Advertisement

Discrete Normal Linear Regression Models

  • Jan de Leeuw
Part of the Lecture Notes in Economics and Mathematical Systems book series (LNE, volume 237)

Abstract

In this paper we continue our study of the Pearsonian approach to discrete multivariate analysis, in which structural properties of the multivariate normal distribution are combined with the essential discreteness of the data into a single comprehensive model. In an earlier publication we studied these ‘block-multinormal’ methods for covariance models. Here we propose a similar approach for the regression model with fixed regressors. Likelihood methods are derived and applied to some examples. We review the related literature and point out some interesting possible generalizations. The effect of continuous misspecification of a discrete model is studied in some detail. Relationships with the optimal scaling approach to multivariate analysis are also investigated.

Keywords

Maximum Likelihood Estimate Conditional Expectation Probit Analysis Royal Statistical Society Optimal Scaling 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Aitchison, J. and Silvey, S.D. The generalization of probit analysis to the case of multiple responses. Biometrika, 1957, 44, 131–140.Google Scholar
  2. Amemiya, T. Regression analysis when the dependent variable is truncated normal. Econometrica, 1973, 41, 997–1016.CrossRefGoogle Scholar
  3. Amemiya, T. Multivariate regression and simultaneous equation models when the dependent variables are truncated normal. Econometrica, 1974, 42, 999–1012.CrossRefGoogle Scholar
  4. Anderson, J.A. Regression and ordered categorical variables. Journal of the Royal Statitiscal Society(B), 1984, 46, 1–30.Google Scholar
  5. Arabmazar, A. and Schmidt, P. An investigation of the robustness of the Tobit estimator to non-normality. Econometrica, 1982, 50, 1055–1063.CrossRefGoogle Scholar
  6. Ashford, J.R. An approach to the analysis of data for semi-quantal responses in biological assay. Biometrics, 1959, 15, 573–581.CrossRefGoogle Scholar
  7. Bliss, C.I. and Stevens, W.L. The calculation of the time-mortality curve. Annals of Applied Biology, 1937, 24, 815–852.CrossRefGoogle Scholar
  8. Boyles, R.A. On the convergence of the EM algorithm. Journal of the Royal Statistical Society, 1983, 45, 47–50.Google Scholar
  9. Burridge, J. A note on maximum likelihood estimation for regression models using grouped data. Journal of the Royal Statistical Society(B), 1981, 43, 41–45.Google Scholar
  10. Curry, H.B. and Schoenberg, I.J. On Polya frequency functions IV: The fundamental spline functions and their limits. Journal d’Analyse Mathematique, 1966, 17, 71–107.CrossRefGoogle Scholar
  11. De Leeuw, J. Models and methods for the analysis of correlation coefficients. Journal of Econometrics, 1983, 22, 113–138.CrossRefGoogle Scholar
  12. De Leeuw, J., Young, F.W., & Takane, Y. Additive structure in qualitative data: an alternating least squares method with optimal scaling features. Psychometrika, 1976, 41, 471–503.CrossRefGoogle Scholar
  13. Dempster, A.P., Laird, N.M., & Rubin, D.B. Maximum likelihood from incomplete data using the EM algorithm. Journal of the Royal Statistical Society(B), 1977, 39, 1–38.Google Scholar
  14. Dempster, A.P. and Rubin, D.B. Rounding error in regression: the appropriateness of Sheppard’s corrections. Journal of the Royal Statistical Society(B), 1983, 45, 51–59.Google Scholar
  15. Don, F.J.H. A note on Sheppard’s corrections for grouping and maximum likelihood estimation. Journal of Multivariate Analysis, 1981, 11, 452–458.CrossRefGoogle Scholar
  16. Dijkstra, T. Some comments on maximum likelihood and partial least squares. Journal of Econometrics, 1983, 22, 67–90.CrossRefGoogle Scholar
  17. Fienberg, S.E. The analysis of cross-classified data. Cambridge (Ma), MIT-Press, 1980.Google Scholar
  18. Finney, D. Probit analysis. Cambridge, Cambridge University Press, 1971.Google Scholar
  19. Fisher, R.A. Statististical methods for research workers. (14 edition). Edinburgh, Oliver and Boyd, 1970.Google Scholar
  20. Gifi, A. Nonlinear multivariate analysis. Leiden, Department of Data Theory, 1981.Google Scholar
  21. Greene, W.H. On the asymptotic bias of the ordinary least squares estimator in the Tobit model. Econometrica, 1981, 49 505–513.CrossRefGoogle Scholar
