Skip to main content

Advertisement

SpringerLink
Log in

Constrained test in linear models with multivariate power exponential distribution

  • Original Paper
  • Published:
Computational Statistics Aims and scope Submit manuscript

Abstract

We investigate the problem of testing equality and inequality constraints on regression coefficients in linear models with multivariate power exponential (MPE) distribution. This distribution has received considerable attention in recent years and provides a useful generalization of the multivariate normal distribution. We examine the performance of the power of the likelihood ratio, Wald and Score tests for grouped data and in the presence of regressors, in small and moderate sample sizes, using Monte Carlo simulations. Additionally, we present a real example to illustrate the performance of the proposed tests under the MPE model.

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

Fig. 1
Fig. 2

Similar content being viewed by others

Notes

  1. Note that the likelihood ratio, score and Wald tests are asymptotically distributed as a mixture of chi-square distributions with weights not depending on null parameters, but possibly depending on correlations. Therefore, in general regression models, the problem of finding the least favorable point in the null hypothesis translates into a problem of searching through a set of correlation coefficients for the least favorable points; see Cysneiros and Paula (2004).

References

  • Atkinson AC (1981) Two graphical displays for outlying and influential observations in regression. Biometrika 68(1):13–20

    Article  MathSciNet  MATH  Google Scholar 

  • Bartholomew DJ (1959a) A test of homogeneity for ordered alternatives, I. Biometrika 46(1/2):36–48

    Article  MathSciNet  MATH  Google Scholar 

  • Bartholomew DJ (1959b) A test of homogeneity for ordered alternatives, II. Biometrika 46(3/4):328–335

    Article  MathSciNet  MATH  Google Scholar 

  • Bohrer R, Chow W (1978) Algorithm AS122. Weights for one-sided multivariate inference. Appl Stat 27(1):100–104

    Article  MATH  Google Scholar 

  • Box GEP, Tiao GC (1973) Bayesian inference in statistical analysis. Addison-Wesley, Boston

    MATH  Google Scholar 

  • Cardoso-Neto J, Paula GA (2001) Wald one-sided test using generalized estimating equations. Comput Stat Data Anal 36(4):475–495

    Article  MathSciNet  MATH  Google Scholar 

  • Chacko VJ (1963) Testing homogeneity against ordered alternatives. Ann Math Stat 34(3):945–956

    Article  MathSciNet  MATH  Google Scholar 

  • Crowder MJ, Hand DJ (1990) Analysis of repeated measurements. Chapman and Hall, London

    MATH  Google Scholar 

  • Cysneiros FJA, Paula GA (2004) One-sided tests in linear models with multivariate \(t\)-distribution. Commun Stat Simul Comput 33(3):747–771

    Article  MathSciNet  MATH  Google Scholar 

  • Cysneiros FJA, Paula GA (2005) Restricted methods in symmetrical linear regression models. Comput Stat Data Anal 49(3):689–708

    Article  MathSciNet  MATH  Google Scholar 

  • Diggle PJ, Liang K, Zeger SL (2002) Analysis of longitudinal data. Clarendon Press, Oxford

    MATH  Google Scholar 

  • Doornik JA (2009) Ox: an object-oriented matrix language, 4th edn. Timberlake Consultants Press, Oxford

    Google Scholar 

  • Fahrmeir L, Klinger J (1994) Estimating and testing generalized linear models under inequality restrictions. Stat Pap 35(1):211–229

    Article  MATH  Google Scholar 

  • Garre FG, Vermut JK, Croon MA (2002) Likelihood-ratio tests for order-restricted log-linear models: a comparison of asymptotic and bootstrap methods. Metodol Cienc Comport 1:1–18

    Google Scholar 

  • Gómez E, Gomez-Villegas MA, Marín JM (1998) A multivariate generalization of the power exponential family of distributions. Commun Stat Theory Methods 27(3):589–600

    Article  MathSciNet  MATH  Google Scholar 

  • Gómez E, Gomez-Villegas MA, Marín JM (2008) Multivariate exponential power distributions as mixtures of normal distributions with Bayesian application. Commun Stat Theory Methods 37(6):972–985

    Article  MATH  Google Scholar 

  • Gouriéroux G, Monfort A (1995) Statistics and econometrics, vol 1, 2. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  • Hubert M, Vandervieren E (2008) An adjusted boxplot for skewed distributions. Comput Stat Data Anal 52(12):5186–5201

    Article  MathSciNet  MATH  Google Scholar 

  • Kodde DA, Palm FC (1986) Wald criteria for jointly testing equality and inequality restrictions. Econometrics 54(5):1243–1248

    Article  MathSciNet  MATH  Google Scholar 

  • Kowalski J, Mendoza-Blanco J, Tu XM, Gleser LJ (1999) On the difference in inference and prediction between the joint and independent \(t\)-error models for seemingly unrelated regressions. Commun Stat Theory Methods 28(9):2119–2140

    Article  MathSciNet  MATH  Google Scholar 

  • Kudô A (1963) A multivariate analogue of the one-sided test. Biometrika 50(3/4):403–418

