lack of fit when sample points are not replicated. Presented at the Joint Statistical Meetings of the American Statistical Association and Biometric Society, Detroit, Mich., Aug. 1981.
• 10: Neter, J., and Wasserman, W. Applied Linear Statistical Models. Richard D. Irwin, Inc., Homewood, I11., 1974.
• 11: Ogawa, J. Distributions and moments of order statistics. In Darhan and Greenbery (Eds.I, Contributions to Order Statistics. Wiley, New York, 1962.
• 12: Schmeiser, B. Random variate generation: A survey. In Oren, Shub, and Roth (Eds.), Simulation with Discrete Models: A State of the Art View, IEEE Press, New York, 1980, pp. 79-104.
• 13: Stephens, M,A. EDF statistics for goodness of fit and some comparisons. J. Amer. Stat. Assoc. 69 (1974), 730-737.
• 14: Alastair J. Walker, An Efficient Method for Generating Discrete Random Variables with General Distributions, ACM Transactions on Mathematical Software (TOMS), v.3 n.3, p.253-256, Sept. 1977 [doi>10.1145/355744.355749]

INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.3 PROBABILITY AND STATISTICS
      Subjects: Random number generation

Keywords:
estimation of inverse functions