The nontruncated marginal of a truncated bivariate normal distribution

Abstract

Inference is considered for the marginal distribution ofX, when (X, Y) has a truncated bivariate normal distribution. TheY variable is truncated, but only theX values are observed. The relationship of this distribution to Azzalini's “skew-normal” distribution is obtained. Method of moments and maximum likelihood estimation are compared for the three-parameter Azzalini distribution. Samples that are uniformative about the skewness of this distribution may occur, even for largen. Profile likelihood methods are employed to describe the uncertainty involved in parameter estimation. A sample of 87 Otis test scores is shown to be well-described by this model.

This is a preview of subscription content, access via your institution.

References

  1. Aitken, M. A. (1964). Correlation in a singly truncated bivariate normal distribution.Psychometrika, 29, 263–270.

    Article  Google Scholar 

  2. Azzalini, A. (1985). A class of distributions which includes the normal ones.Scandinavian Journal of Statistics, 12, 171–178.

    Google Scholar 

  3. Birnbaum, Z. W. (1950). Effect of linear truncation on a multinormal population.The Annals of Mathematical Statistics, 21, 272–279.

    Article  Google Scholar 

  4. Cartinhour, J. (1990). One-dimensional marginal density functions of a truncated multivariate normal density function.Communications in Statistics, Part A—Theory and Methods, 19, 197–203.

    Article  Google Scholar 

  5. Chou, Y-M., & Owen, D. B. (1984). An approximation to the percentiles of a variable of the bivariate normal distribution when the other variable is truncated, with applications.Communications in Statistics, Part A—Theory and Methods, 13, 2535–2547.

    Article  Google Scholar 

  6. Cohen, A. C. (1950). Estimating the mean and variance of normal populations from singly truncated and doubly truncated samples.The Annals of Mathematical Statistics, 21, 557–569.

    Article  Google Scholar 

  7. Cohen, A. C. (1955). Restriction and selection in samples from bivariate normal distributions.Journal of the American Statistical Association, 50, 884–893.

    Article  Google Scholar 

  8. Cohen, A. C. (1991).Truncated and Censored Samples: Theory and Applications. New York, NY: Marcel Dekker.

    Google Scholar 

  9. Dennis, J. E., Gay, D. M., & Welsch, R. E. (1981). An adaptive nonlinear least-squares algorithm.ACM Transactions on Mathematical Software, 7, 348–383.

    Article  Google Scholar 

  10. Henze, N. (1986). A probabilistic representation of the ‘skew-normal’ distribution.Scandinavian Journal of Statistics, 13, 271–275.

    Google Scholar 

  11. Hutchinson, T. P., & Lai, C. D. (1990).Continuous bivariate distributions, emphasizing applications. Adelaide, South Australia: Rambsby Scientific Publishing.

    Google Scholar 

  12. Roberts, H. V. (1988).Data analysis for managers with Minitab. Redwood City, CA: Scientific Press.

    Google Scholar 

  13. S-PLUS reference manual. (1990). Seattle, WA: Statistical Sciences.

  14. Zacks, S., (1981).Parametric statistical inference. Oxford, U.K.: Pergamon Press.

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Richard A. Groeneveld.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Arnold, B.C., Beaver, R.J., Groeneveld, R.A. et al. The nontruncated marginal of a truncated bivariate normal distribution. Psychometrika 58, 471–488 (1993). https://doi.org/10.1007/BF02294652

Download citation

Key words

  • monitored variable
  • profile likelihood
  • screening
  • skew-normal distribution