    Article  MathSciNet  MATH  Google Scholar 

  • Kwitt R, Meerwald P, Uhl A (2009) Color-image watermarking using multivariate power exponential distribution. In: Proceedings of the 16th IEEE international conference on image processing, pp 4245–4248

  • Lange KL, Litte RJA, Taylor JMG (1989) Robust statistical modeling using the \(t\) distribution. J Am Stat Assoc 84(408):881–896

    MathSciNet  Google Scholar 

  • Lindsey JK (1999) Multivariate elliptically contoured distributions for repeated measurements. Biometrics 55(4):1277–1280

    Article  MathSciNet  MATH  Google Scholar 

  • Lindsey JK (2006) Multivariate distributions with correlation matrices for nonlinear repeated measurements. Biometrics 50(3):720–732

    MathSciNet  MATH  Google Scholar 

  • Mahrer JM, Magel RC (1995) A comparison of tests for the k-sample, non-decreasing alternative. Stat Med 14(8):863–871

    Article  Google Scholar 

  • McDonald JM, Diamond I (1990) On the fitting of generalized linear models with non-negativity parameter constraints. Biometrics 46(1):201–206

    Article  MATH  Google Scholar 

  • Nyquist H (1991) Restricted estimation of generalized linear models. Appl Stat 40(1):133–141

    Article  MATH  Google Scholar 

  • Paula GA, Artes R (2000) One-sided test to assess correlation in logistic linear models using estimation equations. Biom J 42(6):701–714

    Article  MathSciNet  MATH  Google Scholar 

  • Paula GA, Sen PK (1995) One-sided tests in generalized linear models with parallel regression lines. Biometrics 51(4):1494–1501

    Article  MATH  Google Scholar 

  • Pilla RS, Qu A (2006) Testing for order-restricted hypothesis in longitudinal data. J R Stat Soc B 68(3):437–455

    Article  MathSciNet  MATH  Google Scholar 

  • Press WH, Teulosky SA, Vetterling WT, Flannery BP (1992) Numerical recipes in C: the art of scientific computing, 2nd edn. Cambridge University Press, New York

    MATH  Google Scholar 

  • Robertson T, Wright FT, Dykstra RL (1988) Order restricted statistical inference. Wiley, New York

    MATH  Google Scholar 

  • Savalli CR, Paula GA, Cysneiros FJA (2006) Assessment of variance components in elliptical linear mixed models. Stat Model 6(1):59–76

    Article  MathSciNet  Google Scholar 

  • Shapiro A (1985) Asymptotic distribution of test statistics in the analysis of moment structures under inequality constraints. Biometrika 72(1):133–144

    Article  MathSciNet  MATH  Google Scholar 

  • Shan G, Young D, Kang L (2014) A new powerful nonparametric rank test for ordered alternative problem. Plos One 9(11):e112924. doi:10.1371/journal.pone.0112924

    Article  Google Scholar 

  • Shin DW, Park CG, Park T (1996) Testing for ordered group effects with repeated measurements. Biometrika 83(3):688–694

    Article  MathSciNet  MATH  Google Scholar 

  • Silvapulle MJ, Silvapulle P (1995) A score test against one-sided alternatives. J Am Stat Assoc 90(429):342–345

    Article  MathSciNet  MATH  Google Scholar 

  • Silvapulle MJ, Silvapulle P, Basawa IV (2002) Test against inequality constraints in semiparametric models. J Stat Plan Inference 107(1–2):307–320

    Article  MathSciNet  MATH  Google Scholar 

  • Terrell GR (2002) The gradient statistic. Comput Sci Stat 34:206–215

    Google Scholar 

  • Terpstra J, Magel R (2003) A new nonparametric test for the ordered alternative problem. J Nonparametr Stat 15(3):289–301

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors wish to thank an Associate Editor and two anonymous referees for their constructive comments on an earlier version of this manuscript which resulted in this improved version. J. Leão would like to thank CAPES for the financial support. H. Saulo gratefully acknowledges CNPq and CAPES for the financial support.

Author information

Authors and Affiliations

  1. Departamento de Estatística, Universidade Federal do Amazonas, Manaus, Brazil

    Jeremias Leão

  2. Departamento de Estatística, Universidade Federal de Pernambuco, Recife, Brazil

    Francisco Cysneiros

  3. Instituto de Matemática e Estatística, Universidade Federal de Goiás, Goiânia, Brazil

    Helton Saulo

  4. Department of Mathematics and Statistics, McMaster University, Hamilton, Canada

    Helton Saulo & N. Balakrishnan

Authors
  1. Jeremias Leão
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Francisco Cysneiros
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Helton Saulo
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. N. Balakrishnan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jeremias Leão.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Leão, J., Cysneiros, F., Saulo, H. et al. Constrained test in linear models with multivariate power exponential distribution. Comput Stat 31, 1569–1592 (2016). https://doi.org/10.1007/s00180-016-0650-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00180-016-0650-x

Keywords

Search

Navigation