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.
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Notes
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
Bartholomew DJ (1959a) A test of homogeneity for ordered alternatives, I. Biometrika 46(1/2):36–48
Bartholomew DJ (1959b) A test of homogeneity for ordered alternatives, II. Biometrika 46(3/4):328–335
Bohrer R, Chow W (1978) Algorithm AS122. Weights for one-sided multivariate inference. Appl Stat 27(1):100–104
Box GEP, Tiao GC (1973) Bayesian inference in statistical analysis. Addison-Wesley, Boston
Cardoso-Neto J, Paula GA (2001) Wald one-sided test using generalized estimating equations. Comput Stat Data Anal 36(4):475–495
Chacko VJ (1963) Testing homogeneity against ordered alternatives. Ann Math Stat 34(3):945–956
Crowder MJ, Hand DJ (1990) Analysis of repeated measurements. Chapman and Hall, London
Cysneiros FJA, Paula GA (2004) One-sided tests in linear models with multivariate \(t\)-distribution. Commun Stat Simul Comput 33(3):747–771
Cysneiros FJA, Paula GA (2005) Restricted methods in symmetrical linear regression models. Comput Stat Data Anal 49(3):689–708
Diggle PJ, Liang K, Zeger SL (2002) Analysis of longitudinal data. Clarendon Press, Oxford
Doornik JA (2009) Ox: an object-oriented matrix language, 4th edn. Timberlake Consultants Press, Oxford
Fahrmeir L, Klinger J (1994) Estimating and testing generalized linear models under inequality restrictions. Stat Pap 35(1):211–229
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
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
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
Gouriéroux G, Monfort A (1995) Statistics and econometrics, vol 1, 2. Cambridge University Press, Cambridge
Hubert M, Vandervieren E (2008) An adjusted boxplot for skewed distributions. Comput Stat Data Anal 52(12):5186–5201
Kodde DA, Palm FC (1986) Wald criteria for jointly testing equality and inequality restrictions. Econometrics 54(5):1243–1248
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
Kudô A (1963) A multivariate analogue of the one-sided test. Biometrika 50(3/4):403–418
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
Lindsey JK (1999) Multivariate elliptically contoured distributions for repeated measurements. Biometrics 55(4):1277–1280
Lindsey JK (2006) Multivariate distributions with correlation matrices for nonlinear repeated measurements. Biometrics 50(3):720–732
Mahrer JM, Magel RC (1995) A comparison of tests for the k-sample, non-decreasing alternative. Stat Med 14(8):863–871
McDonald JM, Diamond I (1990) On the fitting of generalized linear models with non-negativity parameter constraints. Biometrics 46(1):201–206
Nyquist H (1991) Restricted estimation of generalized linear models. Appl Stat 40(1):133–141
Paula GA, Artes R (2000) One-sided test to assess correlation in logistic linear models using estimation equations. Biom J 42(6):701–714
Paula GA, Sen PK (1995) One-sided tests in generalized linear models with parallel regression lines. Biometrics 51(4):1494–1501
Pilla RS, Qu A (2006) Testing for order-restricted hypothesis in longitudinal data. J R Stat Soc B 68(3):437–455
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
Robertson T, Wright FT, Dykstra RL (1988) Order restricted statistical inference. Wiley, New York
Savalli CR, Paula GA, Cysneiros FJA (2006) Assessment of variance components in elliptical linear mixed models. Stat Model 6(1):59–76
Shapiro A (1985) Asymptotic distribution of test statistics in the analysis of moment structures under inequality constraints. Biometrika 72(1):133–144
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
Shin DW, Park CG, Park T (1996) Testing for ordered group effects with repeated measurements. Biometrika 83(3):688–694
Silvapulle MJ, Silvapulle P (1995) A score test against one-sided alternatives. J Am Stat Assoc 90(429):342–345
Silvapulle MJ, Silvapulle P, Basawa IV (2002) Test against inequality constraints in semiparametric models. J Stat Plan Inference 107(1–2):307–320
Terrell GR (2002) The gradient statistic. Comput Sci Stat 34:206–215
Terpstra J, Magel R (2003) A new nonparametric test for the ordered alternative problem. J Nonparametr Stat 15(3):289–301
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.
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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
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DOI: https://doi.org/10.1007/s00180-016-0650-x