  22. Guilford, J.P. Psychometric methods. (2 edition). New York, McGraw-Hill, 1954.Google Scholar
  23. Gurland, J., Lee, I., & Dahm, P.A. Polychotomous quantal response in biological assay. Biometrics, 1960, 16, 382–398.CrossRefGoogle Scholar
  24. Hemelrijk, J. Underlining random variables. Statistica Neerlandica, 1966, 20, 1–8.CrossRefGoogle Scholar
  25. Joreskog, K.G. and Wold, H.O.A. Systems under indirect observation. Amsterdam, North Holland Publishing Company, 1982.Google Scholar
  26. Kendall, M.G. and Stuart, A. The advanced theory of statistics. Volume II, 2 edition, London, Griffin, 1967.Google Scholar
  27. Krein, M.G. and Nudelman, A.A. The Markov moment problem and extrremal problems. Providence (RI), American Mathematical Society, 1977.Google Scholar
  28. Kruskal, J.B. Analysis of factorial experiments by estimating monotone transformations of the data. Journal of the Royal Statistical Society, 1965, 27, 251–263.Google Scholar
  29. Lord, F.M. and Novick, M.R. Statistical theories of mental test scores. Reading (Ma), Addison-Wesley, 1967.Google Scholar
  30. Louis, T.A. Finding the observed information matrix when using the EM Algorithm. Journal of the Royal Statistical Society(B), 1982, 44, 226–233.Google Scholar
  31. McCullagh, P. Regression models for ordinal data. Journal of the Royal Statistical Society, 1980, 42, 109–142.Google Scholar
  32. McKelvey, R.D. and Zavoina, W. A statistical model for the analysis of ordinal level dependent variables. Journal of Mathematical Sociology, 1975, 4, 103–120.CrossRefGoogle Scholar
  33. Manski, C.F. and McFadden, D. Structural analysis of discrete data with econometric applications. Cambridge (Ma), MIT Press, 1981.Google Scholar
  34. Nelson, F.D. On a general computer algorithm for the analysis of models with limited dependent variables. Annals of economic and social measurement, 1976, 5, 493–509.Google Scholar
  35. Nelson, F.D. A test for misspecification in the censored normal model. Econometrica, 1981, 49, 1317–1329.CrossRefGoogle Scholar
  36. Pratt, J.W. Concavity of the log likelihood. Journal of the American Statistical Association, 1981, 76, 103–106.CrossRefGoogle Scholar
  37. Rosett, R.N. and Nelson, F.D. Estimation of the two-limit probit regression model. Econometrica, 1975, 43, 141–146.CrossRefGoogle Scholar
  38. Samejima, F. Estimation of latent ability using a response pattern of graded scores. Psychometric Monograph, no 17, 1969.Google Scholar
  39. Sampford, M.R. and Taylor, J. Censored observations in randomized block experiments. Journal of the Royal Statistical Society, 1959, 21, 214–237.Google Scholar
  40. Stewart, M.B. On least squares estimation when the dependent variable is grouped. Review of economic studies, 1983, 50, 737–753.CrossRefGoogle Scholar
  41. Tobin, J. Estimation of relationships for limited dependent variables. Econometrica, 1958, 26, 24–36.CrossRefGoogle Scholar
  42. Winsberg, S. and Ramsay, J.O. Analysis of pairwise preference data using integrated B-splines. Psychometrika, 1981, 46, 171–186.CrossRefGoogle Scholar
  43. Winship, C. and Mare, R.D. Structural equations and path analysis for discrete data. American Journal of Sociology, 1983, 89, 54–110.CrossRefGoogle Scholar
  44. Wolynetz, M.C. Maximum likelihood estimation in a linear model from confined and censored data. Applied Statistics, 1979, 28, 196–206.Google Scholar
  45. Wu, C.F.J. On the convergence properties of the EM algorithm. The Annals of Statistics, 1983, 11, 95–103.CrossRefGoogle Scholar
  46. Young, F.W., De Leeuw, J., & Takane, Y. Regression with qualitative and quantitative variables: an alternating least squares method with optimal scaling features. Psychometrika, 1976, 40, 505–529.CrossRefGoogle Scholar
  47. Zellner, A. and Lee, T.H. Joint eestimation of relationships involving discrete random variables. Econometrica, 1965, 33, 382–394.CrossRefGoogle Scholar
  48. Ruud, P.A. Sufficient conditions for the consistency of maximum likelihood estimates despite misspecification of distribution in multinomial discrete choice models. Econometrica, 1983, 51, 225–228.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1984

Authors and Affiliations

  • Jan de Leeuw
    • 1
  1. 1.Department of Data TheoryUniversity of LeidenLeidenThe Netherlands

Personalised recommendations