Skip to main content
SpringerLink
Log in

An alternative formulation of Neyman’s smooth goodness of fit tests under composite alternatives

  • Publications
  • Published:
Metrika Aims and scope Submit manuscript

Summary

We propose a natural extension of Neyman’s smooth goodness of fit tests under composite hypotheses. The components of our smooth tests are immediate analogues of the corresponding components in the completely specified null case. We show that, when testing for univariate normality, one of our smooth tests is similar to D’Agostino and Pearson’s (1973) statistic; when testing for multivariate normality, our smooth test for skewness is identical to Mardia’s (1970) measure of multivariate skewness; and, when testing for exponentiality, our simplest smooth test is equivalent to Greenwood’s (1946) statistic.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Barton DE (1953) Neyman’sψ /2 K test of goodness of fit when the null hypothesis is composite. Skand Aktuartioskr 38:1–16

    Google Scholar 

  • Billingsley P (1968) Convergence of probability measures. John Wiley, New York

    MATH  Google Scholar 

  • Cox DR, Hinkley DV (1974) Theoretical statistics. Chapman and Hall, London

    MATH  Google Scholar 

  • Cramér H, Leadbetter MR (1967) Stationary and related stochastic processes. John Wiley, New York

    MATH  Google Scholar 

  • Cramér H (1951) Mathematical methods of statistics. Princeton University Press, Princeton

    Google Scholar 

  • D’Agostino R, Pearson ES (1973) Tests for departure from normality. Empirical results for the distributions ofb 2 and √b 1. Biometrika 60:613–622

    MATH  MathSciNet  Google Scholar 

  • Darling DA (1953) On a class of problems related to the random division of an interval. Ann Math Statist 24:239–253

    MathSciNet  Google Scholar 

  • Durbin J (1973) Weak convergence of the sample distribution function when parameters are estimated. Ann Statist 1:279–290

    MATH  MathSciNet  Google Scholar 

  • Gihman II, Skorohod AV (1974) The theory of stochastic processes I. Springer, Berlin

    MATH  Google Scholar 

  • Greenwood M (1946) The statistical study of infectious diseases. J Roy Statist Soc A109:85–110

    Article  MathSciNet  Google Scholar 

  • Kendall MG, Stuart A (1967) The advanced theory of statistics, vol 2. Hafner, New York

    Google Scholar 

  • Kopecky KJ, Pierce DA (1979) Efficiency of smooth goodness-of-fit tests. J Amer Statist Assoc 74:393–397

    Article  MATH  Google Scholar 

  • Koziol JA (1979) A smooth test for bivariate independence. Sankhya B41:260–269

    MathSciNet  Google Scholar 

  • Koziol JA (1980) A note on limiting distributions for spacings statistics. Z Wahrscheinlichkeitstheorie verw Gebiete 51:55–62

    Article  MATH  MathSciNet  Google Scholar 

  • Koziol JA, Perlman MP (1978) On combining independent chisquared statistics. J Amer Statist Assoc 73:763

    Article  Google Scholar 

  • Loynes RM (1980) The empirical distribution function of residuals from generalized regression. Ann Statist 8:285–298

    MATH  MathSciNet  Google Scholar 

  • Mardia KV (1970) Measures of multivariate skewness and kurtosis with applications. Biometrika 57:519–530

    Article  MATH  MathSciNet  Google Scholar 

  • Mardia KV (1974) Applications of some measures of multivariate skewness and kurtosis in testing normality and robustness studies. Sankhya B36:115–128

    MathSciNet  Google Scholar 

  • Moran PAP (1947) The random division of an interval. J Roy Statist Soc B9:92–93

    Google Scholar 

  • Neyman J (1937) “Smooth” test for goodness of fit. Skand Aktuartioskr 20:149–199

    Google Scholar 

  • Pierce DA, Kopecky KJ (1979) Testing goodness of fit for the distribution of errors in regression models. Biometrika 66:1–5

    Article  MATH  MathSciNet  Google Scholar 

  • Pyke R (1965) Spacings. J Roy Statist Soc B27:395–449

    MathSciNet  Google Scholar 

  • Small NJH (1980) Marginal skewness and kurtosis in testing multivariate normality. Appl Statist 29:85–87

    Article  Google Scholar 

  • Thomas DR, Pierce DA (1979) Neyman’s smooth goodness-of-fit test when the hypothesis is composite. J Amer Statist Assoc 74:441–445

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dept. of Basic and Clinical Research/301N, Scripps Clinic and Research Foundation, 92037, La Jolla, CA, USA

    J. A. Koziol Ph.D.

Authors
  1. J. A. Koziol Ph.D.
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Koziol, J.A. An alternative formulation of Neyman’s smooth goodness of fit tests under composite alternatives. Metrika 34, 17–24 (1987). https://doi.org/10.1007/BF02613126

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02613126

Keywords

Search

Navigation