Advertisement

A Survey of Estimating Distributional Parameters and Sample Sizes from Truncated Samples

  • Saul Blumenthal
Part of the NATO Advanced study Institutes Series book series (ASIC, volume 79)

Summary

Suppose X1,…, XN are independent random variables with common distribution F(x;θ). We observe Xi only if it lies outside a given region R. Thus the number n of observed X’s is a binomial (N,P) variate, where P = 1 — P(X in R). Based on n and the values of the observed X’s, we want to estimate N and θ. Conditional and unconditional maximum likelihood estimators and modifications of these will be discussed. Asymptotic results of both first and second order will be considered, and small sample results also mentioned. Sequential and two stage results will receive brief coverage. A variety of applications will be given.

Key Words

Estimating n sample size estimation asymptotic expansions second order efficiency truncated samples 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Blumenthal, S. (1977). Estimating population size with truncated sampling. Communications in Statistics, A6, 297–308.MathSciNetCrossRefGoogle Scholar
  2. Blumenthal, S. (1980). Stochastic expansions for point estimation from complete, censored and truncated samples. Submitted for publication.Google Scholar
  3. Blumenthal, S., Dahiya, R. C., Gross, A. J. (1978). Estimating the complete sample size from an incomplete Poisson sample. Journal of the American Statistical Association, 73, 182–187.MathSciNetzbMATHCrossRefGoogle Scholar
  4. Blumenthal, S., Marcus, R. (1975a). Estimating the number of items on life test. IEEE Transactions on Reliability R-24, 271–273.Google Scholar
  5. Blumenthal, S., Marcus, R. (1975b). Estimating population size with exponential failure. Journal of the American Statistical Association, 28, 913–922.MathSciNetCrossRefGoogle Scholar
  6. Blumenthal, S., Sanathanan, L.P. (1980). Estimation with truncated inverse binomial sampling. Communications in Statistics, A9, 997–1017.MathSciNetCrossRefGoogle Scholar
  7. Dahiya, R.C. (1980a). An improved method of estimating an integer-parameter by maximum likelihood. The American Statistician, to appear.Google Scholar
  8. Dahiya, R.C. (1980b). Pearson goodness-of-fit test when the sample size is unknown. Unpublished manuscript.Google Scholar
  9. Dahiya, R.C., Gross, A.J. (1973). Estimating the zero class from a truncated Poisson sample. Journal of the American Statistical Association, 68, 731–733.MathSciNetzbMATHCrossRefGoogle Scholar
  10. Goel, A.L., Joglekar, A.M. (1976). Reliability acceptance sampling plans based upon prior distribution, III. Implications and determination of the prior distribution. Technical Report No. 76–3, Department of Industrial Engineering and Operations Research, Syracuse University.Google Scholar
  11. Gross, A.J. (1971). Monotonicity properties of the moments of truncated gamma and Weibull density functions. Techometrics, 13, 851–857.zbMATHCrossRefGoogle Scholar
  12. Hartley, H.O. (1958). Maximum likelihood estimation from incomplete data. Biometrics, 14, 174–194.zbMATHCrossRefGoogle Scholar
  13. Holla, M.S. (1967). Reliability estimation of the truncated exponential model. Techometrics, 9, 332–335.MathSciNetzbMATHCrossRefGoogle Scholar
  14. Jelinski, Z., Moranda, P. (1972). Software reliability research. In Statistical Computer Performance Evaluation, W. Freiberger, ed. Academic Press, New York. Pages 465–484.Google Scholar
  15. Johnson, N.L. (1962). Estimation of sample size. Technometrics, 4, 59–67.MathSciNetzbMATHCrossRefGoogle Scholar
  16. Johnson, N.L. (1967). Sample censoring. ARO-D Rep. 67–2, U. S. Army Research Office - Durham, Durham, N.C.Google Scholar
  17. Johnson, N.L. (1970). A general purpose test of censoring of extreme sample values. In Essays in Probability and Statistics (S. N. Roy Memorial Volume), R. C. Bose, ed. University of North Carolina Press, Chapel Hill, Pages 377– 384.Google Scholar
  18. Johnson, N.L. (1971). Comparison of some tests of sample censoring of extreme values. Australian Journal of Statistics, 13, 1–6.zbMATHCrossRefGoogle Scholar
  19. Johnson, N.L. (1972). Inferences on sample size: sequences of samples. Trabajos de Estadistica, 23, 85–110.zbMATHGoogle Scholar
  20. Johnson, N.L. (1973). Robustness of certain tests of censoring of extreme values. Institute of Statistics mimeo series No. 866, University of North Carolina, Chapel Hill.Google Scholar
  21. Johnson, N.L. (1974a). Robustness of certain tests of censoring of extreme values, II: some exact results for exponential populations. Institute of Statistics mimeo series No. 940, University of North Carolina, Chapel Hill.Google Scholar
  22. Johnson, N.L. (1974b). Estimation of rank order. Institute of Statistics mimeo series No. 931, University of North Carolina, Chapel Hill.Google Scholar
  23. Johnson, N. L. (1976). Completeness comparisons among sequences of samples. Institute of Statistics mimeo series No. 1056, University of North Carolina, Chapel Hill.Google Scholar
  24. Johnson, N. L., Kotz, S. (1969). Discrete Distributions. Houghton Mifflin, Boston.zbMATHGoogle Scholar
  25. Johnson, N. L., Kotz, S. (1970a). Continuous Univariate Distributions-1. Houghton Mifflin, Boston.Google Scholar
  26. Johnson, N. L., Kotz, S. (1970b). Continuous Univariate Distributions-2. Houghton Mifflin, Boston.Google Scholar
  27. Johnson, N. L., Kotz, S. (1972). Distributions in Statistics: Continuous Multivariate Distributions. Wiley, New York.zbMATHGoogle Scholar
  28. Marcus, R., Blumenthal, S. (1974). A sequential screening procedure. Techometries, 16, 229–234.MathSciNetzbMATHCrossRefGoogle Scholar
  29. Moranda, P.B. (1975). Prediction of software reliability during debugging. MDAC paper WD 2471, McDonnell Douglas Astronautics Company-West, Huntington Beach, California.Google Scholar
  30. Nath, G.B. (1975). Unbiased estimates of reliability for the truncated gamma distribution. Scandinavian Actuarial Journal, 1, 181–186.MathSciNetGoogle Scholar
  31. Rao, B.R., Mazumdar, S., Waller, J.H., Li, C.C. (1973). Correlation between the numbers of two types of children in a family. Biometrics, 29, 271–279.CrossRefGoogle Scholar
  32. Sampford, M.R. (1955). The truncated negative binomial distribution. Biometrika, 42, 58–69.MathSciNetzbMATHGoogle Scholar
  33. Sanathanan, L.P. (1972a). Estimating the size of a multinomial population. Annals of Mathematical Statistics, 43, 142–152.MathSciNetzbMATHCrossRefGoogle Scholar
  34. Sanathanan, L.P. (1972b). Models and estimation methods in visual scanning experiments. Technometrics, 14, 813–840.zbMATHCrossRefGoogle Scholar
  35. Sanathanan, L.P. (1977). Estimating the size of a truncated sample. Journal of the American Statistical Association, 72, 669–672.MathSciNetzbMATHCrossRefGoogle Scholar
  36. Smith, W. L. (1957). A note on truncation and sufficient statistics. Annals of Mathematical Statistics, 28, 247–252.MathSciNetzbMATHCrossRefGoogle Scholar
  37. Tukey, J.W. (1949). Sufficiency, truncation, and selection. Annals of Mathematical Statistics, 20, 309–311.MathSciNetzbMATHCrossRefGoogle Scholar
  38. Watson, D., Blumenthal, S. (1980). Estimating the size of a truncated sample. Communications in Statistics, A9, 1535–1550.CrossRefGoogle Scholar
  39. Watson, D., Blumenthal, S., (1981). A two stage procedure for estimating the size of a truncated exponential sample. Australian Journal of Statistics 20, No. 3.Google Scholar
  40. Wittes, J.T. (1970). Estimation of population size: the Bernoulli census. Ph.D. thesis, Harvard University.Google Scholar
  41. Wittes, J.T., Cotton, T., Sidel, V.W. (1974). Capture- recapture methods for assessing the completeness of case ascertainment when using multiple information sources. Journal of Chronic Disease, 27, 25–36.CrossRefGoogle Scholar

Copyright information

© D. Reidel Publishing Company 1981

Authors and Affiliations

  • Saul Blumenthal
    • 1
  1. 1.Department of MathematicsUniversity of Illinois at Urbana-ChampaignUrbanaUSA

Personalised recommendations